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Theorem rngcinvALTV 47251
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 𝐢 = (RngCatALTVβ€˜π‘ˆ)
rngcsectALTV.b 𝐡 = (Baseβ€˜πΆ)
rngcsectALTV.u (πœ‘ β†’ π‘ˆ ∈ 𝑉)
rngcsectALTV.x (πœ‘ β†’ 𝑋 ∈ 𝐡)
rngcsectALTV.y (πœ‘ β†’ π‘Œ ∈ 𝐡)
rngcinvALTV.n 𝑁 = (Invβ€˜πΆ)
Assertion
Ref Expression
rngcinvALTV (πœ‘ β†’ (𝐹(π‘‹π‘π‘Œ)𝐺 ↔ (𝐹 ∈ (𝑋 RngIso π‘Œ) ∧ 𝐺 = ◑𝐹)))

Proof of Theorem rngcinvALTV
StepHypRef Expression
1 rngcsectALTV.b . . 3 𝐡 = (Baseβ€˜πΆ)
2 rngcinvALTV.n . . 3 𝑁 = (Invβ€˜πΆ)
3 rngcsectALTV.u . . . 4 (πœ‘ β†’ π‘ˆ ∈ 𝑉)
4 rngcsectALTV.c . . . . 5 𝐢 = (RngCatALTVβ€˜π‘ˆ)
54rngccatALTV 47248 . . . 4 (π‘ˆ ∈ 𝑉 β†’ 𝐢 ∈ Cat)
63, 5syl 17 . . 3 (πœ‘ β†’ 𝐢 ∈ Cat)
7 rngcsectALTV.x . . 3 (πœ‘ β†’ 𝑋 ∈ 𝐡)
8 rngcsectALTV.y . . 3 (πœ‘ β†’ π‘Œ ∈ 𝐡)
9 eqid 2727 . . 3 (Sectβ€˜πΆ) = (Sectβ€˜πΆ)
101, 2, 6, 7, 8, 9isinv 17728 . 2 (πœ‘ β†’ (𝐹(π‘‹π‘π‘Œ)𝐺 ↔ (𝐹(𝑋(Sectβ€˜πΆ)π‘Œ)𝐺 ∧ 𝐺(π‘Œ(Sectβ€˜πΆ)𝑋)𝐹)))
11 eqid 2727 . . . . . 6 (Baseβ€˜π‘‹) = (Baseβ€˜π‘‹)
124, 1, 3, 7, 8, 11, 9rngcsectALTV 47250 . . . . 5 (πœ‘ β†’ (𝐹(𝑋(Sectβ€˜πΆ)π‘Œ)𝐺 ↔ (𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHom 𝑋) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹)))))
13 df-3an 1087 . . . . 5 ((𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHom 𝑋) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ↔ ((𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))))
1412, 13bitrdi 287 . . . 4 (πœ‘ β†’ (𝐹(𝑋(Sectβ€˜πΆ)π‘Œ)𝐺 ↔ ((𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹)))))
15 eqid 2727 . . . . . 6 (Baseβ€˜π‘Œ) = (Baseβ€˜π‘Œ)
164, 1, 3, 8, 7, 15, 9rngcsectALTV 47250 . . . . 5 (πœ‘ β†’ (𝐺(π‘Œ(Sectβ€˜πΆ)𝑋)𝐹 ↔ (𝐺 ∈ (π‘Œ RngHom 𝑋) ∧ 𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ (𝐹 ∘ 𝐺) = ( I β†Ύ (Baseβ€˜π‘Œ)))))
17 3ancoma 1096 . . . . . 6 ((𝐺 ∈ (π‘Œ RngHom 𝑋) ∧ 𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ (𝐹 ∘ 𝐺) = ( I β†Ύ (Baseβ€˜π‘Œ))) ↔ (𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHom 𝑋) ∧ (𝐹 ∘ 𝐺) = ( I β†Ύ (Baseβ€˜π‘Œ))))
18 df-3an 1087 . . . . . 6 ((𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHom 𝑋) ∧ (𝐹 ∘ 𝐺) = ( I β†Ύ (Baseβ€˜π‘Œ))) ↔ ((𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHom 𝑋)) ∧ (𝐹 ∘ 𝐺) = ( I β†Ύ (Baseβ€˜π‘Œ))))
1917, 18bitri 275 . . . . 5 ((𝐺 ∈ (π‘Œ RngHom 𝑋) ∧ 𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ (𝐹 ∘ 𝐺) = ( I β†Ύ (Baseβ€˜π‘Œ))) ↔ ((𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHom 𝑋)) ∧ (𝐹 ∘ 𝐺) = ( I β†Ύ (Baseβ€˜π‘Œ))))
2016, 19bitrdi 287 . . . 4 (πœ‘ β†’ (𝐺(π‘Œ(Sectβ€˜πΆ)𝑋)𝐹 ↔ ((𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHom 𝑋)) ∧ (𝐹 ∘ 𝐺) = ( I β†Ύ (Baseβ€˜π‘Œ)))))
2114, 20anbi12d 630 . . 3 (πœ‘ β†’ ((𝐹(𝑋(Sectβ€˜πΆ)π‘Œ)𝐺 ∧ 𝐺(π‘Œ(Sectβ€˜πΆ)𝑋)𝐹) ↔ (((𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ ((𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHom 𝑋)) ∧ (𝐹 ∘ 𝐺) = ( I β†Ύ (Baseβ€˜π‘Œ))))))
22 anandi 675 . . 3 ((((𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ ((𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHom 𝑋)) ∧ (𝐹 ∘ 𝐺) = ( I β†Ύ (Baseβ€˜π‘Œ)))) ↔ ((((𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ (𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHom 𝑋))) ∧ (((𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ (𝐹 ∘ 𝐺) = ( I β†Ύ (Baseβ€˜π‘Œ)))))
2321, 22bitrdi 287 . 2 (πœ‘ β†’ ((𝐹(𝑋(Sectβ€˜πΆ)π‘Œ)𝐺 ∧ 𝐺(π‘Œ(Sectβ€˜πΆ)𝑋)𝐹) ↔ ((((𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ (𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHom 𝑋))) ∧ (((𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ (𝐹 ∘ 𝐺) = ( I β†Ύ (Baseβ€˜π‘Œ))))))
24 simplrl 776 . . . . . 6 (((((𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ (𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHom 𝑋))) ∧ (((𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ (𝐹 ∘ 𝐺) = ( I β†Ύ (Baseβ€˜π‘Œ)))) β†’ 𝐹 ∈ (𝑋 RngHom π‘Œ))
2524adantl 481 . . . . 5 ((πœ‘ ∧ ((((𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ (𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHom 𝑋))) ∧ (((𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ (𝐹 ∘ 𝐺) = ( I β†Ύ (Baseβ€˜π‘Œ))))) β†’ 𝐹 ∈ (𝑋 RngHom π‘Œ))
2611, 15rnghmf 20369 . . . . . . . . . 10 (𝐹 ∈ (𝑋 RngHom π‘Œ) β†’ 𝐹:(Baseβ€˜π‘‹)⟢(Baseβ€˜π‘Œ))
2715, 11rnghmf 20369 . . . . . . . . . 10 (𝐺 ∈ (π‘Œ RngHom 𝑋) β†’ 𝐺:(Baseβ€˜π‘Œ)⟢(Baseβ€˜π‘‹))
2826, 27anim12i 612 . . . . . . . . 9 ((𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHom 𝑋)) β†’ (𝐹:(Baseβ€˜π‘‹)⟢(Baseβ€˜π‘Œ) ∧ 𝐺:(Baseβ€˜π‘Œ)⟢(Baseβ€˜π‘‹)))
2928ad2antlr 726 . . . . . . . 8 (((((𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ (𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHom 𝑋))) ∧ (((𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ (𝐹 ∘ 𝐺) = ( I β†Ύ (Baseβ€˜π‘Œ)))) β†’ (𝐹:(Baseβ€˜π‘‹)⟢(Baseβ€˜π‘Œ) ∧ 𝐺:(Baseβ€˜π‘Œ)⟢(Baseβ€˜π‘‹)))
30 simpr 484 . . . . . . . . 9 ((((𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ (𝐹 ∘ 𝐺) = ( I β†Ύ (Baseβ€˜π‘Œ))) β†’ (𝐹 ∘ 𝐺) = ( I β†Ύ (Baseβ€˜π‘Œ)))
3130adantl 481 . . . . . . . 8 (((((𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ (𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHom 𝑋))) ∧ (((𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ (𝐹 ∘ 𝐺) = ( I β†Ύ (Baseβ€˜π‘Œ)))) β†’ (𝐹 ∘ 𝐺) = ( I β†Ύ (Baseβ€˜π‘Œ)))
32 simpr 484 . . . . . . . . 9 (((𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) β†’ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹)))
3332ad2antrl 727 . . . . . . . 8 (((((𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ (𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHom 𝑋))) ∧ (((𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ (𝐹 ∘ 𝐺) = ( I β†Ύ (Baseβ€˜π‘Œ)))) β†’ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹)))
3429, 31, 33jca32 515 . . . . . . 7 (((((𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ (𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHom 𝑋))) ∧ (((𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ (𝐹 ∘ 𝐺) = ( I β†Ύ (Baseβ€˜π‘Œ)))) β†’ ((𝐹:(Baseβ€˜π‘‹)⟢(Baseβ€˜π‘Œ) ∧ 𝐺:(Baseβ€˜π‘Œ)⟢(Baseβ€˜π‘‹)) ∧ ((𝐹 ∘ 𝐺) = ( I β†Ύ (Baseβ€˜π‘Œ)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹)))))
3534adantl 481 . . . . . 6 ((πœ‘ ∧ ((((𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ (𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHom 𝑋))) ∧ (((𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ (𝐹 ∘ 𝐺) = ( I β†Ύ (Baseβ€˜π‘Œ))))) β†’ ((𝐹:(Baseβ€˜π‘‹)⟢(Baseβ€˜π‘Œ) ∧ 𝐺:(Baseβ€˜π‘Œ)⟢(Baseβ€˜π‘‹)) ∧ ((𝐹 ∘ 𝐺) = ( I β†Ύ (Baseβ€˜π‘Œ)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹)))))
36 fcof1o 7299 . . . . . . 7 (((𝐹:(Baseβ€˜π‘‹)⟢(Baseβ€˜π‘Œ) ∧ 𝐺:(Baseβ€˜π‘Œ)⟢(Baseβ€˜π‘‹)) ∧ ((𝐹 ∘ 𝐺) = ( I β†Ύ (Baseβ€˜π‘Œ)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹)))) β†’ (𝐹:(Baseβ€˜π‘‹)–1-1-ontoβ†’(Baseβ€˜π‘Œ) ∧ ◑𝐹 = 𝐺))
37 eqcom 2734 . . . . . . . 8 (◑𝐹 = 𝐺 ↔ 𝐺 = ◑𝐹)
3837anbi2i 622 . . . . . . 7 ((𝐹:(Baseβ€˜π‘‹)–1-1-ontoβ†’(Baseβ€˜π‘Œ) ∧ ◑𝐹 = 𝐺) ↔ (𝐹:(Baseβ€˜π‘‹)–1-1-ontoβ†’(Baseβ€˜π‘Œ) ∧ 𝐺 = ◑𝐹))
3936, 38sylib 217 . . . . . 6 (((𝐹:(Baseβ€˜π‘‹)⟢(Baseβ€˜π‘Œ) ∧ 𝐺:(Baseβ€˜π‘Œ)⟢(Baseβ€˜π‘‹)) ∧ ((𝐹 ∘ 𝐺) = ( I β†Ύ (Baseβ€˜π‘Œ)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹)))) β†’ (𝐹:(Baseβ€˜π‘‹)–1-1-ontoβ†’(Baseβ€˜π‘Œ) ∧ 𝐺 = ◑𝐹))
4035, 39syl 17 . . . . 5 ((πœ‘ ∧ ((((𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ (𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHom 𝑋))) ∧ (((𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ (𝐹 ∘ 𝐺) = ( I β†Ύ (Baseβ€˜π‘Œ))))) β†’ (𝐹:(Baseβ€˜π‘‹)–1-1-ontoβ†’(Baseβ€˜π‘Œ) ∧ 𝐺 = ◑𝐹))
41 anass 468 . . . . 5 (((𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ 𝐹:(Baseβ€˜π‘‹)–1-1-ontoβ†’(Baseβ€˜π‘Œ)) ∧ 𝐺 = ◑𝐹) ↔ (𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ (𝐹:(Baseβ€˜π‘‹)–1-1-ontoβ†’(Baseβ€˜π‘Œ) ∧ 𝐺 = ◑𝐹)))
4225, 40, 41sylanbrc 582 . . . 4 ((πœ‘ ∧ ((((𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ (𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHom 𝑋))) ∧ (((𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ (𝐹 ∘ 𝐺) = ( I β†Ύ (Baseβ€˜π‘Œ))))) β†’ ((𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ 𝐹:(Baseβ€˜π‘‹)–1-1-ontoβ†’(Baseβ€˜π‘Œ)) ∧ 𝐺 = ◑𝐹))
4311, 15isrngim2 20374 . . . . . . 7 ((𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝐹 ∈ (𝑋 RngIso π‘Œ) ↔ (𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ 𝐹:(Baseβ€˜π‘‹)–1-1-ontoβ†’(Baseβ€˜π‘Œ))))
447, 8, 43syl2anc 583 . . . . . 6 (πœ‘ β†’ (𝐹 ∈ (𝑋 RngIso π‘Œ) ↔ (𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ 𝐹:(Baseβ€˜π‘‹)–1-1-ontoβ†’(Baseβ€˜π‘Œ))))
4544anbi1d 629 . . . . 5 (πœ‘ β†’ ((𝐹 ∈ (𝑋 RngIso π‘Œ) ∧ 𝐺 = ◑𝐹) ↔ ((𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ 𝐹:(Baseβ€˜π‘‹)–1-1-ontoβ†’(Baseβ€˜π‘Œ)) ∧ 𝐺 = ◑𝐹)))
4645adantr 480 . . . 4 ((πœ‘ ∧ ((((𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ (𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHom 𝑋))) ∧ (((𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ (𝐹 ∘ 𝐺) = ( I β†Ύ (Baseβ€˜π‘Œ))))) β†’ ((𝐹 ∈ (𝑋 RngIso π‘Œ) ∧ 𝐺 = ◑𝐹) ↔ ((𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ 𝐹:(Baseβ€˜π‘‹)–1-1-ontoβ†’(Baseβ€˜π‘Œ)) ∧ 𝐺 = ◑𝐹)))
4742, 46mpbird 257 . . 3 ((πœ‘ ∧ ((((𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ (𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHom 𝑋))) ∧ (((𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ (𝐹 ∘ 𝐺) = ( I β†Ύ (Baseβ€˜π‘Œ))))) β†’ (𝐹 ∈ (𝑋 RngIso π‘Œ) ∧ 𝐺 = ◑𝐹))
4811, 15rngimrnghm 20376 . . . . . 6 (𝐹 ∈ (𝑋 RngIso π‘Œ) β†’ 𝐹 ∈ (𝑋 RngHom π‘Œ))
4948ad2antrl 727 . . . . 5 ((πœ‘ ∧ (𝐹 ∈ (𝑋 RngIso π‘Œ) ∧ 𝐺 = ◑𝐹)) β†’ 𝐹 ∈ (𝑋 RngHom π‘Œ))
50 isrngim 20366 . . . . . . . . . . 11 ((𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝐹 ∈ (𝑋 RngIso π‘Œ) ↔ (𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ ◑𝐹 ∈ (π‘Œ RngHom 𝑋))))
517, 8, 50syl2anc 583 . . . . . . . . . 10 (πœ‘ β†’ (𝐹 ∈ (𝑋 RngIso π‘Œ) ↔ (𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ ◑𝐹 ∈ (π‘Œ RngHom 𝑋))))
52 eleq1 2816 . . . . . . . . . . . 12 (◑𝐹 = 𝐺 β†’ (◑𝐹 ∈ (π‘Œ RngHom 𝑋) ↔ 𝐺 ∈ (π‘Œ RngHom 𝑋)))
5352eqcoms 2735 . . . . . . . . . . 11 (𝐺 = ◑𝐹 β†’ (◑𝐹 ∈ (π‘Œ RngHom 𝑋) ↔ 𝐺 ∈ (π‘Œ RngHom 𝑋)))
5453anbi2d 628 . . . . . . . . . 10 (𝐺 = ◑𝐹 β†’ ((𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ ◑𝐹 ∈ (π‘Œ RngHom 𝑋)) ↔ (𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHom 𝑋))))
5551, 54sylan9bbr 510 . . . . . . . . 9 ((𝐺 = ◑𝐹 ∧ πœ‘) β†’ (𝐹 ∈ (𝑋 RngIso π‘Œ) ↔ (𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHom 𝑋))))
56 simpr 484 . . . . . . . . 9 ((𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHom 𝑋)) β†’ 𝐺 ∈ (π‘Œ RngHom 𝑋))
5755, 56syl6bi 253 . . . . . . . 8 ((𝐺 = ◑𝐹 ∧ πœ‘) β†’ (𝐹 ∈ (𝑋 RngIso π‘Œ) β†’ 𝐺 ∈ (π‘Œ RngHom 𝑋)))
5857com12 32 . . . . . . 7 (𝐹 ∈ (𝑋 RngIso π‘Œ) β†’ ((𝐺 = ◑𝐹 ∧ πœ‘) β†’ 𝐺 ∈ (π‘Œ RngHom 𝑋)))
5958expdimp 452 . . . . . 6 ((𝐹 ∈ (𝑋 RngIso π‘Œ) ∧ 𝐺 = ◑𝐹) β†’ (πœ‘ β†’ 𝐺 ∈ (π‘Œ RngHom 𝑋)))
6059impcom 407 . . . . 5 ((πœ‘ ∧ (𝐹 ∈ (𝑋 RngIso π‘Œ) ∧ 𝐺 = ◑𝐹)) β†’ 𝐺 ∈ (π‘Œ RngHom 𝑋))
61 coeq1 5854 . . . . . . 7 (𝐺 = ◑𝐹 β†’ (𝐺 ∘ 𝐹) = (◑𝐹 ∘ 𝐹))
6261ad2antll 728 . . . . . 6 ((πœ‘ ∧ (𝐹 ∈ (𝑋 RngIso π‘Œ) ∧ 𝐺 = ◑𝐹)) β†’ (𝐺 ∘ 𝐹) = (◑𝐹 ∘ 𝐹))
6311, 15rngimf1o 20375 . . . . . . . 8 (𝐹 ∈ (𝑋 RngIso π‘Œ) β†’ 𝐹:(Baseβ€˜π‘‹)–1-1-ontoβ†’(Baseβ€˜π‘Œ))
6463ad2antrl 727 . . . . . . 7 ((πœ‘ ∧ (𝐹 ∈ (𝑋 RngIso π‘Œ) ∧ 𝐺 = ◑𝐹)) β†’ 𝐹:(Baseβ€˜π‘‹)–1-1-ontoβ†’(Baseβ€˜π‘Œ))
65 f1ococnv1 6862 . . . . . . 7 (𝐹:(Baseβ€˜π‘‹)–1-1-ontoβ†’(Baseβ€˜π‘Œ) β†’ (◑𝐹 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹)))
6664, 65syl 17 . . . . . 6 ((πœ‘ ∧ (𝐹 ∈ (𝑋 RngIso π‘Œ) ∧ 𝐺 = ◑𝐹)) β†’ (◑𝐹 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹)))
6762, 66eqtrd 2767 . . . . 5 ((πœ‘ ∧ (𝐹 ∈ (𝑋 RngIso π‘Œ) ∧ 𝐺 = ◑𝐹)) β†’ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹)))
6849, 60, 67jca31 514 . . . 4 ((πœ‘ ∧ (𝐹 ∈ (𝑋 RngIso π‘Œ) ∧ 𝐺 = ◑𝐹)) β†’ ((𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))))
6951biimpcd 248 . . . . . . 7 (𝐹 ∈ (𝑋 RngIso π‘Œ) β†’ (πœ‘ β†’ (𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ ◑𝐹 ∈ (π‘Œ RngHom 𝑋))))
7069adantr 480 . . . . . 6 ((𝐹 ∈ (𝑋 RngIso π‘Œ) ∧ 𝐺 = ◑𝐹) β†’ (πœ‘ β†’ (𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ ◑𝐹 ∈ (π‘Œ RngHom 𝑋))))
7170impcom 407 . . . . 5 ((πœ‘ ∧ (𝐹 ∈ (𝑋 RngIso π‘Œ) ∧ 𝐺 = ◑𝐹)) β†’ (𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ ◑𝐹 ∈ (π‘Œ RngHom 𝑋)))
72 eleq1 2816 . . . . . . 7 (𝐺 = ◑𝐹 β†’ (𝐺 ∈ (π‘Œ RngHom 𝑋) ↔ ◑𝐹 ∈ (π‘Œ RngHom 𝑋)))
7372ad2antll 728 . . . . . 6 ((πœ‘ ∧ (𝐹 ∈ (𝑋 RngIso π‘Œ) ∧ 𝐺 = ◑𝐹)) β†’ (𝐺 ∈ (π‘Œ RngHom 𝑋) ↔ ◑𝐹 ∈ (π‘Œ RngHom 𝑋)))
7473anbi2d 628 . . . . 5 ((πœ‘ ∧ (𝐹 ∈ (𝑋 RngIso π‘Œ) ∧ 𝐺 = ◑𝐹)) β†’ ((𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHom 𝑋)) ↔ (𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ ◑𝐹 ∈ (π‘Œ RngHom 𝑋))))
7571, 74mpbird 257 . . . 4 ((πœ‘ ∧ (𝐹 ∈ (𝑋 RngIso π‘Œ) ∧ 𝐺 = ◑𝐹)) β†’ (𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHom 𝑋)))
76 coeq2 5855 . . . . . . 7 (𝐺 = ◑𝐹 β†’ (𝐹 ∘ 𝐺) = (𝐹 ∘ ◑𝐹))
7776ad2antll 728 . . . . . 6 ((πœ‘ ∧ (𝐹 ∈ (𝑋 RngIso π‘Œ) ∧ 𝐺 = ◑𝐹)) β†’ (𝐹 ∘ 𝐺) = (𝐹 ∘ ◑𝐹))
78 f1ococnv2 6860 . . . . . . 7 (𝐹:(Baseβ€˜π‘‹)–1-1-ontoβ†’(Baseβ€˜π‘Œ) β†’ (𝐹 ∘ ◑𝐹) = ( I β†Ύ (Baseβ€˜π‘Œ)))
7964, 78syl 17 . . . . . 6 ((πœ‘ ∧ (𝐹 ∈ (𝑋 RngIso π‘Œ) ∧ 𝐺 = ◑𝐹)) β†’ (𝐹 ∘ ◑𝐹) = ( I β†Ύ (Baseβ€˜π‘Œ)))
8077, 79eqtrd 2767 . . . . 5 ((πœ‘ ∧ (𝐹 ∈ (𝑋 RngIso π‘Œ) ∧ 𝐺 = ◑𝐹)) β†’ (𝐹 ∘ 𝐺) = ( I β†Ύ (Baseβ€˜π‘Œ)))
8175, 67, 80jca31 514 . . . 4 ((πœ‘ ∧ (𝐹 ∈ (𝑋 RngIso π‘Œ) ∧ 𝐺 = ◑𝐹)) β†’ (((𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ (𝐹 ∘ 𝐺) = ( I β†Ύ (Baseβ€˜π‘Œ))))
8268, 75, 81jca31 514 . . 3 ((πœ‘ ∧ (𝐹 ∈ (𝑋 RngIso π‘Œ) ∧ 𝐺 = ◑𝐹)) β†’ ((((𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ (𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHom 𝑋))) ∧ (((𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ (𝐹 ∘ 𝐺) = ( I β†Ύ (Baseβ€˜π‘Œ)))))
8347, 82impbida 800 . 2 (πœ‘ β†’ (((((𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ (𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHom 𝑋))) ∧ (((𝐹 ∈ (𝑋 RngHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ (𝐹 ∘ 𝐺) = ( I β†Ύ (Baseβ€˜π‘Œ)))) ↔ (𝐹 ∈ (𝑋 RngIso π‘Œ) ∧ 𝐺 = ◑𝐹)))
8410, 23, 833bitrd 305 1 (πœ‘ β†’ (𝐹(π‘‹π‘π‘Œ)𝐺 ↔ (𝐹 ∈ (𝑋 RngIso π‘Œ) ∧ 𝐺 = ◑𝐹)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099   class class class wbr 5142   I cid 5569  β—‘ccnv 5671   β†Ύ cres 5674   ∘ ccom 5676  βŸΆwf 6538  β€“1-1-ontoβ†’wf1o 6541  β€˜cfv 6542  (class class class)co 7414  Basecbs 17165  Catccat 17629  Sectcsect 17712  Invcinv 17713   RngHom crnghm 20355   RngIso crngim 20356  RngCatALTVcrngcALTV 47238
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7732  ax-cnex 11180  ax-resscn 11181  ax-1cn 11182  ax-icn 11183  ax-addcl 11184  ax-addrcl 11185  ax-mulcl 11186  ax-mulrcl 11187  ax-mulcom 11188  ax-addass 11189  ax-mulass 11190  ax-distr 11191  ax-i2m1 11192  ax-1ne0 11193  ax-1rid 11194  ax-rnegex 11195  ax-rrecex 11196  ax-cnre 11197  ax-pre-lttri 11198  ax-pre-lttrn 11199  ax-pre-ltadd 11200  ax-pre-mulgt0 11201
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7863  df-1st 7985  df-2nd 7986  df-frecs 8278  df-wrecs 8309  df-recs 8383  df-rdg 8422  df-1o 8478  df-er 8716  df-map 8836  df-en 8954  df-dom 8955  df-sdom 8956  df-fin 8957  df-pnf 11266  df-mnf 11267  df-xr 11268  df-ltxr 11269  df-le 11270  df-sub 11462  df-neg 11463  df-nn 12229  df-2 12291  df-3 12292  df-4 12293  df-5 12294  df-6 12295  df-7 12296  df-8 12297  df-9 12298  df-n0 12489  df-z 12575  df-dec 12694  df-uz 12839  df-fz 13503  df-struct 17101  df-sets 17118  df-slot 17136  df-ndx 17148  df-base 17166  df-plusg 17231  df-hom 17242  df-cco 17243  df-0g 17408  df-cat 17633  df-cid 17634  df-sect 17715  df-inv 17716  df-mgm 18585  df-mgmhm 18637  df-sgrp 18664  df-mnd 18680  df-mhm 18725  df-grp 18878  df-ghm 19152  df-abl 19722  df-mgp 20059  df-rng 20077  df-rnghm 20357  df-rngim 20358  df-rngcALTV 47239
This theorem is referenced by:  rngcisoALTV  47252
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