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Theorem invcoisoid 17743
Description: The inverse of an isomorphism composed with the isomorphism is the identity. (Contributed by AV, 5-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)
invcoisoid.o ⚬ = (βŸ¨π‘‹, π‘ŒβŸ©(compβ€˜πΆ)𝑋)
Assertion
Ref Expression
invcoisoid (πœ‘ β†’ (((π‘‹π‘π‘Œ)β€˜πΉ) ⚬ 𝐹) = ( 1 β€˜π‘‹))

Proof of Theorem invcoisoid
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, 7invisoinvr 17742 . . 3 (πœ‘ β†’ 𝐹(π‘‹π‘π‘Œ)((π‘‹π‘π‘Œ)β€˜πΉ))
9 eqid 2730 . . . . 5 (Sectβ€˜πΆ) = (Sectβ€˜πΆ)
101, 3, 4, 5, 6, 9isinv 17711 . . . 4 (πœ‘ β†’ (𝐹(π‘‹π‘π‘Œ)((π‘‹π‘π‘Œ)β€˜πΉ) ↔ (𝐹(𝑋(Sectβ€˜πΆ)π‘Œ)((π‘‹π‘π‘Œ)β€˜πΉ) ∧ ((π‘‹π‘π‘Œ)β€˜πΉ)(π‘Œ(Sectβ€˜πΆ)𝑋)𝐹)))
11 simpl 481 . . . 4 ((𝐹(𝑋(Sectβ€˜πΆ)π‘Œ)((π‘‹π‘π‘Œ)β€˜πΉ) ∧ ((π‘‹π‘π‘Œ)β€˜πΉ)(π‘Œ(Sectβ€˜πΆ)𝑋)𝐹) β†’ 𝐹(𝑋(Sectβ€˜πΆ)π‘Œ)((π‘‹π‘π‘Œ)β€˜πΉ))
1210, 11syl6bi 252 . . 3 (πœ‘ β†’ (𝐹(π‘‹π‘π‘Œ)((π‘‹π‘π‘Œ)β€˜πΉ) β†’ 𝐹(𝑋(Sectβ€˜πΆ)π‘Œ)((π‘‹π‘π‘Œ)β€˜πΉ)))
138, 12mpd 15 . 2 (πœ‘ β†’ 𝐹(𝑋(Sectβ€˜πΆ)π‘Œ)((π‘‹π‘π‘Œ)β€˜πΉ))
14 eqid 2730 . . . 4 (Hom β€˜πΆ) = (Hom β€˜πΆ)
15 eqid 2730 . . . 4 (compβ€˜πΆ) = (compβ€˜πΆ)
16 invcoisoid.1 . . . 4 1 = (Idβ€˜πΆ)
171, 14, 2, 4, 5, 6isohom 17727 . . . . 5 (πœ‘ β†’ (π‘‹πΌπ‘Œ) βŠ† (𝑋(Hom β€˜πΆ)π‘Œ))
1817, 7sseldd 3982 . . . 4 (πœ‘ β†’ 𝐹 ∈ (𝑋(Hom β€˜πΆ)π‘Œ))
191, 14, 2, 4, 6, 5isohom 17727 . . . . 5 (πœ‘ β†’ (π‘ŒπΌπ‘‹) βŠ† (π‘Œ(Hom β€˜πΆ)𝑋))
201, 3, 4, 5, 6, 2invf 17719 . . . . . 6 (πœ‘ β†’ (π‘‹π‘π‘Œ):(π‘‹πΌπ‘Œ)⟢(π‘ŒπΌπ‘‹))
2120, 7ffvelcdmd 7086 . . . . 5 (πœ‘ β†’ ((π‘‹π‘π‘Œ)β€˜πΉ) ∈ (π‘ŒπΌπ‘‹))
2219, 21sseldd 3982 . . . 4 (πœ‘ β†’ ((π‘‹π‘π‘Œ)β€˜πΉ) ∈ (π‘Œ(Hom β€˜πΆ)𝑋))
231, 14, 15, 16, 9, 4, 5, 6, 18, 22issect2 17705 . . 3 (πœ‘ β†’ (𝐹(𝑋(Sectβ€˜πΆ)π‘Œ)((π‘‹π‘π‘Œ)β€˜πΉ) ↔ (((π‘‹π‘π‘Œ)β€˜πΉ)(βŸ¨π‘‹, π‘ŒβŸ©(compβ€˜πΆ)𝑋)𝐹) = ( 1 β€˜π‘‹)))
24 invcoisoid.o . . . . . . 7 ⚬ = (βŸ¨π‘‹, π‘ŒβŸ©(compβ€˜πΆ)𝑋)
2524a1i 11 . . . . . 6 (πœ‘ β†’ ⚬ = (βŸ¨π‘‹, π‘ŒβŸ©(compβ€˜πΆ)𝑋))
2625eqcomd 2736 . . . . 5 (πœ‘ β†’ (βŸ¨π‘‹, π‘ŒβŸ©(compβ€˜πΆ)𝑋) = ⚬ )
2726oveqd 7428 . . . 4 (πœ‘ β†’ (((π‘‹π‘π‘Œ)β€˜πΉ)(βŸ¨π‘‹, π‘ŒβŸ©(compβ€˜πΆ)𝑋)𝐹) = (((π‘‹π‘π‘Œ)β€˜πΉ) ⚬ 𝐹))
2827eqeq1d 2732 . . 3 (πœ‘ β†’ ((((π‘‹π‘π‘Œ)β€˜πΉ)(βŸ¨π‘‹, π‘ŒβŸ©(compβ€˜πΆ)𝑋)𝐹) = ( 1 β€˜π‘‹) ↔ (((π‘‹π‘π‘Œ)β€˜πΉ) ⚬ 𝐹) = ( 1 β€˜π‘‹)))
2923, 28bitrd 278 . 2 (πœ‘ β†’ (𝐹(𝑋(Sectβ€˜πΆ)π‘Œ)((π‘‹π‘π‘Œ)β€˜πΉ) ↔ (((π‘‹π‘π‘Œ)β€˜πΉ) ⚬ 𝐹) = ( 1 β€˜π‘‹)))
3013, 29mpbid 231 1 (πœ‘ β†’ (((π‘‹π‘π‘Œ)β€˜πΉ) ⚬ 𝐹) = ( 1 β€˜π‘‹))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1539   ∈ wcel 2104  βŸ¨cop 4633   class class class wbr 5147  β€˜cfv 6542  (class class class)co 7411  Basecbs 17148  Hom chom 17212  compcco 17213  Catccat 17612  Idccid 17613  Sectcsect 17695  Invcinv 17696  Isociso 17697
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  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 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7977  df-2nd 7978  df-cat 17616  df-cid 17617  df-sect 17698  df-inv 17699  df-iso 17700
This theorem is referenced by:  rcaninv  17745
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