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Theorem isocoinvid 17767
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 17764 . . 3 (πœ‘ β†’ ((π‘‹π‘π‘Œ)β€˜πΉ)(π‘Œπ‘π‘‹)𝐹)
9 eqid 2727 . . . . 5 (Sectβ€˜πΆ) = (Sectβ€˜πΆ)
101, 3, 4, 6, 5, 9isinv 17734 . . . 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 482 . . . 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, 11biimtrdi 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 2727 . . . 4 (Hom β€˜πΆ) = (Hom β€˜πΆ)
15 eqid 2727 . . . 4 (compβ€˜πΆ) = (compβ€˜πΆ)
16 invcoisoid.1 . . . 4 1 = (Idβ€˜πΆ)
171, 14, 2, 4, 6, 5isohom 17750 . . . . 5 (πœ‘ β†’ (π‘ŒπΌπ‘‹) βŠ† (π‘Œ(Hom β€˜πΆ)𝑋))
181, 3, 4, 5, 6, 2invf 17742 . . . . . 6 (πœ‘ β†’ (π‘‹π‘π‘Œ):(π‘‹πΌπ‘Œ)⟢(π‘ŒπΌπ‘‹))
1918, 7ffvelcdmd 7089 . . . . 5 (πœ‘ β†’ ((π‘‹π‘π‘Œ)β€˜πΉ) ∈ (π‘ŒπΌπ‘‹))
2017, 19sseldd 3979 . . . 4 (πœ‘ β†’ ((π‘‹π‘π‘Œ)β€˜πΉ) ∈ (π‘Œ(Hom β€˜πΆ)𝑋))
211, 14, 2, 4, 5, 6isohom 17750 . . . . 5 (πœ‘ β†’ (π‘‹πΌπ‘Œ) βŠ† (𝑋(Hom β€˜πΆ)π‘Œ))
2221, 7sseldd 3979 . . . 4 (πœ‘ β†’ 𝐹 ∈ (𝑋(Hom β€˜πΆ)π‘Œ))
231, 14, 15, 16, 9, 4, 6, 5, 20, 22issect2 17728 . . 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 2733 . . . . 5 (πœ‘ β†’ (βŸ¨π‘Œ, π‘‹βŸ©(compβ€˜πΆ)π‘Œ) = ⚬ )
2726oveqd 7431 . . . 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 2729 . . 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 279 . 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 395   = wceq 1534   ∈ wcel 2099  βŸ¨cop 4630   class class class wbr 5142  β€˜cfv 6542  (class class class)co 7414  Basecbs 17171  Hom chom 17235  compcco 17236  Catccat 17635  Idccid 17636  Sectcsect 17718  Invcinv 17719  Isociso 17720
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7987  df-2nd 7988  df-cat 17639  df-cid 17640  df-sect 17721  df-inv 17722  df-iso 17723
This theorem is referenced by: (None)
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