ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- mfderiv_const
@[simp, mfld_simps]
theorem mfderiv_const :
    mfderiv I I' (fun _ : M => c) x = (0 : TangentSpace I x →L[𝕜] TangentSpace I' c) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] (I' : ModelWithCorners 𝕜 E' H') {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] [SmoothManifoldWithCorners I' M'] {x : M} {c : M'}
Goal: mfderiv I I' (fun x => c) x = 0
-- mfderivWithin_const
theorem mfderivWithin_const (hxs : UniqueMDiffWithinAt I s x) :
    mfderivWithin I I' (fun _ : M => c) s x = (0 : TangentSpace I x →L[𝕜] TangentSpace I' c) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] (I' : ModelWithCorners 𝕜 E' H') {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] [SmoothManifoldWithCorners I' M'] {s : Set M} {x : M} {c : M'} (hxs : UniqueMDiffWithinAt I s x)
Goal: mfderivWithin I I' (fun x => c) s x = 0
-- hasMFDerivAt_fst
theorem hasMFDerivAt_fst (x : M × M') :
    HasMFDerivAt (I.prod I') I Prod.fst x
      (ContinuousLinearMap.fst 𝕜 (TangentSpace I x.1) (TangentSpace I' x.2)) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] (I' : ModelWithCorners 𝕜 E' H') {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] [SmoothManifoldWithCorners I' M'] (x : M × M')
Goal: HasMFDerivAt (I.prod I') I Prod.fst x (ContinuousLinearMap.fst 𝕜 (TangentSpace I x.1) (TangentSpace I' x.2))
-- hasMFDerivWithinAt_fst
theorem hasMFDerivWithinAt_fst (s : Set (M × M')) (x : M × M') :
    HasMFDerivWithinAt (I.prod I') I Prod.fst s x
      (ContinuousLinearMap.fst 𝕜 (TangentSpace I x.1) (TangentSpace I' x.2)) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] (I' : ModelWithCorners 𝕜 E' H') {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] [SmoothManifoldWithCorners I' M'] (s : Set (M × M')) (x : M × M')
Goal: HasMFDerivWithinAt (I.prod I') I Prod.fst s x (ContinuousLinearMap.fst 𝕜 (TangentSpace I x.1) (TangentSpace I' x.2))
-- mdifferentiableAt_fst
theorem mdifferentiableAt_fst {x : M × M'} : MDifferentiableAt (I.prod I') I Prod.fst x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] (I' : ModelWithCorners 𝕜 E' H') {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] [SmoothManifoldWithCorners I' M'] {x : M × M'}
Goal: MDifferentiableAt (I.prod I') I Prod.fst x
-- mdifferentiableWithinAt_fst
theorem mdifferentiableWithinAt_fst {s : Set (M × M')} {x : M × M'} :
    MDifferentiableWithinAt (I.prod I') I Prod.fst s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] (I' : ModelWithCorners 𝕜 E' H') {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] [SmoothManifoldWithCorners I' M'] {s : Set (M × M')} {x : M × M'}
Goal: MDifferentiableWithinAt (I.prod I') I Prod.fst s x
-- mdifferentiable_fst
theorem mdifferentiable_fst : MDifferentiable (I.prod I') I (Prod.fst : M × M' → M) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] (I' : ModelWithCorners 𝕜 E' H') {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] [SmoothManifoldWithCorners I' M']
Goal: MDifferentiable (I.prod I') I Prod.fst
-- mdifferentiableOn_fst
theorem mdifferentiableOn_fst {s : Set (M × M')} : MDifferentiableOn (I.prod I') I Prod.fst s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] (I' : ModelWithCorners 𝕜 E' H') {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] [SmoothManifoldWithCorners I' M'] {s : Set (M × M')}
Goal: MDifferentiableOn (I.prod I') I Prod.fst s
-- mfderiv_fst
@[simp, mfld_simps]
theorem mfderiv_fst {x : M × M'} :
    mfderiv (I.prod I') I Prod.fst x =
      ContinuousLinearMap.fst 𝕜 (TangentSpace I x.1) (TangentSpace I' x.2) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] (I' : ModelWithCorners 𝕜 E' H') {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] [SmoothManifoldWithCorners I' M'] {x : M × M'}
Goal: mfderiv (I.prod I') I Prod.fst x = ContinuousLinearMap.fst 𝕜 (TangentSpace I x.1) (TangentSpace I' x.2)
-- mfderivWithin_fst
theorem mfderivWithin_fst {s : Set (M × M')} {x : M × M'}
    (hxs : UniqueMDiffWithinAt (I.prod I') s x) :
    mfderivWithin (I.prod I') I Prod.fst s x =
      ContinuousLinearMap.fst 𝕜 (TangentSpace I x.1) (TangentSpace I' x.2) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] (I' : ModelWithCorners 𝕜 E' H') {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] [SmoothManifoldWithCorners I' M'] {s : Set (M × M')} {x : M × M'} (hxs : UniqueMDiffWithinAt (I.prod I') s x)
Goal: mfderivWithin (I.prod I') I Prod.fst s x = ContinuousLinearMap.fst 𝕜 (TangentSpace I x.1) (TangentSpace I' x.2)
-- tangentMap_prod_fst
@[simp, mfld_simps]
theorem tangentMap_prod_fst {p : TangentBundle (I.prod I') (M × M')} :
    tangentMap (I.prod I') I Prod.fst p = ⟨p.proj.1, p.2.1⟩ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] (I' : ModelWithCorners 𝕜 E' H') {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] [SmoothManifoldWithCorners I' M'] {p : TangentBundle (I.prod I') (M × M')}
Goal: tangentMap (I.prod I') I Prod.fst p = { proj := p.proj.1, snd := p.snd.1 }
-- tangentMapWithin_prod_fst
theorem tangentMapWithin_prod_fst {s : Set (M × M')} {p : TangentBundle (I.prod I') (M × M')}
    (hs : UniqueMDiffWithinAt (I.prod I') s p.proj) :
    tangentMapWithin (I.prod I') I Prod.fst s p = ⟨p.proj.1, p.2.1⟩ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] (I' : ModelWithCorners 𝕜 E' H') {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] [SmoothManifoldWithCorners I' M'] {s : Set (M × M')} {p : TangentBundle (I.prod I') (M × M')} (hs : UniqueMDiffWithinAt (I.prod I') s p.proj)
Goal: tangentMapWithin (I.prod I') I Prod.fst s p = { proj := p.proj.1, snd := p.snd.1 }
-- hasMFDerivAt_snd
theorem hasMFDerivAt_snd (x : M × M') :
    HasMFDerivAt (I.prod I') I' Prod.snd x
      (ContinuousLinearMap.snd 𝕜 (TangentSpace I x.1) (TangentSpace I' x.2)) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] (I' : ModelWithCorners 𝕜 E' H') {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] [SmoothManifoldWithCorners I' M'] (x : M × M')
Goal: HasMFDerivAt (I.prod I') I' Prod.snd x (ContinuousLinearMap.snd 𝕜 (TangentSpace I x.1) (TangentSpace I' x.2))
-- hasMFDerivWithinAt_snd
theorem hasMFDerivWithinAt_snd (s : Set (M × M')) (x : M × M') :
    HasMFDerivWithinAt (I.prod I') I' Prod.snd s x
      (ContinuousLinearMap.snd 𝕜 (TangentSpace I x.1) (TangentSpace I' x.2)) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] (I' : ModelWithCorners 𝕜 E' H') {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] [SmoothManifoldWithCorners I' M'] (s : Set (M × M')) (x : M × M')
Goal: HasMFDerivWithinAt (I.prod I') I' Prod.snd s x (ContinuousLinearMap.snd 𝕜 (TangentSpace I x.1) (TangentSpace I' x.2))
-- mdifferentiableAt_snd
theorem mdifferentiableAt_snd {x : M × M'} : MDifferentiableAt (I.prod I') I' Prod.snd x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] (I' : ModelWithCorners 𝕜 E' H') {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] [SmoothManifoldWithCorners I' M'] {x : M × M'}
Goal: MDifferentiableAt (I.prod I') I' Prod.snd x
-- mdifferentiableWithinAt_snd
theorem mdifferentiableWithinAt_snd {s : Set (M × M')} {x : M × M'} :
    MDifferentiableWithinAt (I.prod I') I' Prod.snd s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] (I' : ModelWithCorners 𝕜 E' H') {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] [SmoothManifoldWithCorners I' M'] {s : Set (M × M')} {x : M × M'}
Goal: MDifferentiableWithinAt (I.prod I') I' Prod.snd s x
-- mdifferentiable_snd
theorem mdifferentiable_snd : MDifferentiable (I.prod I') I' (Prod.snd : M × M' → M') 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] (I' : ModelWithCorners 𝕜 E' H') {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] [SmoothManifoldWithCorners I' M']
Goal: MDifferentiable (I.prod I') I' Prod.snd
-- mdifferentiableOn_snd
theorem mdifferentiableOn_snd {s : Set (M × M')} : MDifferentiableOn (I.prod I') I' Prod.snd s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] (I' : ModelWithCorners 𝕜 E' H') {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] [SmoothManifoldWithCorners I' M'] {s : Set (M × M')}
Goal: MDifferentiableOn (I.prod I') I' Prod.snd s
-- mfderiv_snd
@[simp, mfld_simps]
theorem mfderiv_snd {x : M × M'} :
    mfderiv (I.prod I') I' Prod.snd x =
      ContinuousLinearMap.snd 𝕜 (TangentSpace I x.1) (TangentSpace I' x.2) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] (I' : ModelWithCorners 𝕜 E' H') {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] [SmoothManifoldWithCorners I' M'] {x : M × M'}
Goal: mfderiv (I.prod I') I' Prod.snd x = ContinuousLinearMap.snd 𝕜 (TangentSpace I x.1) (TangentSpace I' x.2)
-- mfderivWithin_snd
theorem mfderivWithin_snd {s : Set (M × M')} {x : M × M'}
    (hxs : UniqueMDiffWithinAt (I.prod I') s x) :
    mfderivWithin (I.prod I') I' Prod.snd s x =
      ContinuousLinearMap.snd 𝕜 (TangentSpace I x.1) (TangentSpace I' x.2) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] (I' : ModelWithCorners 𝕜 E' H') {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] [SmoothManifoldWithCorners I' M'] {s : Set (M × M')} {x : M × M'} (hxs : UniqueMDiffWithinAt (I.prod I') s x)
Goal: mfderivWithin (I.prod I') I' Prod.snd s x = ContinuousLinearMap.snd 𝕜 (TangentSpace I x.1) (TangentSpace I' x.2)
-- tangentMap_prod_snd
@[simp, mfld_simps]
theorem tangentMap_prod_snd {p : TangentBundle (I.prod I') (M × M')} :
    tangentMap (I.prod I') I' Prod.snd p = ⟨p.proj.2, p.2.2⟩ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] (I' : ModelWithCorners 𝕜 E' H') {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] [SmoothManifoldWithCorners I' M'] {p : TangentBundle (I.prod I') (M × M')}
Goal: tangentMap (I.prod I') I' Prod.snd p = { proj := p.proj.2, snd := p.snd.2 }
-- tangentMapWithin_prod_snd
theorem tangentMapWithin_prod_snd {s : Set (M × M')} {p : TangentBundle (I.prod I') (M × M')}
    (hs : UniqueMDiffWithinAt (I.prod I') s p.proj) :
    tangentMapWithin (I.prod I') I' Prod.snd s p = ⟨p.proj.2, p.2.2⟩ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] (I' : ModelWithCorners 𝕜 E' H') {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] [SmoothManifoldWithCorners I' M'] {s : Set (M × M')} {p : TangentBundle (I.prod I') (M × M')} (hs : UniqueMDiffWithinAt (I.prod I') s p.proj)
Goal: tangentMapWithin (I.prod I') I' Prod.snd s p = { proj := p.proj.2, snd := p.snd.2 }
-- MDifferentiableAt.mfderiv_prod
theorem MDifferentiableAt.mfderiv_prod {f : M → M'} {g : M → M''} {x : M}
    (hf : MDifferentiableAt I I' f x) (hg : MDifferentiableAt I I'' g x) :
    mfderiv I (I'.prod I'') (fun x => (f x, g x)) x =
      (mfderiv I I' f x).prod (mfderiv I I'' g x) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] [SmoothManifoldWithCorners I' M'] {E'' : Type u_8} [NormedAddCommGroup E''] [NormedSpace 𝕜 E''] {H'' : Type u_9} [TopologicalSpace H''] {I'' : ModelWithCorners 𝕜 E'' H''} {M'' : Type u_10} [TopologicalSpace M''] [ChartedSpace H'' M''] [SmoothManifoldWithCorners I'' M''] {f : M → M'} {g : M → M''} {x : M} (hf : MDifferentiableAt I I' f x) (hg : MDifferentiableAt I I'' g x)
Goal: mfderiv I (I'.prod I'') (fun x => (f x, g x)) x = (mfderiv I I' f x).prod (mfderiv I I'' g x)
-- mfderiv_prod_left
theorem mfderiv_prod_left {x₀ : M} {y₀ : M'} :
    mfderiv I (I.prod I') (fun x => (x, y₀)) x₀ =
      ContinuousLinearMap.inl 𝕜 (TangentSpace I x₀) (TangentSpace I' y₀) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] (I' : ModelWithCorners 𝕜 E' H') {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] [SmoothManifoldWithCorners I' M'] {x₀ : M} {y₀ : M'}
Goal: mfderiv I (I.prod I') (fun x => (x, y₀)) x₀ = ContinuousLinearMap.inl 𝕜 (TangentSpace I x₀) (TangentSpace I' y₀)
-- mfderiv_prod_right
theorem mfderiv_prod_right {x₀ : M} {y₀ : M'} :
    mfderiv I' (I.prod I') (fun y => (x₀, y)) y₀ =
      ContinuousLinearMap.inr 𝕜 (TangentSpace I x₀) (TangentSpace I' y₀) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] (I' : ModelWithCorners 𝕜 E' H') {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] [SmoothManifoldWithCorners I' M'] {x₀ : M} {y₀ : M'}
Goal: mfderiv I' (I.prod I') (fun y => (x₀, y)) y₀ = ContinuousLinearMap.inr 𝕜 (TangentSpace I x₀) (TangentSpace I' y₀)
-- mfderiv_prod_eq_add
theorem mfderiv_prod_eq_add {f : M × M' → M''} {p : M × M'}
    (hf : MDifferentiableAt (I.prod I') I'' f p) :
    mfderiv (I.prod I') I'' f p =
      show E × E' →L[𝕜] E'' from
        mfderiv (I.prod I') I'' (fun z : M × M' => f (z.1, p.2)) p +
          mfderiv (I.prod I') I'' (fun z : M × M' => f (p.1, z.2)) p 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] (I' : ModelWithCorners 𝕜 E' H') {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] [SmoothManifoldWithCorners I' M'] {E'' : Type u_8} [NormedAddCommGroup E''] [NormedSpace 𝕜 E''] {H'' : Type u_9} [TopologicalSpace H''] (I'' : ModelWithCorners 𝕜 E'' H'') {M'' : Type u_10} [TopologicalSpace M''] [ChartedSpace H'' M''] [SmoothManifoldWithCorners I'' M''] {f : M × M' → M''} {p : M × M'} (hf : MDifferentiableAt (I.prod I') I'' f p)
Goal: mfderiv (I.prod I') I'' f p =
  let_fun this := mfderiv (I.prod I') I'' (fun z => f (z.1, p.2)) p + mfderiv (I.prod I') I'' (fun z => f (p.1, z.2)) p;
  this
-- HasMFDerivAt.add
theorem HasMFDerivAt.add (hf : HasMFDerivAt I 𝓘(𝕜, E') f z f')
    (hg : HasMFDerivAt I 𝓘(𝕜, E') g z g') : HasMFDerivAt I 𝓘(𝕜, E') (f + g) z (f' + g') 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {z : M} {f g : M → E'} {f' g' : TangentSpace I z →L[𝕜] E'} (hf : HasMFDerivAt I 𝓘(𝕜, E') f z f') (hg : HasMFDerivAt I 𝓘(𝕜, E') g z g')
Goal: HasMFDerivAt I 𝓘(𝕜, E') (f + g) z (f' + g')
-- MDifferentiableAt.add
theorem MDifferentiableAt.add (hf : MDifferentiableAt I 𝓘(𝕜, E') f z)
    (hg : MDifferentiableAt I 𝓘(𝕜, E') g z) : MDifferentiableAt I 𝓘(𝕜, E') (f + g) z 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {z : M} {f g : M → E'} (hf : MDifferentiableAt I 𝓘(𝕜, E') f z) (hg : MDifferentiableAt I 𝓘(𝕜, E') g z)
Goal: MDifferentiableAt I 𝓘(𝕜, E') (f + g) z
-- MDifferentiable.add
theorem MDifferentiable.add (hf : MDifferentiable I 𝓘(𝕜, E') f)
    (hg : MDifferentiable I 𝓘(𝕜, E') g) : MDifferentiable I 𝓘(𝕜, E') (f + g) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {f g : M → E'} (hf : MDifferentiable I 𝓘(𝕜, E') f) (hg : MDifferentiable I 𝓘(𝕜, E') g)
Goal: MDifferentiable I 𝓘(𝕜, E') (f + g)
-- mfderiv_add
theorem mfderiv_add (hf : MDifferentiableAt I 𝓘(𝕜, E') f z)
    (hg : MDifferentiableAt I 𝓘(𝕜, E') g z) :
    (by exact mfderiv I 𝓘(𝕜, E') (f + g) z : TangentSpace I z →L[𝕜] E') =
      (by exact mfderiv I 𝓘(𝕜, E') f z) + (by exact mfderiv I 𝓘(𝕜, E') g z) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {z : M} {f g : M → E'} (hf : MDifferentiableAt I 𝓘(𝕜, E') f z) (hg : MDifferentiableAt I 𝓘(𝕜, E') g z)
Goal: mfderiv I 𝓘(𝕜, E') (f + g) z = mfderiv I 𝓘(𝕜, E') f z + mfderiv I 𝓘(𝕜, E') g z