Theorem: AffineMap.continuous_iff
File: Mathlib/Topology/Algebra/Affine.lean

tactic: constructor
state_before: R : Type u_1
E : Type u_2
F : Type u_3
inst✝⁷ : AddCommGroup E
inst✝⁶ : TopologicalSpace E
inst✝⁵ : AddCommGroup F
inst✝⁴ : TopologicalSpace F
inst✝³ : TopologicalAddGroup F
inst✝² : Ring R
inst✝¹ : Module R E
inst✝ : Module R F
f : E →ᵃ[R] F
⊢ Continuous ⇑f ↔ Continuous ⇑f.linear
state_after: case mp
R : Type u_1
E : Type u_2
F : Type u_3
inst✝⁷ : AddCommGroup E
inst✝⁶ : TopologicalSpace E
inst✝⁵ : AddCommGroup F
inst✝⁴ : TopologicalSpace F
inst✝³ : TopologicalAddGroup F
inst✝² : Ring R
inst✝¹ : Module R E
inst✝ : Module R F
f : E →ᵃ[R] F
⊢ Continuous ⇑f → Continuous ⇑f.linear

case mpr
R : Type u_1
E : Type u_2
F : Type u_3
inst✝⁷ : AddCommGroup E
inst✝⁶ : TopologicalSpace E
inst✝⁵ : AddCommGroup F
inst✝⁴ : TopologicalSpace F
inst✝³ : TopologicalAddGroup F
inst✝² : Ring R
inst✝¹ : Module R E
inst✝ : Module R F
f : E →ᵃ[R] F
⊢ Continuous ⇑f.linear → Continuous ⇑f

tactic: intro hc
state_before: case mp
R : Type u_1
E : Type u_2
F : Type u_3
inst✝⁷ : AddCommGroup E
inst✝⁶ : TopologicalSpace E
inst✝⁵ : AddCommGroup F
inst✝⁴ : TopologicalSpace F
inst✝³ : TopologicalAddGroup F
inst✝² : Ring R
inst✝¹ : Module R E
inst✝ : Module R F
f : E →ᵃ[R] F
⊢ Continuous ⇑f → Continuous ⇑f.linear
state_after: case mp
R : Type u_1
E : Type u_2
F : Type u_3
inst✝⁷ : AddCommGroup E
inst✝⁶ : TopologicalSpace E
inst✝⁵ : AddCommGroup F
inst✝⁴ : TopologicalSpace F
inst✝³ : TopologicalAddGroup F
inst✝² : Ring R
inst✝¹ : Module R E
inst✝ : Module R F
f : E →ᵃ[R] F
hc : Continuous ⇑f
⊢ Continuous ⇑f.linear

tactic: rw [decomp' f]
state_before: case mp
R : Type u_1
E : Type u_2
F : Type u_3
inst✝⁷ : AddCommGroup E
inst✝⁶ : TopologicalSpace E
inst✝⁵ : AddCommGroup F
inst✝⁴ : TopologicalSpace F
inst✝³ : TopologicalAddGroup F
inst✝² : Ring R
inst✝¹ : Module R E
inst✝ : Module R F
f : E →ᵃ[R] F
hc : Continuous ⇑f
⊢ Continuous ⇑f.linear
state_after: case mp
R : Type u_1
E : Type u_2
F : Type u_3
inst✝⁷ : AddCommGroup E
inst✝⁶ : TopologicalSpace E
inst✝⁵ : AddCommGroup F
inst✝⁴ : TopologicalSpace F
inst✝³ : TopologicalAddGroup F
inst✝² : Ring R
inst✝¹ : Module R E
inst✝ : Module R F
f : E →ᵃ[R] F
hc : Continuous ⇑f
⊢ Continuous (⇑f - fun x => f 0)

tactic: exact hc.sub continuous_const
state_before: case mp
R : Type u_1
E : Type u_2
F : Type u_3
inst✝⁷ : AddCommGroup E
inst✝⁶ : TopologicalSpace E
inst✝⁵ : AddCommGroup F
inst✝⁴ : TopologicalSpace F
inst✝³ : TopologicalAddGroup F
inst✝² : Ring R
inst✝¹ : Module R E
inst✝ : Module R F
f : E →ᵃ[R] F
hc : Continuous ⇑f
⊢ Continuous (⇑f - fun x => f 0)
state_after: no goals

tactic: intro hc
state_before: case mpr
R : Type u_1
E : Type u_2
F : Type u_3
inst✝⁷ : AddCommGroup E
inst✝⁶ : TopologicalSpace E
inst✝⁵ : AddCommGroup F
inst✝⁴ : TopologicalSpace F
inst✝³ : TopologicalAddGroup F
inst✝² : Ring R
inst✝¹ : Module R E
inst✝ : Module R F
f : E →ᵃ[R] F
⊢ Continuous ⇑f.linear → Continuous ⇑f
state_after: case mpr
R : Type u_1
E : Type u_2
F : Type u_3
inst✝⁷ : AddCommGroup E
inst✝⁶ : TopologicalSpace E
inst✝⁵ : AddCommGroup F
inst✝⁴ : TopologicalSpace F
inst✝³ : TopologicalAddGroup F
inst✝² : Ring R
inst✝¹ : Module R E
inst✝ : Module R F
f : E →ᵃ[R] F
hc : Continuous ⇑f.linear
⊢ Continuous ⇑f

tactic: rw [decomp f]
state_before: case mpr
R : Type u_1
E : Type u_2
F : Type u_3
inst✝⁷ : AddCommGroup E
inst✝⁶ : TopologicalSpace E
inst✝⁵ : AddCommGroup F
inst✝⁴ : TopologicalSpace F
inst✝³ : TopologicalAddGroup F
inst✝² : Ring R
inst✝¹ : Module R E
inst✝ : Module R F
f : E →ᵃ[R] F
hc : Continuous ⇑f.linear
⊢ Continuous ⇑f
state_after: case mpr
R : Type u_1
E : Type u_2
F : Type u_3
inst✝⁷ : AddCommGroup E
inst✝⁶ : TopologicalSpace E
inst✝⁵ : AddCommGroup F
inst✝⁴ : TopologicalSpace F
inst✝³ : TopologicalAddGroup F
inst✝² : Ring R
inst✝¹ : Module R E
inst✝ : Module R F
f : E →ᵃ[R] F
hc : Continuous ⇑f.linear
⊢ Continuous (⇑f.linear + fun x => f 0)

tactic: exact hc.add continuous_const
state_before: case mpr
R : Type u_1
E : Type u_2
F : Type u_3
inst✝⁷ : AddCommGroup E
inst✝⁶ : TopologicalSpace E
inst✝⁵ : AddCommGroup F
inst✝⁴ : TopologicalSpace F
inst✝³ : TopologicalAddGroup F
inst✝² : Ring R
inst✝¹ : Module R E
inst✝ : Module R F
f : E →ᵃ[R] F
hc : Continuous ⇑f.linear
⊢ Continuous (⇑f.linear + fun x => f 0)
state_after: no goals