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Theorem ringcinvALTV 47480
Description: An inverse in the category of rings is the converse operation. (Contributed by AV, 14-Feb-2020.) (New usage is discouraged.)
Hypotheses
Ref Expression
ringcsectALTV.c 𝐢 = (RingCatALTVβ€˜π‘ˆ)
ringcsectALTV.b 𝐡 = (Baseβ€˜πΆ)
ringcsectALTV.u (πœ‘ β†’ π‘ˆ ∈ 𝑉)
ringcsectALTV.x (πœ‘ β†’ 𝑋 ∈ 𝐡)
ringcsectALTV.y (πœ‘ β†’ π‘Œ ∈ 𝐡)
ringcinvALTV.n 𝑁 = (Invβ€˜πΆ)
Assertion
Ref Expression
ringcinvALTV (πœ‘ β†’ (𝐹(π‘‹π‘π‘Œ)𝐺 ↔ (𝐹 ∈ (𝑋 RingIso π‘Œ) ∧ 𝐺 = ◑𝐹)))

Proof of Theorem ringcinvALTV
StepHypRef Expression
1 ringcsectALTV.b . . 3 𝐡 = (Baseβ€˜πΆ)
2 ringcinvALTV.n . . 3 𝑁 = (Invβ€˜πΆ)
3 ringcsectALTV.u . . . 4 (πœ‘ β†’ π‘ˆ ∈ 𝑉)
4 ringcsectALTV.c . . . . 5 𝐢 = (RingCatALTVβ€˜π‘ˆ)
54ringccatALTV 47477 . . . 4 (π‘ˆ ∈ 𝑉 β†’ 𝐢 ∈ Cat)
63, 5syl 17 . . 3 (πœ‘ β†’ 𝐢 ∈ Cat)
7 ringcsectALTV.x . . 3 (πœ‘ β†’ 𝑋 ∈ 𝐡)
8 ringcsectALTV.y . . 3 (πœ‘ β†’ π‘Œ ∈ 𝐡)
9 eqid 2725 . . 3 (Sectβ€˜πΆ) = (Sectβ€˜πΆ)
101, 2, 6, 7, 8, 9isinv 17737 . 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 2725 . . . . . 6 (Baseβ€˜π‘‹) = (Baseβ€˜π‘‹)
124, 1, 3, 7, 8, 11, 9ringcsectALTV 47479 . . . . 5 (πœ‘ β†’ (𝐹(𝑋(Sectβ€˜πΆ)π‘Œ)𝐺 ↔ (𝐹 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RingHom 𝑋) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹)))))
13 df-3an 1086 . . . . 5 ((𝐹 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RingHom 𝑋) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ↔ ((𝐹 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RingHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))))
1412, 13bitrdi 286 . . . 4 (πœ‘ β†’ (𝐹(𝑋(Sectβ€˜πΆ)π‘Œ)𝐺 ↔ ((𝐹 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RingHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹)))))
15 eqid 2725 . . . . . 6 (Baseβ€˜π‘Œ) = (Baseβ€˜π‘Œ)
164, 1, 3, 8, 7, 15, 9ringcsectALTV 47479 . . . . 5 (πœ‘ β†’ (𝐺(π‘Œ(Sectβ€˜πΆ)𝑋)𝐹 ↔ (𝐺 ∈ (π‘Œ RingHom 𝑋) ∧ 𝐹 ∈ (𝑋 RingHom π‘Œ) ∧ (𝐹 ∘ 𝐺) = ( I β†Ύ (Baseβ€˜π‘Œ)))))
17 3ancoma 1095 . . . . . 6 ((𝐺 ∈ (π‘Œ RingHom 𝑋) ∧ 𝐹 ∈ (𝑋 RingHom π‘Œ) ∧ (𝐹 ∘ 𝐺) = ( I β†Ύ (Baseβ€˜π‘Œ))) ↔ (𝐹 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RingHom 𝑋) ∧ (𝐹 ∘ 𝐺) = ( I β†Ύ (Baseβ€˜π‘Œ))))
18 df-3an 1086 . . . . . 6 ((𝐹 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RingHom 𝑋) ∧ (𝐹 ∘ 𝐺) = ( I β†Ύ (Baseβ€˜π‘Œ))) ↔ ((𝐹 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RingHom 𝑋)) ∧ (𝐹 ∘ 𝐺) = ( I β†Ύ (Baseβ€˜π‘Œ))))
1917, 18bitri 274 . . . . 5 ((𝐺 ∈ (π‘Œ RingHom 𝑋) ∧ 𝐹 ∈ (𝑋 RingHom π‘Œ) ∧ (𝐹 ∘ 𝐺) = ( I β†Ύ (Baseβ€˜π‘Œ))) ↔ ((𝐹 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RingHom 𝑋)) ∧ (𝐹 ∘ 𝐺) = ( I β†Ύ (Baseβ€˜π‘Œ))))
2016, 19bitrdi 286 . . . 4 (πœ‘ β†’ (𝐺(π‘Œ(Sectβ€˜πΆ)𝑋)𝐹 ↔ ((𝐹 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RingHom 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 RingHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RingHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ ((𝐹 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RingHom 𝑋)) ∧ (𝐹 ∘ 𝐺) = ( I β†Ύ (Baseβ€˜π‘Œ))))))
22 anandi 674 . . 3 ((((𝐹 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RingHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ ((𝐹 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RingHom 𝑋)) ∧ (𝐹 ∘ 𝐺) = ( I β†Ύ (Baseβ€˜π‘Œ)))) ↔ ((((𝐹 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RingHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ (𝐹 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RingHom 𝑋))) ∧ (((𝐹 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RingHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ (𝐹 ∘ 𝐺) = ( I β†Ύ (Baseβ€˜π‘Œ)))))
2321, 22bitrdi 286 . 2 (πœ‘ β†’ ((𝐹(𝑋(Sectβ€˜πΆ)π‘Œ)𝐺 ∧ 𝐺(π‘Œ(Sectβ€˜πΆ)𝑋)𝐹) ↔ ((((𝐹 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RingHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ (𝐹 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RingHom 𝑋))) ∧ (((𝐹 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RingHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ (𝐹 ∘ 𝐺) = ( I β†Ύ (Baseβ€˜π‘Œ))))))
24 simplrl 775 . . . . . 6 (((((𝐹 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RingHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ (𝐹 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RingHom 𝑋))) ∧ (((𝐹 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RingHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ (𝐹 ∘ 𝐺) = ( I β†Ύ (Baseβ€˜π‘Œ)))) β†’ 𝐹 ∈ (𝑋 RingHom π‘Œ))
2524adantl 480 . . . . 5 ((πœ‘ ∧ ((((𝐹 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RingHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ (𝐹 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RingHom 𝑋))) ∧ (((𝐹 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RingHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ (𝐹 ∘ 𝐺) = ( I β†Ύ (Baseβ€˜π‘Œ))))) β†’ 𝐹 ∈ (𝑋 RingHom π‘Œ))
2611, 15rhmf 20423 . . . . . . . . . 10 (𝐹 ∈ (𝑋 RingHom π‘Œ) β†’ 𝐹:(Baseβ€˜π‘‹)⟢(Baseβ€˜π‘Œ))
2715, 11rhmf 20423 . . . . . . . . . 10 (𝐺 ∈ (π‘Œ RingHom 𝑋) β†’ 𝐺:(Baseβ€˜π‘Œ)⟢(Baseβ€˜π‘‹))
2826, 27anim12i 611 . . . . . . . . 9 ((𝐹 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RingHom 𝑋)) β†’ (𝐹:(Baseβ€˜π‘‹)⟢(Baseβ€˜π‘Œ) ∧ 𝐺:(Baseβ€˜π‘Œ)⟢(Baseβ€˜π‘‹)))
2928ad2antlr 725 . . . . . . . 8 (((((𝐹 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RingHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ (𝐹 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RingHom 𝑋))) ∧ (((𝐹 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RingHom 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 483 . . . . . . . . 9 ((((𝐹 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RingHom 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 480 . . . . . . . 8 (((((𝐹 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RingHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ (𝐹 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RingHom 𝑋))) ∧ (((𝐹 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RingHom 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 483 . . . . . . . . 9 (((𝐹 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RingHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) β†’ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹)))
3332ad2antrl 726 . . . . . . . 8 (((((𝐹 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RingHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ (𝐹 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RingHom 𝑋))) ∧ (((𝐹 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RingHom 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 514 . . . . . . 7 (((((𝐹 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RingHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ (𝐹 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RingHom 𝑋))) ∧ (((𝐹 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RingHom 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 480 . . . . . 6 ((πœ‘ ∧ ((((𝐹 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RingHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ (𝐹 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RingHom 𝑋))) ∧ (((𝐹 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RingHom 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 2732 . . . . . . . 8 (◑𝐹 = 𝐺 ↔ 𝐺 = ◑𝐹)
3837anbi2i 621 . . . . . . 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 RingHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RingHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ (𝐹 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RingHom 𝑋))) ∧ (((𝐹 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RingHom 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 467 . . . . 5 (((𝐹 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐹:(Baseβ€˜π‘‹)–1-1-ontoβ†’(Baseβ€˜π‘Œ)) ∧ 𝐺 = ◑𝐹) ↔ (𝐹 ∈ (𝑋 RingHom π‘Œ) ∧ (𝐹:(Baseβ€˜π‘‹)–1-1-ontoβ†’(Baseβ€˜π‘Œ) ∧ 𝐺 = ◑𝐹)))
4225, 40, 41sylanbrc 581 . . . 4 ((πœ‘ ∧ ((((𝐹 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RingHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ (𝐹 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RingHom 𝑋))) ∧ (((𝐹 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RingHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ (𝐹 ∘ 𝐺) = ( I β†Ύ (Baseβ€˜π‘Œ))))) β†’ ((𝐹 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐹:(Baseβ€˜π‘‹)–1-1-ontoβ†’(Baseβ€˜π‘Œ)) ∧ 𝐺 = ◑𝐹))
4311, 15isrim 20430 . . . . . . 7 (𝐹 ∈ (𝑋 RingIso π‘Œ) ↔ (𝐹 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐹:(Baseβ€˜π‘‹)–1-1-ontoβ†’(Baseβ€˜π‘Œ)))
4443a1i 11 . . . . . 6 (πœ‘ β†’ (𝐹 ∈ (𝑋 RingIso π‘Œ) ↔ (𝐹 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐹:(Baseβ€˜π‘‹)–1-1-ontoβ†’(Baseβ€˜π‘Œ))))
4544anbi1d 629 . . . . 5 (πœ‘ β†’ ((𝐹 ∈ (𝑋 RingIso π‘Œ) ∧ 𝐺 = ◑𝐹) ↔ ((𝐹 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐹:(Baseβ€˜π‘‹)–1-1-ontoβ†’(Baseβ€˜π‘Œ)) ∧ 𝐺 = ◑𝐹)))
4645adantr 479 . . . 4 ((πœ‘ ∧ ((((𝐹 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RingHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ (𝐹 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RingHom 𝑋))) ∧ (((𝐹 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RingHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ (𝐹 ∘ 𝐺) = ( I β†Ύ (Baseβ€˜π‘Œ))))) β†’ ((𝐹 ∈ (𝑋 RingIso π‘Œ) ∧ 𝐺 = ◑𝐹) ↔ ((𝐹 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐹:(Baseβ€˜π‘‹)–1-1-ontoβ†’(Baseβ€˜π‘Œ)) ∧ 𝐺 = ◑𝐹)))
4742, 46mpbird 256 . . 3 ((πœ‘ ∧ ((((𝐹 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RingHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ (𝐹 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RingHom 𝑋))) ∧ (((𝐹 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RingHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ (𝐹 ∘ 𝐺) = ( I β†Ύ (Baseβ€˜π‘Œ))))) β†’ (𝐹 ∈ (𝑋 RingIso π‘Œ) ∧ 𝐺 = ◑𝐹))
48 rimrhm 20434 . . . . . 6 (𝐹 ∈ (𝑋 RingIso π‘Œ) β†’ 𝐹 ∈ (𝑋 RingHom π‘Œ))
4948ad2antrl 726 . . . . 5 ((πœ‘ ∧ (𝐹 ∈ (𝑋 RingIso π‘Œ) ∧ 𝐺 = ◑𝐹)) β†’ 𝐹 ∈ (𝑋 RingHom π‘Œ))
50 isrim0 20421 . . . . . . . . 9 (𝐹 ∈ (𝑋 RingIso π‘Œ) ↔ (𝐹 ∈ (𝑋 RingHom π‘Œ) ∧ ◑𝐹 ∈ (π‘Œ RingHom 𝑋)))
5150simprbi 495 . . . . . . . 8 (𝐹 ∈ (𝑋 RingIso π‘Œ) β†’ ◑𝐹 ∈ (π‘Œ RingHom 𝑋))
52 eleq1 2813 . . . . . . . 8 (𝐺 = ◑𝐹 β†’ (𝐺 ∈ (π‘Œ RingHom 𝑋) ↔ ◑𝐹 ∈ (π‘Œ RingHom 𝑋)))
5351, 52syl5ibrcom 246 . . . . . . 7 (𝐹 ∈ (𝑋 RingIso π‘Œ) β†’ (𝐺 = ◑𝐹 β†’ 𝐺 ∈ (π‘Œ RingHom 𝑋)))
5453imp 405 . . . . . 6 ((𝐹 ∈ (𝑋 RingIso π‘Œ) ∧ 𝐺 = ◑𝐹) β†’ 𝐺 ∈ (π‘Œ RingHom 𝑋))
5554adantl 480 . . . . 5 ((πœ‘ ∧ (𝐹 ∈ (𝑋 RingIso π‘Œ) ∧ 𝐺 = ◑𝐹)) β†’ 𝐺 ∈ (π‘Œ RingHom 𝑋))
56 coeq1 5855 . . . . . . 7 (𝐺 = ◑𝐹 β†’ (𝐺 ∘ 𝐹) = (◑𝐹 ∘ 𝐹))
5756ad2antll 727 . . . . . 6 ((πœ‘ ∧ (𝐹 ∈ (𝑋 RingIso π‘Œ) ∧ 𝐺 = ◑𝐹)) β†’ (𝐺 ∘ 𝐹) = (◑𝐹 ∘ 𝐹))
5811, 15rimf1o 20432 . . . . . . . 8 (𝐹 ∈ (𝑋 RingIso π‘Œ) β†’ 𝐹:(Baseβ€˜π‘‹)–1-1-ontoβ†’(Baseβ€˜π‘Œ))
5958ad2antrl 726 . . . . . . 7 ((πœ‘ ∧ (𝐹 ∈ (𝑋 RingIso π‘Œ) ∧ 𝐺 = ◑𝐹)) β†’ 𝐹:(Baseβ€˜π‘‹)–1-1-ontoβ†’(Baseβ€˜π‘Œ))
60 f1ococnv1 6861 . . . . . . 7 (𝐹:(Baseβ€˜π‘‹)–1-1-ontoβ†’(Baseβ€˜π‘Œ) β†’ (◑𝐹 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹)))
6159, 60syl 17 . . . . . 6 ((πœ‘ ∧ (𝐹 ∈ (𝑋 RingIso π‘Œ) ∧ 𝐺 = ◑𝐹)) β†’ (◑𝐹 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹)))
6257, 61eqtrd 2765 . . . . 5 ((πœ‘ ∧ (𝐹 ∈ (𝑋 RingIso π‘Œ) ∧ 𝐺 = ◑𝐹)) β†’ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹)))
6349, 55, 62jca31 513 . . . 4 ((πœ‘ ∧ (𝐹 ∈ (𝑋 RingIso π‘Œ) ∧ 𝐺 = ◑𝐹)) β†’ ((𝐹 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RingHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))))
6450biimpi 215 . . . . . 6 (𝐹 ∈ (𝑋 RingIso π‘Œ) β†’ (𝐹 ∈ (𝑋 RingHom π‘Œ) ∧ ◑𝐹 ∈ (π‘Œ RingHom 𝑋)))
6564ad2antrl 726 . . . . 5 ((πœ‘ ∧ (𝐹 ∈ (𝑋 RingIso π‘Œ) ∧ 𝐺 = ◑𝐹)) β†’ (𝐹 ∈ (𝑋 RingHom π‘Œ) ∧ ◑𝐹 ∈ (π‘Œ RingHom 𝑋)))
6652ad2antll 727 . . . . . 6 ((πœ‘ ∧ (𝐹 ∈ (𝑋 RingIso π‘Œ) ∧ 𝐺 = ◑𝐹)) β†’ (𝐺 ∈ (π‘Œ RingHom 𝑋) ↔ ◑𝐹 ∈ (π‘Œ RingHom 𝑋)))
6766anbi2d 628 . . . . 5 ((πœ‘ ∧ (𝐹 ∈ (𝑋 RingIso π‘Œ) ∧ 𝐺 = ◑𝐹)) β†’ ((𝐹 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RingHom 𝑋)) ↔ (𝐹 ∈ (𝑋 RingHom π‘Œ) ∧ ◑𝐹 ∈ (π‘Œ RingHom 𝑋))))
6865, 67mpbird 256 . . . 4 ((πœ‘ ∧ (𝐹 ∈ (𝑋 RingIso π‘Œ) ∧ 𝐺 = ◑𝐹)) β†’ (𝐹 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RingHom 𝑋)))
69 coeq2 5856 . . . . . . 7 (𝐺 = ◑𝐹 β†’ (𝐹 ∘ 𝐺) = (𝐹 ∘ ◑𝐹))
7069ad2antll 727 . . . . . 6 ((πœ‘ ∧ (𝐹 ∈ (𝑋 RingIso π‘Œ) ∧ 𝐺 = ◑𝐹)) β†’ (𝐹 ∘ 𝐺) = (𝐹 ∘ ◑𝐹))
71 f1ococnv2 6859 . . . . . . 7 (𝐹:(Baseβ€˜π‘‹)–1-1-ontoβ†’(Baseβ€˜π‘Œ) β†’ (𝐹 ∘ ◑𝐹) = ( I β†Ύ (Baseβ€˜π‘Œ)))
7259, 71syl 17 . . . . . 6 ((πœ‘ ∧ (𝐹 ∈ (𝑋 RingIso π‘Œ) ∧ 𝐺 = ◑𝐹)) β†’ (𝐹 ∘ ◑𝐹) = ( I β†Ύ (Baseβ€˜π‘Œ)))
7370, 72eqtrd 2765 . . . . 5 ((πœ‘ ∧ (𝐹 ∈ (𝑋 RingIso π‘Œ) ∧ 𝐺 = ◑𝐹)) β†’ (𝐹 ∘ 𝐺) = ( I β†Ύ (Baseβ€˜π‘Œ)))
7468, 62, 73jca31 513 . . . 4 ((πœ‘ ∧ (𝐹 ∈ (𝑋 RingIso π‘Œ) ∧ 𝐺 = ◑𝐹)) β†’ (((𝐹 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RingHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ (𝐹 ∘ 𝐺) = ( I β†Ύ (Baseβ€˜π‘Œ))))
7563, 68, 74jca31 513 . . 3 ((πœ‘ ∧ (𝐹 ∈ (𝑋 RingIso π‘Œ) ∧ 𝐺 = ◑𝐹)) β†’ ((((𝐹 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RingHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ (𝐹 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RingHom 𝑋))) ∧ (((𝐹 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RingHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ (𝐹 ∘ 𝐺) = ( I β†Ύ (Baseβ€˜π‘Œ)))))
7647, 75impbida 799 . 2 (πœ‘ β†’ (((((𝐹 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RingHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ (𝐹 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RingHom 𝑋))) ∧ (((𝐹 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RingHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹))) ∧ (𝐹 ∘ 𝐺) = ( I β†Ύ (Baseβ€˜π‘Œ)))) ↔ (𝐹 ∈ (𝑋 RingIso π‘Œ) ∧ 𝐺 = ◑𝐹)))
7710, 23, 763bitrd 304 1 (πœ‘ β†’ (𝐹(π‘‹π‘π‘Œ)𝐺 ↔ (𝐹 ∈ (𝑋 RingIso π‘Œ) ∧ 𝐺 = ◑𝐹)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   class class class wbr 5144   I cid 5570  β—‘ccnv 5672   β†Ύ cres 5675   ∘ ccom 5677  βŸΆwf 6539  β€“1-1-ontoβ†’wf1o 6542  β€˜cfv 6543  (class class class)co 7413  Basecbs 17174  Catccat 17638  Sectcsect 17721  Invcinv 17722   RingHom crh 20407   RingIso crs 20408  RingCatALTVcringcALTV 47457
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 2696  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7735  ax-cnex 11189  ax-resscn 11190  ax-1cn 11191  ax-icn 11192  ax-addcl 11193  ax-addrcl 11194  ax-mulcl 11195  ax-mulrcl 11196  ax-mulcom 11197  ax-addass 11198  ax-mulass 11199  ax-distr 11200  ax-i2m1 11201  ax-1ne0 11202  ax-1rid 11203  ax-rnegex 11204  ax-rrecex 11205  ax-cnre 11206  ax-pre-lttri 11207  ax-pre-lttrn 11208  ax-pre-ltadd 11209  ax-pre-mulgt0 11210
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 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3961  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-uni 4905  df-iun 4994  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5571  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-we 5630  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7369  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7866  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-er 8718  df-map 8840  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-pnf 11275  df-mnf 11276  df-xr 11277  df-ltxr 11278  df-le 11279  df-sub 11471  df-neg 11472  df-nn 12238  df-2 12300  df-3 12301  df-4 12302  df-5 12303  df-6 12304  df-7 12305  df-8 12306  df-9 12307  df-n0 12498  df-z 12584  df-dec 12703  df-uz 12848  df-fz 13512  df-struct 17110  df-sets 17127  df-slot 17145  df-ndx 17157  df-base 17175  df-plusg 17240  df-hom 17251  df-cco 17252  df-0g 17417  df-cat 17642  df-cid 17643  df-sect 17724  df-inv 17725  df-mgm 18594  df-sgrp 18673  df-mnd 18689  df-mhm 18734  df-grp 18892  df-ghm 19167  df-mgp 20074  df-ur 20121  df-ring 20174  df-rhm 20410  df-rim 20411  df-ringcALTV 47458
This theorem is referenced by:  ringcisoALTV  47481
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