Theorem ringcinvALTV 45502

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	⊢ (𝜑 → 𝑈 ∈ 𝑉)
ringcsectALTV.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
ringcsectALTV.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
ringcinvALTV.n	⊢ 𝑁 = (Inv‘𝐶)

Assertion

Ref	Expression
ringcinvALTV	⊢ (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 ↔ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = ◡𝐹)))

Proof of Theorem ringcinvALTV

Step	Hyp	Ref	Expression
1		ringcsectALTV.b	. . 3 ⊢ 𝐵 = (Base‘𝐶)
2		ringcinvALTV.n	. . 3 ⊢ 𝑁 = (Inv‘𝐶)
3		ringcsectALTV.u	. . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
4		ringcsectALTV.c	. . . . 5 ⊢ 𝐶 = (RingCatALTV‘𝑈)
5	4	ringccatALTV 45499	. . . 4 ⊢ (𝑈 ∈ 𝑉 → 𝐶 ∈ Cat)
6	3, 5	syl 17	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
7		ringcsectALTV.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
8		ringcsectALTV.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
9		eqid 2738	. . 3 ⊢ (Sect‘𝐶) = (Sect‘𝐶)
10	1, 2, 6, 7, 8, 9	isinv 17389	. 2 ⊢ (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 ↔ (𝐹(𝑋(Sect‘𝐶)𝑌)𝐺 ∧ 𝐺(𝑌(Sect‘𝐶)𝑋)𝐹)))
11		eqid 2738	. . . . . 6 ⊢ (Base‘𝑋) = (Base‘𝑋)
12	4, 1, 3, 7, 8, 11, 9	ringcsectALTV 45501	. . . . 5 ⊢ (𝜑 → (𝐹(𝑋(Sect‘𝐶)𝑌)𝐺 ↔ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋)))))
13		df-3an 1087	. . . . 5 ⊢ ((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))) ↔ ((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))))
14	12, 13	bitrdi 286	. . . 4 ⊢ (𝜑 → (𝐹(𝑋(Sect‘𝐶)𝑌)𝐺 ↔ ((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋)))))
15		eqid 2738	. . . . . 6 ⊢ (Base‘𝑌) = (Base‘𝑌)
16	4, 1, 3, 8, 7, 15, 9	ringcsectALTV 45501	. . . . 5 ⊢ (𝜑 → (𝐺(𝑌(Sect‘𝐶)𝑋)𝐹 ↔ (𝐺 ∈ (𝑌 RingHom 𝑋) ∧ 𝐹 ∈ (𝑋 RingHom 𝑌) ∧ (𝐹 ∘ 𝐺) = ( I ↾ (Base‘𝑌)))))
17		3ancoma 1096	. . . . . 6 ⊢ ((𝐺 ∈ (𝑌 RingHom 𝑋) ∧ 𝐹 ∈ (𝑋 RingHom 𝑌) ∧ (𝐹 ∘ 𝐺) = ( I ↾ (Base‘𝑌))) ↔ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋) ∧ (𝐹 ∘ 𝐺) = ( I ↾ (Base‘𝑌))))
18		df-3an 1087	. . . . . 6 ⊢ ((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋) ∧ (𝐹 ∘ 𝐺) = ( I ↾ (Base‘𝑌))) ↔ ((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐹 ∘ 𝐺) = ( I ↾ (Base‘𝑌))))
19	17, 18	bitri 274	. . . . 5 ⊢ ((𝐺 ∈ (𝑌 RingHom 𝑋) ∧ 𝐹 ∈ (𝑋 RingHom 𝑌) ∧ (𝐹 ∘ 𝐺) = ( I ↾ (Base‘𝑌))) ↔ ((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐹 ∘ 𝐺) = ( I ↾ (Base‘𝑌))))
20	16, 19	bitrdi 286	. . . 4 ⊢ (𝜑 → (𝐺(𝑌(Sect‘𝐶)𝑋)𝐹 ↔ ((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐹 ∘ 𝐺) = ( I ↾ (Base‘𝑌)))))
21	14, 20	anbi12d 630	. . 3 ⊢ (𝜑 → ((𝐹(𝑋(Sect‘𝐶)𝑌)𝐺 ∧ 𝐺(𝑌(Sect‘𝐶)𝑋)𝐹) ↔ (((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))) ∧ ((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐹 ∘ 𝐺) = ( I ↾ (Base‘𝑌))))))
22		anandi 672	. . 3 ⊢ ((((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))) ∧ ((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐹 ∘ 𝐺) = ( I ↾ (Base‘𝑌)))) ↔ ((((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋))) ∧ (((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∘ 𝐺) = ( I ↾ (Base‘𝑌)))))
23	21, 22	bitrdi 286	. 2 ⊢ (𝜑 → ((𝐹(𝑋(Sect‘𝐶)𝑌)𝐺 ∧ 𝐺(𝑌(Sect‘𝐶)𝑋)𝐹) ↔ ((((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋))) ∧ (((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∘ 𝐺) = ( I ↾ (Base‘𝑌))))))
24		simplrl 773	. . . . . 6 ⊢ (((((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋))) ∧ (((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∘ 𝐺) = ( I ↾ (Base‘𝑌)))) → 𝐹 ∈ (𝑋 RingHom 𝑌))
25	24	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ ((((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋))) ∧ (((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∘ 𝐺) = ( I ↾ (Base‘𝑌))))) → 𝐹 ∈ (𝑋 RingHom 𝑌))
26	11, 15	rhmf 19885	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝑋 RingHom 𝑌) → 𝐹:(Base‘𝑋)⟶(Base‘𝑌))
27	15, 11	rhmf 19885	. . . . . . . . . 10 ⊢ (𝐺 ∈ (𝑌 RingHom 𝑋) → 𝐺:(Base‘𝑌)⟶(Base‘𝑋))
28	26, 27	anim12i 612	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) → (𝐹:(Base‘𝑋)⟶(Base‘𝑌) ∧ 𝐺:(Base‘𝑌)⟶(Base‘𝑋)))
29	28	ad2antlr 723	. . . . . . . 8 ⊢ (((((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋))) ∧ (((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∘ 𝐺) = ( I ↾ (Base‘𝑌)))) → (𝐹:(Base‘𝑋)⟶(Base‘𝑌) ∧ 𝐺:(Base‘𝑌)⟶(Base‘𝑋)))
30		simpr 484	. . . . . . . . . 10 ⊢ ((((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∘ 𝐺) = ( I ↾ (Base‘𝑌))) → (𝐹 ∘ 𝐺) = ( I ↾ (Base‘𝑌)))
31	30	adantl 481	. . . . . . . . 9 ⊢ (((((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋))) ∧ (((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∘ 𝐺) = ( I ↾ (Base‘𝑌)))) → (𝐹 ∘ 𝐺) = ( I ↾ (Base‘𝑌)))
32		simpr 484	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))) → (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋)))
33	32	ad2antrl 724	. . . . . . . . 9 ⊢ (((((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋))) ∧ (((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∘ 𝐺) = ( I ↾ (Base‘𝑌)))) → (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋)))
34	31, 33	jca 511	. . . . . . . 8 ⊢ (((((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋))) ∧ (((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∘ 𝐺) = ( I ↾ (Base‘𝑌)))) → ((𝐹 ∘ 𝐺) = ( I ↾ (Base‘𝑌)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))))
35	29, 34	jca 511	. . . . . . 7 ⊢ (((((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋))) ∧ (((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∘ 𝐺) = ( I ↾ (Base‘𝑌)))) → ((𝐹:(Base‘𝑋)⟶(Base‘𝑌) ∧ 𝐺:(Base‘𝑌)⟶(Base‘𝑋)) ∧ ((𝐹 ∘ 𝐺) = ( I ↾ (Base‘𝑌)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋)))))
36	35	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ ((((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋))) ∧ (((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∘ 𝐺) = ( I ↾ (Base‘𝑌))))) → ((𝐹:(Base‘𝑋)⟶(Base‘𝑌) ∧ 𝐺:(Base‘𝑌)⟶(Base‘𝑋)) ∧ ((𝐹 ∘ 𝐺) = ( I ↾ (Base‘𝑌)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋)))))
37		fcof1o 7148	. . . . . . 7 ⊢ (((𝐹:(Base‘𝑋)⟶(Base‘𝑌) ∧ 𝐺:(Base‘𝑌)⟶(Base‘𝑋)) ∧ ((𝐹 ∘ 𝐺) = ( I ↾ (Base‘𝑌)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋)))) → (𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌) ∧ ◡𝐹 = 𝐺))
38		eqcom 2745	. . . . . . . 8 ⊢ (◡𝐹 = 𝐺 ↔ 𝐺 = ◡𝐹)
39	38	anbi2i 622	. . . . . . 7 ⊢ ((𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌) ∧ ◡𝐹 = 𝐺) ↔ (𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌) ∧ 𝐺 = ◡𝐹))
40	37, 39	sylib 217	. . . . . 6 ⊢ (((𝐹:(Base‘𝑋)⟶(Base‘𝑌) ∧ 𝐺:(Base‘𝑌)⟶(Base‘𝑋)) ∧ ((𝐹 ∘ 𝐺) = ( I ↾ (Base‘𝑌)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋)))) → (𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌) ∧ 𝐺 = ◡𝐹))
41	36, 40	syl 17	. . . . 5 ⊢ ((𝜑 ∧ ((((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋))) ∧ (((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∘ 𝐺) = ( I ↾ (Base‘𝑌))))) → (𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌) ∧ 𝐺 = ◡𝐹))
42		anass 468	. . . . 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)))
43	25, 41, 42	sylanbrc 582	. . . 4 ⊢ ((𝜑 ∧ ((((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋))) ∧ (((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∘ 𝐺) = ( I ↾ (Base‘𝑌))))) → ((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌)) ∧ 𝐺 = ◡𝐹))
44	7, 8	jca 511	. . . . . . 7 ⊢ (𝜑 → (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵))
45	11, 15	isrim 19892	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝐹 ∈ (𝑋 RingIso 𝑌) ↔ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌))))
46	44, 45	syl 17	. . . . . 6 ⊢ (𝜑 → (𝐹 ∈ (𝑋 RingIso 𝑌) ↔ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌))))
47	46	anbi1d 629	. . . . 5 ⊢ (𝜑 → ((𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = ◡𝐹) ↔ ((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌)) ∧ 𝐺 = ◡𝐹)))
48	47	adantr 480	. . . 4 ⊢ ((𝜑 ∧ ((((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋))) ∧ (((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∘ 𝐺) = ( I ↾ (Base‘𝑌))))) → ((𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = ◡𝐹) ↔ ((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌)) ∧ 𝐺 = ◡𝐹)))
49	43, 48	mpbird 256	. . 3 ⊢ ((𝜑 ∧ ((((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋))) ∧ (((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∘ 𝐺) = ( I ↾ (Base‘𝑌))))) → (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = ◡𝐹))
50	11, 15	rimrhm 19894	. . . . . 6 ⊢ (𝐹 ∈ (𝑋 RingIso 𝑌) → 𝐹 ∈ (𝑋 RingHom 𝑌))
51	50	ad2antrl 724	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = ◡𝐹)) → 𝐹 ∈ (𝑋 RingHom 𝑌))
52		isrim0 19882	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝐹 ∈ (𝑋 RingIso 𝑌) ↔ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ ◡𝐹 ∈ (𝑌 RingHom 𝑋))))
53	44, 52	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 ∈ (𝑋 RingIso 𝑌) ↔ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ ◡𝐹 ∈ (𝑌 RingHom 𝑋))))
54		eleq1 2826	. . . . . . . . . . . 12 ⊢ (◡𝐹 = 𝐺 → (◡𝐹 ∈ (𝑌 RingHom 𝑋) ↔ 𝐺 ∈ (𝑌 RingHom 𝑋)))
55	54	eqcoms 2746	. . . . . . . . . . 11 ⊢ (𝐺 = ◡𝐹 → (◡𝐹 ∈ (𝑌 RingHom 𝑋) ↔ 𝐺 ∈ (𝑌 RingHom 𝑋)))
56	55	anbi2d 628	. . . . . . . . . 10 ⊢ (𝐺 = ◡𝐹 → ((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ ◡𝐹 ∈ (𝑌 RingHom 𝑋)) ↔ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋))))
57	53, 56	sylan9bbr 510	. . . . . . . . 9 ⊢ ((𝐺 = ◡𝐹 ∧ 𝜑) → (𝐹 ∈ (𝑋 RingIso 𝑌) ↔ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋))))
58		simpr 484	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) → 𝐺 ∈ (𝑌 RingHom 𝑋))
59	57, 58	syl6bi 252	. . . . . . . 8 ⊢ ((𝐺 = ◡𝐹 ∧ 𝜑) → (𝐹 ∈ (𝑋 RingIso 𝑌) → 𝐺 ∈ (𝑌 RingHom 𝑋)))
60	59	com12 32	. . . . . . 7 ⊢ (𝐹 ∈ (𝑋 RingIso 𝑌) → ((𝐺 = ◡𝐹 ∧ 𝜑) → 𝐺 ∈ (𝑌 RingHom 𝑋)))
61	60	expdimp 452	. . . . . 6 ⊢ ((𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = ◡𝐹) → (𝜑 → 𝐺 ∈ (𝑌 RingHom 𝑋)))
62	61	impcom 407	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = ◡𝐹)) → 𝐺 ∈ (𝑌 RingHom 𝑋))
63		coeq1 5755	. . . . . . 7 ⊢ (𝐺 = ◡𝐹 → (𝐺 ∘ 𝐹) = (◡𝐹 ∘ 𝐹))
64	63	ad2antll 725	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = ◡𝐹)) → (𝐺 ∘ 𝐹) = (◡𝐹 ∘ 𝐹))
65	11, 15	rimf1o 19893	. . . . . . . 8 ⊢ (𝐹 ∈ (𝑋 RingIso 𝑌) → 𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌))
66	65	ad2antrl 724	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = ◡𝐹)) → 𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌))
67		f1ococnv1 6728	. . . . . . 7 ⊢ (𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌) → (◡𝐹 ∘ 𝐹) = ( I ↾ (Base‘𝑋)))
68	66, 67	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = ◡𝐹)) → (◡𝐹 ∘ 𝐹) = ( I ↾ (Base‘𝑋)))
69	64, 68	eqtrd 2778	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = ◡𝐹)) → (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋)))
70	51, 62, 69	jca31 514	. . . 4 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = ◡𝐹)) → ((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))))
71	53	biimpcd 248	. . . . . . 7 ⊢ (𝐹 ∈ (𝑋 RingIso 𝑌) → (𝜑 → (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ ◡𝐹 ∈ (𝑌 RingHom 𝑋))))
72	71	adantr 480	. . . . . 6 ⊢ ((𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = ◡𝐹) → (𝜑 → (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ ◡𝐹 ∈ (𝑌 RingHom 𝑋))))
73	72	impcom 407	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = ◡𝐹)) → (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ ◡𝐹 ∈ (𝑌 RingHom 𝑋)))
74		eleq1 2826	. . . . . . 7 ⊢ (𝐺 = ◡𝐹 → (𝐺 ∈ (𝑌 RingHom 𝑋) ↔ ◡𝐹 ∈ (𝑌 RingHom 𝑋)))
75	74	ad2antll 725	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = ◡𝐹)) → (𝐺 ∈ (𝑌 RingHom 𝑋) ↔ ◡𝐹 ∈ (𝑌 RingHom 𝑋)))
76	75	anbi2d 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))))
77	73, 76	mpbird 256	. . . 4 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = ◡𝐹)) → (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)))
78		coeq2 5756	. . . . . . 7 ⊢ (𝐺 = ◡𝐹 → (𝐹 ∘ 𝐺) = (𝐹 ∘ ◡𝐹))
79	78	ad2antll 725	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = ◡𝐹)) → (𝐹 ∘ 𝐺) = (𝐹 ∘ ◡𝐹))
80		f1ococnv2 6726	. . . . . . 7 ⊢ (𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌) → (𝐹 ∘ ◡𝐹) = ( I ↾ (Base‘𝑌)))
81	66, 80	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = ◡𝐹)) → (𝐹 ∘ ◡𝐹) = ( I ↾ (Base‘𝑌)))
82	79, 81	eqtrd 2778	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = ◡𝐹)) → (𝐹 ∘ 𝐺) = ( I ↾ (Base‘𝑌)))
83	77, 69, 82	jca31 514	. . . 4 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = ◡𝐹)) → (((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∘ 𝐺) = ( I ↾ (Base‘𝑌))))
84	70, 77, 83	jca31 514	. . 3 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = ◡𝐹)) → ((((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋))) ∧ (((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∘ 𝐺) = ( I ↾ (Base‘𝑌)))))
85	49, 84	impbida 797	. 2 ⊢ (𝜑 → (((((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋))) ∧ (((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∘ 𝐺) = ( I ↾ (Base‘𝑌)))) ↔ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = ◡𝐹)))
86	10, 23, 85	3bitrd 304	1 ⊢ (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 ↔ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = ◡𝐹)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108   class class class wbr 5070   I cid 5479  ^◡ccnv 5579   ↾ cres 5582   ∘ ccom 5584  ⟶wf 6414  –1-1-onto→wf1o 6417  ‘cfv 6418  (class class class)co 7255  Basecbs 16840  Catccat 17290  Sectcsect 17373  Invcinv 17374   RingHom crh 19871   RingIso crs 19872  RingCatALTVcringcALTV 45450

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-er 8456  df-map 8575  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-2 11966  df-3 11967  df-4 11968  df-5 11969  df-6 11970  df-7 11971  df-8 11972  df-9 11973  df-n0 12164  df-z 12250  df-dec 12367  df-uz 12512  df-fz 13169  df-struct 16776  df-sets 16793  df-slot 16811  df-ndx 16823  df-base 16841  df-plusg 16901  df-hom 16912  df-cco 16913  df-0g 17069  df-cat 17294  df-cid 17295  df-sect 17376  df-inv 17377  df-mgm 18241  df-sgrp 18290  df-mnd 18301  df-mhm 18345  df-grp 18495  df-ghm 18747  df-mgp 19636  df-ur 19653  df-ring 19700  df-rnghom 19874  df-rngiso 19875  df-ringcALTV 45452

This theorem is referenced by:  ringcisoALTV  45503