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Theorem rngcinv 42408
Description: An inverse in the category of non-unital rings is the converse operation. (Contributed by AV, 28-Feb-2020.)
Hypotheses
Ref Expression
rngcsect.c 𝐶 = (RngCat‘𝑈)
rngcsect.b 𝐵 = (Base‘𝐶)
rngcsect.u (𝜑𝑈𝑉)
rngcsect.x (𝜑𝑋𝐵)
rngcsect.y (𝜑𝑌𝐵)
rngcinv.n 𝑁 = (Inv‘𝐶)
Assertion
Ref Expression
rngcinv (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 ↔ (𝐹 ∈ (𝑋 RngIsom 𝑌) ∧ 𝐺 = 𝐹)))

Proof of Theorem rngcinv
StepHypRef Expression
1 rngcsect.b . . 3 𝐵 = (Base‘𝐶)
2 rngcinv.n . . 3 𝑁 = (Inv‘𝐶)
3 rngcsect.u . . . 4 (𝜑𝑈𝑉)
4 rngcsect.c . . . . 5 𝐶 = (RngCat‘𝑈)
54rngccat 42405 . . . 4 (𝑈𝑉𝐶 ∈ Cat)
63, 5syl 17 . . 3 (𝜑𝐶 ∈ Cat)
7 rngcsect.x . . 3 (𝜑𝑋𝐵)
8 rngcsect.y . . 3 (𝜑𝑌𝐵)
9 eqid 2724 . . 3 (Sect‘𝐶) = (Sect‘𝐶)
101, 2, 6, 7, 8, 9isinv 16542 . 2 (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 ↔ (𝐹(𝑋(Sect‘𝐶)𝑌)𝐺𝐺(𝑌(Sect‘𝐶)𝑋)𝐹)))
11 eqid 2724 . . . . . 6 (Base‘𝑋) = (Base‘𝑋)
124, 1, 3, 7, 8, 11, 9rngcsect 42407 . . . . 5 (𝜑 → (𝐹(𝑋(Sect‘𝐶)𝑌)𝐺 ↔ (𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋)))))
13 df-3an 1074 . . . . 5 ((𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋))) ↔ ((𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋))))
1412, 13syl6bb 276 . . . 4 (𝜑 → (𝐹(𝑋(Sect‘𝐶)𝑌)𝐺 ↔ ((𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋)))))
15 eqid 2724 . . . . . 6 (Base‘𝑌) = (Base‘𝑌)
164, 1, 3, 8, 7, 15, 9rngcsect 42407 . . . . 5 (𝜑 → (𝐺(𝑌(Sect‘𝐶)𝑋)𝐹 ↔ (𝐺 ∈ (𝑌 RngHomo 𝑋) ∧ 𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ (𝐹𝐺) = ( I ↾ (Base‘𝑌)))))
17 3ancoma 1084 . . . . . 6 ((𝐺 ∈ (𝑌 RngHomo 𝑋) ∧ 𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ (𝐹𝐺) = ( I ↾ (Base‘𝑌))) ↔ (𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋) ∧ (𝐹𝐺) = ( I ↾ (Base‘𝑌))))
18 df-3an 1074 . . . . . 6 ((𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋) ∧ (𝐹𝐺) = ( I ↾ (Base‘𝑌))) ↔ ((𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋)) ∧ (𝐹𝐺) = ( I ↾ (Base‘𝑌))))
1917, 18bitri 264 . . . . 5 ((𝐺 ∈ (𝑌 RngHomo 𝑋) ∧ 𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ (𝐹𝐺) = ( I ↾ (Base‘𝑌))) ↔ ((𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋)) ∧ (𝐹𝐺) = ( I ↾ (Base‘𝑌))))
2016, 19syl6bb 276 . . . 4 (𝜑 → (𝐺(𝑌(Sect‘𝐶)𝑋)𝐹 ↔ ((𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋)) ∧ (𝐹𝐺) = ( I ↾ (Base‘𝑌)))))
2114, 20anbi12d 749 . . 3 (𝜑 → ((𝐹(𝑋(Sect‘𝐶)𝑌)𝐺𝐺(𝑌(Sect‘𝐶)𝑋)𝐹) ↔ (((𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋))) ∧ ((𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋)) ∧ (𝐹𝐺) = ( I ↾ (Base‘𝑌))))))
22 anandi 906 . . 3 ((((𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋))) ∧ ((𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋)) ∧ (𝐹𝐺) = ( I ↾ (Base‘𝑌)))) ↔ ((((𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋))) ∧ (((𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 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 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋))) ∧ (((𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹𝐺) = ( I ↾ (Base‘𝑌))))))
24 simplrl 819 . . . . . 6 (((((𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋))) ∧ (((𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹𝐺) = ( I ↾ (Base‘𝑌)))) → 𝐹 ∈ (𝑋 RngHomo 𝑌))
2524adantl 473 . . . . 5 ((𝜑 ∧ ((((𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋))) ∧ (((𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹𝐺) = ( I ↾ (Base‘𝑌))))) → 𝐹 ∈ (𝑋 RngHomo 𝑌))
2611, 15rnghmf 42326 . . . . . . . . . 10 (𝐹 ∈ (𝑋 RngHomo 𝑌) → 𝐹:(Base‘𝑋)⟶(Base‘𝑌))
2715, 11rnghmf 42326 . . . . . . . . . 10 (𝐺 ∈ (𝑌 RngHomo 𝑋) → 𝐺:(Base‘𝑌)⟶(Base‘𝑋))
2826, 27anim12i 591 . . . . . . . . 9 ((𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋)) → (𝐹:(Base‘𝑋)⟶(Base‘𝑌) ∧ 𝐺:(Base‘𝑌)⟶(Base‘𝑋)))
2928ad2antlr 765 . . . . . . . 8 (((((𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋))) ∧ (((𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 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 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹𝐺) = ( I ↾ (Base‘𝑌))) → (𝐹𝐺) = ( I ↾ (Base‘𝑌)))
3130adantl 473 . . . . . . . 8 (((((𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋))) ∧ (((𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹𝐺) = ( I ↾ (Base‘𝑌)))) → (𝐹𝐺) = ( I ↾ (Base‘𝑌)))
32 simpr 479 . . . . . . . . 9 (((𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋))) → (𝐺𝐹) = ( I ↾ (Base‘𝑋)))
3332ad2antrl 766 . . . . . . . 8 (((((𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋))) ∧ (((𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹𝐺) = ( I ↾ (Base‘𝑌)))) → (𝐺𝐹) = ( I ↾ (Base‘𝑋)))
3429, 31, 33jca32 559 . . . . . . 7 (((((𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋))) ∧ (((𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 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 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋))) ∧ (((𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 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 6666 . . . . . . 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 2731 . . . . . . . 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 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋))) ∧ (((𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 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 RngHomo 𝑌) ∧ 𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌)) ∧ 𝐺 = 𝐹) ↔ (𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ (𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌) ∧ 𝐺 = 𝐹)))
4225, 40, 41sylanbrc 701 . . . 4 ((𝜑 ∧ ((((𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋))) ∧ (((𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹𝐺) = ( I ↾ (Base‘𝑌))))) → ((𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌)) ∧ 𝐺 = 𝐹))
4311, 15isrngim 42331 . . . . . . 7 ((𝑋𝐵𝑌𝐵) → (𝐹 ∈ (𝑋 RngIsom 𝑌) ↔ (𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌))))
447, 8, 43syl2anc 696 . . . . . 6 (𝜑 → (𝐹 ∈ (𝑋 RngIsom 𝑌) ↔ (𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌))))
4544anbi1d 743 . . . . 5 (𝜑 → ((𝐹 ∈ (𝑋 RngIsom 𝑌) ∧ 𝐺 = 𝐹) ↔ ((𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌)) ∧ 𝐺 = 𝐹)))
4645adantr 472 . . . 4 ((𝜑 ∧ ((((𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋))) ∧ (((𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹𝐺) = ( I ↾ (Base‘𝑌))))) → ((𝐹 ∈ (𝑋 RngIsom 𝑌) ∧ 𝐺 = 𝐹) ↔ ((𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌)) ∧ 𝐺 = 𝐹)))
4742, 46mpbird 247 . . 3 ((𝜑 ∧ ((((𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋))) ∧ (((𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹𝐺) = ( I ↾ (Base‘𝑌))))) → (𝐹 ∈ (𝑋 RngIsom 𝑌) ∧ 𝐺 = 𝐹))
4811, 15rngimrnghm 42333 . . . . . 6 (𝐹 ∈ (𝑋 RngIsom 𝑌) → 𝐹 ∈ (𝑋 RngHomo 𝑌))
4948ad2antrl 766 . . . . 5 ((𝜑 ∧ (𝐹 ∈ (𝑋 RngIsom 𝑌) ∧ 𝐺 = 𝐹)) → 𝐹 ∈ (𝑋 RngHomo 𝑌))
50 isrngisom 42323 . . . . . . . . . . 11 ((𝑋𝐵𝑌𝐵) → (𝐹 ∈ (𝑋 RngIsom 𝑌) ↔ (𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐹 ∈ (𝑌 RngHomo 𝑋))))
517, 8, 50syl2anc 696 . . . . . . . . . 10 (𝜑 → (𝐹 ∈ (𝑋 RngIsom 𝑌) ↔ (𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐹 ∈ (𝑌 RngHomo 𝑋))))
52 eleq1 2791 . . . . . . . . . . . 12 (𝐹 = 𝐺 → (𝐹 ∈ (𝑌 RngHomo 𝑋) ↔ 𝐺 ∈ (𝑌 RngHomo 𝑋)))
5352eqcoms 2732 . . . . . . . . . . 11 (𝐺 = 𝐹 → (𝐹 ∈ (𝑌 RngHomo 𝑋) ↔ 𝐺 ∈ (𝑌 RngHomo 𝑋)))
5453anbi2d 742 . . . . . . . . . 10 (𝐺 = 𝐹 → ((𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐹 ∈ (𝑌 RngHomo 𝑋)) ↔ (𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋))))
5551, 54sylan9bbr 739 . . . . . . . . 9 ((𝐺 = 𝐹𝜑) → (𝐹 ∈ (𝑋 RngIsom 𝑌) ↔ (𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋))))
56 simpr 479 . . . . . . . . 9 ((𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋)) → 𝐺 ∈ (𝑌 RngHomo 𝑋))
5755, 56syl6bi 243 . . . . . . . 8 ((𝐺 = 𝐹𝜑) → (𝐹 ∈ (𝑋 RngIsom 𝑌) → 𝐺 ∈ (𝑌 RngHomo 𝑋)))
5857com12 32 . . . . . . 7 (𝐹 ∈ (𝑋 RngIsom 𝑌) → ((𝐺 = 𝐹𝜑) → 𝐺 ∈ (𝑌 RngHomo 𝑋)))
5958expdimp 452 . . . . . 6 ((𝐹 ∈ (𝑋 RngIsom 𝑌) ∧ 𝐺 = 𝐹) → (𝜑𝐺 ∈ (𝑌 RngHomo 𝑋)))
6059impcom 445 . . . . 5 ((𝜑 ∧ (𝐹 ∈ (𝑋 RngIsom 𝑌) ∧ 𝐺 = 𝐹)) → 𝐺 ∈ (𝑌 RngHomo 𝑋))
61 coeq1 5387 . . . . . . 7 (𝐺 = 𝐹 → (𝐺𝐹) = (𝐹𝐹))
6261ad2antll 767 . . . . . 6 ((𝜑 ∧ (𝐹 ∈ (𝑋 RngIsom 𝑌) ∧ 𝐺 = 𝐹)) → (𝐺𝐹) = (𝐹𝐹))
6311, 15rngimf1o 42332 . . . . . . . 8 (𝐹 ∈ (𝑋 RngIsom 𝑌) → 𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌))
6463ad2antrl 766 . . . . . . 7 ((𝜑 ∧ (𝐹 ∈ (𝑋 RngIsom 𝑌) ∧ 𝐺 = 𝐹)) → 𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌))
65 f1ococnv1 6278 . . . . . . 7 (𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌) → (𝐹𝐹) = ( I ↾ (Base‘𝑋)))
6664, 65syl 17 . . . . . 6 ((𝜑 ∧ (𝐹 ∈ (𝑋 RngIsom 𝑌) ∧ 𝐺 = 𝐹)) → (𝐹𝐹) = ( I ↾ (Base‘𝑋)))
6762, 66eqtrd 2758 . . . . 5 ((𝜑 ∧ (𝐹 ∈ (𝑋 RngIsom 𝑌) ∧ 𝐺 = 𝐹)) → (𝐺𝐹) = ( I ↾ (Base‘𝑋)))
6849, 60, 67jca31 558 . . . 4 ((𝜑 ∧ (𝐹 ∈ (𝑋 RngIsom 𝑌) ∧ 𝐺 = 𝐹)) → ((𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋))))
6951biimpcd 239 . . . . . . 7 (𝐹 ∈ (𝑋 RngIsom 𝑌) → (𝜑 → (𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐹 ∈ (𝑌 RngHomo 𝑋))))
7069adantr 472 . . . . . 6 ((𝐹 ∈ (𝑋 RngIsom 𝑌) ∧ 𝐺 = 𝐹) → (𝜑 → (𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐹 ∈ (𝑌 RngHomo 𝑋))))
7170impcom 445 . . . . 5 ((𝜑 ∧ (𝐹 ∈ (𝑋 RngIsom 𝑌) ∧ 𝐺 = 𝐹)) → (𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐹 ∈ (𝑌 RngHomo 𝑋)))
72 eleq1 2791 . . . . . . 7 (𝐺 = 𝐹 → (𝐺 ∈ (𝑌 RngHomo 𝑋) ↔ 𝐹 ∈ (𝑌 RngHomo 𝑋)))
7372ad2antll 767 . . . . . 6 ((𝜑 ∧ (𝐹 ∈ (𝑋 RngIsom 𝑌) ∧ 𝐺 = 𝐹)) → (𝐺 ∈ (𝑌 RngHomo 𝑋) ↔ 𝐹 ∈ (𝑌 RngHomo 𝑋)))
7473anbi2d 742 . . . . 5 ((𝜑 ∧ (𝐹 ∈ (𝑋 RngIsom 𝑌) ∧ 𝐺 = 𝐹)) → ((𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋)) ↔ (𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐹 ∈ (𝑌 RngHomo 𝑋))))
7571, 74mpbird 247 . . . 4 ((𝜑 ∧ (𝐹 ∈ (𝑋 RngIsom 𝑌) ∧ 𝐺 = 𝐹)) → (𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋)))
76 coeq2 5388 . . . . . . 7 (𝐺 = 𝐹 → (𝐹𝐺) = (𝐹𝐹))
7776ad2antll 767 . . . . . 6 ((𝜑 ∧ (𝐹 ∈ (𝑋 RngIsom 𝑌) ∧ 𝐺 = 𝐹)) → (𝐹𝐺) = (𝐹𝐹))
78 f1ococnv2 6276 . . . . . . 7 (𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌) → (𝐹𝐹) = ( I ↾ (Base‘𝑌)))
7964, 78syl 17 . . . . . 6 ((𝜑 ∧ (𝐹 ∈ (𝑋 RngIsom 𝑌) ∧ 𝐺 = 𝐹)) → (𝐹𝐹) = ( I ↾ (Base‘𝑌)))
8077, 79eqtrd 2758 . . . . 5 ((𝜑 ∧ (𝐹 ∈ (𝑋 RngIsom 𝑌) ∧ 𝐺 = 𝐹)) → (𝐹𝐺) = ( I ↾ (Base‘𝑌)))
8175, 67, 80jca31 558 . . . 4 ((𝜑 ∧ (𝐹 ∈ (𝑋 RngIsom 𝑌) ∧ 𝐺 = 𝐹)) → (((𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹𝐺) = ( I ↾ (Base‘𝑌))))
8268, 75, 81jca31 558 . . 3 ((𝜑 ∧ (𝐹 ∈ (𝑋 RngIsom 𝑌) ∧ 𝐺 = 𝐹)) → ((((𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋))) ∧ (((𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹𝐺) = ( I ↾ (Base‘𝑌)))))
8347, 82impbida 913 . 2 (𝜑 → (((((𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋))) ∧ (((𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹𝐺) = ( I ↾ (Base‘𝑌)))) ↔ (𝐹 ∈ (𝑋 RngIsom 𝑌) ∧ 𝐺 = 𝐹)))
8410, 23, 833bitrd 294 1 (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 ↔ (𝐹 ∈ (𝑋 RngIsom 𝑌) ∧ 𝐺 = 𝐹)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1596  wcel 2103   class class class wbr 4760   I cid 5127  ccnv 5217  cres 5220  ccom 5222  wf 5997  1-1-ontowf1o 6000  cfv 6001  (class class class)co 6765  Basecbs 15980  Catccat 16447  Sectcsect 16526  Invcinv 16527   RngHomo crngh 42312   RngIsom crngs 42313  RngCatcrngc 42384
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066  ax-cnex 10105  ax-resscn 10106  ax-1cn 10107  ax-icn 10108  ax-addcl 10109  ax-addrcl 10110  ax-mulcl 10111  ax-mulrcl 10112  ax-mulcom 10113  ax-addass 10114  ax-mulass 10115  ax-distr 10116  ax-i2m1 10117  ax-1ne0 10118  ax-1rid 10119  ax-rnegex 10120  ax-rrecex 10121  ax-cnre 10122  ax-pre-lttri 10123  ax-pre-lttrn 10124  ax-pre-ltadd 10125  ax-pre-mulgt0 10126
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1599  df-fal 1602  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-nel 3000  df-ral 3019  df-rex 3020  df-reu 3021  df-rmo 3022  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-pss 3696  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-tp 4290  df-op 4292  df-uni 4545  df-int 4584  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-tr 4861  df-id 5128  df-eprel 5133  df-po 5139  df-so 5140  df-fr 5177  df-we 5179  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-pred 5793  df-ord 5839  df-on 5840  df-lim 5841  df-suc 5842  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-riota 6726  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-om 7183  df-1st 7285  df-2nd 7286  df-wrecs 7527  df-recs 7588  df-rdg 7626  df-1o 7680  df-oadd 7684  df-er 7862  df-map 7976  df-pm 7977  df-ixp 8026  df-en 8073  df-dom 8074  df-sdom 8075  df-fin 8076  df-pnf 10189  df-mnf 10190  df-xr 10191  df-ltxr 10192  df-le 10193  df-sub 10381  df-neg 10382  df-nn 11134  df-2 11192  df-3 11193  df-4 11194  df-5 11195  df-6 11196  df-7 11197  df-8 11198  df-9 11199  df-n0 11406  df-z 11491  df-dec 11607  df-uz 11801  df-fz 12441  df-struct 15982  df-ndx 15983  df-slot 15984  df-base 15986  df-sets 15987  df-ress 15988  df-plusg 16077  df-hom 16089  df-cco 16090  df-0g 16225  df-cat 16451  df-cid 16452  df-homf 16453  df-sect 16529  df-inv 16530  df-ssc 16592  df-resc 16593  df-subc 16594  df-estrc 16885  df-mgm 17364  df-sgrp 17406  df-mnd 17417  df-mhm 17457  df-grp 17547  df-ghm 17780  df-abl 18317  df-mgp 18611  df-mgmhm 42206  df-rng0 42302  df-rnghomo 42314  df-rngisom 42315  df-rngc 42386
This theorem is referenced by:  rngciso  42409
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