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Theorem isocoinvid 17742
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 17739 . . 3 (πœ‘ β†’ ((π‘‹π‘π‘Œ)β€˜πΉ)(π‘Œπ‘π‘‹)𝐹)
9 eqid 2732 . . . . 5 (Sectβ€˜πΆ) = (Sectβ€˜πΆ)
101, 3, 4, 6, 5, 9isinv 17709 . . . 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 2732 . . . 4 (Hom β€˜πΆ) = (Hom β€˜πΆ)
15 eqid 2732 . . . 4 (compβ€˜πΆ) = (compβ€˜πΆ)
16 invcoisoid.1 . . . 4 1 = (Idβ€˜πΆ)
171, 14, 2, 4, 6, 5isohom 17725 . . . . 5 (πœ‘ β†’ (π‘ŒπΌπ‘‹) βŠ† (π‘Œ(Hom β€˜πΆ)𝑋))
181, 3, 4, 5, 6, 2invf 17717 . . . . . 6 (πœ‘ β†’ (π‘‹π‘π‘Œ):(π‘‹πΌπ‘Œ)⟢(π‘ŒπΌπ‘‹))
1918, 7ffvelcdmd 7087 . . . . 5 (πœ‘ β†’ ((π‘‹π‘π‘Œ)β€˜πΉ) ∈ (π‘ŒπΌπ‘‹))
2017, 19sseldd 3983 . . . 4 (πœ‘ β†’ ((π‘‹π‘π‘Œ)β€˜πΉ) ∈ (π‘Œ(Hom β€˜πΆ)𝑋))
211, 14, 2, 4, 5, 6isohom 17725 . . . . 5 (πœ‘ β†’ (π‘‹πΌπ‘Œ) βŠ† (𝑋(Hom β€˜πΆ)π‘Œ))
2221, 7sseldd 3983 . . . 4 (πœ‘ β†’ 𝐹 ∈ (𝑋(Hom β€˜πΆ)π‘Œ))
231, 14, 15, 16, 9, 4, 6, 5, 20, 22issect2 17703 . . 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 2738 . . . . 5 (πœ‘ β†’ (βŸ¨π‘Œ, π‘‹βŸ©(compβ€˜πΆ)π‘Œ) = ⚬ )
2726oveqd 7428 . . . 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 2734 . . 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 1541   ∈ wcel 2106  βŸ¨cop 4634   class class class wbr 5148  β€˜cfv 6543  (class class class)co 7411  Basecbs 17146  Hom chom 17210  compcco 17211  Catccat 17610  Idccid 17611  Sectcsect 17693  Invcinv 17694  Isociso 17695
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  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 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7977  df-2nd 7978  df-cat 17614  df-cid 17615  df-sect 17696  df-inv 17697  df-iso 17698
This theorem is referenced by: (None)
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