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Theorem ringcinv 20567
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 20559 . . . 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 2726 . . 3 (Sectβ€˜πΆ) = (Sectβ€˜πΆ)
101, 2, 6, 7, 8, 9isinv 17716 . 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, 9ringcsect 20566 . . . . 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 287 . . . 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 2726 . . . . . 6 (Baseβ€˜π‘Œ) = (Baseβ€˜π‘Œ)
164, 1, 3, 8, 7, 15, 9ringcsect 20566 . . . . 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 275 . . . . 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 287 . . . 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 673 . . 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 287 . 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 774 . . . . . 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 481 . . . . 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 20387 . . . . . . . . . 10 (𝐹 ∈ (𝑋 RingHom π‘Œ) β†’ 𝐹:(Baseβ€˜π‘‹)⟢(Baseβ€˜π‘Œ))
2715, 11rhmf 20387 . . . . . . . . . 10 (𝐺 ∈ (π‘Œ RingHom 𝑋) β†’ 𝐺:(Baseβ€˜π‘Œ)⟢(Baseβ€˜π‘‹))
2826, 27anim12i 612 . . . . . . . . 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 724 . . . . . . . 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 484 . . . . . . . . 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 481 . . . . . . . 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 484 . . . . . . . . 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 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†’ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹)))
3429, 31, 33jca32 515 . . . . . . 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 481 . . . . . 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 7290 . . . . . . 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 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 468 . . . . 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 582 . . . 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 20394 . . . . . . 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 480 . . . 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 257 . . 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 20398 . . . . . 6 (𝐹 ∈ (𝑋 RingIso π‘Œ) β†’ 𝐹 ∈ (𝑋 RingHom π‘Œ))
4948ad2antrl 725 . . . . 5 ((πœ‘ ∧ (𝐹 ∈ (𝑋 RingIso π‘Œ) ∧ 𝐺 = ◑𝐹)) β†’ 𝐹 ∈ (𝑋 RingHom π‘Œ))
50 isrim0 20385 . . . . . . . . 9 (𝐹 ∈ (𝑋 RingIso π‘Œ) ↔ (𝐹 ∈ (𝑋 RingHom π‘Œ) ∧ ◑𝐹 ∈ (π‘Œ RingHom 𝑋)))
5150simprbi 496 . . . . . . . 8 (𝐹 ∈ (𝑋 RingIso π‘Œ) β†’ ◑𝐹 ∈ (π‘Œ RingHom 𝑋))
52 eleq1 2815 . . . . . . . 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 406 . . . . . 6 ((𝐹 ∈ (𝑋 RingIso π‘Œ) ∧ 𝐺 = ◑𝐹) β†’ 𝐺 ∈ (π‘Œ RingHom 𝑋))
5554adantl 481 . . . . 5 ((πœ‘ ∧ (𝐹 ∈ (𝑋 RingIso π‘Œ) ∧ 𝐺 = ◑𝐹)) β†’ 𝐺 ∈ (π‘Œ RingHom 𝑋))
56 coeq1 5851 . . . . . . 7 (𝐺 = ◑𝐹 β†’ (𝐺 ∘ 𝐹) = (◑𝐹 ∘ 𝐹))
5756ad2antll 726 . . . . . 6 ((πœ‘ ∧ (𝐹 ∈ (𝑋 RingIso π‘Œ) ∧ 𝐺 = ◑𝐹)) β†’ (𝐺 ∘ 𝐹) = (◑𝐹 ∘ 𝐹))
5811, 15rimf1o 20396 . . . . . . . 8 (𝐹 ∈ (𝑋 RingIso π‘Œ) β†’ 𝐹:(Baseβ€˜π‘‹)–1-1-ontoβ†’(Baseβ€˜π‘Œ))
5958ad2antrl 725 . . . . . . 7 ((πœ‘ ∧ (𝐹 ∈ (𝑋 RingIso π‘Œ) ∧ 𝐺 = ◑𝐹)) β†’ 𝐹:(Baseβ€˜π‘‹)–1-1-ontoβ†’(Baseβ€˜π‘Œ))
60 f1ococnv1 6856 . . . . . . 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 2766 . . . . 5 ((πœ‘ ∧ (𝐹 ∈ (𝑋 RingIso π‘Œ) ∧ 𝐺 = ◑𝐹)) β†’ (𝐺 ∘ 𝐹) = ( I β†Ύ (Baseβ€˜π‘‹)))
6349, 55, 62jca31 514 . . . 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 725 . . . . 5 ((πœ‘ ∧ (𝐹 ∈ (𝑋 RingIso π‘Œ) ∧ 𝐺 = ◑𝐹)) β†’ (𝐹 ∈ (𝑋 RingHom π‘Œ) ∧ ◑𝐹 ∈ (π‘Œ RingHom 𝑋)))
6652ad2antll 726 . . . . . 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 257 . . . 4 ((πœ‘ ∧ (𝐹 ∈ (𝑋 RingIso π‘Œ) ∧ 𝐺 = ◑𝐹)) β†’ (𝐹 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐺 ∈ (π‘Œ RingHom 𝑋)))
69 coeq2 5852 . . . . . . 7 (𝐺 = ◑𝐹 β†’ (𝐹 ∘ 𝐺) = (𝐹 ∘ ◑𝐹))
7069ad2antll 726 . . . . . 6 ((πœ‘ ∧ (𝐹 ∈ (𝑋 RingIso π‘Œ) ∧ 𝐺 = ◑𝐹)) β†’ (𝐹 ∘ 𝐺) = (𝐹 ∘ ◑𝐹))
71 f1ococnv2 6854 . . . . . . 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 2766 . . . . 5 ((πœ‘ ∧ (𝐹 ∈ (𝑋 RingIso π‘Œ) ∧ 𝐺 = ◑𝐹)) β†’ (𝐹 ∘ 𝐺) = ( I β†Ύ (Baseβ€˜π‘Œ)))
7468, 62, 73jca31 514 . . . 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 514 . . 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 798 . 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 305 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 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 6533  β€“1-1-ontoβ†’wf1o 6536  β€˜cfv 6537  (class class class)co 7405  Basecbs 17153  Catccat 17617  Sectcsect 17700  Invcinv 17701   RingHom crh 20371   RingIso crs 20372  RingCatcringc 20541
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 7722  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
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 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-1st 7974  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-1o 8467  df-er 8705  df-map 8824  df-pm 8825  df-ixp 8894  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-3 12280  df-4 12281  df-5 12282  df-6 12283  df-7 12284  df-8 12285  df-9 12286  df-n0 12477  df-z 12563  df-dec 12682  df-uz 12827  df-fz 13491  df-struct 17089  df-sets 17106  df-slot 17124  df-ndx 17136  df-base 17154  df-ress 17183  df-plusg 17219  df-hom 17230  df-cco 17231  df-0g 17396  df-cat 17621  df-cid 17622  df-homf 17623  df-sect 17703  df-inv 17704  df-ssc 17766  df-resc 17767  df-subc 17768  df-estrc 18086  df-mgm 18573  df-sgrp 18652  df-mnd 18668  df-mhm 18713  df-grp 18866  df-ghm 19139  df-mgp 20040  df-ur 20087  df-ring 20140  df-rhm 20374  df-rim 20375  df-ringc 20542
This theorem is referenced by:  ringciso  20568
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