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Theorem rngcinv 20577

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 20574	. . . 4 ⊢ (𝑈 ∈ 𝑉 → 𝐶 ∈ Cat)
6	3, 5	syl 17	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
7		rngcsect.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
8		rngcsect.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
9		eqid 2728	. . 3 ⊢ (Sect‘𝐶) = (Sect‘𝐶)
10	1, 2, 6, 7, 8, 9	isinv 17750	. 2 ⊢ (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 ↔ (𝐹(𝑋(Sect‘𝐶)𝑌)𝐺 ∧ 𝐺(𝑌(Sect‘𝐶)𝑋)𝐹)))
11		eqid 2728	. . . . . 6 ⊢ (Base‘𝑋) = (Base‘𝑋)
12	4, 1, 3, 7, 8, 11, 9	rngcsect 20576	. . . . 5 ⊢ (𝜑 → (𝐹(𝑋(Sect‘𝐶)𝑌)𝐺 ↔ (𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋)))))
13		df-3an 1086	. . . . 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 286	. . . 4 ⊢ (𝜑 → (𝐹(𝑋(Sect‘𝐶)𝑌)𝐺 ↔ ((𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋)))))
15		eqid 2728	. . . . . 6 ⊢ (Base‘𝑌) = (Base‘𝑌)
16	4, 1, 3, 8, 7, 15, 9	rngcsect 20576	. . . . 5 ⊢ (𝜑 → (𝐺(𝑌(Sect‘𝐶)𝑋)𝐹 ↔ (𝐺 ∈ (𝑌 RngHom 𝑋) ∧ 𝐹 ∈ (𝑋 RngHom 𝑌) ∧ (𝐹 ∘ 𝐺) = ( I ↾ (Base‘𝑌)))))
17		3ancoma 1095	. . . . . 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 1086	. . . . . 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 274	. . . . 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 286	. . . 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 630	. . 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 674	. . 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 286	. 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 775	. . . . . 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 480	. . . . 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 20394	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝑋 RngHom 𝑌) → 𝐹:(Base‘𝑋)⟶(Base‘𝑌))
27	15, 11	rnghmf 20394	. . . . . . . . . 10 ⊢ (𝐺 ∈ (𝑌 RngHom 𝑋) → 𝐺:(Base‘𝑌)⟶(Base‘𝑋))
28	26, 27	anim12i 611	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋)) → (𝐹:(Base‘𝑋)⟶(Base‘𝑌) ∧ 𝐺:(Base‘𝑌)⟶(Base‘𝑋)))
29	28	ad2antlr 725	. . . . . . . 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 483	. . . . . . . . 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 480	. . . . . . . 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 483	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))) → (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋)))
33	32	ad2antrl 726	. . . . . . . 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 514	. . . . . . 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 480	. . . . . 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 7311	. . . . . . 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 2735	. . . . . . . 8 ⊢ (^◡𝐹 = 𝐺 ↔ 𝐺 = ^◡𝐹)
38	37	anbi2i 621	. . . . . . 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 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))
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 467	. . . . 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 581	. . . 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 20399	. . . . . . 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 582	. . . . . 6 ⊢ (𝜑 → (𝐹 ∈ (𝑋 RngIso 𝑌) ↔ (𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌))))
45	44	anbi1d 629	. . . . 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 479	. . . 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 256	. . 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 20401	. . . . . 6 ⊢ (𝐹 ∈ (𝑋 RngIso 𝑌) → 𝐹 ∈ (𝑋 RngHom 𝑌))
49	48	ad2antrl 726	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RngIso 𝑌) ∧ 𝐺 = ^◡𝐹)) → 𝐹 ∈ (𝑋 RngHom 𝑌))
50		isrngim 20391	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝐹 ∈ (𝑋 RngIso 𝑌) ↔ (𝐹 ∈ (𝑋 RngHom 𝑌) ∧ ^◡𝐹 ∈ (𝑌 RngHom 𝑋))))
51	7, 8, 50	syl2anc 582	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 ∈ (𝑋 RngIso 𝑌) ↔ (𝐹 ∈ (𝑋 RngHom 𝑌) ∧ ^◡𝐹 ∈ (𝑌 RngHom 𝑋))))
52		eleq1 2817	. . . . . . . . . . . 12 ⊢ (^◡𝐹 = 𝐺 → (^◡𝐹 ∈ (𝑌 RngHom 𝑋) ↔ 𝐺 ∈ (𝑌 RngHom 𝑋)))
53	52	eqcoms 2736	. . . . . . . . . . 11 ⊢ (𝐺 = ^◡𝐹 → (^◡𝐹 ∈ (𝑌 RngHom 𝑋) ↔ 𝐺 ∈ (𝑌 RngHom 𝑋)))
54	53	anbi2d 628	. . . . . . . . . 10 ⊢ (𝐺 = ^◡𝐹 → ((𝐹 ∈ (𝑋 RngHom 𝑌) ∧ ^◡𝐹 ∈ (𝑌 RngHom 𝑋)) ↔ (𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋))))
55	51, 54	sylan9bbr 509	. . . . . . . . 9 ⊢ ((𝐺 = ^◡𝐹 ∧ 𝜑) → (𝐹 ∈ (𝑋 RngIso 𝑌) ↔ (𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋))))
56		simpr 483	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋)) → 𝐺 ∈ (𝑌 RngHom 𝑋))
57	55, 56	biimtrdi 252	. . . . . . . 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 451	. . . . . 6 ⊢ ((𝐹 ∈ (𝑋 RngIso 𝑌) ∧ 𝐺 = ^◡𝐹) → (𝜑 → 𝐺 ∈ (𝑌 RngHom 𝑋)))
60	59	impcom 406	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RngIso 𝑌) ∧ 𝐺 = ^◡𝐹)) → 𝐺 ∈ (𝑌 RngHom 𝑋))
61		coeq1 5864	. . . . . . 7 ⊢ (𝐺 = ^◡𝐹 → (𝐺 ∘ 𝐹) = (^◡𝐹 ∘ 𝐹))
62	61	ad2antll 727	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RngIso 𝑌) ∧ 𝐺 = ^◡𝐹)) → (𝐺 ∘ 𝐹) = (^◡𝐹 ∘ 𝐹))
63	11, 15	rngimf1o 20400	. . . . . . . 8 ⊢ (𝐹 ∈ (𝑋 RngIso 𝑌) → 𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌))
64	63	ad2antrl 726	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RngIso 𝑌) ∧ 𝐺 = ^◡𝐹)) → 𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌))
65		f1ococnv1 6873	. . . . . . 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 2768	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RngIso 𝑌) ∧ 𝐺 = ^◡𝐹)) → (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋)))
68	49, 60, 67	jca31 513	. . . 4 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RngIso 𝑌) ∧ 𝐺 = ^◡𝐹)) → ((𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))))
69	51	biimpcd 248	. . . . . . 7 ⊢ (𝐹 ∈ (𝑋 RngIso 𝑌) → (𝜑 → (𝐹 ∈ (𝑋 RngHom 𝑌) ∧ ^◡𝐹 ∈ (𝑌 RngHom 𝑋))))
70	69	adantr 479	. . . . . 6 ⊢ ((𝐹 ∈ (𝑋 RngIso 𝑌) ∧ 𝐺 = ^◡𝐹) → (𝜑 → (𝐹 ∈ (𝑋 RngHom 𝑌) ∧ ^◡𝐹 ∈ (𝑌 RngHom 𝑋))))
71	70	impcom 406	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RngIso 𝑌) ∧ 𝐺 = ^◡𝐹)) → (𝐹 ∈ (𝑋 RngHom 𝑌) ∧ ^◡𝐹 ∈ (𝑌 RngHom 𝑋)))
72		eleq1 2817	. . . . . . 7 ⊢ (𝐺 = ^◡𝐹 → (𝐺 ∈ (𝑌 RngHom 𝑋) ↔ ^◡𝐹 ∈ (𝑌 RngHom 𝑋)))
73	72	ad2antll 727	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RngIso 𝑌) ∧ 𝐺 = ^◡𝐹)) → (𝐺 ∈ (𝑌 RngHom 𝑋) ↔ ^◡𝐹 ∈ (𝑌 RngHom 𝑋)))
74	73	anbi2d 628	. . . . 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 256	. . . 4 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RngIso 𝑌) ∧ 𝐺 = ^◡𝐹)) → (𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋)))
76		coeq2 5865	. . . . . . 7 ⊢ (𝐺 = ^◡𝐹 → (𝐹 ∘ 𝐺) = (𝐹 ∘ ^◡𝐹))
77	76	ad2antll 727	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RngIso 𝑌) ∧ 𝐺 = ^◡𝐹)) → (𝐹 ∘ 𝐺) = (𝐹 ∘ ^◡𝐹))
78		f1ococnv2 6871	. . . . . . 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 2768	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RngIso 𝑌) ∧ 𝐺 = ^◡𝐹)) → (𝐹 ∘ 𝐺) = ( I ↾ (Base‘𝑌)))
81	75, 67, 80	jca31 513	. . . 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 513	. . 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 799	. 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 304	1 ⊢ (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 ↔ (𝐹 ∈ (𝑋 RngIso 𝑌) ∧ 𝐺 = ^◡𝐹)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 class class class wbr 5152 I cid 5579 ^◡ccnv 5681 ↾ cres 5684 ∘ ccom 5686 ⟶wf 6549 –1-1-onto→wf1o 6552 ‘cfv 6553 (class class class)co 7426 Basecbs 17187 Catccat 17651 Sectcsect 17734 Invcinv 17735 RngHom crnghm 20380 RngIso crngim 20381 RngCatcrngc 20556

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-map 8853 df-pm 8854 df-ixp 8923 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-fz 13525 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-hom 17264 df-cco 17265 df-0g 17430 df-cat 17655 df-cid 17656 df-homf 17657 df-sect 17737 df-inv 17738 df-ssc 17800 df-resc 17801 df-subc 17802 df-estrc 18120 df-mgm 18607 df-mgmhm 18659 df-sgrp 18686 df-mnd 18702 df-mhm 18747 df-grp 18900 df-ghm 19175 df-abl 19745 df-mgp 20082 df-rng 20100 df-rnghm 20382 df-rngim 20383 df-rngc 20557

This theorem is referenced by: rngciso 20578

W3C validator