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Theorem isocoinvid 17603
Description: The inverse of an isomorphism composed with the isomorphism is the identity. (Contributed by AV, 10-Apr-2020.)
Hypotheses
Ref Expression
invisoinv.b 𝐡 = (Baseβ€˜πΆ)
invisoinv.i 𝐼 = (Isoβ€˜πΆ)
invisoinv.n 𝑁 = (Invβ€˜πΆ)
invisoinv.c (πœ‘ β†’ 𝐢 ∈ Cat)
invisoinv.x (πœ‘ β†’ 𝑋 ∈ 𝐡)
invisoinv.y (πœ‘ β†’ π‘Œ ∈ 𝐡)
invisoinv.f (πœ‘ β†’ 𝐹 ∈ (π‘‹πΌπ‘Œ))
invcoisoid.1 1 = (Idβ€˜πΆ)
isocoinvid.o ⚬ = (βŸ¨π‘Œ, π‘‹βŸ©(compβ€˜πΆ)π‘Œ)
Assertion
Ref Expression
isocoinvid (πœ‘ β†’ (𝐹 ⚬ ((π‘‹π‘π‘Œ)β€˜πΉ)) = ( 1 β€˜π‘Œ))

Proof of Theorem isocoinvid
StepHypRef Expression
1 invisoinv.b . . . 4 𝐡 = (Baseβ€˜πΆ)
2 invisoinv.i . . . 4 𝐼 = (Isoβ€˜πΆ)
3 invisoinv.n . . . 4 𝑁 = (Invβ€˜πΆ)
4 invisoinv.c . . . 4 (πœ‘ β†’ 𝐢 ∈ Cat)
5 invisoinv.x . . . 4 (πœ‘ β†’ 𝑋 ∈ 𝐡)
6 invisoinv.y . . . 4 (πœ‘ β†’ π‘Œ ∈ 𝐡)
7 invisoinv.f . . . 4 (πœ‘ β†’ 𝐹 ∈ (π‘‹πΌπ‘Œ))
81, 2, 3, 4, 5, 6, 7invisoinvl 17600 . . 3 (πœ‘ β†’ ((π‘‹π‘π‘Œ)β€˜πΉ)(π‘Œπ‘π‘‹)𝐹)
9 eqid 2736 . . . . 5 (Sectβ€˜πΆ) = (Sectβ€˜πΆ)
101, 3, 4, 6, 5, 9isinv 17570 . . . 4 (πœ‘ β†’ (((π‘‹π‘π‘Œ)β€˜πΉ)(π‘Œπ‘π‘‹)𝐹 ↔ (((π‘‹π‘π‘Œ)β€˜πΉ)(π‘Œ(Sectβ€˜πΆ)𝑋)𝐹 ∧ 𝐹(𝑋(Sectβ€˜πΆ)π‘Œ)((π‘‹π‘π‘Œ)β€˜πΉ))))
11 simpl 483 . . . 4 ((((π‘‹π‘π‘Œ)β€˜πΉ)(π‘Œ(Sectβ€˜πΆ)𝑋)𝐹 ∧ 𝐹(𝑋(Sectβ€˜πΆ)π‘Œ)((π‘‹π‘π‘Œ)β€˜πΉ)) β†’ ((π‘‹π‘π‘Œ)β€˜πΉ)(π‘Œ(Sectβ€˜πΆ)𝑋)𝐹)
1210, 11syl6bi 252 . . 3 (πœ‘ β†’ (((π‘‹π‘π‘Œ)β€˜πΉ)(π‘Œπ‘π‘‹)𝐹 β†’ ((π‘‹π‘π‘Œ)β€˜πΉ)(π‘Œ(Sectβ€˜πΆ)𝑋)𝐹))
138, 12mpd 15 . 2 (πœ‘ β†’ ((π‘‹π‘π‘Œ)β€˜πΉ)(π‘Œ(Sectβ€˜πΆ)𝑋)𝐹)
14 eqid 2736 . . . 4 (Hom β€˜πΆ) = (Hom β€˜πΆ)
15 eqid 2736 . . . 4 (compβ€˜πΆ) = (compβ€˜πΆ)
16 invcoisoid.1 . . . 4 1 = (Idβ€˜πΆ)
171, 14, 2, 4, 6, 5isohom 17586 . . . . 5 (πœ‘ β†’ (π‘ŒπΌπ‘‹) βŠ† (π‘Œ(Hom β€˜πΆ)𝑋))
181, 3, 4, 5, 6, 2invf 17578 . . . . . 6 (πœ‘ β†’ (π‘‹π‘π‘Œ):(π‘‹πΌπ‘Œ)⟢(π‘ŒπΌπ‘‹))
1918, 7ffvelcdmd 7019 . . . . 5 (πœ‘ β†’ ((π‘‹π‘π‘Œ)β€˜πΉ) ∈ (π‘ŒπΌπ‘‹))
2017, 19sseldd 3933 . . . 4 (πœ‘ β†’ ((π‘‹π‘π‘Œ)β€˜πΉ) ∈ (π‘Œ(Hom β€˜πΆ)𝑋))
211, 14, 2, 4, 5, 6isohom 17586 . . . . 5 (πœ‘ β†’ (π‘‹πΌπ‘Œ) βŠ† (𝑋(Hom β€˜πΆ)π‘Œ))
2221, 7sseldd 3933 . . . 4 (πœ‘ β†’ 𝐹 ∈ (𝑋(Hom β€˜πΆ)π‘Œ))
231, 14, 15, 16, 9, 4, 6, 5, 20, 22issect2 17564 . . 3 (πœ‘ β†’ (((π‘‹π‘π‘Œ)β€˜πΉ)(π‘Œ(Sectβ€˜πΆ)𝑋)𝐹 ↔ (𝐹(βŸ¨π‘Œ, π‘‹βŸ©(compβ€˜πΆ)π‘Œ)((π‘‹π‘π‘Œ)β€˜πΉ)) = ( 1 β€˜π‘Œ)))
24 isocoinvid.o . . . . . . 7 ⚬ = (βŸ¨π‘Œ, π‘‹βŸ©(compβ€˜πΆ)π‘Œ)
2524a1i 11 . . . . . 6 (πœ‘ β†’ ⚬ = (βŸ¨π‘Œ, π‘‹βŸ©(compβ€˜πΆ)π‘Œ))
2625eqcomd 2742 . . . . 5 (πœ‘ β†’ (βŸ¨π‘Œ, π‘‹βŸ©(compβ€˜πΆ)π‘Œ) = ⚬ )
2726oveqd 7355 . . . 4 (πœ‘ β†’ (𝐹(βŸ¨π‘Œ, π‘‹βŸ©(compβ€˜πΆ)π‘Œ)((π‘‹π‘π‘Œ)β€˜πΉ)) = (𝐹 ⚬ ((π‘‹π‘π‘Œ)β€˜πΉ)))
2827eqeq1d 2738 . . 3 (πœ‘ β†’ ((𝐹(βŸ¨π‘Œ, π‘‹βŸ©(compβ€˜πΆ)π‘Œ)((π‘‹π‘π‘Œ)β€˜πΉ)) = ( 1 β€˜π‘Œ) ↔ (𝐹 ⚬ ((π‘‹π‘π‘Œ)β€˜πΉ)) = ( 1 β€˜π‘Œ)))
2923, 28bitrd 278 . 2 (πœ‘ β†’ (((π‘‹π‘π‘Œ)β€˜πΉ)(π‘Œ(Sectβ€˜πΆ)𝑋)𝐹 ↔ (𝐹 ⚬ ((π‘‹π‘π‘Œ)β€˜πΉ)) = ( 1 β€˜π‘Œ)))
3013, 29mpbid 231 1 (πœ‘ β†’ (𝐹 ⚬ ((π‘‹π‘π‘Œ)β€˜πΉ)) = ( 1 β€˜π‘Œ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   = wceq 1540   ∈ wcel 2105  βŸ¨cop 4580   class class class wbr 5093  β€˜cfv 6480  (class class class)co 7338  Basecbs 17010  Hom chom 17071  compcco 17072  Catccat 17471  Idccid 17472  Sectcsect 17554  Invcinv 17555  Isociso 17556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5230  ax-sep 5244  ax-nul 5251  ax-pow 5309  ax-pr 5373  ax-un 7651
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3349  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4271  df-if 4475  df-pw 4550  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4854  df-iun 4944  df-br 5094  df-opab 5156  df-mpt 5177  df-id 5519  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6432  df-fun 6482  df-fn 6483  df-f 6484  df-f1 6485  df-fo 6486  df-f1o 6487  df-fv 6488  df-riota 7294  df-ov 7341  df-oprab 7342  df-mpo 7343  df-1st 7900  df-2nd 7901  df-cat 17475  df-cid 17476  df-sect 17557  df-inv 17558  df-iso 17559
This theorem is referenced by: (None)
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