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Theorem rngcinv 20800

Description: An inverse in the category of non-unital rings is the converse operation. (Contributed by AV, 28-Feb-2020.)

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

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

**Proof of Theorem rngcinv**
Step	Hyp	Ref	Expression
1		rngcsect.b	. . 3 ⊢ 𝐵 = (Base‘𝐶)
2		rngcinv.n	. . 3 ⊢ 𝑁 = (Inv‘𝐶)
3		rngcsect.u	. . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
4		rngcsect.c	. . . . 5 ⊢ 𝐶 = (RngCat‘𝑈)
5	4	rngccat 20797	. . . 4 ⊢ (𝑈 ∈ 𝑉 → 𝐶 ∈ Cat)
6	3, 5	syl 18	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
7		rngcsect.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
8		rngcsect.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
9		eqid 2760	. . 3 ⊢ (Sect‘𝐶) = (Sect‘𝐶)
10	1, 2, 6, 7, 8, 9	isinv 17850	. 2 ⊢ (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 ↔ (𝐹(𝑋(Sect‘𝐶)𝑌)𝐺 ∧ 𝐺(𝑌(Sect‘𝐶)𝑋)𝐹)))
11		eqid 2760	. . . . . 6 ⊢ (Base‘𝑋) = (Base‘𝑋)
12	4, 1, 3, 7, 8, 11, 9	rngcsect 20799	. . . . 5 ⊢ (𝜑 → (𝐹(𝑋(Sect‘𝐶)𝑌)𝐺 ↔ (𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋)))))
13		df-3an 1105	. . . . 5 ⊢ ((𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))) ↔ ((𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))))
14	12, 13	bitrdi 290	. . . 4 ⊢ (𝜑 → (𝐹(𝑋(Sect‘𝐶)𝑌)𝐺 ↔ ((𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋)))))
15		eqid 2760	. . . . . 6 ⊢ (Base‘𝑌) = (Base‘𝑌)
16	4, 1, 3, 8, 7, 15, 9	rngcsect 20799	. . . . 5 ⊢ (𝜑 → (𝐺(𝑌(Sect‘𝐶)𝑋)𝐹 ↔ (𝐺 ∈ (𝑌 RngHom 𝑋) ∧ 𝐹 ∈ (𝑋 RngHom 𝑌) ∧ (𝐹 ∘ 𝐺) = ( I ↾ (Base‘𝑌)))))
17		3ancoma 1115	. . . . . 6 ⊢ ((𝐺 ∈ (𝑌 RngHom 𝑋) ∧ 𝐹 ∈ (𝑋 RngHom 𝑌) ∧ (𝐹 ∘ 𝐺) = ( I ↾ (Base‘𝑌))) ↔ (𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋) ∧ (𝐹 ∘ 𝐺) = ( I ↾ (Base‘𝑌))))
18		df-3an 1105	. . . . . 6 ⊢ ((𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋) ∧ (𝐹 ∘ 𝐺) = ( I ↾ (Base‘𝑌))) ↔ ((𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋)) ∧ (𝐹 ∘ 𝐺) = ( I ↾ (Base‘𝑌))))
19	17, 18	bitri 278	. . . . 5 ⊢ ((𝐺 ∈ (𝑌 RngHom 𝑋) ∧ 𝐹 ∈ (𝑋 RngHom 𝑌) ∧ (𝐹 ∘ 𝐺) = ( I ↾ (Base‘𝑌))) ↔ ((𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋)) ∧ (𝐹 ∘ 𝐺) = ( I ↾ (Base‘𝑌))))
20	16, 19	bitrdi 290	. . . 4 ⊢ (𝜑 → (𝐺(𝑌(Sect‘𝐶)𝑋)𝐹 ↔ ((𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋)) ∧ (𝐹 ∘ 𝐺) = ( I ↾ (Base‘𝑌)))))
21	14, 20	anbi12d 644	. . 3 ⊢ (𝜑 → ((𝐹(𝑋(Sect‘𝐶)𝑌)𝐺 ∧ 𝐺(𝑌(Sect‘𝐶)𝑋)𝐹) ↔ (((𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))) ∧ ((𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋)) ∧ (𝐹 ∘ 𝐺) = ( I ↾ (Base‘𝑌))))))
22		anandi 689	. . 3 ⊢ ((((𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))) ∧ ((𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋)) ∧ (𝐹 ∘ 𝐺) = ( I ↾ (Base‘𝑌)))) ↔ ((((𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋))) ∧ (((𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∘ 𝐺) = ( I ↾ (Base‘𝑌)))))
23	21, 22	bitrdi 290	. 2 ⊢ (𝜑 → ((𝐹(𝑋(Sect‘𝐶)𝑌)𝐺 ∧ 𝐺(𝑌(Sect‘𝐶)𝑋)𝐹) ↔ ((((𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋))) ∧ (((𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∘ 𝐺) = ( I ↾ (Base‘𝑌))))))
24		simplrl 789	. . . . . 6 ⊢ (((((𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋))) ∧ (((𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∘ 𝐺) = ( I ↾ (Base‘𝑌)))) → 𝐹 ∈ (𝑋 RngHom 𝑌))
25	24	adantl 487	. . . . 5 ⊢ ((𝜑 ∧ ((((𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋))) ∧ (((𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∘ 𝐺) = ( I ↾ (Base‘𝑌))))) → 𝐹 ∈ (𝑋 RngHom 𝑌))
26	11, 15	rnghmf 20590	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝑋 RngHom 𝑌) → 𝐹:(Base‘𝑋)⟶(Base‘𝑌))
27	15, 11	rnghmf 20590	. . . . . . . . . 10 ⊢ (𝐺 ∈ (𝑌 RngHom 𝑋) → 𝐺:(Base‘𝑌)⟶(Base‘𝑋))
28	26, 27	anim12i 625	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋)) → (𝐹:(Base‘𝑋)⟶(Base‘𝑌) ∧ 𝐺:(Base‘𝑌)⟶(Base‘𝑋)))
29	28	ad2antlr 740	. . . . . . . 8 ⊢ (((((𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋))) ∧ (((𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∘ 𝐺) = ( I ↾ (Base‘𝑌)))) → (𝐹:(Base‘𝑋)⟶(Base‘𝑌) ∧ 𝐺:(Base‘𝑌)⟶(Base‘𝑋)))
30		simpr 490	. . . . . . . . 9 ⊢ ((((𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∘ 𝐺) = ( I ↾ (Base‘𝑌))) → (𝐹 ∘ 𝐺) = ( I ↾ (Base‘𝑌)))
31	30	adantl 487	. . . . . . . 8 ⊢ (((((𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋))) ∧ (((𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∘ 𝐺) = ( I ↾ (Base‘𝑌)))) → (𝐹 ∘ 𝐺) = ( I ↾ (Base‘𝑌)))
32		simpr 490	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))) → (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋)))
33	32	ad2antrl 741	. . . . . . . 8 ⊢ (((((𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋))) ∧ (((𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∘ 𝐺) = ( I ↾ (Base‘𝑌)))) → (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋)))
34	29, 31, 33	jca32 525	. . . . . . 7 ⊢ (((((𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋))) ∧ (((𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 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 487	. . . . . 6 ⊢ ((𝜑 ∧ ((((𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋))) ∧ (((𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 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 7298	. . . . . . 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 2767	. . . . . . . 8 ⊢ (^◡𝐹 = 𝐺 ↔ 𝐺 = ^◡𝐹)
38	37	anbi2i 635	. . . . . . 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 221	. . . . . 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 18	. . . . 5 ⊢ ((𝜑 ∧ ((((𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋))) ∧ (((𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∘ 𝐺) = ( I ↾ (Base‘𝑌))))) → (𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌) ∧ 𝐺 = ^◡𝐹))
41		anass 474	. . . . 5 ⊢ (((𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌)) ∧ 𝐺 = ^◡𝐹) ↔ (𝐹 ∈ (𝑋 RngHom 𝑌) ∧ (𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌) ∧ 𝐺 = ^◡𝐹)))
42	25, 40, 41	sylanbrc 595	. . . 4 ⊢ ((𝜑 ∧ ((((𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋))) ∧ (((𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∘ 𝐺) = ( I ↾ (Base‘𝑌))))) → ((𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌)) ∧ 𝐺 = ^◡𝐹))
43	11, 15	isrngim2 20595	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝐹 ∈ (𝑋 RngIso 𝑌) ↔ (𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌))))
44	7, 8, 43	syl2anc 596	. . . . . 6 ⊢ (𝜑 → (𝐹 ∈ (𝑋 RngIso 𝑌) ↔ (𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌))))
45	44	anbi1d 643	. . . . 5 ⊢ (𝜑 → ((𝐹 ∈ (𝑋 RngIso 𝑌) ∧ 𝐺 = ^◡𝐹) ↔ ((𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌)) ∧ 𝐺 = ^◡𝐹)))
46	45	adantr 486	. . . 4 ⊢ ((𝜑 ∧ ((((𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋))) ∧ (((𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∘ 𝐺) = ( I ↾ (Base‘𝑌))))) → ((𝐹 ∈ (𝑋 RngIso 𝑌) ∧ 𝐺 = ^◡𝐹) ↔ ((𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌)) ∧ 𝐺 = ^◡𝐹)))
47	42, 46	mpbird 260	. . 3 ⊢ ((𝜑 ∧ ((((𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋))) ∧ (((𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∘ 𝐺) = ( I ↾ (Base‘𝑌))))) → (𝐹 ∈ (𝑋 RngIso 𝑌) ∧ 𝐺 = ^◡𝐹))
48	11, 15	rngimrnghm 20597	. . . . . 6 ⊢ (𝐹 ∈ (𝑋 RngIso 𝑌) → 𝐹 ∈ (𝑋 RngHom 𝑌))
49	48	ad2antrl 741	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RngIso 𝑌) ∧ 𝐺 = ^◡𝐹)) → 𝐹 ∈ (𝑋 RngHom 𝑌))
50		isrngim 20587	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝐹 ∈ (𝑋 RngIso 𝑌) ↔ (𝐹 ∈ (𝑋 RngHom 𝑌) ∧ ^◡𝐹 ∈ (𝑌 RngHom 𝑋))))
51	7, 8, 50	syl2anc 596	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 ∈ (𝑋 RngIso 𝑌) ↔ (𝐹 ∈ (𝑋 RngHom 𝑌) ∧ ^◡𝐹 ∈ (𝑌 RngHom 𝑋))))
52		eleq1 2848	. . . . . . . . . . . 12 ⊢ (^◡𝐹 = 𝐺 → (^◡𝐹 ∈ (𝑌 RngHom 𝑋) ↔ 𝐺 ∈ (𝑌 RngHom 𝑋)))
53	52	eqcoms 2768	. . . . . . . . . . 11 ⊢ (𝐺 = ^◡𝐹 → (^◡𝐹 ∈ (𝑌 RngHom 𝑋) ↔ 𝐺 ∈ (𝑌 RngHom 𝑋)))
54	53	anbi2d 642	. . . . . . . . . 10 ⊢ (𝐺 = ^◡𝐹 → ((𝐹 ∈ (𝑋 RngHom 𝑌) ∧ ^◡𝐹 ∈ (𝑌 RngHom 𝑋)) ↔ (𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋))))
55	51, 54	sylan9bbr 520	. . . . . . . . 9 ⊢ ((𝐺 = ^◡𝐹 ∧ 𝜑) → (𝐹 ∈ (𝑋 RngIso 𝑌) ↔ (𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋))))
56		simpr 490	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋)) → 𝐺 ∈ (𝑌 RngHom 𝑋))
57	55, 56	biimtrdi 256	. . . . . . . 8 ⊢ ((𝐺 = ^◡𝐹 ∧ 𝜑) → (𝐹 ∈ (𝑋 RngIso 𝑌) → 𝐺 ∈ (𝑌 RngHom 𝑋)))
58	57	com12 33	. . . . . . 7 ⊢ (𝐹 ∈ (𝑋 RngIso 𝑌) → ((𝐺 = ^◡𝐹 ∧ 𝜑) → 𝐺 ∈ (𝑌 RngHom 𝑋)))
59	58	expdimp 458	. . . . . 6 ⊢ ((𝐹 ∈ (𝑋 RngIso 𝑌) ∧ 𝐺 = ^◡𝐹) → (𝜑 → 𝐺 ∈ (𝑌 RngHom 𝑋)))
60	59	impcom 413	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RngIso 𝑌) ∧ 𝐺 = ^◡𝐹)) → 𝐺 ∈ (𝑌 RngHom 𝑋))
61		coeq1 5837	. . . . . . 7 ⊢ (𝐺 = ^◡𝐹 → (𝐺 ∘ 𝐹) = (^◡𝐹 ∘ 𝐹))
62	61	ad2antll 742	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RngIso 𝑌) ∧ 𝐺 = ^◡𝐹)) → (𝐺 ∘ 𝐹) = (^◡𝐹 ∘ 𝐹))
63	11, 15	rngimf1o 20596	. . . . . . . 8 ⊢ (𝐹 ∈ (𝑋 RngIso 𝑌) → 𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌))
64	63	ad2antrl 741	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RngIso 𝑌) ∧ 𝐺 = ^◡𝐹)) → 𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌))
65		f1ococnv1 6848	. . . . . . 7 ⊢ (𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌) → (^◡𝐹 ∘ 𝐹) = ( I ↾ (Base‘𝑋)))
66	64, 65	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RngIso 𝑌) ∧ 𝐺 = ^◡𝐹)) → (^◡𝐹 ∘ 𝐹) = ( I ↾ (Base‘𝑋)))
67	62, 66	eqtrd 2795	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RngIso 𝑌) ∧ 𝐺 = ^◡𝐹)) → (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋)))
68	49, 60, 67	jca31 524	. . . 4 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RngIso 𝑌) ∧ 𝐺 = ^◡𝐹)) → ((𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))))
69	51	biimpcd 252	. . . . . . 7 ⊢ (𝐹 ∈ (𝑋 RngIso 𝑌) → (𝜑 → (𝐹 ∈ (𝑋 RngHom 𝑌) ∧ ^◡𝐹 ∈ (𝑌 RngHom 𝑋))))
70	69	adantr 486	. . . . . 6 ⊢ ((𝐹 ∈ (𝑋 RngIso 𝑌) ∧ 𝐺 = ^◡𝐹) → (𝜑 → (𝐹 ∈ (𝑋 RngHom 𝑌) ∧ ^◡𝐹 ∈ (𝑌 RngHom 𝑋))))
71	70	impcom 413	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RngIso 𝑌) ∧ 𝐺 = ^◡𝐹)) → (𝐹 ∈ (𝑋 RngHom 𝑌) ∧ ^◡𝐹 ∈ (𝑌 RngHom 𝑋)))
72		eleq1 2848	. . . . . . 7 ⊢ (𝐺 = ^◡𝐹 → (𝐺 ∈ (𝑌 RngHom 𝑋) ↔ ^◡𝐹 ∈ (𝑌 RngHom 𝑋)))
73	72	ad2antll 742	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RngIso 𝑌) ∧ 𝐺 = ^◡𝐹)) → (𝐺 ∈ (𝑌 RngHom 𝑋) ↔ ^◡𝐹 ∈ (𝑌 RngHom 𝑋)))
74	73	anbi2d 642	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RngIso 𝑌) ∧ 𝐺 = ^◡𝐹)) → ((𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋)) ↔ (𝐹 ∈ (𝑋 RngHom 𝑌) ∧ ^◡𝐹 ∈ (𝑌 RngHom 𝑋))))
75	71, 74	mpbird 260	. . . 4 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RngIso 𝑌) ∧ 𝐺 = ^◡𝐹)) → (𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋)))
76		coeq2 5838	. . . . . . 7 ⊢ (𝐺 = ^◡𝐹 → (𝐹 ∘ 𝐺) = (𝐹 ∘ ^◡𝐹))
77	76	ad2antll 742	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RngIso 𝑌) ∧ 𝐺 = ^◡𝐹)) → (𝐹 ∘ 𝐺) = (𝐹 ∘ ^◡𝐹))
78		f1ococnv2 6846	. . . . . . 7 ⊢ (𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌) → (𝐹 ∘ ^◡𝐹) = ( I ↾ (Base‘𝑌)))
79	64, 78	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RngIso 𝑌) ∧ 𝐺 = ^◡𝐹)) → (𝐹 ∘ ^◡𝐹) = ( I ↾ (Base‘𝑌)))
80	77, 79	eqtrd 2795	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RngIso 𝑌) ∧ 𝐺 = ^◡𝐹)) → (𝐹 ∘ 𝐺) = ( I ↾ (Base‘𝑌)))
81	75, 67, 80	jca31 524	. . . 4 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RngIso 𝑌) ∧ 𝐺 = ^◡𝐹)) → (((𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∘ 𝐺) = ( I ↾ (Base‘𝑌))))
82	68, 75, 81	jca31 524	. . 3 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RngIso 𝑌) ∧ 𝐺 = ^◡𝐹)) → ((((𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋))) ∧ (((𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∘ 𝐺) = ( I ↾ (Base‘𝑌)))))
83	47, 82	impbida 813	. 2 ⊢ (𝜑 → (((((𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋))) ∧ (((𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))) ∧ (𝐹 ∘ 𝐺) = ( I ↾ (Base‘𝑌)))) ↔ (𝐹 ∈ (𝑋 RngIso 𝑌) ∧ 𝐺 = ^◡𝐹)))
84	10, 23, 83	3bitrd 308	1 ⊢ (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 ↔ (𝐹 ∈ (𝑋 RngIso 𝑌) ∧ 𝐺 = ^◡𝐹)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 class class class wbr 5103 I cid 5549 ^◡ccnv 5654 ↾ cres 5657 ∘ ccom 5659 ⟶wf 6529 –1-1-onto→wf1o 6532 ‘cfv 6533 (class class class)co 7414 Basecbs 17302 Catccat 17753 Sectcsect 17834 Invcinv 17835 RngHom crnghm 20576 RngIso crngim 20577 RngCatcrngc 20779

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7737 ax-cnex 11181 ax-resscn 11182 ax-1cn 11183 ax-icn 11184 ax-addcl 11185 ax-addrcl 11186 ax-mulcl 11187 ax-mulrcl 11188 ax-mulcom 11189 ax-addass 11190 ax-mulass 11191 ax-distr 11192 ax-i2m1 11193 ax-1ne0 11194 ax-1rid 11195 ax-rnegex 11196 ax-rrecex 11197 ax-cnre 11198 ax-pre-lttri 11199 ax-pre-lttrn 11200 ax-pre-ltadd 11201 ax-pre-mulgt0 11202

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7371 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7864 df-1st 7987 df-2nd 7988 df-frecs 8281 df-wrecs 8312 df-recs 8361 df-rdg 8400 df-1o 8456 df-er 8697 df-map 8829 df-pm 8830 df-ixp 8906 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-pnf 11270 df-mnf 11271 df-xr 11272 df-ltxr 11273 df-le 11274 df-sub 11468 df-neg 11469 df-nn 12259 df-2 12328 df-3 12329 df-4 12330 df-5 12331 df-6 12332 df-7 12333 df-8 12334 df-9 12335 df-n0 12530 df-z 12617 df-dec 12738 df-uz 12889 df-fz 13563 df-struct 17240 df-sets 17257 df-slot 17275 df-ndx 17287 df-base 17303 df-ress 17324 df-plusg 17356 df-hom 17367 df-cco 17368 df-0g 17527 df-cat 17757 df-cid 17758 df-homf 17759 df-sect 17837 df-inv 17838 df-ssc 17900 df-resc 17901 df-subc 17902 df-estrc 18212 df-mgm 18731 df-mgmhm 18795 df-sgrp 18822 df-mnd 18838 df-mhm 18892 df-grp 19061 df-ghm 19342 df-abl 19911 df-mgp 20275 df-rng 20289 df-rnghm 20578 df-rngim 20579 df-rngc 20780

This theorem is used by: rngciso 20801

W3C validator