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Theorem rngcinvALTV 46365
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 RngIsom 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 46362 . . . 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 2737 . . 3 (Sectβ€˜πΆ) = (Sectβ€˜πΆ)
101, 2, 6, 7, 8, 9isinv 17650 . 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 2737 . . . . . 6 (Baseβ€˜π‘‹) = (Baseβ€˜π‘‹)
124, 1, 3, 7, 8, 11, 9rngcsectALTV 46364 . . . . 5 (πœ‘ β†’ (𝐹(𝑋(Sectβ€˜πΆ)π‘Œ)𝐺 ↔ (𝐹 ∈ (𝑋 RngHomo π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHomo 𝑋) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹)))))
13 df-3an 1090 . . . . 5 ((𝐹 ∈ (𝑋 RngHomo π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHomo 𝑋) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ↔ ((𝐹 ∈ (𝑋 RngHomo π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHomo 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 RngHomo π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHomo 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹)))))
15 eqid 2737 . . . . . 6 (Baseβ€˜π‘Œ) = (Baseβ€˜π‘Œ)
164, 1, 3, 8, 7, 15, 9rngcsectALTV 46364 . . . . 5 (πœ‘ β†’ (𝐺(π‘Œ(Sectβ€˜πΆ)𝑋)𝐹 ↔ (𝐺 ∈ (π‘Œ RngHomo 𝑋) ∧ 𝐹 ∈ (𝑋 RngHomo π‘Œ) ∧ (𝐹 ∘ 𝐺) = ( I β†Ύ (Baseβ€˜π‘Œ)))))
17 3ancoma 1099 . . . . . 6 ((𝐺 ∈ (π‘Œ RngHomo 𝑋) ∧ 𝐹 ∈ (𝑋 RngHomo π‘Œ) ∧ (𝐹 ∘ 𝐺) = ( I β†Ύ (Baseβ€˜π‘Œ))) ↔ (𝐹 ∈ (𝑋 RngHomo π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHomo 𝑋) ∧ (𝐹 ∘ 𝐺) = ( I β†Ύ (Baseβ€˜π‘Œ))))
18 df-3an 1090 . . . . . 6 ((𝐹 ∈ (𝑋 RngHomo π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHomo 𝑋) ∧ (𝐹 ∘ 𝐺) = ( I β†Ύ (Baseβ€˜π‘Œ))) ↔ ((𝐹 ∈ (𝑋 RngHomo π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHomo 𝑋)) ∧ (𝐹 ∘ 𝐺) = ( I β†Ύ (Baseβ€˜π‘Œ))))
1917, 18bitri 275 . . . . 5 ((𝐺 ∈ (π‘Œ RngHomo 𝑋) ∧ 𝐹 ∈ (𝑋 RngHomo π‘Œ) ∧ (𝐹 ∘ 𝐺) = ( I β†Ύ (Baseβ€˜π‘Œ))) ↔ ((𝐹 ∈ (𝑋 RngHomo π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHomo 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 RngHomo π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHomo 𝑋)) ∧ (𝐹 ∘ 𝐺) = ( I β†Ύ (Baseβ€˜π‘Œ)))))
2114, 20anbi12d 632 . . 3 (πœ‘ β†’ ((𝐹(𝑋(Sectβ€˜πΆ)π‘Œ)𝐺 ∧ 𝐺(π‘Œ(Sectβ€˜πΆ)𝑋)𝐹) ↔ (((𝐹 ∈ (𝑋 RngHomo π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHomo 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ ((𝐹 ∈ (𝑋 RngHomo π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHomo 𝑋)) ∧ (𝐹 ∘ 𝐺) = ( I β†Ύ (Baseβ€˜π‘Œ))))))
22 anandi 675 . . 3 ((((𝐹 ∈ (𝑋 RngHomo π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHomo 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ ((𝐹 ∈ (𝑋 RngHomo π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHomo 𝑋)) ∧ (𝐹 ∘ 𝐺) = ( I β†Ύ (Baseβ€˜π‘Œ)))) ↔ ((((𝐹 ∈ (𝑋 RngHomo π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHomo 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ (𝐹 ∈ (𝑋 RngHomo π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHomo 𝑋))) ∧ (((𝐹 ∈ (𝑋 RngHomo π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHomo 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 RngHomo π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHomo 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ (𝐹 ∈ (𝑋 RngHomo π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHomo 𝑋))) ∧ (((𝐹 ∈ (𝑋 RngHomo π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHomo 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ (𝐹 ∘ 𝐺) = ( I β†Ύ (Baseβ€˜π‘Œ))))))
24 simplrl 776 . . . . . 6 (((((𝐹 ∈ (𝑋 RngHomo π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHomo 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ (𝐹 ∈ (𝑋 RngHomo π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHomo 𝑋))) ∧ (((𝐹 ∈ (𝑋 RngHomo π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHomo 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ (𝐹 ∘ 𝐺) = ( I β†Ύ (Baseβ€˜π‘Œ)))) β†’ 𝐹 ∈ (𝑋 RngHomo π‘Œ))
2524adantl 483 . . . . 5 ((πœ‘ ∧ ((((𝐹 ∈ (𝑋 RngHomo π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHomo 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ (𝐹 ∈ (𝑋 RngHomo π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHomo 𝑋))) ∧ (((𝐹 ∈ (𝑋 RngHomo π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHomo 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ (𝐹 ∘ 𝐺) = ( I β†Ύ (Baseβ€˜π‘Œ))))) β†’ 𝐹 ∈ (𝑋 RngHomo π‘Œ))
2611, 15rnghmf 46271 . . . . . . . . . 10 (𝐹 ∈ (𝑋 RngHomo π‘Œ) β†’ 𝐹:(Baseβ€˜π‘‹)⟢(Baseβ€˜π‘Œ))
2715, 11rnghmf 46271 . . . . . . . . . 10 (𝐺 ∈ (π‘Œ RngHomo 𝑋) β†’ 𝐺:(Baseβ€˜π‘Œ)⟢(Baseβ€˜π‘‹))
2826, 27anim12i 614 . . . . . . . . 9 ((𝐹 ∈ (𝑋 RngHomo π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHomo 𝑋)) β†’ (𝐹:(Baseβ€˜π‘‹)⟢(Baseβ€˜π‘Œ) ∧ 𝐺:(Baseβ€˜π‘Œ)⟢(Baseβ€˜π‘‹)))
2928ad2antlr 726 . . . . . . . 8 (((((𝐹 ∈ (𝑋 RngHomo π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHomo 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ (𝐹 ∈ (𝑋 RngHomo π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHomo 𝑋))) ∧ (((𝐹 ∈ (𝑋 RngHomo π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHomo 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 486 . . . . . . . . 9 ((((𝐹 ∈ (𝑋 RngHomo π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHomo 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 483 . . . . . . . 8 (((((𝐹 ∈ (𝑋 RngHomo π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHomo 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ (𝐹 ∈ (𝑋 RngHomo π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHomo 𝑋))) ∧ (((𝐹 ∈ (𝑋 RngHomo π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHomo 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 486 . . . . . . . . 9 (((𝐹 ∈ (𝑋 RngHomo π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHomo 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) β†’ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹)))
3332ad2antrl 727 . . . . . . . 8 (((((𝐹 ∈ (𝑋 RngHomo π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHomo 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ (𝐹 ∈ (𝑋 RngHomo π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHomo 𝑋))) ∧ (((𝐹 ∈ (𝑋 RngHomo π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHomo 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 517 . . . . . . 7 (((((𝐹 ∈ (𝑋 RngHomo π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHomo 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ (𝐹 ∈ (𝑋 RngHomo π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHomo 𝑋))) ∧ (((𝐹 ∈ (𝑋 RngHomo π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHomo 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 483 . . . . . 6 ((πœ‘ ∧ ((((𝐹 ∈ (𝑋 RngHomo π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHomo 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ (𝐹 ∈ (𝑋 RngHomo π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHomo 𝑋))) ∧ (((𝐹 ∈ (𝑋 RngHomo π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHomo 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 7247 . . . . . . 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 2744 . . . . . . . 8 (◑𝐹 = 𝐺 ↔ 𝐺 = ◑𝐹)
3837anbi2i 624 . . . . . . 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 RngHomo π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHomo 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ (𝐹 ∈ (𝑋 RngHomo π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHomo 𝑋))) ∧ (((𝐹 ∈ (𝑋 RngHomo π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHomo 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 470 . . . . 5 (((𝐹 ∈ (𝑋 RngHomo π‘Œ) ∧ 𝐹:(Baseβ€˜π‘‹)–1-1-ontoβ†’(Baseβ€˜π‘Œ)) ∧ 𝐺 = ◑𝐹) ↔ (𝐹 ∈ (𝑋 RngHomo π‘Œ) ∧ (𝐹:(Baseβ€˜π‘‹)–1-1-ontoβ†’(Baseβ€˜π‘Œ) ∧ 𝐺 = ◑𝐹)))
4225, 40, 41sylanbrc 584 . . . 4 ((πœ‘ ∧ ((((𝐹 ∈ (𝑋 RngHomo π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHomo 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ (𝐹 ∈ (𝑋 RngHomo π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHomo 𝑋))) ∧ (((𝐹 ∈ (𝑋 RngHomo π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHomo 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ (𝐹 ∘ 𝐺) = ( I β†Ύ (Baseβ€˜π‘Œ))))) β†’ ((𝐹 ∈ (𝑋 RngHomo π‘Œ) ∧ 𝐹:(Baseβ€˜π‘‹)–1-1-ontoβ†’(Baseβ€˜π‘Œ)) ∧ 𝐺 = ◑𝐹))
4311, 15isrngim 46276 . . . . . . 7 ((𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝐹 ∈ (𝑋 RngIsom π‘Œ) ↔ (𝐹 ∈ (𝑋 RngHomo π‘Œ) ∧ 𝐹:(Baseβ€˜π‘‹)–1-1-ontoβ†’(Baseβ€˜π‘Œ))))
447, 8, 43syl2anc 585 . . . . . 6 (πœ‘ β†’ (𝐹 ∈ (𝑋 RngIsom π‘Œ) ↔ (𝐹 ∈ (𝑋 RngHomo π‘Œ) ∧ 𝐹:(Baseβ€˜π‘‹)–1-1-ontoβ†’(Baseβ€˜π‘Œ))))
4544anbi1d 631 . . . . 5 (πœ‘ β†’ ((𝐹 ∈ (𝑋 RngIsom π‘Œ) ∧ 𝐺 = ◑𝐹) ↔ ((𝐹 ∈ (𝑋 RngHomo π‘Œ) ∧ 𝐹:(Baseβ€˜π‘‹)–1-1-ontoβ†’(Baseβ€˜π‘Œ)) ∧ 𝐺 = ◑𝐹)))
4645adantr 482 . . . 4 ((πœ‘ ∧ ((((𝐹 ∈ (𝑋 RngHomo π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHomo 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ (𝐹 ∈ (𝑋 RngHomo π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHomo 𝑋))) ∧ (((𝐹 ∈ (𝑋 RngHomo π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHomo 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ (𝐹 ∘ 𝐺) = ( I β†Ύ (Baseβ€˜π‘Œ))))) β†’ ((𝐹 ∈ (𝑋 RngIsom π‘Œ) ∧ 𝐺 = ◑𝐹) ↔ ((𝐹 ∈ (𝑋 RngHomo π‘Œ) ∧ 𝐹:(Baseβ€˜π‘‹)–1-1-ontoβ†’(Baseβ€˜π‘Œ)) ∧ 𝐺 = ◑𝐹)))
4742, 46mpbird 257 . . 3 ((πœ‘ ∧ ((((𝐹 ∈ (𝑋 RngHomo π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHomo 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ (𝐹 ∈ (𝑋 RngHomo π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHomo 𝑋))) ∧ (((𝐹 ∈ (𝑋 RngHomo π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHomo 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ (𝐹 ∘ 𝐺) = ( I β†Ύ (Baseβ€˜π‘Œ))))) β†’ (𝐹 ∈ (𝑋 RngIsom π‘Œ) ∧ 𝐺 = ◑𝐹))
4811, 15rngimrnghm 46278 . . . . . 6 (𝐹 ∈ (𝑋 RngIsom π‘Œ) β†’ 𝐹 ∈ (𝑋 RngHomo π‘Œ))
4948ad2antrl 727 . . . . 5 ((πœ‘ ∧ (𝐹 ∈ (𝑋 RngIsom π‘Œ) ∧ 𝐺 = ◑𝐹)) β†’ 𝐹 ∈ (𝑋 RngHomo π‘Œ))
50 isrngisom 46268 . . . . . . . . . . 11 ((𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝐹 ∈ (𝑋 RngIsom π‘Œ) ↔ (𝐹 ∈ (𝑋 RngHomo π‘Œ) ∧ ◑𝐹 ∈ (π‘Œ RngHomo 𝑋))))
517, 8, 50syl2anc 585 . . . . . . . . . 10 (πœ‘ β†’ (𝐹 ∈ (𝑋 RngIsom π‘Œ) ↔ (𝐹 ∈ (𝑋 RngHomo π‘Œ) ∧ ◑𝐹 ∈ (π‘Œ RngHomo 𝑋))))
52 eleq1 2826 . . . . . . . . . . . 12 (◑𝐹 = 𝐺 β†’ (◑𝐹 ∈ (π‘Œ RngHomo 𝑋) ↔ 𝐺 ∈ (π‘Œ RngHomo 𝑋)))
5352eqcoms 2745 . . . . . . . . . . 11 (𝐺 = ◑𝐹 β†’ (◑𝐹 ∈ (π‘Œ RngHomo 𝑋) ↔ 𝐺 ∈ (π‘Œ RngHomo 𝑋)))
5453anbi2d 630 . . . . . . . . . 10 (𝐺 = ◑𝐹 β†’ ((𝐹 ∈ (𝑋 RngHomo π‘Œ) ∧ ◑𝐹 ∈ (π‘Œ RngHomo 𝑋)) ↔ (𝐹 ∈ (𝑋 RngHomo π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHomo 𝑋))))
5551, 54sylan9bbr 512 . . . . . . . . 9 ((𝐺 = ◑𝐹 ∧ πœ‘) β†’ (𝐹 ∈ (𝑋 RngIsom π‘Œ) ↔ (𝐹 ∈ (𝑋 RngHomo π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHomo 𝑋))))
56 simpr 486 . . . . . . . . 9 ((𝐹 ∈ (𝑋 RngHomo π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHomo 𝑋)) β†’ 𝐺 ∈ (π‘Œ RngHomo 𝑋))
5755, 56syl6bi 253 . . . . . . . 8 ((𝐺 = ◑𝐹 ∧ πœ‘) β†’ (𝐹 ∈ (𝑋 RngIsom π‘Œ) β†’ 𝐺 ∈ (π‘Œ RngHomo 𝑋)))
5857com12 32 . . . . . . 7 (𝐹 ∈ (𝑋 RngIsom π‘Œ) β†’ ((𝐺 = ◑𝐹 ∧ πœ‘) β†’ 𝐺 ∈ (π‘Œ RngHomo 𝑋)))
5958expdimp 454 . . . . . 6 ((𝐹 ∈ (𝑋 RngIsom π‘Œ) ∧ 𝐺 = ◑𝐹) β†’ (πœ‘ β†’ 𝐺 ∈ (π‘Œ RngHomo 𝑋)))
6059impcom 409 . . . . 5 ((πœ‘ ∧ (𝐹 ∈ (𝑋 RngIsom π‘Œ) ∧ 𝐺 = ◑𝐹)) β†’ 𝐺 ∈ (π‘Œ RngHomo 𝑋))
61 coeq1 5818 . . . . . . 7 (𝐺 = ◑𝐹 β†’ (𝐺 ∘ 𝐹) = (◑𝐹 ∘ 𝐹))
6261ad2antll 728 . . . . . 6 ((πœ‘ ∧ (𝐹 ∈ (𝑋 RngIsom π‘Œ) ∧ 𝐺 = ◑𝐹)) β†’ (𝐺 ∘ 𝐹) = (◑𝐹 ∘ 𝐹))
6311, 15rngimf1o 46277 . . . . . . . 8 (𝐹 ∈ (𝑋 RngIsom π‘Œ) β†’ 𝐹:(Baseβ€˜π‘‹)–1-1-ontoβ†’(Baseβ€˜π‘Œ))
6463ad2antrl 727 . . . . . . 7 ((πœ‘ ∧ (𝐹 ∈ (𝑋 RngIsom π‘Œ) ∧ 𝐺 = ◑𝐹)) β†’ 𝐹:(Baseβ€˜π‘‹)–1-1-ontoβ†’(Baseβ€˜π‘Œ))
65 f1ococnv1 6818 . . . . . . 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 RngIsom π‘Œ) ∧ 𝐺 = ◑𝐹)) β†’ (◑𝐹 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹)))
6762, 66eqtrd 2777 . . . . 5 ((πœ‘ ∧ (𝐹 ∈ (𝑋 RngIsom π‘Œ) ∧ 𝐺 = ◑𝐹)) β†’ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹)))
6849, 60, 67jca31 516 . . . 4 ((πœ‘ ∧ (𝐹 ∈ (𝑋 RngIsom π‘Œ) ∧ 𝐺 = ◑𝐹)) β†’ ((𝐹 ∈ (𝑋 RngHomo π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHomo 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))))
6951biimpcd 249 . . . . . . 7 (𝐹 ∈ (𝑋 RngIsom π‘Œ) β†’ (πœ‘ β†’ (𝐹 ∈ (𝑋 RngHomo π‘Œ) ∧ ◑𝐹 ∈ (π‘Œ RngHomo 𝑋))))
7069adantr 482 . . . . . 6 ((𝐹 ∈ (𝑋 RngIsom π‘Œ) ∧ 𝐺 = ◑𝐹) β†’ (πœ‘ β†’ (𝐹 ∈ (𝑋 RngHomo π‘Œ) ∧ ◑𝐹 ∈ (π‘Œ RngHomo 𝑋))))
7170impcom 409 . . . . 5 ((πœ‘ ∧ (𝐹 ∈ (𝑋 RngIsom π‘Œ) ∧ 𝐺 = ◑𝐹)) β†’ (𝐹 ∈ (𝑋 RngHomo π‘Œ) ∧ ◑𝐹 ∈ (π‘Œ RngHomo 𝑋)))
72 eleq1 2826 . . . . . . 7 (𝐺 = ◑𝐹 β†’ (𝐺 ∈ (π‘Œ RngHomo 𝑋) ↔ ◑𝐹 ∈ (π‘Œ RngHomo 𝑋)))
7372ad2antll 728 . . . . . 6 ((πœ‘ ∧ (𝐹 ∈ (𝑋 RngIsom π‘Œ) ∧ 𝐺 = ◑𝐹)) β†’ (𝐺 ∈ (π‘Œ RngHomo 𝑋) ↔ ◑𝐹 ∈ (π‘Œ RngHomo 𝑋)))
7473anbi2d 630 . . . . 5 ((πœ‘ ∧ (𝐹 ∈ (𝑋 RngIsom π‘Œ) ∧ 𝐺 = ◑𝐹)) β†’ ((𝐹 ∈ (𝑋 RngHomo π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHomo 𝑋)) ↔ (𝐹 ∈ (𝑋 RngHomo π‘Œ) ∧ ◑𝐹 ∈ (π‘Œ RngHomo 𝑋))))
7571, 74mpbird 257 . . . 4 ((πœ‘ ∧ (𝐹 ∈ (𝑋 RngIsom π‘Œ) ∧ 𝐺 = ◑𝐹)) β†’ (𝐹 ∈ (𝑋 RngHomo π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHomo 𝑋)))
76 coeq2 5819 . . . . . . 7 (𝐺 = ◑𝐹 β†’ (𝐹 ∘ 𝐺) = (𝐹 ∘ ◑𝐹))
7776ad2antll 728 . . . . . 6 ((πœ‘ ∧ (𝐹 ∈ (𝑋 RngIsom π‘Œ) ∧ 𝐺 = ◑𝐹)) β†’ (𝐹 ∘ 𝐺) = (𝐹 ∘ ◑𝐹))
78 f1ococnv2 6816 . . . . . . 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 RngIsom π‘Œ) ∧ 𝐺 = ◑𝐹)) β†’ (𝐹 ∘ ◑𝐹) = ( I β†Ύ (Baseβ€˜π‘Œ)))
8077, 79eqtrd 2777 . . . . 5 ((πœ‘ ∧ (𝐹 ∈ (𝑋 RngIsom π‘Œ) ∧ 𝐺 = ◑𝐹)) β†’ (𝐹 ∘ 𝐺) = ( I β†Ύ (Baseβ€˜π‘Œ)))
8175, 67, 80jca31 516 . . . 4 ((πœ‘ ∧ (𝐹 ∈ (𝑋 RngIsom π‘Œ) ∧ 𝐺 = ◑𝐹)) β†’ (((𝐹 ∈ (𝑋 RngHomo π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHomo 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ (𝐹 ∘ 𝐺) = ( I β†Ύ (Baseβ€˜π‘Œ))))
8268, 75, 81jca31 516 . . 3 ((πœ‘ ∧ (𝐹 ∈ (𝑋 RngIsom π‘Œ) ∧ 𝐺 = ◑𝐹)) β†’ ((((𝐹 ∈ (𝑋 RngHomo π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHomo 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ (𝐹 ∈ (𝑋 RngHomo π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHomo 𝑋))) ∧ (((𝐹 ∈ (𝑋 RngHomo π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHomo 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 RngHomo π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHomo 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ (𝐹 ∈ (𝑋 RngHomo π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHomo 𝑋))) ∧ (((𝐹 ∈ (𝑋 RngHomo π‘Œ) ∧ 𝐺 ∈ (π‘Œ RngHomo 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ (𝐹 ∘ 𝐺) = ( I β†Ύ (Baseβ€˜π‘Œ)))) ↔ (𝐹 ∈ (𝑋 RngIsom π‘Œ) ∧ 𝐺 = ◑𝐹)))
8410, 23, 833bitrd 305 1 (πœ‘ β†’ (𝐹(π‘‹π‘π‘Œ)𝐺 ↔ (𝐹 ∈ (𝑋 RngIsom π‘Œ) ∧ 𝐺 = ◑𝐹)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   class class class wbr 5110   I cid 5535  β—‘ccnv 5637   β†Ύ cres 5640   ∘ ccom 5642  βŸΆwf 6497  β€“1-1-ontoβ†’wf1o 6500  β€˜cfv 6501  (class class class)co 7362  Basecbs 17090  Catccat 17551  Sectcsect 17634  Invcinv 17635   RngHomo crngh 46257   RngIsom crngs 46258  RngCatALTVcrngcALTV 46330
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-cnex 11114  ax-resscn 11115  ax-1cn 11116  ax-icn 11117  ax-addcl 11118  ax-addrcl 11119  ax-mulcl 11120  ax-mulrcl 11121  ax-mulcom 11122  ax-addass 11123  ax-mulass 11124  ax-distr 11125  ax-i2m1 11126  ax-1ne0 11127  ax-1rid 11128  ax-rnegex 11129  ax-rrecex 11130  ax-cnre 11131  ax-pre-lttri 11132  ax-pre-lttrn 11133  ax-pre-ltadd 11134  ax-pre-mulgt0 11135
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-er 8655  df-map 8774  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-pnf 11198  df-mnf 11199  df-xr 11200  df-ltxr 11201  df-le 11202  df-sub 11394  df-neg 11395  df-nn 12161  df-2 12223  df-3 12224  df-4 12225  df-5 12226  df-6 12227  df-7 12228  df-8 12229  df-9 12230  df-n0 12421  df-z 12507  df-dec 12626  df-uz 12771  df-fz 13432  df-struct 17026  df-sets 17043  df-slot 17061  df-ndx 17073  df-base 17091  df-plusg 17153  df-hom 17164  df-cco 17165  df-0g 17330  df-cat 17555  df-cid 17556  df-sect 17637  df-inv 17638  df-mgm 18504  df-sgrp 18553  df-mnd 18564  df-mhm 18608  df-grp 18758  df-ghm 19013  df-abl 19572  df-mgp 19904  df-mgmhm 46147  df-rng 46247  df-rnghomo 46259  df-rngisom 46260  df-rngcALTV 46332
This theorem is referenced by:  rngcisoALTV  46366
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