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Theorem rngcinvALTV 42522
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 42519 . . . 4 (𝑈𝑉𝐶 ∈ Cat)
63, 5syl 17 . . 3 (𝜑𝐶 ∈ Cat)
7 rngcsectALTV.x . . 3 (𝜑𝑋𝐵)
8 rngcsectALTV.y . . 3 (𝜑𝑌𝐵)
9 eqid 2761 . . 3 (Sect‘𝐶) = (Sect‘𝐶)
101, 2, 6, 7, 8, 9isinv 16642 . 2 (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 ↔ (𝐹(𝑋(Sect‘𝐶)𝑌)𝐺𝐺(𝑌(Sect‘𝐶)𝑋)𝐹)))
11 eqid 2761 . . . . . 6 (Base‘𝑋) = (Base‘𝑋)
124, 1, 3, 7, 8, 11, 9rngcsectALTV 42521 . . . . 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 2761 . . . . . 6 (Base‘𝑌) = (Base‘𝑌)
164, 1, 3, 8, 7, 15, 9rngcsectALTV 42521 . . . . 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 42428 . . . . . . . . . 10 (𝐹 ∈ (𝑋 RngHomo 𝑌) → 𝐹:(Base‘𝑋)⟶(Base‘𝑌))
2715, 11rnghmf 42428 . . . . . . . . . 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 6716 . . . . . . 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 2768 . . . . . . . 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 42433 . . . . . . 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 42435 . . . . . 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 42425 . . . . . . . . . . 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 2828 . . . . . . . . . . . 12 (𝐹 = 𝐺 → (𝐹 ∈ (𝑌 RngHomo 𝑋) ↔ 𝐺 ∈ (𝑌 RngHomo 𝑋)))
5352eqcoms 2769 . . . . . . . . . . 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 5436 . . . . . . 7 (𝐺 = 𝐹 → (𝐺𝐹) = (𝐹𝐹))
6261ad2antll 767 . . . . . 6 ((𝜑 ∧ (𝐹 ∈ (𝑋 RngIsom 𝑌) ∧ 𝐺 = 𝐹)) → (𝐺𝐹) = (𝐹𝐹))
6311, 15rngimf1o 42434 . . . . . . . 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 6328 . . . . . . 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 2795 . . . . 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 2828 . . . . . . 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 5437 . . . . . . 7 (𝐺 = 𝐹 → (𝐹𝐺) = (𝐹𝐹))
7776ad2antll 767 . . . . . 6 ((𝜑 ∧ (𝐹 ∈ (𝑋 RngIsom 𝑌) ∧ 𝐺 = 𝐹)) → (𝐹𝐺) = (𝐹𝐹))
78 f1ococnv2 6326 . . . . . . 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 2795 . . . . 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 1632  wcel 2140   class class class wbr 4805   I cid 5174  ccnv 5266  cres 5269  ccom 5271  wf 6046  1-1-ontowf1o 6049  cfv 6050  (class class class)co 6815  Basecbs 16080  Catccat 16547  Sectcsect 16626  Invcinv 16627   RngHomo crngh 42414   RngIsom crngs 42415  RngCatALTVcrngcALTV 42487
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-rep 4924  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056  ax-un 7116  ax-cnex 10205  ax-resscn 10206  ax-1cn 10207  ax-icn 10208  ax-addcl 10209  ax-addrcl 10210  ax-mulcl 10211  ax-mulrcl 10212  ax-mulcom 10213  ax-addass 10214  ax-mulass 10215  ax-distr 10216  ax-i2m1 10217  ax-1ne0 10218  ax-1rid 10219  ax-rnegex 10220  ax-rrecex 10221  ax-cnre 10222  ax-pre-lttri 10223  ax-pre-lttrn 10224  ax-pre-ltadd 10225  ax-pre-mulgt0 10226
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-nel 3037  df-ral 3056  df-rex 3057  df-reu 3058  df-rmo 3059  df-rab 3060  df-v 3343  df-sbc 3578  df-csb 3676  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-pss 3732  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-tp 4327  df-op 4329  df-uni 4590  df-int 4629  df-iun 4675  df-br 4806  df-opab 4866  df-mpt 4883  df-tr 4906  df-id 5175  df-eprel 5180  df-po 5188  df-so 5189  df-fr 5226  df-we 5228  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-pred 5842  df-ord 5888  df-on 5889  df-lim 5890  df-suc 5891  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-f1 6055  df-fo 6056  df-f1o 6057  df-fv 6058  df-riota 6776  df-ov 6818  df-oprab 6819  df-mpt2 6820  df-om 7233  df-1st 7335  df-2nd 7336  df-wrecs 7578  df-recs 7639  df-rdg 7677  df-1o 7731  df-oadd 7735  df-er 7914  df-map 8028  df-en 8125  df-dom 8126  df-sdom 8127  df-fin 8128  df-pnf 10289  df-mnf 10290  df-xr 10291  df-ltxr 10292  df-le 10293  df-sub 10481  df-neg 10482  df-nn 11234  df-2 11292  df-3 11293  df-4 11294  df-5 11295  df-6 11296  df-7 11297  df-8 11298  df-9 11299  df-n0 11506  df-z 11591  df-dec 11707  df-uz 11901  df-fz 12541  df-struct 16082  df-ndx 16083  df-slot 16084  df-base 16086  df-sets 16087  df-plusg 16177  df-hom 16189  df-cco 16190  df-0g 16325  df-cat 16551  df-cid 16552  df-sect 16629  df-inv 16630  df-mgm 17464  df-sgrp 17506  df-mnd 17517  df-mhm 17557  df-grp 17647  df-ghm 17880  df-abl 18417  df-mgp 18711  df-mgmhm 42308  df-rng0 42404  df-rnghomo 42416  df-rngisom 42417  df-rngcALTV 42489
This theorem is referenced by:  rngcisoALTV  42523
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