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Theorem rngcinv 20530
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 𝐢 = (RngCatβ€˜π‘ˆ)
rngcsect.b 𝐡 = (Baseβ€˜πΆ)
rngcsect.u (πœ‘ β†’ π‘ˆ ∈ 𝑉)
rngcsect.x (πœ‘ β†’ 𝑋 ∈ 𝐡)
rngcsect.y (πœ‘ β†’ π‘Œ ∈ 𝐡)
rngcinv.n 𝑁 = (Invβ€˜πΆ)
Assertion
Ref Expression
rngcinv (πœ‘ β†’ (𝐹(π‘‹π‘π‘Œ)𝐺 ↔ (𝐹 ∈ (𝑋 RngIso π‘Œ) ∧ 𝐺 = ◑𝐹)))

Proof of Theorem rngcinv
StepHypRef Expression
1 rngcsect.b . . 3 𝐡 = (Baseβ€˜πΆ)
2 rngcinv.n . . 3 𝑁 = (Invβ€˜πΆ)
3 rngcsect.u . . . 4 (πœ‘ β†’ π‘ˆ ∈ 𝑉)
4 rngcsect.c . . . . 5 𝐢 = (RngCatβ€˜π‘ˆ)
54rngccat 20527 . . . 4 (π‘ˆ ∈ 𝑉 β†’ 𝐢 ∈ Cat)
63, 5syl 17 . . 3 (πœ‘ β†’ 𝐢 ∈ Cat)
7 rngcsect.x . . 3 (πœ‘ β†’ 𝑋 ∈ 𝐡)
8 rngcsect.y . . 3 (πœ‘ β†’ π‘Œ ∈ 𝐡)
9 eqid 2726 . . 3 (Sectβ€˜πΆ) = (Sectβ€˜πΆ)
101, 2, 6, 7, 8, 9isinv 17713 . 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 2726 . . . . . 6 (Baseβ€˜π‘‹) = (Baseβ€˜π‘‹)
124, 1, 3, 7, 8, 11, 9rngcsect 20529 . . . . 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 1086 . . . . 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 2726 . . . . . 6 (Baseβ€˜π‘Œ) = (Baseβ€˜π‘Œ)
164, 1, 3, 8, 7, 15, 9rngcsect 20529 . . . . 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 1095 . . . . . 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 1086 . . . . . 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 673 . . 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 774 . . . . . 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 20347 . . . . . . . . . 10 (𝐹 ∈ (𝑋 RngHom π‘Œ) β†’ 𝐹:(Baseβ€˜π‘‹)⟢(Baseβ€˜π‘Œ))
2715, 11rnghmf 20347 . . . . . . . . . 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 724 . . . . . . . 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 725 . . . . . . . 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 7289 . . . . . . 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 2733 . . . . . . . 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 20352 . . . . . . 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 20354 . . . . . 6 (𝐹 ∈ (𝑋 RngIso π‘Œ) β†’ 𝐹 ∈ (𝑋 RngHom π‘Œ))
4948ad2antrl 725 . . . . 5 ((πœ‘ ∧ (𝐹 ∈ (𝑋 RngIso π‘Œ) ∧ 𝐺 = ◑𝐹)) β†’ 𝐹 ∈ (𝑋 RngHom π‘Œ))
50 isrngim 20344 . . . . . . . . . . 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 2815 . . . . . . . . . . . 12 (◑𝐹 = 𝐺 β†’ (◑𝐹 ∈ (π‘Œ RngHom 𝑋) ↔ 𝐺 ∈ (π‘Œ RngHom 𝑋)))
5352eqcoms 2734 . . . . . . . . . . 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, 56biimtrdi 252 . . . . . . . 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 5850 . . . . . . 7 (𝐺 = ◑𝐹 β†’ (𝐺 ∘ 𝐹) = (◑𝐹 ∘ 𝐹))
6261ad2antll 726 . . . . . 6 ((πœ‘ ∧ (𝐹 ∈ (𝑋 RngIso π‘Œ) ∧ 𝐺 = ◑𝐹)) β†’ (𝐺 ∘ 𝐹) = (◑𝐹 ∘ 𝐹))
6311, 15rngimf1o 20353 . . . . . . . 8 (𝐹 ∈ (𝑋 RngIso π‘Œ) β†’ 𝐹:(Baseβ€˜π‘‹)–1-1-ontoβ†’(Baseβ€˜π‘Œ))
6463ad2antrl 725 . . . . . . 7 ((πœ‘ ∧ (𝐹 ∈ (𝑋 RngIso π‘Œ) ∧ 𝐺 = ◑𝐹)) β†’ 𝐹:(Baseβ€˜π‘‹)–1-1-ontoβ†’(Baseβ€˜π‘Œ))
65 f1ococnv1 6855 . . . . . . 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 2766 . . . . 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 2815 . . . . . . 7 (𝐺 = ◑𝐹 β†’ (𝐺 ∈ (π‘Œ RngHom 𝑋) ↔ ◑𝐹 ∈ (π‘Œ RngHom 𝑋)))
7372ad2antll 726 . . . . . 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 5851 . . . . . . 7 (𝐺 = ◑𝐹 β†’ (𝐹 ∘ 𝐺) = (𝐹 ∘ ◑𝐹))
7776ad2antll 726 . . . . . 6 ((πœ‘ ∧ (𝐹 ∈ (𝑋 RngIso π‘Œ) ∧ 𝐺 = ◑𝐹)) β†’ (𝐹 ∘ 𝐺) = (𝐹 ∘ ◑𝐹))
78 f1ococnv2 6853 . . . . . . 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 2766 . . . . 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 798 . 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 1084   = wceq 1533   ∈ wcel 2098   class class class wbr 5141   I cid 5566  β—‘ccnv 5668   β†Ύ cres 5671   ∘ ccom 5673  βŸΆwf 6532  β€“1-1-ontoβ†’wf1o 6535  β€˜cfv 6536  (class class class)co 7404  Basecbs 17150  Catccat 17614  Sectcsect 17697  Invcinv 17698   RngHom crnghm 20333   RngIso crngim 20334  RngCatcrngc 20509
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6293  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8264  df-wrecs 8295  df-recs 8369  df-rdg 8408  df-1o 8464  df-er 8702  df-map 8821  df-pm 8822  df-ixp 8891  df-en 8939  df-dom 8940  df-sdom 8941  df-fin 8942  df-pnf 11251  df-mnf 11252  df-xr 11253  df-ltxr 11254  df-le 11255  df-sub 11447  df-neg 11448  df-nn 12214  df-2 12276  df-3 12277  df-4 12278  df-5 12279  df-6 12280  df-7 12281  df-8 12282  df-9 12283  df-n0 12474  df-z 12560  df-dec 12679  df-uz 12824  df-fz 13488  df-struct 17086  df-sets 17103  df-slot 17121  df-ndx 17133  df-base 17151  df-ress 17180  df-plusg 17216  df-hom 17227  df-cco 17228  df-0g 17393  df-cat 17618  df-cid 17619  df-homf 17620  df-sect 17700  df-inv 17701  df-ssc 17763  df-resc 17764  df-subc 17765  df-estrc 18083  df-mgm 18570  df-mgmhm 18622  df-sgrp 18649  df-mnd 18665  df-mhm 18710  df-grp 18863  df-ghm 19136  df-abl 19700  df-mgp 20037  df-rng 20055  df-rnghm 20335  df-rngim 20336  df-rngc 20510
This theorem is referenced by:  rngciso  20531
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