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Theorem ringcinv 42540
Description: An inverse in the category of unital rings is the converse operation. (Contributed by AV, 14-Feb-2020.)
Hypotheses
Ref Expression
ringcsect.c 𝐶 = (RingCat‘𝑈)
ringcsect.b 𝐵 = (Base‘𝐶)
ringcsect.u (𝜑𝑈𝑉)
ringcsect.x (𝜑𝑋𝐵)
ringcsect.y (𝜑𝑌𝐵)
ringcinv.n 𝑁 = (Inv‘𝐶)
Assertion
Ref Expression
ringcinv (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 ↔ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = 𝐹)))

Proof of Theorem ringcinv
StepHypRef Expression
1 ringcsect.b . . 3 𝐵 = (Base‘𝐶)
2 ringcinv.n . . 3 𝑁 = (Inv‘𝐶)
3 ringcsect.u . . . 4 (𝜑𝑈𝑉)
4 ringcsect.c . . . . 5 𝐶 = (RingCat‘𝑈)
54ringccat 42532 . . . 4 (𝑈𝑉𝐶 ∈ Cat)
63, 5syl 17 . . 3 (𝜑𝐶 ∈ Cat)
7 ringcsect.x . . 3 (𝜑𝑋𝐵)
8 ringcsect.y . . 3 (𝜑𝑌𝐵)
9 eqid 2758 . . 3 (Sect‘𝐶) = (Sect‘𝐶)
101, 2, 6, 7, 8, 9isinv 16619 . 2 (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 ↔ (𝐹(𝑋(Sect‘𝐶)𝑌)𝐺𝐺(𝑌(Sect‘𝐶)𝑋)𝐹)))
11 eqid 2758 . . . . . 6 (Base‘𝑋) = (Base‘𝑋)
124, 1, 3, 7, 8, 11, 9ringcsect 42539 . . . . 5 (𝜑 → (𝐹(𝑋(Sect‘𝐶)𝑌)𝐺 ↔ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋)))))
13 df-3an 1074 . . . . 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 276 . . . 4 (𝜑 → (𝐹(𝑋(Sect‘𝐶)𝑌)𝐺 ↔ ((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋)))))
15 eqid 2758 . . . . . 6 (Base‘𝑌) = (Base‘𝑌)
164, 1, 3, 8, 7, 15, 9ringcsect 42539 . . . . 5 (𝜑 → (𝐺(𝑌(Sect‘𝐶)𝑋)𝐹 ↔ (𝐺 ∈ (𝑌 RingHom 𝑋) ∧ 𝐹 ∈ (𝑋 RingHom 𝑌) ∧ (𝐹𝐺) = ( I ↾ (Base‘𝑌)))))
17 3ancoma 1084 . . . . . 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 1074 . . . . . 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 264 . . . . 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 276 . . . 4 (𝜑 → (𝐺(𝑌(Sect‘𝐶)𝑋)𝐹 ↔ ((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐹𝐺) = ( I ↾ (Base‘𝑌)))))
2114, 20anbi12d 749 . . 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 906 . . 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 276 . 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 819 . . . . . 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 473 . . . . 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 18926 . . . . . . . . . 10 (𝐹 ∈ (𝑋 RingHom 𝑌) → 𝐹:(Base‘𝑋)⟶(Base‘𝑌))
2715, 11rhmf 18926 . . . . . . . . . 10 (𝐺 ∈ (𝑌 RingHom 𝑋) → 𝐺:(Base‘𝑌)⟶(Base‘𝑋))
2826, 27anim12i 591 . . . . . . . . 9 ((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) → (𝐹:(Base‘𝑋)⟶(Base‘𝑌) ∧ 𝐺:(Base‘𝑌)⟶(Base‘𝑋)))
2928ad2antlr 765 . . . . . . . 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 479 . . . . . . . . 9 ((((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹𝐺) = ( I ↾ (Base‘𝑌))) → (𝐹𝐺) = ( I ↾ (Base‘𝑌)))
3130adantl 473 . . . . . . . 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)))
32 simpr 479 . . . . . . . . 9 (((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋))) → (𝐺𝐹) = ( I ↾ (Base‘𝑋)))
3332ad2antrl 766 . . . . . . . 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)))) → (𝐺𝐹) = ( I ↾ (Base‘𝑋)))
3429, 31, 33jca32 559 . . . . . . 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)))))
3534adantl 473 . . . . . 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)))))
36 fcof1o 6712 . . . . . . 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))
37 eqcom 2765 . . . . . . . 8 (𝐹 = 𝐺𝐺 = 𝐹)
3837anbi2i 732 . . . . . . 7 ((𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌) ∧ 𝐹 = 𝐺) ↔ (𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌) ∧ 𝐺 = 𝐹))
3936, 38sylib 208 . . . . . 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))
4035, 39syl 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))
41 anass 684 . . . . 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)))
4225, 40, 41sylanbrc 701 . . . 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))
4311, 15isrim 18933 . . . . . . 7 ((𝑋𝐵𝑌𝐵) → (𝐹 ∈ (𝑋 RingIso 𝑌) ↔ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌))))
447, 8, 43syl2anc 696 . . . . . 6 (𝜑 → (𝐹 ∈ (𝑋 RingIso 𝑌) ↔ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌))))
4544anbi1d 743 . . . . 5 (𝜑 → ((𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = 𝐹) ↔ ((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌)) ∧ 𝐺 = 𝐹)))
4645adantr 472 . . . 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)))
4742, 46mpbird 247 . . 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))
4811, 15rimrhm 18935 . . . . . 6 (𝐹 ∈ (𝑋 RingIso 𝑌) → 𝐹 ∈ (𝑋 RingHom 𝑌))
4948ad2antrl 766 . . . . 5 ((𝜑 ∧ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = 𝐹)) → 𝐹 ∈ (𝑋 RingHom 𝑌))
50 isrim0 18923 . . . . . . . . . . 11 ((𝑋𝐵𝑌𝐵) → (𝐹 ∈ (𝑋 RingIso 𝑌) ↔ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐹 ∈ (𝑌 RingHom 𝑋))))
517, 8, 50syl2anc 696 . . . . . . . . . 10 (𝜑 → (𝐹 ∈ (𝑋 RingIso 𝑌) ↔ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐹 ∈ (𝑌 RingHom 𝑋))))
52 eleq1 2825 . . . . . . . . . . . 12 (𝐹 = 𝐺 → (𝐹 ∈ (𝑌 RingHom 𝑋) ↔ 𝐺 ∈ (𝑌 RingHom 𝑋)))
5352eqcoms 2766 . . . . . . . . . . 11 (𝐺 = 𝐹 → (𝐹 ∈ (𝑌 RingHom 𝑋) ↔ 𝐺 ∈ (𝑌 RingHom 𝑋)))
5453anbi2d 742 . . . . . . . . . 10 (𝐺 = 𝐹 → ((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐹 ∈ (𝑌 RingHom 𝑋)) ↔ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋))))
5551, 54sylan9bbr 739 . . . . . . . . 9 ((𝐺 = 𝐹𝜑) → (𝐹 ∈ (𝑋 RingIso 𝑌) ↔ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋))))
56 simpr 479 . . . . . . . . 9 ((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) → 𝐺 ∈ (𝑌 RingHom 𝑋))
5755, 56syl6bi 243 . . . . . . . 8 ((𝐺 = 𝐹𝜑) → (𝐹 ∈ (𝑋 RingIso 𝑌) → 𝐺 ∈ (𝑌 RingHom 𝑋)))
5857com12 32 . . . . . . 7 (𝐹 ∈ (𝑋 RingIso 𝑌) → ((𝐺 = 𝐹𝜑) → 𝐺 ∈ (𝑌 RingHom 𝑋)))
5958expdimp 452 . . . . . 6 ((𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = 𝐹) → (𝜑𝐺 ∈ (𝑌 RingHom 𝑋)))
6059impcom 445 . . . . 5 ((𝜑 ∧ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = 𝐹)) → 𝐺 ∈ (𝑌 RingHom 𝑋))
61 coeq1 5433 . . . . . . 7 (𝐺 = 𝐹 → (𝐺𝐹) = (𝐹𝐹))
6261ad2antll 767 . . . . . 6 ((𝜑 ∧ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = 𝐹)) → (𝐺𝐹) = (𝐹𝐹))
6311, 15rimf1o 18934 . . . . . . . 8 (𝐹 ∈ (𝑋 RingIso 𝑌) → 𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌))
6463ad2antrl 766 . . . . . . 7 ((𝜑 ∧ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = 𝐹)) → 𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌))
65 f1ococnv1 6324 . . . . . . 7 (𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌) → (𝐹𝐹) = ( I ↾ (Base‘𝑋)))
6664, 65syl 17 . . . . . 6 ((𝜑 ∧ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = 𝐹)) → (𝐹𝐹) = ( I ↾ (Base‘𝑋)))
6762, 66eqtrd 2792 . . . . 5 ((𝜑 ∧ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = 𝐹)) → (𝐺𝐹) = ( I ↾ (Base‘𝑋)))
6849, 60, 67jca31 558 . . . 4 ((𝜑 ∧ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = 𝐹)) → ((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋))))
6951biimpcd 239 . . . . . . 7 (𝐹 ∈ (𝑋 RingIso 𝑌) → (𝜑 → (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐹 ∈ (𝑌 RingHom 𝑋))))
7069adantr 472 . . . . . 6 ((𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = 𝐹) → (𝜑 → (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐹 ∈ (𝑌 RingHom 𝑋))))
7170impcom 445 . . . . 5 ((𝜑 ∧ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = 𝐹)) → (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐹 ∈ (𝑌 RingHom 𝑋)))
72 eleq1 2825 . . . . . . 7 (𝐺 = 𝐹 → (𝐺 ∈ (𝑌 RingHom 𝑋) ↔ 𝐹 ∈ (𝑌 RingHom 𝑋)))
7372ad2antll 767 . . . . . 6 ((𝜑 ∧ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = 𝐹)) → (𝐺 ∈ (𝑌 RingHom 𝑋) ↔ 𝐹 ∈ (𝑌 RingHom 𝑋)))
7473anbi2d 742 . . . . 5 ((𝜑 ∧ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = 𝐹)) → ((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ↔ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐹 ∈ (𝑌 RingHom 𝑋))))
7571, 74mpbird 247 . . . 4 ((𝜑 ∧ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = 𝐹)) → (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)))
76 coeq2 5434 . . . . . . 7 (𝐺 = 𝐹 → (𝐹𝐺) = (𝐹𝐹))
7776ad2antll 767 . . . . . 6 ((𝜑 ∧ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = 𝐹)) → (𝐹𝐺) = (𝐹𝐹))
78 f1ococnv2 6322 . . . . . . 7 (𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌) → (𝐹𝐹) = ( I ↾ (Base‘𝑌)))
7964, 78syl 17 . . . . . 6 ((𝜑 ∧ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = 𝐹)) → (𝐹𝐹) = ( I ↾ (Base‘𝑌)))
8077, 79eqtrd 2792 . . . . 5 ((𝜑 ∧ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = 𝐹)) → (𝐹𝐺) = ( I ↾ (Base‘𝑌)))
8175, 67, 80jca31 558 . . . 4 ((𝜑 ∧ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = 𝐹)) → (((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹𝐺) = ( I ↾ (Base‘𝑌))))
8268, 75, 81jca31 558 . . 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)))))
8347, 82impbida 913 . 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)))
8410, 23, 833bitrd 294 1 (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 ↔ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = 𝐹)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1630  wcel 2137   class class class wbr 4802   I cid 5171  ccnv 5263  cres 5266  ccom 5268  wf 6043  1-1-ontowf1o 6046  cfv 6047  (class class class)co 6811  Basecbs 16057  Catccat 16524  Sectcsect 16603  Invcinv 16604   RingHom crh 18912   RingIso crs 18913  RingCatcringc 42511
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-rep 4921  ax-sep 4931  ax-nul 4939  ax-pow 4990  ax-pr 5053  ax-un 7112  ax-cnex 10182  ax-resscn 10183  ax-1cn 10184  ax-icn 10185  ax-addcl 10186  ax-addrcl 10187  ax-mulcl 10188  ax-mulrcl 10189  ax-mulcom 10190  ax-addass 10191  ax-mulass 10192  ax-distr 10193  ax-i2m1 10194  ax-1ne0 10195  ax-1rid 10196  ax-rnegex 10197  ax-rrecex 10198  ax-cnre 10199  ax-pre-lttri 10200  ax-pre-lttrn 10201  ax-pre-ltadd 10202  ax-pre-mulgt0 10203
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1633  df-fal 1636  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-nel 3034  df-ral 3053  df-rex 3054  df-reu 3055  df-rmo 3056  df-rab 3057  df-v 3340  df-sbc 3575  df-csb 3673  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-pss 3729  df-nul 4057  df-if 4229  df-pw 4302  df-sn 4320  df-pr 4322  df-tp 4324  df-op 4326  df-uni 4587  df-int 4626  df-iun 4672  df-br 4803  df-opab 4863  df-mpt 4880  df-tr 4903  df-id 5172  df-eprel 5177  df-po 5185  df-so 5186  df-fr 5223  df-we 5225  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-rn 5275  df-res 5276  df-ima 5277  df-pred 5839  df-ord 5885  df-on 5886  df-lim 5887  df-suc 5888  df-iota 6010  df-fun 6049  df-fn 6050  df-f 6051  df-f1 6052  df-fo 6053  df-f1o 6054  df-fv 6055  df-riota 6772  df-ov 6814  df-oprab 6815  df-mpt2 6816  df-om 7229  df-1st 7331  df-2nd 7332  df-wrecs 7574  df-recs 7635  df-rdg 7673  df-1o 7727  df-oadd 7731  df-er 7909  df-map 8023  df-pm 8024  df-ixp 8073  df-en 8120  df-dom 8121  df-sdom 8122  df-fin 8123  df-pnf 10266  df-mnf 10267  df-xr 10268  df-ltxr 10269  df-le 10270  df-sub 10458  df-neg 10459  df-nn 11211  df-2 11269  df-3 11270  df-4 11271  df-5 11272  df-6 11273  df-7 11274  df-8 11275  df-9 11276  df-n0 11483  df-z 11568  df-dec 11684  df-uz 11878  df-fz 12518  df-struct 16059  df-ndx 16060  df-slot 16061  df-base 16063  df-sets 16064  df-ress 16065  df-plusg 16154  df-hom 16166  df-cco 16167  df-0g 16302  df-cat 16528  df-cid 16529  df-homf 16530  df-sect 16606  df-inv 16607  df-ssc 16669  df-resc 16670  df-subc 16671  df-estrc 16962  df-mgm 17441  df-sgrp 17483  df-mnd 17494  df-mhm 17534  df-grp 17624  df-ghm 17857  df-mgp 18688  df-ur 18700  df-ring 18747  df-rnghom 18915  df-rngiso 18916  df-ringc 42513
This theorem is referenced by:  ringciso  42541
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