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Theorem rngcinvALTV 41281
Description: An inverse in the category of non-unital rings is the converse operation. (Contributed by AV, 28-Feb-2020.) (New usage is discouraged.)
Hypotheses
Ref Expression
rngcsectALTV.c 𝐶 = (RngCatALTV‘𝑈)
rngcsectALTV.b 𝐵 = (Base‘𝐶)
rngcsectALTV.u (𝜑𝑈𝑉)
rngcsectALTV.x (𝜑𝑋𝐵)
rngcsectALTV.y (𝜑𝑌𝐵)
rngcinvALTV.n 𝑁 = (Inv‘𝐶)
Assertion
Ref Expression
rngcinvALTV (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 ↔ (𝐹 ∈ (𝑋 RngIsom 𝑌) ∧ 𝐺 = 𝐹)))

Proof of Theorem rngcinvALTV
StepHypRef Expression
1 rngcsectALTV.b . . 3 𝐵 = (Base‘𝐶)
2 rngcinvALTV.n . . 3 𝑁 = (Inv‘𝐶)
3 rngcsectALTV.u . . . 4 (𝜑𝑈𝑉)
4 rngcsectALTV.c . . . . 5 𝐶 = (RngCatALTV‘𝑈)
54rngccatALTV 41278 . . . 4 (𝑈𝑉𝐶 ∈ Cat)
63, 5syl 17 . . 3 (𝜑𝐶 ∈ Cat)
7 rngcsectALTV.x . . 3 (𝜑𝑋𝐵)
8 rngcsectALTV.y . . 3 (𝜑𝑌𝐵)
9 eqid 2621 . . 3 (Sect‘𝐶) = (Sect‘𝐶)
101, 2, 6, 7, 8, 9isinv 16341 . 2 (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 ↔ (𝐹(𝑋(Sect‘𝐶)𝑌)𝐺𝐺(𝑌(Sect‘𝐶)𝑋)𝐹)))
11 eqid 2621 . . . . . 6 (Base‘𝑋) = (Base‘𝑋)
124, 1, 3, 7, 8, 11, 9rngcsectALTV 41280 . . . . 5 (𝜑 → (𝐹(𝑋(Sect‘𝐶)𝑌)𝐺 ↔ (𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋)))))
13 df-3an 1038 . . . . 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 2621 . . . . . 6 (Base‘𝑌) = (Base‘𝑌)
164, 1, 3, 8, 7, 15, 9rngcsectALTV 41280 . . . . 5 (𝜑 → (𝐺(𝑌(Sect‘𝐶)𝑋)𝐹 ↔ (𝐺 ∈ (𝑌 RngHomo 𝑋) ∧ 𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ (𝐹𝐺) = ( I ↾ (Base‘𝑌)))))
17 3ancoma 1043 . . . . . 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 1038 . . . . . 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 746 . . 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 870 . . 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 799 . . . . . 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 482 . . . . 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 41187 . . . . . . . . . 10 (𝐹 ∈ (𝑋 RngHomo 𝑌) → 𝐹:(Base‘𝑋)⟶(Base‘𝑌))
2715, 11rnghmf 41187 . . . . . . . . . 10 (𝐺 ∈ (𝑌 RngHomo 𝑋) → 𝐺:(Base‘𝑌)⟶(Base‘𝑋))
2826, 27anim12i 589 . . . . . . . . 9 ((𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋)) → (𝐹:(Base‘𝑋)⟶(Base‘𝑌) ∧ 𝐺:(Base‘𝑌)⟶(Base‘𝑋)))
2928ad2antlr 762 . . . . . . . 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 477 . . . . . . . . 9 ((((𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹𝐺) = ( I ↾ (Base‘𝑌))) → (𝐹𝐺) = ( I ↾ (Base‘𝑌)))
3130adantl 482 . . . . . . . 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 477 . . . . . . . . 9 (((𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋)) ∧ (𝐺𝐹) = ( I ↾ (Base‘𝑋))) → (𝐺𝐹) = ( I ↾ (Base‘𝑋)))
3332ad2antrl 763 . . . . . . . 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 557 . . . . . . 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 482 . . . . . 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 6505 . . . . . . 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 2628 . . . . . . . 8 (𝐹 = 𝐺𝐺 = 𝐹)
3837anbi2i 729 . . . . . . 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 680 . . . . 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 697 . . . 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 41192 . . . . . . 7 ((𝑋𝐵𝑌𝐵) → (𝐹 ∈ (𝑋 RngIsom 𝑌) ↔ (𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌))))
447, 8, 43syl2anc 692 . . . . . 6 (𝜑 → (𝐹 ∈ (𝑋 RngIsom 𝑌) ↔ (𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌))))
4544anbi1d 740 . . . . 5 (𝜑 → ((𝐹 ∈ (𝑋 RngIsom 𝑌) ∧ 𝐺 = 𝐹) ↔ ((𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌)) ∧ 𝐺 = 𝐹)))
4645adantr 481 . . . 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 41194 . . . . . 6 (𝐹 ∈ (𝑋 RngIsom 𝑌) → 𝐹 ∈ (𝑋 RngHomo 𝑌))
4948ad2antrl 763 . . . . 5 ((𝜑 ∧ (𝐹 ∈ (𝑋 RngIsom 𝑌) ∧ 𝐺 = 𝐹)) → 𝐹 ∈ (𝑋 RngHomo 𝑌))
50 isrngisom 41184 . . . . . . . . . . 11 ((𝑋𝐵𝑌𝐵) → (𝐹 ∈ (𝑋 RngIsom 𝑌) ↔ (𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐹 ∈ (𝑌 RngHomo 𝑋))))
517, 8, 50syl2anc 692 . . . . . . . . . 10 (𝜑 → (𝐹 ∈ (𝑋 RngIsom 𝑌) ↔ (𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐹 ∈ (𝑌 RngHomo 𝑋))))
52 eleq1 2686 . . . . . . . . . . . 12 (𝐹 = 𝐺 → (𝐹 ∈ (𝑌 RngHomo 𝑋) ↔ 𝐺 ∈ (𝑌 RngHomo 𝑋)))
5352eqcoms 2629 . . . . . . . . . . 11 (𝐺 = 𝐹 → (𝐹 ∈ (𝑌 RngHomo 𝑋) ↔ 𝐺 ∈ (𝑌 RngHomo 𝑋)))
5453anbi2d 739 . . . . . . . . . 10 (𝐺 = 𝐹 → ((𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐹 ∈ (𝑌 RngHomo 𝑋)) ↔ (𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋))))
5551, 54sylan9bbr 736 . . . . . . . . 9 ((𝐺 = 𝐹𝜑) → (𝐹 ∈ (𝑋 RngIsom 𝑌) ↔ (𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋))))
56 simpr 477 . . . . . . . . 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 453 . . . . . 6 ((𝐹 ∈ (𝑋 RngIsom 𝑌) ∧ 𝐺 = 𝐹) → (𝜑𝐺 ∈ (𝑌 RngHomo 𝑋)))
6059impcom 446 . . . . 5 ((𝜑 ∧ (𝐹 ∈ (𝑋 RngIsom 𝑌) ∧ 𝐺 = 𝐹)) → 𝐺 ∈ (𝑌 RngHomo 𝑋))
61 coeq1 5239 . . . . . . 7 (𝐺 = 𝐹 → (𝐺𝐹) = (𝐹𝐹))
6261ad2antll 764 . . . . . 6 ((𝜑 ∧ (𝐹 ∈ (𝑋 RngIsom 𝑌) ∧ 𝐺 = 𝐹)) → (𝐺𝐹) = (𝐹𝐹))
6311, 15rngimf1o 41193 . . . . . . . 8 (𝐹 ∈ (𝑋 RngIsom 𝑌) → 𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌))
6463ad2antrl 763 . . . . . . 7 ((𝜑 ∧ (𝐹 ∈ (𝑋 RngIsom 𝑌) ∧ 𝐺 = 𝐹)) → 𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌))
65 f1ococnv1 6122 . . . . . . 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 2655 . . . . 5 ((𝜑 ∧ (𝐹 ∈ (𝑋 RngIsom 𝑌) ∧ 𝐺 = 𝐹)) → (𝐺𝐹) = ( I ↾ (Base‘𝑋)))
6849, 60, 67jca31 556 . . . 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 481 . . . . . 6 ((𝐹 ∈ (𝑋 RngIsom 𝑌) ∧ 𝐺 = 𝐹) → (𝜑 → (𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐹 ∈ (𝑌 RngHomo 𝑋))))
7170impcom 446 . . . . 5 ((𝜑 ∧ (𝐹 ∈ (𝑋 RngIsom 𝑌) ∧ 𝐺 = 𝐹)) → (𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐹 ∈ (𝑌 RngHomo 𝑋)))
72 eleq1 2686 . . . . . . 7 (𝐺 = 𝐹 → (𝐺 ∈ (𝑌 RngHomo 𝑋) ↔ 𝐹 ∈ (𝑌 RngHomo 𝑋)))
7372ad2antll 764 . . . . . 6 ((𝜑 ∧ (𝐹 ∈ (𝑋 RngIsom 𝑌) ∧ 𝐺 = 𝐹)) → (𝐺 ∈ (𝑌 RngHomo 𝑋) ↔ 𝐹 ∈ (𝑌 RngHomo 𝑋)))
7473anbi2d 739 . . . . 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 5240 . . . . . . 7 (𝐺 = 𝐹 → (𝐹𝐺) = (𝐹𝐹))
7776ad2antll 764 . . . . . 6 ((𝜑 ∧ (𝐹 ∈ (𝑋 RngIsom 𝑌) ∧ 𝐺 = 𝐹)) → (𝐹𝐺) = (𝐹𝐹))
78 f1ococnv2 6120 . . . . . . 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 2655 . . . . 5 ((𝜑 ∧ (𝐹 ∈ (𝑋 RngIsom 𝑌) ∧ 𝐺 = 𝐹)) → (𝐹𝐺) = ( I ↾ (Base‘𝑌)))
8175, 67, 80jca31 556 . . . 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 556 . . 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 876 . 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 384  w3a 1036   = wceq 1480  wcel 1987   class class class wbr 4613   I cid 4984  ccnv 5073  cres 5076  ccom 5078  wf 5843  1-1-ontowf1o 5846  cfv 5847  (class class class)co 6604  Basecbs 15781  Catccat 16246  Sectcsect 16325  Invcinv 16326   RngHomo crngh 41173   RngIsom crngs 41174  RngCatALTVcrngcALTV 41246
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-map 7804  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-2 11023  df-3 11024  df-4 11025  df-5 11026  df-6 11027  df-7 11028  df-8 11029  df-9 11030  df-n0 11237  df-z 11322  df-dec 11438  df-uz 11632  df-fz 12269  df-struct 15783  df-ndx 15784  df-slot 15785  df-base 15786  df-sets 15787  df-plusg 15875  df-hom 15887  df-cco 15888  df-0g 16023  df-cat 16250  df-cid 16251  df-sect 16328  df-inv 16329  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-mhm 17256  df-grp 17346  df-ghm 17579  df-abl 18117  df-mgp 18411  df-mgmhm 41067  df-rng0 41163  df-rnghomo 41175  df-rngisom 41176  df-rngcALTV 41248
This theorem is referenced by:  rngcisoALTV  41282
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