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Theorem ringcinv 20671

Description: An inverse in the category of unital rings is the converse operation. (Contributed by AV, 14-Feb-2020.)

**Hypotheses**
Ref	Expression
ringcsect.c	⊢ 𝐶 = (RingCat‘𝑈)
ringcsect.b	⊢ 𝐵 = (Base‘𝐶)
ringcsect.u	⊢ (𝜑 → 𝑈 ∈ 𝑉)
ringcsect.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
ringcsect.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
ringcinv.n	⊢ 𝑁 = (Inv‘𝐶)

**Assertion**
Ref	Expression
ringcinv	⊢ (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 ↔ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = ^◡𝐹)))

**Proof of Theorem ringcinv**
Step	Hyp	Ref	Expression
1		ringcsect.b	. . 3 ⊢ 𝐵 = (Base‘𝐶)
2		ringcinv.n	. . 3 ⊢ 𝑁 = (Inv‘𝐶)
3		ringcsect.u	. . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
4		ringcsect.c	. . . . 5 ⊢ 𝐶 = (RingCat‘𝑈)
5	4	ringccat 20663	. . . 4 ⊢ (𝑈 ∈ 𝑉 → 𝐶 ∈ Cat)
6	3, 5	syl 17	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
7		ringcsect.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
8		ringcsect.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	ringcsect 20670	. . . . 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	ringcsect 20670	. . . . 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)))

Colors of variables: wff setvar class

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 RingCatcringc 20645

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-pm 8869 df-ixp 8938 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-ress 17275 df-plusg 17310 df-hom 17321 df-cco 17322 df-0g 17486 df-cat 17711 df-cid 17712 df-homf 17713 df-sect 17791 df-inv 17792 df-ssc 17854 df-resc 17855 df-subc 17856 df-estrc 18167 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-ringc 20646

This theorem is referenced by: ringciso 20672

W3C validator