Theorem: IsLocalization.localization_localization_isLocalization_of_has_all_units
File: Mathlib/RingTheory/Localization/LocalizationLocalization.lean

tactic: convert localization_localization_isLocalization M N T using 1
state_before: R : Type u_3
inst✝⁹ : CommRing R
M : Submonoid R
S : Type u_1
inst✝⁸ : CommRing S
inst✝⁷ : Algebra R S
P : Type ?u.238314
inst✝⁶ : CommRing P
N : Submonoid S
T : Type u_2
inst✝⁵ : CommRing T
inst✝⁴ : Algebra R T
inst✝³ : Algebra S T
inst✝² : IsScalarTower R S T
inst✝¹ : IsLocalization M S
inst✝ : IsLocalization N T
H : ∀ (x : S), IsUnit x → x ∈ N
⊢ IsLocalization (Submonoid.comap (algebraMap R S) N) T

tactic: dsimp [localizationLocalizationSubmodule]
state_before: case h.e'_3
R : Type u_3
inst✝⁹ : CommRing R
M : Submonoid R
S : Type u_1
inst✝⁸ : CommRing S
inst✝⁷ : Algebra R S
P : Type ?u.238314
inst✝⁶ : CommRing P
N : Submonoid S
T : Type u_2
inst✝⁵ : CommRing T
inst✝⁴ : Algebra R T
inst✝³ : Algebra S T
inst✝² : IsScalarTower R S T
inst✝¹ : IsLocalization M S
inst✝ : IsLocalization N T
H : ∀ (x : S), IsUnit x → x ∈ N
⊢ Submonoid.comap (algebraMap R S) N = localizationLocalizationSubmodule M N

tactic: congr
state_before: case h.e'_3
R : Type u_3
inst✝⁹ : CommRing R
M : Submonoid R
S : Type u_1
inst✝⁸ : CommRing S
inst✝⁷ : Algebra R S
P : Type ?u.238314
inst✝⁶ : CommRing P
N : Submonoid S
T : Type u_2
inst✝⁵ : CommRing T
inst✝⁴ : Algebra R T
inst✝³ : Algebra S T
inst✝² : IsScalarTower R S T
inst✝¹ : IsLocalization M S
inst✝ : IsLocalization N T
H : ∀ (x : S), IsUnit x → x ∈ N
⊢ Submonoid.comap (algebraMap R S) N = Submonoid.comap (algebraMap R S) (N ⊔ Submonoid.map (algebraMap R S) M)

tactic: symm
state_before: case h.e'_3.e_S
R : Type u_3
inst✝⁹ : CommRing R
M : Submonoid R
S : Type u_1
inst✝⁸ : CommRing S
inst✝⁷ : Algebra R S
P : Type ?u.238314
inst✝⁶ : CommRing P
N : Submonoid S
T : Type u_2
inst✝⁵ : CommRing T
inst✝⁴ : Algebra R T
inst✝³ : Algebra S T
inst✝² : IsScalarTower R S T
inst✝¹ : IsLocalization M S
inst✝ : IsLocalization N T
H : ∀ (x : S), IsUnit x → x ∈ N
⊢ N = N ⊔ Submonoid.map (algebraMap R S) M

tactic: rw [sup_eq_left]
state_before: case h.e'_3.e_S
R : Type u_3
inst✝⁹ : CommRing R
M : Submonoid R
S : Type u_1
inst✝⁸ : CommRing S
inst✝⁷ : Algebra R S
P : Type ?u.238314
inst✝⁶ : CommRing P
N : Submonoid S
T : Type u_2
inst✝⁵ : CommRing T
inst✝⁴ : Algebra R T
inst✝³ : Algebra S T
inst✝² : IsScalarTower R S T
inst✝¹ : IsLocalization M S
inst✝ : IsLocalization N T
H : ∀ (x : S), IsUnit x → x ∈ N
⊢ N ⊔ Submonoid.map (algebraMap R S) M = N

tactic: rintro _ ⟨x, hx, rfl⟩
state_before: case h.e'_3.e_S
R : Type u_3
inst✝⁹ : CommRing R
M : Submonoid R
S : Type u_1
inst✝⁸ : CommRing S
inst✝⁷ : Algebra R S
P : Type ?u.238314
inst✝⁶ : CommRing P
N : Submonoid S
T : Type u_2
inst✝⁵ : CommRing T
inst✝⁴ : Algebra R T
inst✝³ : Algebra S T
inst✝² : IsScalarTower R S T
inst✝¹ : IsLocalization M S
inst✝ : IsLocalization N T
H : ∀ (x : S), IsUnit x → x ∈ N
⊢ Submonoid.map (algebraMap R S) M ≤ N

tactic: exact H _ (IsLocalization.map_units _ ⟨x, hx⟩)
state_before: case h.e'_3.e_S.intro.intro
R : Type u_3
inst✝⁹ : CommRing R
M : Submonoid R
S : Type u_1
inst✝⁸ : CommRing S
inst✝⁷ : Algebra R S
P : Type ?u.238314
inst✝⁶ : CommRing P
N : Submonoid S
T : Type u_2
inst✝⁵ : CommRing T
inst✝⁴ : Algebra R T
inst✝³ : Algebra S T
inst✝² : IsScalarTower R S T
inst✝¹ : IsLocalization M S
inst✝ : IsLocalization N T
H : ∀ (x : S), IsUnit x → x ∈ N
x : R
hx : x ∈ ↑M
⊢ ↑(algebraMap R S) x ∈ N