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Theorem ringcinvALTV 43827
Description: An inverse in the category of rings is the converse operation. (Contributed by AV, 14-Feb-2020.) (New usage is discouraged.)
Hypotheses
Ref Expression
ringcsectALTV.c 𝐶 = (RingCatALTV‘𝑈)
ringcsectALTV.b 𝐵 = (Base‘𝐶)
ringcsectALTV.u (𝜑𝑈𝑉)
ringcsectALTV.x (𝜑𝑋𝐵)
ringcsectALTV.y (𝜑𝑌𝐵)
ringcinvALTV.n 𝑁 = (Inv‘𝐶)
Assertion
Ref Expression
ringcinvALTV (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 ↔ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = 𝐹)))

Proof of Theorem ringcinvALTV
StepHypRef Expression
1 ringcsectALTV.b . . 3 𝐵 = (Base‘𝐶)
2 ringcinvALTV.n . . 3 𝑁 = (Inv‘𝐶)
3 ringcsectALTV.u . . . 4 (𝜑𝑈𝑉)
4 ringcsectALTV.c . . . . 5 𝐶 = (RingCatALTV‘𝑈)
54ringccatALTV 43824 . . . 4 (𝑈𝑉𝐶 ∈ Cat)
63, 5syl 17 . . 3 (𝜑𝐶 ∈ Cat)
7 ringcsectALTV.x . . 3 (𝜑𝑋𝐵)
8 ringcsectALTV.y . . 3 (𝜑𝑌𝐵)
9 eqid 2797 . . 3 (Sect‘𝐶) = (Sect‘𝐶)
101, 2, 6, 7, 8, 9isinv 16863 . 2 (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 ↔ (𝐹(𝑋(Sect‘𝐶)𝑌)𝐺𝐺(𝑌(Sect‘𝐶)𝑋)𝐹)))
11 eqid 2797 . . . . . 6 (Base‘𝑋) = (Base‘𝑋)
124, 1, 3, 7, 8, 11, 9ringcsectALTV 43826 . . . . 5 (𝜑 → (𝐹(𝑋(Sect‘𝐶)𝑌)𝐺 ↔ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋)))))
13 df-3an 1082 . . . . 5 ((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋))) ↔ ((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋))))
1412, 13syl6bb 288 . . . 4 (𝜑 → (𝐹(𝑋(Sect‘𝐶)𝑌)𝐺 ↔ ((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋)))))
15 eqid 2797 . . . . . 6 (Base‘𝑌) = (Base‘𝑌)
164, 1, 3, 8, 7, 15, 9ringcsectALTV 43826 . . . . 5 (𝜑 → (𝐺(𝑌(Sect‘𝐶)𝑋)𝐹 ↔ (𝐺 ∈ (𝑌 RingHom 𝑋) ∧ 𝐹 ∈ (𝑋 RingHom 𝑌) ∧ (𝐹𝐺) = ( I ↾ (Base‘𝑌)))))
17 3ancoma 1091 . . . . . 6 ((𝐺 ∈ (𝑌 RingHom 𝑋) ∧ 𝐹 ∈ (𝑋 RingHom 𝑌) ∧ (𝐹𝐺) = ( I ↾ (Base‘𝑌))) ↔ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋) ∧ (𝐹𝐺) = ( I ↾ (Base‘𝑌))))
18 df-3an 1082 . . . . . 6 ((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋) ∧ (𝐹𝐺) = ( I ↾ (Base‘𝑌))) ↔ ((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐹𝐺) = ( I ↾ (Base‘𝑌))))
1917, 18bitri 276 . . . . 5 ((𝐺 ∈ (𝑌 RingHom 𝑋) ∧ 𝐹 ∈ (𝑋 RingHom 𝑌) ∧ (𝐹𝐺) = ( I ↾ (Base‘𝑌))) ↔ ((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐹𝐺) = ( I ↾ (Base‘𝑌))))
2016, 19syl6bb 288 . . . 4 (𝜑 → (𝐺(𝑌(Sect‘𝐶)𝑋)𝐹 ↔ ((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐹𝐺) = ( I ↾ (Base‘𝑌)))))
2114, 20anbi12d 630 . . 3 (𝜑 → ((𝐹(𝑋(Sect‘𝐶)𝑌)𝐺𝐺(𝑌(Sect‘𝐶)𝑋)𝐹) ↔ (((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋))) ∧ ((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐹𝐺) = ( I ↾ (Base‘𝑌))))))
22 anandi 672 . . 3 ((((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋))) ∧ ((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐹𝐺) = ( I ↾ (Base‘𝑌)))) ↔ ((((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋))) ∧ (((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹𝐺) = ( I ↾ (Base‘𝑌)))))
2321, 22syl6bb 288 . 2 (𝜑 → ((𝐹(𝑋(Sect‘𝐶)𝑌)𝐺𝐺(𝑌(Sect‘𝐶)𝑋)𝐹) ↔ ((((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋))) ∧ (((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹𝐺) = ( I ↾ (Base‘𝑌))))))
24 simplrl 773 . . . . . 6 (((((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋))) ∧ (((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹𝐺) = ( I ↾ (Base‘𝑌)))) → 𝐹 ∈ (𝑋 RingHom 𝑌))
2524adantl 482 . . . . 5 ((𝜑 ∧ ((((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋))) ∧ (((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹𝐺) = ( I ↾ (Base‘𝑌))))) → 𝐹 ∈ (𝑋 RingHom 𝑌))
2611, 15rhmf 19172 . . . . . . . . . 10 (𝐹 ∈ (𝑋 RingHom 𝑌) → 𝐹:(Base‘𝑋)⟶(Base‘𝑌))
2715, 11rhmf 19172 . . . . . . . . . 10 (𝐺 ∈ (𝑌 RingHom 𝑋) → 𝐺:(Base‘𝑌)⟶(Base‘𝑋))
2826, 27anim12i 612 . . . . . . . . 9 ((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) → (𝐹:(Base‘𝑋)⟶(Base‘𝑌) ∧ 𝐺:(Base‘𝑌)⟶(Base‘𝑋)))
2928ad2antlr 723 . . . . . . . 8 (((((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋))) ∧ (((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹𝐺) = ( I ↾ (Base‘𝑌)))) → (𝐹:(Base‘𝑋)⟶(Base‘𝑌) ∧ 𝐺:(Base‘𝑌)⟶(Base‘𝑋)))
30 simpr 485 . . . . . . . . . 10 ((((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹𝐺) = ( I ↾ (Base‘𝑌))) → (𝐹𝐺) = ( I ↾ (Base‘𝑌)))
3130adantl 482 . . . . . . . . 9 (((((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋))) ∧ (((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹𝐺) = ( I ↾ (Base‘𝑌)))) → (𝐹𝐺) = ( I ↾ (Base‘𝑌)))
32 simpr 485 . . . . . . . . . 10 (((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋))) → (𝐺𝐹) = ( I ↾ (Base‘𝑋)))
3332ad2antrl 724 . . . . . . . . 9 (((((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋))) ∧ (((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹𝐺) = ( I ↾ (Base‘𝑌)))) → (𝐺𝐹) = ( I ↾ (Base‘𝑋)))
3431, 33jca 512 . . . . . . . 8 (((((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋))) ∧ (((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹𝐺) = ( I ↾ (Base‘𝑌)))) → ((𝐹𝐺) = ( I ↾ (Base‘𝑌)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋))))
3529, 34jca 512 . . . . . . 7 (((((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋))) ∧ (((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹𝐺) = ( I ↾ (Base‘𝑌)))) → ((𝐹:(Base‘𝑋)⟶(Base‘𝑌) ∧ 𝐺:(Base‘𝑌)⟶(Base‘𝑋)) ∧ ((𝐹𝐺) = ( I ↾ (Base‘𝑌)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋)))))
3635adantl 482 . . . . . 6 ((𝜑 ∧ ((((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋))) ∧ (((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹𝐺) = ( I ↾ (Base‘𝑌))))) → ((𝐹:(Base‘𝑋)⟶(Base‘𝑌) ∧ 𝐺:(Base‘𝑌)⟶(Base‘𝑋)) ∧ ((𝐹𝐺) = ( I ↾ (Base‘𝑌)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋)))))
37 fcof1o 6924 . . . . . . 7 (((𝐹:(Base‘𝑋)⟶(Base‘𝑌) ∧ 𝐺:(Base‘𝑌)⟶(Base‘𝑋)) ∧ ((𝐹𝐺) = ( I ↾ (Base‘𝑌)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋)))) → (𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌) ∧ 𝐹 = 𝐺))
38 eqcom 2804 . . . . . . . 8 (𝐹 = 𝐺𝐺 = 𝐹)
3938anbi2i 622 . . . . . . 7 ((𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌) ∧ 𝐹 = 𝐺) ↔ (𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌) ∧ 𝐺 = 𝐹))
4037, 39sylib 219 . . . . . 6 (((𝐹:(Base‘𝑋)⟶(Base‘𝑌) ∧ 𝐺:(Base‘𝑌)⟶(Base‘𝑋)) ∧ ((𝐹𝐺) = ( I ↾ (Base‘𝑌)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋)))) → (𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌) ∧ 𝐺 = 𝐹))
4136, 40syl 17 . . . . 5 ((𝜑 ∧ ((((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋))) ∧ (((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹𝐺) = ( I ↾ (Base‘𝑌))))) → (𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌) ∧ 𝐺 = 𝐹))
42 anass 469 . . . . 5 (((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌)) ∧ 𝐺 = 𝐹) ↔ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ (𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌) ∧ 𝐺 = 𝐹)))
4325, 41, 42sylanbrc 583 . . . 4 ((𝜑 ∧ ((((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋))) ∧ (((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹𝐺) = ( I ↾ (Base‘𝑌))))) → ((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌)) ∧ 𝐺 = 𝐹))
447, 8jca 512 . . . . . . 7 (𝜑 → (𝑋𝐵𝑌𝐵))
4511, 15isrim 19179 . . . . . . 7 ((𝑋𝐵𝑌𝐵) → (𝐹 ∈ (𝑋 RingIso 𝑌) ↔ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌))))
4644, 45syl 17 . . . . . 6 (𝜑 → (𝐹 ∈ (𝑋 RingIso 𝑌) ↔ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌))))
4746anbi1d 629 . . . . 5 (𝜑 → ((𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = 𝐹) ↔ ((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌)) ∧ 𝐺 = 𝐹)))
4847adantr 481 . . . 4 ((𝜑 ∧ ((((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋))) ∧ (((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹𝐺) = ( I ↾ (Base‘𝑌))))) → ((𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = 𝐹) ↔ ((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌)) ∧ 𝐺 = 𝐹)))
4943, 48mpbird 258 . . 3 ((𝜑 ∧ ((((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋))) ∧ (((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹𝐺) = ( I ↾ (Base‘𝑌))))) → (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = 𝐹))
5011, 15rimrhm 19181 . . . . . 6 (𝐹 ∈ (𝑋 RingIso 𝑌) → 𝐹 ∈ (𝑋 RingHom 𝑌))
5150ad2antrl 724 . . . . 5 ((𝜑 ∧ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = 𝐹)) → 𝐹 ∈ (𝑋 RingHom 𝑌))
52 isrim0 19169 . . . . . . . . . . 11 ((𝑋𝐵𝑌𝐵) → (𝐹 ∈ (𝑋 RingIso 𝑌) ↔ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐹 ∈ (𝑌 RingHom 𝑋))))
5344, 52syl 17 . . . . . . . . . 10 (𝜑 → (𝐹 ∈ (𝑋 RingIso 𝑌) ↔ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐹 ∈ (𝑌 RingHom 𝑋))))
54 eleq1 2872 . . . . . . . . . . . 12 (𝐹 = 𝐺 → (𝐹 ∈ (𝑌 RingHom 𝑋) ↔ 𝐺 ∈ (𝑌 RingHom 𝑋)))
5554eqcoms 2805 . . . . . . . . . . 11 (𝐺 = 𝐹 → (𝐹 ∈ (𝑌 RingHom 𝑋) ↔ 𝐺 ∈ (𝑌 RingHom 𝑋)))
5655anbi2d 628 . . . . . . . . . 10 (𝐺 = 𝐹 → ((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐹 ∈ (𝑌 RingHom 𝑋)) ↔ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋))))
5753, 56sylan9bbr 511 . . . . . . . . 9 ((𝐺 = 𝐹𝜑) → (𝐹 ∈ (𝑋 RingIso 𝑌) ↔ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋))))
58 simpr 485 . . . . . . . . 9 ((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) → 𝐺 ∈ (𝑌 RingHom 𝑋))
5957, 58syl6bi 254 . . . . . . . 8 ((𝐺 = 𝐹𝜑) → (𝐹 ∈ (𝑋 RingIso 𝑌) → 𝐺 ∈ (𝑌 RingHom 𝑋)))
6059com12 32 . . . . . . 7 (𝐹 ∈ (𝑋 RingIso 𝑌) → ((𝐺 = 𝐹𝜑) → 𝐺 ∈ (𝑌 RingHom 𝑋)))
6160expdimp 453 . . . . . 6 ((𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = 𝐹) → (𝜑𝐺 ∈ (𝑌 RingHom 𝑋)))
6261impcom 408 . . . . 5 ((𝜑 ∧ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = 𝐹)) → 𝐺 ∈ (𝑌 RingHom 𝑋))
63 coeq1 5621 . . . . . . 7 (𝐺 = 𝐹 → (𝐺𝐹) = (𝐹𝐹))
6463ad2antll 725 . . . . . 6 ((𝜑 ∧ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = 𝐹)) → (𝐺𝐹) = (𝐹𝐹))
6511, 15rimf1o 19180 . . . . . . . 8 (𝐹 ∈ (𝑋 RingIso 𝑌) → 𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌))
6665ad2antrl 724 . . . . . . 7 ((𝜑 ∧ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = 𝐹)) → 𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌))
67 f1ococnv1 6518 . . . . . . 7 (𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌) → (𝐹𝐹) = ( I ↾ (Base‘𝑋)))
6866, 67syl 17 . . . . . 6 ((𝜑 ∧ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = 𝐹)) → (𝐹𝐹) = ( I ↾ (Base‘𝑋)))
6964, 68eqtrd 2833 . . . . 5 ((𝜑 ∧ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = 𝐹)) → (𝐺𝐹) = ( I ↾ (Base‘𝑋)))
7051, 62, 69jca31 515 . . . 4 ((𝜑 ∧ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = 𝐹)) → ((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋))))
7153biimpcd 250 . . . . . . 7 (𝐹 ∈ (𝑋 RingIso 𝑌) → (𝜑 → (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐹 ∈ (𝑌 RingHom 𝑋))))
7271adantr 481 . . . . . 6 ((𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = 𝐹) → (𝜑 → (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐹 ∈ (𝑌 RingHom 𝑋))))
7372impcom 408 . . . . 5 ((𝜑 ∧ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = 𝐹)) → (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐹 ∈ (𝑌 RingHom 𝑋)))
74 eleq1 2872 . . . . . . 7 (𝐺 = 𝐹 → (𝐺 ∈ (𝑌 RingHom 𝑋) ↔ 𝐹 ∈ (𝑌 RingHom 𝑋)))
7574ad2antll 725 . . . . . 6 ((𝜑 ∧ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = 𝐹)) → (𝐺 ∈ (𝑌 RingHom 𝑋) ↔ 𝐹 ∈ (𝑌 RingHom 𝑋)))
7675anbi2d 628 . . . . 5 ((𝜑 ∧ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = 𝐹)) → ((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ↔ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐹 ∈ (𝑌 RingHom 𝑋))))
7773, 76mpbird 258 . . . 4 ((𝜑 ∧ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = 𝐹)) → (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)))
78 coeq2 5622 . . . . . . 7 (𝐺 = 𝐹 → (𝐹𝐺) = (𝐹𝐹))
7978ad2antll 725 . . . . . 6 ((𝜑 ∧ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = 𝐹)) → (𝐹𝐺) = (𝐹𝐹))
80 f1ococnv2 6516 . . . . . . 7 (𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌) → (𝐹𝐹) = ( I ↾ (Base‘𝑌)))
8166, 80syl 17 . . . . . 6 ((𝜑 ∧ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = 𝐹)) → (𝐹𝐹) = ( I ↾ (Base‘𝑌)))
8279, 81eqtrd 2833 . . . . 5 ((𝜑 ∧ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = 𝐹)) → (𝐹𝐺) = ( I ↾ (Base‘𝑌)))
8377, 69, 82jca31 515 . . . 4 ((𝜑 ∧ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = 𝐹)) → (((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹𝐺) = ( I ↾ (Base‘𝑌))))
8470, 77, 83jca31 515 . . 3 ((𝜑 ∧ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = 𝐹)) → ((((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋))) ∧ (((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹𝐺) = ( I ↾ (Base‘𝑌)))))
8549, 84impbida 797 . 2 (𝜑 → (((((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋))) ∧ (((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹𝐺) = ( I ↾ (Base‘𝑌)))) ↔ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = 𝐹)))
8610, 23, 853bitrd 306 1 (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 ↔ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = 𝐹)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1080   = wceq 1525  wcel 2083   class class class wbr 4968   I cid 5354  ccnv 5449  cres 5452  ccom 5454  wf 6228  1-1-ontowf1o 6231  cfv 6232  (class class class)co 7023  Basecbs 16316  Catccat 16768  Sectcsect 16847  Invcinv 16848   RingHom crh 19158   RingIso crs 19159  RingCatALTVcringcALTV 43775
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-rep 5088  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326  ax-cnex 10446  ax-resscn 10447  ax-1cn 10448  ax-icn 10449  ax-addcl 10450  ax-addrcl 10451  ax-mulcl 10452  ax-mulrcl 10453  ax-mulcom 10454  ax-addass 10455  ax-mulass 10456  ax-distr 10457  ax-i2m1 10458  ax-1ne0 10459  ax-1rid 10460  ax-rnegex 10461  ax-rrecex 10462  ax-cnre 10463  ax-pre-lttri 10464  ax-pre-lttrn 10465  ax-pre-ltadd 10466  ax-pre-mulgt0 10467
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-nel 3093  df-ral 3112  df-rex 3113  df-reu 3114  df-rmo 3115  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-pss 3882  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-tp 4483  df-op 4485  df-uni 4752  df-int 4789  df-iun 4833  df-br 4969  df-opab 5031  df-mpt 5048  df-tr 5071  df-id 5355  df-eprel 5360  df-po 5369  df-so 5370  df-fr 5409  df-we 5411  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-pred 6030  df-ord 6076  df-on 6077  df-lim 6078  df-suc 6079  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-f1 6237  df-fo 6238  df-f1o 6239  df-fv 6240  df-riota 6984  df-ov 7026  df-oprab 7027  df-mpo 7028  df-om 7444  df-1st 7552  df-2nd 7553  df-wrecs 7805  df-recs 7867  df-rdg 7905  df-1o 7960  df-oadd 7964  df-er 8146  df-map 8265  df-en 8365  df-dom 8366  df-sdom 8367  df-fin 8368  df-pnf 10530  df-mnf 10531  df-xr 10532  df-ltxr 10533  df-le 10534  df-sub 10725  df-neg 10726  df-nn 11493  df-2 11554  df-3 11555  df-4 11556  df-5 11557  df-6 11558  df-7 11559  df-8 11560  df-9 11561  df-n0 11752  df-z 11836  df-dec 11953  df-uz 12098  df-fz 12747  df-struct 16318  df-ndx 16319  df-slot 16320  df-base 16322  df-sets 16323  df-plusg 16411  df-hom 16422  df-cco 16423  df-0g 16548  df-cat 16772  df-cid 16773  df-sect 16850  df-inv 16851  df-mgm 17685  df-sgrp 17727  df-mnd 17738  df-mhm 17778  df-grp 17868  df-ghm 18101  df-mgp 18934  df-ur 18946  df-ring 18993  df-rnghom 19161  df-rngiso 19162  df-ringcALTV 43777
This theorem is referenced by:  ringcisoALTV  43828
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