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Theorem isocoinvid 16812
Description: The inverse of an isomorphism composed with the isomorphism is the identity. (Contributed by AV, 10-Apr-2020.)
Hypotheses
Ref Expression
invisoinv.b 𝐵 = (Base‘𝐶)
invisoinv.i 𝐼 = (Iso‘𝐶)
invisoinv.n 𝑁 = (Inv‘𝐶)
invisoinv.c (𝜑𝐶 ∈ Cat)
invisoinv.x (𝜑𝑋𝐵)
invisoinv.y (𝜑𝑌𝐵)
invisoinv.f (𝜑𝐹 ∈ (𝑋𝐼𝑌))
invcoisoid.1 1 = (Id‘𝐶)
isocoinvid.o = (⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)
Assertion
Ref Expression
isocoinvid (𝜑 → (𝐹 ((𝑋𝑁𝑌)‘𝐹)) = ( 1𝑌))

Proof of Theorem isocoinvid
StepHypRef Expression
1 invisoinv.b . . . 4 𝐵 = (Base‘𝐶)
2 invisoinv.i . . . 4 𝐼 = (Iso‘𝐶)
3 invisoinv.n . . . 4 𝑁 = (Inv‘𝐶)
4 invisoinv.c . . . 4 (𝜑𝐶 ∈ Cat)
5 invisoinv.x . . . 4 (𝜑𝑋𝐵)
6 invisoinv.y . . . 4 (𝜑𝑌𝐵)
7 invisoinv.f . . . 4 (𝜑𝐹 ∈ (𝑋𝐼𝑌))
81, 2, 3, 4, 5, 6, 7invisoinvl 16809 . . 3 (𝜑 → ((𝑋𝑁𝑌)‘𝐹)(𝑌𝑁𝑋)𝐹)
9 eqid 2825 . . . . 5 (Sect‘𝐶) = (Sect‘𝐶)
101, 3, 4, 6, 5, 9isinv 16779 . . . 4 (𝜑 → (((𝑋𝑁𝑌)‘𝐹)(𝑌𝑁𝑋)𝐹 ↔ (((𝑋𝑁𝑌)‘𝐹)(𝑌(Sect‘𝐶)𝑋)𝐹𝐹(𝑋(Sect‘𝐶)𝑌)((𝑋𝑁𝑌)‘𝐹))))
11 simpl 476 . . . 4 ((((𝑋𝑁𝑌)‘𝐹)(𝑌(Sect‘𝐶)𝑋)𝐹𝐹(𝑋(Sect‘𝐶)𝑌)((𝑋𝑁𝑌)‘𝐹)) → ((𝑋𝑁𝑌)‘𝐹)(𝑌(Sect‘𝐶)𝑋)𝐹)
1210, 11syl6bi 245 . . 3 (𝜑 → (((𝑋𝑁𝑌)‘𝐹)(𝑌𝑁𝑋)𝐹 → ((𝑋𝑁𝑌)‘𝐹)(𝑌(Sect‘𝐶)𝑋)𝐹))
138, 12mpd 15 . 2 (𝜑 → ((𝑋𝑁𝑌)‘𝐹)(𝑌(Sect‘𝐶)𝑋)𝐹)
14 eqid 2825 . . . 4 (Hom ‘𝐶) = (Hom ‘𝐶)
15 eqid 2825 . . . 4 (comp‘𝐶) = (comp‘𝐶)
16 invcoisoid.1 . . . 4 1 = (Id‘𝐶)
171, 14, 2, 4, 6, 5isohom 16795 . . . . 5 (𝜑 → (𝑌𝐼𝑋) ⊆ (𝑌(Hom ‘𝐶)𝑋))
181, 3, 4, 5, 6, 2invf 16787 . . . . . 6 (𝜑 → (𝑋𝑁𝑌):(𝑋𝐼𝑌)⟶(𝑌𝐼𝑋))
1918, 7ffvelrnd 6614 . . . . 5 (𝜑 → ((𝑋𝑁𝑌)‘𝐹) ∈ (𝑌𝐼𝑋))
2017, 19sseldd 3828 . . . 4 (𝜑 → ((𝑋𝑁𝑌)‘𝐹) ∈ (𝑌(Hom ‘𝐶)𝑋))
211, 14, 2, 4, 5, 6isohom 16795 . . . . 5 (𝜑 → (𝑋𝐼𝑌) ⊆ (𝑋(Hom ‘𝐶)𝑌))
2221, 7sseldd 3828 . . . 4 (𝜑𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
231, 14, 15, 16, 9, 4, 6, 5, 20, 22issect2 16773 . . 3 (𝜑 → (((𝑋𝑁𝑌)‘𝐹)(𝑌(Sect‘𝐶)𝑋)𝐹 ↔ (𝐹(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)((𝑋𝑁𝑌)‘𝐹)) = ( 1𝑌)))
24 isocoinvid.o . . . . . . 7 = (⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)
2524a1i 11 . . . . . 6 (𝜑 = (⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌))
2625eqcomd 2831 . . . . 5 (𝜑 → (⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌) = )
2726oveqd 6927 . . . 4 (𝜑 → (𝐹(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)((𝑋𝑁𝑌)‘𝐹)) = (𝐹 ((𝑋𝑁𝑌)‘𝐹)))
2827eqeq1d 2827 . . 3 (𝜑 → ((𝐹(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)((𝑋𝑁𝑌)‘𝐹)) = ( 1𝑌) ↔ (𝐹 ((𝑋𝑁𝑌)‘𝐹)) = ( 1𝑌)))
2923, 28bitrd 271 . 2 (𝜑 → (((𝑋𝑁𝑌)‘𝐹)(𝑌(Sect‘𝐶)𝑋)𝐹 ↔ (𝐹 ((𝑋𝑁𝑌)‘𝐹)) = ( 1𝑌)))
3013, 29mpbid 224 1 (𝜑 → (𝐹 ((𝑋𝑁𝑌)‘𝐹)) = ( 1𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1656  wcel 2164  cop 4405   class class class wbr 4875  cfv 6127  (class class class)co 6910  Basecbs 16229  Hom chom 16323  compcco 16324  Catccat 16684  Idccid 16685  Sectcsect 16763  Invcinv 16764  Isociso 16765
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-riota 6871  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-1st 7433  df-2nd 7434  df-cat 16688  df-cid 16689  df-sect 16766  df-inv 16767  df-iso 16768
This theorem is referenced by: (None)
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