Theorem ringcinvALTV 48226

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 48223	. . . 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 2737	. . 3 ⊢ (Sect‘𝐶) = (Sect‘𝐶)
10	1, 2, 6, 7, 8, 9	isinv 17804	. 2 ⊢ (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 ↔ (𝐹(𝑋(Sect‘𝐶)𝑌)𝐺 ∧ 𝐺(𝑌(Sect‘𝐶)𝑋)𝐹)))
11		eqid 2737	. . . . . 6 ⊢ (Base‘𝑋) = (Base‘𝑋)
12	4, 1, 3, 7, 8, 11, 9	ringcsectALTV 48225	. . . . 5 ⊢ (𝜑 → (𝐹(𝑋(Sect‘𝐶)𝑌)𝐺 ↔ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋)))))
13		df-3an 1089	. . . . 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 287	. . . 4 ⊢ (𝜑 → (𝐹(𝑋(Sect‘𝐶)𝑌)𝐺 ↔ ((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋)))))
15		eqid 2737	. . . . . 6 ⊢ (Base‘𝑌) = (Base‘𝑌)
16	4, 1, 3, 8, 7, 15, 9	ringcsectALTV 48225	. . . . 5 ⊢ (𝜑 → (𝐺(𝑌(Sect‘𝐶)𝑋)𝐹 ↔ (𝐺 ∈ (𝑌 RingHom 𝑋) ∧ 𝐹 ∈ (𝑋 RingHom 𝑌) ∧ (𝐹 ∘ 𝐺) = ( I ↾ (Base‘𝑌)))))
17		3ancoma 1098	. . . . . 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 1089	. . . . . 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 275	. . . . 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 287	. . . 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 632	. . 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 676	. . 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 287	. 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 777	. . . . . 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 20485	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝑋 RingHom 𝑌) → 𝐹:(Base‘𝑋)⟶(Base‘𝑌))
27	15, 11	rhmf 20485	. . . . . . . . . 10 ⊢ (𝐺 ∈ (𝑌 RingHom 𝑋) → 𝐺:(Base‘𝑌)⟶(Base‘𝑋))
28	26, 27	anim12i 613	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) → (𝐹:(Base‘𝑋)⟶(Base‘𝑌) ∧ 𝐺:(Base‘𝑌)⟶(Base‘𝑋)))
29	28	ad2antlr 727	. . . . . . . 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	. . . . . . . . 9 ⊢ ((((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	. . . . . . . 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)))
32		simpr 484	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))) → (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋)))
33	32	ad2antrl 728	. . . . . . . 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)))) → (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋)))
34	29, 31, 33	jca32 515	. . . . . . 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)))))
35	34	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)))))
36		fcof1o 7316	. . . . . . 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))
37		eqcom 2744	. . . . . . . 8 ⊢ (◡𝐹 = 𝐺 ↔ 𝐺 = ◡𝐹)
38	37	anbi2i 623	. . . . . . 7 ⊢ ((𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌) ∧ ◡𝐹 = 𝐺) ↔ (𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌) ∧ 𝐺 = ◡𝐹))
39	36, 38	sylib 218	. . . . . 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))
40	35, 39	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))
41		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)))
42	25, 40, 41	sylanbrc 583	. . . 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))
43	11, 15	isrim 20492	. . . . . . 7 ⊢ (𝐹 ∈ (𝑋 RingIso 𝑌) ↔ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌)))
44	43	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝐹 ∈ (𝑋 RingIso 𝑌) ↔ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌))))
45	44	anbi1d 631	. . . . 5 ⊢ (𝜑 → ((𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = ◡𝐹) ↔ ((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌)) ∧ 𝐺 = ◡𝐹)))
46	45	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)))
47	42, 46	mpbird 257	. . 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))
48		rimrhm 20496	. . . . . 6 ⊢ (𝐹 ∈ (𝑋 RingIso 𝑌) → 𝐹 ∈ (𝑋 RingHom 𝑌))
49	48	ad2antrl 728	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = ◡𝐹)) → 𝐹 ∈ (𝑋 RingHom 𝑌))
50		isrim0 20483	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝑋 RingIso 𝑌) ↔ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ ◡𝐹 ∈ (𝑌 RingHom 𝑋)))
51	50	simprbi 496	. . . . . . . 8 ⊢ (𝐹 ∈ (𝑋 RingIso 𝑌) → ◡𝐹 ∈ (𝑌 RingHom 𝑋))
52		eleq1 2829	. . . . . . . 8 ⊢ (𝐺 = ◡𝐹 → (𝐺 ∈ (𝑌 RingHom 𝑋) ↔ ◡𝐹 ∈ (𝑌 RingHom 𝑋)))
53	51, 52	syl5ibrcom 247	. . . . . . 7 ⊢ (𝐹 ∈ (𝑋 RingIso 𝑌) → (𝐺 = ◡𝐹 → 𝐺 ∈ (𝑌 RingHom 𝑋)))
54	53	imp 406	. . . . . 6 ⊢ ((𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = ◡𝐹) → 𝐺 ∈ (𝑌 RingHom 𝑋))
55	54	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = ◡𝐹)) → 𝐺 ∈ (𝑌 RingHom 𝑋))
56		coeq1 5868	. . . . . . 7 ⊢ (𝐺 = ◡𝐹 → (𝐺 ∘ 𝐹) = (◡𝐹 ∘ 𝐹))
57	56	ad2antll 729	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = ◡𝐹)) → (𝐺 ∘ 𝐹) = (◡𝐹 ∘ 𝐹))
58	11, 15	rimf1o 20494	. . . . . . . 8 ⊢ (𝐹 ∈ (𝑋 RingIso 𝑌) → 𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌))
59	58	ad2antrl 728	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = ◡𝐹)) → 𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌))
60		f1ococnv1 6877	. . . . . . 7 ⊢ (𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌) → (◡𝐹 ∘ 𝐹) = ( I ↾ (Base‘𝑋)))
61	59, 60	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = ◡𝐹)) → (◡𝐹 ∘ 𝐹) = ( I ↾ (Base‘𝑋)))
62	57, 61	eqtrd 2777	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = ◡𝐹)) → (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋)))
63	49, 55, 62	jca31 514	. . . 4 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = ◡𝐹)) → ((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))))
64	50	biimpi 216	. . . . . 6 ⊢ (𝐹 ∈ (𝑋 RingIso 𝑌) → (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ ◡𝐹 ∈ (𝑌 RingHom 𝑋)))
65	64	ad2antrl 728	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = ◡𝐹)) → (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ ◡𝐹 ∈ (𝑌 RingHom 𝑋)))
66	52	ad2antll 729	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = ◡𝐹)) → (𝐺 ∈ (𝑌 RingHom 𝑋) ↔ ◡𝐹 ∈ (𝑌 RingHom 𝑋)))
67	66	anbi2d 630	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = ◡𝐹)) → ((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ↔ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ ◡𝐹 ∈ (𝑌 RingHom 𝑋))))
68	65, 67	mpbird 257	. . . 4 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = ◡𝐹)) → (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)))
69		coeq2 5869	. . . . . . 7 ⊢ (𝐺 = ◡𝐹 → (𝐹 ∘ 𝐺) = (𝐹 ∘ ◡𝐹))
70	69	ad2antll 729	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = ◡𝐹)) → (𝐹 ∘ 𝐺) = (𝐹 ∘ ◡𝐹))
71		f1ococnv2 6875	. . . . . . 7 ⊢ (𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌) → (𝐹 ∘ ◡𝐹) = ( I ↾ (Base‘𝑌)))
72	59, 71	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = ◡𝐹)) → (𝐹 ∘ ◡𝐹) = ( I ↾ (Base‘𝑌)))
73	70, 72	eqtrd 2777	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = ◡𝐹)) → (𝐹 ∘ 𝐺) = ( I ↾ (Base‘𝑌)))
74	68, 62, 73	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))))
75	63, 68, 74	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)))))
76	47, 75	impbida 801	. 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)))
77	10, 23, 76	3bitrd 305	1 ⊢ (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 ↔ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = ◡𝐹)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108   class class class wbr 5143   I cid 5577  ^◡ccnv 5684   ↾ cres 5687   ∘ ccom 5689  ⟶wf 6557  –1-1-onto→wf1o 6560  ‘cfv 6561  (class class class)co 7431  Basecbs 17247  Catccat 17707  Sectcsect 17788  Invcinv 17789   RingHom crh 20469   RingIso crs 20470  RingCatALTVcringcALTV 48203

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-fz 13548  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-plusg 17310  df-hom 17321  df-cco 17322  df-0g 17486  df-cat 17711  df-cid 17712  df-sect 17791  df-inv 17792  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-mhm 18796  df-grp 18954  df-ghm 19231  df-mgp 20138  df-ur 20179  df-ring 20232  df-rhm 20472  df-rim 20473  df-ringcALTV 48204

This theorem is referenced by:  ringcisoALTV  48227