rngcinvALTV - Mathbox for Alexander van der Vekens

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Theorem rngcinvALTV 48251

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

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

**Proof of Theorem rngcinvALTV**
Step	Hyp	Ref	Expression
1		rngcsectALTV.b	. . 3 ⊢ 𝐵 = (Base‘𝐶)
2		rngcinvALTV.n	. . 3 ⊢ 𝑁 = (Inv‘𝐶)
3		rngcsectALTV.u	. . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
4		rngcsectALTV.c	. . . . 5 ⊢ 𝐶 = (RngCatALTV‘𝑈)
5	4	rngccatALTV 48248	. . . 4 ⊢ (𝑈 ∈ 𝑉 → 𝐶 ∈ Cat)
6	3, 5	syl 17	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
7		rngcsectALTV.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
8		rngcsectALTV.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
9		eqid 2735	. . 3 ⊢ (Sect‘𝐶) = (Sect‘𝐶)
10	1, 2, 6, 7, 8, 9	isinv 17773	. 2 ⊢ (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 ↔ (𝐹(𝑋(Sect‘𝐶)𝑌)𝐺 ∧ 𝐺(𝑌(Sect‘𝐶)𝑋)𝐹)))
11		eqid 2735	. . . . . 6 ⊢ (Base‘𝑋) = (Base‘𝑋)
12	4, 1, 3, 7, 8, 11, 9	rngcsectALTV 48250	. . . . 5 ⊢ (𝜑 → (𝐹(𝑋(Sect‘𝐶)𝑌)𝐺 ↔ (𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋)))))
13		df-3an 1088	. . . . 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 287	. . . 4 ⊢ (𝜑 → (𝐹(𝑋(Sect‘𝐶)𝑌)𝐺 ↔ ((𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋)))))
15		eqid 2735	. . . . . 6 ⊢ (Base‘𝑌) = (Base‘𝑌)
16	4, 1, 3, 8, 7, 15, 9	rngcsectALTV 48250	. . . . 5 ⊢ (𝜑 → (𝐺(𝑌(Sect‘𝐶)𝑋)𝐹 ↔ (𝐺 ∈ (𝑌 RngHom 𝑋) ∧ 𝐹 ∈ (𝑋 RngHom 𝑌) ∧ (𝐹 ∘ 𝐺) = ( I ↾ (Base‘𝑌)))))
17		3ancoma 1097	. . . . . 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 1088	. . . . . 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 275	. . . . 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 287	. . . 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 632	. . 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 676	. . 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 287	. 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 776	. . . . . 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 481	. . . . 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 20408	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝑋 RngHom 𝑌) → 𝐹:(Base‘𝑋)⟶(Base‘𝑌))
27	15, 11	rnghmf 20408	. . . . . . . . . 10 ⊢ (𝐺 ∈ (𝑌 RngHom 𝑋) → 𝐺:(Base‘𝑌)⟶(Base‘𝑋))
28	26, 27	anim12i 613	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋)) → (𝐹:(Base‘𝑋)⟶(Base‘𝑌) ∧ 𝐺:(Base‘𝑌)⟶(Base‘𝑋)))
29	28	ad2antlr 727	. . . . . . . 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 484	. . . . . . . . 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 481	. . . . . . . 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 484	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))) → (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋)))
33	32	ad2antrl 728	. . . . . . . 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 515	. . . . . . 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 481	. . . . . 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 7289	. . . . . . 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 2742	. . . . . . . 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 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 468	. . . . 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 583	. . . 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 20413	. . . . . . 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 584	. . . . . 6 ⊢ (𝜑 → (𝐹 ∈ (𝑋 RngIso 𝑌) ↔ (𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌))))
45	44	anbi1d 631	. . . . 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 480	. . . 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 257	. . 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 20415	. . . . . 6 ⊢ (𝐹 ∈ (𝑋 RngIso 𝑌) → 𝐹 ∈ (𝑋 RngHom 𝑌))
49	48	ad2antrl 728	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RngIso 𝑌) ∧ 𝐺 = ^◡𝐹)) → 𝐹 ∈ (𝑋 RngHom 𝑌))
50		isrngim 20405	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝐹 ∈ (𝑋 RngIso 𝑌) ↔ (𝐹 ∈ (𝑋 RngHom 𝑌) ∧ ^◡𝐹 ∈ (𝑌 RngHom 𝑋))))
51	7, 8, 50	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 ∈ (𝑋 RngIso 𝑌) ↔ (𝐹 ∈ (𝑋 RngHom 𝑌) ∧ ^◡𝐹 ∈ (𝑌 RngHom 𝑋))))
52		eleq1 2822	. . . . . . . . . . . 12 ⊢ (^◡𝐹 = 𝐺 → (^◡𝐹 ∈ (𝑌 RngHom 𝑋) ↔ 𝐺 ∈ (𝑌 RngHom 𝑋)))
53	52	eqcoms 2743	. . . . . . . . . . 11 ⊢ (𝐺 = ^◡𝐹 → (^◡𝐹 ∈ (𝑌 RngHom 𝑋) ↔ 𝐺 ∈ (𝑌 RngHom 𝑋)))
54	53	anbi2d 630	. . . . . . . . . 10 ⊢ (𝐺 = ^◡𝐹 → ((𝐹 ∈ (𝑋 RngHom 𝑌) ∧ ^◡𝐹 ∈ (𝑌 RngHom 𝑋)) ↔ (𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋))))
55	51, 54	sylan9bbr 510	. . . . . . . . 9 ⊢ ((𝐺 = ^◡𝐹 ∧ 𝜑) → (𝐹 ∈ (𝑋 RngIso 𝑌) ↔ (𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋))))
56		simpr 484	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋)) → 𝐺 ∈ (𝑌 RngHom 𝑋))
57	55, 56	biimtrdi 253	. . . . . . . 8 ⊢ ((𝐺 = ^◡𝐹 ∧ 𝜑) → (𝐹 ∈ (𝑋 RngIso 𝑌) → 𝐺 ∈ (𝑌 RngHom 𝑋)))
58	57	com12 32	. . . . . . 7 ⊢ (𝐹 ∈ (𝑋 RngIso 𝑌) → ((𝐺 = ^◡𝐹 ∧ 𝜑) → 𝐺 ∈ (𝑌 RngHom 𝑋)))
59	58	expdimp 452	. . . . . 6 ⊢ ((𝐹 ∈ (𝑋 RngIso 𝑌) ∧ 𝐺 = ^◡𝐹) → (𝜑 → 𝐺 ∈ (𝑌 RngHom 𝑋)))
60	59	impcom 407	. . . . 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 729	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RngIso 𝑌) ∧ 𝐺 = ^◡𝐹)) → (𝐺 ∘ 𝐹) = (^◡𝐹 ∘ 𝐹))
63	11, 15	rngimf1o 20414	. . . . . . . 8 ⊢ (𝐹 ∈ (𝑋 RngIso 𝑌) → 𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌))
64	63	ad2antrl 728	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RngIso 𝑌) ∧ 𝐺 = ^◡𝐹)) → 𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌))
65		f1ococnv1 6847	. . . . . . 7 ⊢ (𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌) → (^◡𝐹 ∘ 𝐹) = ( I ↾ (Base‘𝑋)))
66	64, 65	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RngIso 𝑌) ∧ 𝐺 = ^◡𝐹)) → (^◡𝐹 ∘ 𝐹) = ( I ↾ (Base‘𝑋)))
67	62, 66	eqtrd 2770	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RngIso 𝑌) ∧ 𝐺 = ^◡𝐹)) → (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋)))
68	49, 60, 67	jca31 514	. . . 4 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RngIso 𝑌) ∧ 𝐺 = ^◡𝐹)) → ((𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))))
69	51	biimpcd 249	. . . . . . 7 ⊢ (𝐹 ∈ (𝑋 RngIso 𝑌) → (𝜑 → (𝐹 ∈ (𝑋 RngHom 𝑌) ∧ ^◡𝐹 ∈ (𝑌 RngHom 𝑋))))
70	69	adantr 480	. . . . . 6 ⊢ ((𝐹 ∈ (𝑋 RngIso 𝑌) ∧ 𝐺 = ^◡𝐹) → (𝜑 → (𝐹 ∈ (𝑋 RngHom 𝑌) ∧ ^◡𝐹 ∈ (𝑌 RngHom 𝑋))))
71	70	impcom 407	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RngIso 𝑌) ∧ 𝐺 = ^◡𝐹)) → (𝐹 ∈ (𝑋 RngHom 𝑌) ∧ ^◡𝐹 ∈ (𝑌 RngHom 𝑋)))
72		eleq1 2822	. . . . . . 7 ⊢ (𝐺 = ^◡𝐹 → (𝐺 ∈ (𝑌 RngHom 𝑋) ↔ ^◡𝐹 ∈ (𝑌 RngHom 𝑋)))
73	72	ad2antll 729	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RngIso 𝑌) ∧ 𝐺 = ^◡𝐹)) → (𝐺 ∈ (𝑌 RngHom 𝑋) ↔ ^◡𝐹 ∈ (𝑌 RngHom 𝑋)))
74	73	anbi2d 630	. . . . 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 257	. . . 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 729	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RngIso 𝑌) ∧ 𝐺 = ^◡𝐹)) → (𝐹 ∘ 𝐺) = (𝐹 ∘ ^◡𝐹))
78		f1ococnv2 6845	. . . . . . 7 ⊢ (𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌) → (𝐹 ∘ ^◡𝐹) = ( I ↾ (Base‘𝑌)))
79	64, 78	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RngIso 𝑌) ∧ 𝐺 = ^◡𝐹)) → (𝐹 ∘ ^◡𝐹) = ( I ↾ (Base‘𝑌)))
80	77, 79	eqtrd 2770	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RngIso 𝑌) ∧ 𝐺 = ^◡𝐹)) → (𝐹 ∘ 𝐺) = ( I ↾ (Base‘𝑌)))
81	75, 67, 80	jca31 514	. . . 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 514	. . 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 800	. 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 305	1 ⊢ (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 ↔ (𝐹 ∈ (𝑋 RngIso 𝑌) ∧ 𝐺 = ^◡𝐹)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2108 class class class wbr 5119 I cid 5547 ^◡ccnv 5653 ↾ cres 5656 ∘ ccom 5658 ⟶wf 6527 –1-1-onto→wf1o 6530 ‘cfv 6531 (class class class)co 7405 Basecbs 17228 Catccat 17676 Sectcsect 17757 Invcinv 17758 RngHom crnghm 20394 RngIso crngim 20395 RngCatALTVcrngcALTV 48238

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 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205 ax-pre-mulgt0 11206

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-tp 4606 df-op 4608 df-uni 4884 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7862 df-1st 7988 df-2nd 7989 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8719 df-map 8842 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-sub 11468 df-neg 11469 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12502 df-z 12589 df-dec 12709 df-uz 12853 df-fz 13525 df-struct 17166 df-sets 17183 df-slot 17201 df-ndx 17213 df-base 17229 df-plusg 17284 df-hom 17295 df-cco 17296 df-0g 17455 df-cat 17680 df-cid 17681 df-sect 17760 df-inv 17761 df-mgm 18618 df-mgmhm 18670 df-sgrp 18697 df-mnd 18713 df-mhm 18761 df-grp 18919 df-ghm 19196 df-abl 19764 df-mgp 20101 df-rng 20113 df-rnghm 20396 df-rngim 20397 df-rngcALTV 48239

This theorem is referenced by: rngcisoALTV 48252

W3C validator