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Theorem invcoisoid 17739
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 17738 . . 3 (πœ‘ β†’ 𝐹(π‘‹π‘π‘Œ)((π‘‹π‘π‘Œ)β€˜πΉ))
9 eqid 2733 . . . . 5 (Sectβ€˜πΆ) = (Sectβ€˜πΆ)
101, 3, 4, 5, 6, 9isinv 17707 . . . 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 484 . . . 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 253 . . 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 2733 . . . 4 (Hom β€˜πΆ) = (Hom β€˜πΆ)
15 eqid 2733 . . . 4 (compβ€˜πΆ) = (compβ€˜πΆ)
16 invcoisoid.1 . . . 4 1 = (Idβ€˜πΆ)
171, 14, 2, 4, 5, 6isohom 17723 . . . . 5 (πœ‘ β†’ (π‘‹πΌπ‘Œ) βŠ† (𝑋(Hom β€˜πΆ)π‘Œ))
1817, 7sseldd 3984 . . . 4 (πœ‘ β†’ 𝐹 ∈ (𝑋(Hom β€˜πΆ)π‘Œ))
191, 14, 2, 4, 6, 5isohom 17723 . . . . 5 (πœ‘ β†’ (π‘ŒπΌπ‘‹) βŠ† (π‘Œ(Hom β€˜πΆ)𝑋))
201, 3, 4, 5, 6, 2invf 17715 . . . . . 6 (πœ‘ β†’ (π‘‹π‘π‘Œ):(π‘‹πΌπ‘Œ)⟢(π‘ŒπΌπ‘‹))
2120, 7ffvelcdmd 7088 . . . . 5 (πœ‘ β†’ ((π‘‹π‘π‘Œ)β€˜πΉ) ∈ (π‘ŒπΌπ‘‹))
2219, 21sseldd 3984 . . . 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 17701 . . 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 2739 . . . . 5 (πœ‘ β†’ (βŸ¨π‘‹, π‘ŒβŸ©(compβ€˜πΆ)𝑋) = ⚬ )
2726oveqd 7426 . . . 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 2735 . . 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 279 . 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 397   = wceq 1542   ∈ wcel 2107  βŸ¨cop 4635   class class class wbr 5149  β€˜cfv 6544  (class class class)co 7409  Basecbs 17144  Hom chom 17208  compcco 17209  Catccat 17608  Idccid 17609  Sectcsect 17691  Invcinv 17692  Isociso 17693
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-cat 17612  df-cid 17613  df-sect 17694  df-inv 17695  df-iso 17696
This theorem is referenced by:  rcaninv  17741
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