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Theorem ringcinv 20618
Description: An inverse in the category of unital rings is the converse operation. (Contributed by AV, 14-Feb-2020.)
Hypotheses
Ref Expression
ringcsect.c 𝐢 = (RingCatβ€˜π‘ˆ)
ringcsect.b 𝐡 = (Baseβ€˜πΆ)
ringcsect.u (πœ‘ β†’ π‘ˆ ∈ 𝑉)
ringcsect.x (πœ‘ β†’ 𝑋 ∈ 𝐡)
ringcsect.y (πœ‘ β†’ π‘Œ ∈ 𝐡)
ringcinv.n 𝑁 = (Invβ€˜πΆ)
Assertion
Ref Expression
ringcinv (πœ‘ β†’ (𝐹(π‘‹π‘π‘Œ)𝐺 ↔ (𝐹 ∈ (𝑋 RingIso π‘Œ) ∧ 𝐺 = ◑𝐹)))

Proof of Theorem ringcinv
StepHypRef Expression
1 ringcsect.b . . 3 𝐡 = (Baseβ€˜πΆ)
2 ringcinv.n . . 3 𝑁 = (Invβ€˜πΆ)
3 ringcsect.u . . . 4 (πœ‘ β†’ π‘ˆ ∈ 𝑉)
4 ringcsect.c . . . . 5 𝐢 = (RingCatβ€˜π‘ˆ)
54ringccat 20610 . . . 4 (π‘ˆ ∈ 𝑉 β†’ 𝐢 ∈ Cat)
63, 5syl 17 . . 3 (πœ‘ β†’ 𝐢 ∈ Cat)
7 ringcsect.x . . 3 (πœ‘ β†’ 𝑋 ∈ 𝐡)
8 ringcsect.y . . 3 (πœ‘ β†’ π‘Œ ∈ 𝐡)
9 eqid 2728 . . 3 (Sectβ€˜πΆ) = (Sectβ€˜πΆ)
101, 2, 6, 7, 8, 9isinv 17752 . 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 2728 . . . . . 6 (Baseβ€˜π‘‹) = (Baseβ€˜π‘‹)
124, 1, 3, 7, 8, 11, 9ringcsect 20617 . . . . 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 2728 . . . . . 6 (Baseβ€˜π‘Œ) = (Baseβ€˜π‘Œ)
164, 1, 3, 8, 7, 15, 9ringcsect 20617 . . . . 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 20438 . . . . . . . . . 10 (𝐹 ∈ (𝑋 RingHom π‘Œ) β†’ 𝐹:(Baseβ€˜π‘‹)⟢(Baseβ€˜π‘Œ))
2715, 11rhmf 20438 . . . . . . . . . 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 7311 . . . . . . 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 2735 . . . . . . . 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 20445 . . . . . . 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 20449 . . . . . 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 20436 . . . . . . . . 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 2817 . . . . . . . 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 5864 . . . . . . 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 20447 . . . . . . . 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 6873 . . . . . . 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 2768 . . . . 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 5865 . . . . . . 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 6871 . . . . . . 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 2768 . . . . 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 5152   I cid 5579  β—‘ccnv 5681   β†Ύ cres 5684   ∘ ccom 5686  βŸΆwf 6549  β€“1-1-ontoβ†’wf1o 6552  β€˜cfv 6553  (class class class)co 7426  Basecbs 17189  Catccat 17653  Sectcsect 17736  Invcinv 17737   RingHom crh 20422   RingIso crs 20423  RingCatcringc 20592
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 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7748  ax-cnex 11204  ax-resscn 11205  ax-1cn 11206  ax-icn 11207  ax-addcl 11208  ax-addrcl 11209  ax-mulcl 11210  ax-mulrcl 11211  ax-mulcom 11212  ax-addass 11213  ax-mulass 11214  ax-distr 11215  ax-i2m1 11216  ax-1ne0 11217  ax-1rid 11218  ax-rnegex 11219  ax-rrecex 11220  ax-cnre 11221  ax-pre-lttri 11222  ax-pre-lttrn 11223  ax-pre-ltadd 11224  ax-pre-mulgt0 11225
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 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-tp 4637  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7879  df-1st 8001  df-2nd 8002  df-frecs 8295  df-wrecs 8326  df-recs 8400  df-rdg 8439  df-1o 8495  df-er 8733  df-map 8855  df-pm 8856  df-ixp 8925  df-en 8973  df-dom 8974  df-sdom 8975  df-fin 8976  df-pnf 11290  df-mnf 11291  df-xr 11292  df-ltxr 11293  df-le 11294  df-sub 11486  df-neg 11487  df-nn 12253  df-2 12315  df-3 12316  df-4 12317  df-5 12318  df-6 12319  df-7 12320  df-8 12321  df-9 12322  df-n0 12513  df-z 12599  df-dec 12718  df-uz 12863  df-fz 13527  df-struct 17125  df-sets 17142  df-slot 17160  df-ndx 17172  df-base 17190  df-ress 17219  df-plusg 17255  df-hom 17266  df-cco 17267  df-0g 17432  df-cat 17657  df-cid 17658  df-homf 17659  df-sect 17739  df-inv 17740  df-ssc 17802  df-resc 17803  df-subc 17804  df-estrc 18122  df-mgm 18609  df-sgrp 18688  df-mnd 18704  df-mhm 18749  df-grp 18907  df-ghm 19182  df-mgp 20089  df-ur 20136  df-ring 20189  df-rhm 20425  df-rim 20426  df-ringc 20593
This theorem is referenced by:  ringciso  20619
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