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Theorem rngcinv 20721

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 20718	. . . 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 2769	. . 3 ⊢ (Sect‘𝐶) = (Sect‘𝐶)
10	1, 2, 6, 7, 8, 9	isinv 17816	. 2 ⊢ (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 ↔ (𝐹(𝑋(Sect‘𝐶)𝑌)𝐺 ∧ 𝐺(𝑌(Sect‘𝐶)𝑋)𝐹)))
11		eqid 2769	. . . . . 6 ⊢ (Base‘𝑋) = (Base‘𝑋)
12	4, 1, 3, 7, 8, 11, 9	rngcsect 20720	. . . . 5 ⊢ (𝜑 → (𝐹(𝑋(Sect‘𝐶)𝑌)𝐺 ↔ (𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋)))))
13		df-3an 1103	. . . . 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 2769	. . . . . 6 ⊢ (Base‘𝑌) = (Base‘𝑌)
16	4, 1, 3, 8, 7, 15, 9	rngcsect 20720	. . . . 5 ⊢ (𝜑 → (𝐺(𝑌(Sect‘𝐶)𝑋)𝐹 ↔ (𝐺 ∈ (𝑌 RngHom 𝑋) ∧ 𝐹 ∈ (𝑋 RngHom 𝑌) ∧ (𝐹 ∘ 𝐺) = ( I ↾ (Base‘𝑌)))))
17		3ancoma 1113	. . . . . 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 1103	. . . . . 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 643	. . 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 688	. . 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 788	. . . . . 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 486	. . . . 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 20529	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝑋 RngHom 𝑌) → 𝐹:(Base‘𝑋)⟶(Base‘𝑌))
27	15, 11	rnghmf 20529	. . . . . . . . . 10 ⊢ (𝐺 ∈ (𝑌 RngHom 𝑋) → 𝐺:(Base‘𝑌)⟶(Base‘𝑋))
28	26, 27	anim12i 624	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋)) → (𝐹:(Base‘𝑋)⟶(Base‘𝑌) ∧ 𝐺:(Base‘𝑌)⟶(Base‘𝑋)))
29	28	ad2antlr 739	. . . . . . . 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 489	. . . . . . . . 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 486	. . . . . . . 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 489	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋))) → (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋)))
33	32	ad2antrl 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)))) → (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋)))
34	29, 31, 33	jca32 524	. . . . . . 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 486	. . . . . 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 7295	. . . . . . 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 2776	. . . . . . . 8 ⊢ (^◡𝐹 = 𝐺 ↔ 𝐺 = ^◡𝐹)
38	37	anbi2i 634	. . . . . . 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 473	. . . . 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 594	. . . 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 20534	. . . . . . 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 595	. . . . . 6 ⊢ (𝜑 → (𝐹 ∈ (𝑋 RngIso 𝑌) ↔ (𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌))))
45	44	anbi1d 642	. . . . 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 485	. . . 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 20536	. . . . . 6 ⊢ (𝐹 ∈ (𝑋 RngIso 𝑌) → 𝐹 ∈ (𝑋 RngHom 𝑌))
49	48	ad2antrl 740	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RngIso 𝑌) ∧ 𝐺 = ^◡𝐹)) → 𝐹 ∈ (𝑋 RngHom 𝑌))
50		isrngim 20526	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝐹 ∈ (𝑋 RngIso 𝑌) ↔ (𝐹 ∈ (𝑋 RngHom 𝑌) ∧ ^◡𝐹 ∈ (𝑌 RngHom 𝑋))))
51	7, 8, 50	syl2anc 595	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 ∈ (𝑋 RngIso 𝑌) ↔ (𝐹 ∈ (𝑋 RngHom 𝑌) ∧ ^◡𝐹 ∈ (𝑌 RngHom 𝑋))))
52		eleq1 2857	. . . . . . . . . . . 12 ⊢ (^◡𝐹 = 𝐺 → (^◡𝐹 ∈ (𝑌 RngHom 𝑋) ↔ 𝐺 ∈ (𝑌 RngHom 𝑋)))
53	52	eqcoms 2777	. . . . . . . . . . 11 ⊢ (𝐺 = ^◡𝐹 → (^◡𝐹 ∈ (𝑌 RngHom 𝑋) ↔ 𝐺 ∈ (𝑌 RngHom 𝑋)))
54	53	anbi2d 641	. . . . . . . . . 10 ⊢ (𝐺 = ^◡𝐹 → ((𝐹 ∈ (𝑋 RngHom 𝑌) ∧ ^◡𝐹 ∈ (𝑌 RngHom 𝑋)) ↔ (𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋))))
55	51, 54	sylan9bbr 519	. . . . . . . . 9 ⊢ ((𝐺 = ^◡𝐹 ∧ 𝜑) → (𝐹 ∈ (𝑋 RngIso 𝑌) ↔ (𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋))))
56		simpr 489	. . . . . . . . 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 457	. . . . . 6 ⊢ ((𝐹 ∈ (𝑋 RngIso 𝑌) ∧ 𝐺 = ^◡𝐹) → (𝜑 → 𝐺 ∈ (𝑌 RngHom 𝑋)))
60	59	impcom 412	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RngIso 𝑌) ∧ 𝐺 = ^◡𝐹)) → 𝐺 ∈ (𝑌 RngHom 𝑋))
61		coeq1 5844	. . . . . . 7 ⊢ (𝐺 = ^◡𝐹 → (𝐺 ∘ 𝐹) = (^◡𝐹 ∘ 𝐹))
62	61	ad2antll 741	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RngIso 𝑌) ∧ 𝐺 = ^◡𝐹)) → (𝐺 ∘ 𝐹) = (^◡𝐹 ∘ 𝐹))
63	11, 15	rngimf1o 20535	. . . . . . . 8 ⊢ (𝐹 ∈ (𝑋 RngIso 𝑌) → 𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌))
64	63	ad2antrl 740	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RngIso 𝑌) ∧ 𝐺 = ^◡𝐹)) → 𝐹:(Base‘𝑋)–1-1-onto→(Base‘𝑌))
65		f1ococnv1 6851	. . . . . . 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 2804	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RngIso 𝑌) ∧ 𝐺 = ^◡𝐹)) → (𝐺 ∘ 𝐹) = ( I ↾ (Base‘𝑋)))
68	49, 60, 67	jca31 523	. . . 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 485	. . . . . 6 ⊢ ((𝐹 ∈ (𝑋 RngIso 𝑌) ∧ 𝐺 = ^◡𝐹) → (𝜑 → (𝐹 ∈ (𝑋 RngHom 𝑌) ∧ ^◡𝐹 ∈ (𝑌 RngHom 𝑋))))
71	70	impcom 412	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RngIso 𝑌) ∧ 𝐺 = ^◡𝐹)) → (𝐹 ∈ (𝑋 RngHom 𝑌) ∧ ^◡𝐹 ∈ (𝑌 RngHom 𝑋)))
72		eleq1 2857	. . . . . . 7 ⊢ (𝐺 = ^◡𝐹 → (𝐺 ∈ (𝑌 RngHom 𝑋) ↔ ^◡𝐹 ∈ (𝑌 RngHom 𝑋)))
73	72	ad2antll 741	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RngIso 𝑌) ∧ 𝐺 = ^◡𝐹)) → (𝐺 ∈ (𝑌 RngHom 𝑋) ↔ ^◡𝐹 ∈ (𝑌 RngHom 𝑋)))
74	73	anbi2d 641	. . . . 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 5845	. . . . . . 7 ⊢ (𝐺 = ^◡𝐹 → (𝐹 ∘ 𝐺) = (𝐹 ∘ ^◡𝐹))
77	76	ad2antll 741	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RngIso 𝑌) ∧ 𝐺 = ^◡𝐹)) → (𝐹 ∘ 𝐺) = (𝐹 ∘ ^◡𝐹))
78		f1ococnv2 6849	. . . . . . 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 2804	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RngIso 𝑌) ∧ 𝐺 = ^◡𝐹)) → (𝐹 ∘ 𝐺) = ( I ↾ (Base‘𝑌)))
81	75, 67, 80	jca31 523	. . . 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 523	. . 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 812	. 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

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 ∧ w3a 1101 = wceq 1567 ∈ wcel 2149 class class class wbr 5113 I cid 5556 ^◡ccnv 5661 ↾ cres 5664 ∘ ccom 5666 ⟶wf 6533 –1-1-onto→wf1o 6536 ‘cfv 6537 (class class class)co 7411 Basecbs 17268 Catccat 17719 Sectcsect 17800 Invcinv 17801 RngHom crnghm 20515 RngIso crngim 20516 RngCatcrngc 20700

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5242 ax-sep 5261 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733 ax-cnex 11155 ax-resscn 11156 ax-1cn 11157 ax-icn 11158 ax-addcl 11159 ax-addrcl 11160 ax-mulcl 11161 ax-mulrcl 11162 ax-mulcom 11163 ax-addass 11164 ax-mulass 11165 ax-distr 11166 ax-i2m1 11167 ax-1ne0 11168 ax-1rid 11169 ax-rnegex 11170 ax-rrecex 11171 ax-cnre 11172 ax-pre-lttri 11173 ax-pre-lttrn 11174 ax-pre-ltadd 11175 ax-pre-mulgt0 11176

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-tp 4599 df-op 4601 df-uni 4877 df-iun 4962 df-br 5114 df-opab 5178 df-mpt 5197 df-tr 5223 df-id 5557 df-eprel 5562 df-po 5570 df-so 5571 df-fr 5615 df-we 5617 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7368 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7862 df-1st 7985 df-2nd 7986 df-frecs 8277 df-wrecs 8308 df-recs 8357 df-rdg 8396 df-1o 8452 df-er 8693 df-map 8825 df-pm 8826 df-ixp 8895 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-pnf 11244 df-mnf 11245 df-xr 11246 df-ltxr 11247 df-le 11248 df-sub 11442 df-neg 11443 df-nn 12233 df-2 12302 df-3 12303 df-4 12304 df-5 12305 df-6 12306 df-7 12307 df-8 12308 df-9 12309 df-n0 12504 df-z 12591 df-dec 12711 df-uz 12862 df-fz 13535 df-struct 17206 df-sets 17223 df-slot 17241 df-ndx 17253 df-base 17269 df-ress 17290 df-plusg 17322 df-hom 17333 df-cco 17334 df-0g 17493 df-cat 17723 df-cid 17724 df-homf 17725 df-sect 17803 df-inv 17804 df-ssc 17866 df-resc 17867 df-subc 17868 df-estrc 18178 df-mgm 18697 df-mgmhm 18749 df-sgrp 18776 df-mnd 18792 df-mhm 18840 df-grp 19002 df-ghm 19283 df-abl 19852 df-mgp 20216 df-rng 20230 df-rnghm 20517 df-rngim 20518 df-rngc 20701

This theorem is referenced by: rngciso 20722

W3C validator