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Theorem isocoinvid 16385
Description: The inverse of an isomorphism composed with the isomorphism is the identity. (Contributed by AV, 10-Apr-2017.)
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 16382 . . 3 (𝜑 → ((𝑋𝑁𝑌)‘𝐹)(𝑌𝑁𝑋)𝐹)
9 eqid 2621 . . . . 5 (Sect‘𝐶) = (Sect‘𝐶)
101, 3, 4, 6, 5, 9isinv 16352 . . . 4 (𝜑 → (((𝑋𝑁𝑌)‘𝐹)(𝑌𝑁𝑋)𝐹 ↔ (((𝑋𝑁𝑌)‘𝐹)(𝑌(Sect‘𝐶)𝑋)𝐹𝐹(𝑋(Sect‘𝐶)𝑌)((𝑋𝑁𝑌)‘𝐹))))
11 simpl 473 . . . 4 ((((𝑋𝑁𝑌)‘𝐹)(𝑌(Sect‘𝐶)𝑋)𝐹𝐹(𝑋(Sect‘𝐶)𝑌)((𝑋𝑁𝑌)‘𝐹)) → ((𝑋𝑁𝑌)‘𝐹)(𝑌(Sect‘𝐶)𝑋)𝐹)
1210, 11syl6bi 243 . . 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 2621 . . . 4 (Hom ‘𝐶) = (Hom ‘𝐶)
15 eqid 2621 . . . 4 (comp‘𝐶) = (comp‘𝐶)
16 invcoisoid.1 . . . 4 1 = (Id‘𝐶)
171, 14, 2, 4, 6, 5isohom 16368 . . . . 5 (𝜑 → (𝑌𝐼𝑋) ⊆ (𝑌(Hom ‘𝐶)𝑋))
181, 3, 4, 5, 6, 2invf 16360 . . . . . 6 (𝜑 → (𝑋𝑁𝑌):(𝑋𝐼𝑌)⟶(𝑌𝐼𝑋))
1918, 7ffvelrnd 6321 . . . . 5 (𝜑 → ((𝑋𝑁𝑌)‘𝐹) ∈ (𝑌𝐼𝑋))
2017, 19sseldd 3588 . . . 4 (𝜑 → ((𝑋𝑁𝑌)‘𝐹) ∈ (𝑌(Hom ‘𝐶)𝑋))
211, 14, 2, 4, 5, 6isohom 16368 . . . . 5 (𝜑 → (𝑋𝐼𝑌) ⊆ (𝑋(Hom ‘𝐶)𝑌))
2221, 7sseldd 3588 . . . 4 (𝜑𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
231, 14, 15, 16, 9, 4, 6, 5, 20, 22issect2 16346 . . 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 2627 . . . . 5 (𝜑 → (⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌) = )
2726oveqd 6627 . . . 4 (𝜑 → (𝐹(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)((𝑋𝑁𝑌)‘𝐹)) = (𝐹 ((𝑋𝑁𝑌)‘𝐹)))
2827eqeq1d 2623 . . 3 (𝜑 → ((𝐹(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)((𝑋𝑁𝑌)‘𝐹)) = ( 1𝑌) ↔ (𝐹 ((𝑋𝑁𝑌)‘𝐹)) = ( 1𝑌)))
2923, 28bitrd 268 . 2 (𝜑 → (((𝑋𝑁𝑌)‘𝐹)(𝑌(Sect‘𝐶)𝑋)𝐹 ↔ (𝐹 ((𝑋𝑁𝑌)‘𝐹)) = ( 1𝑌)))
3013, 29mpbid 222 1 (𝜑 → (𝐹 ((𝑋𝑁𝑌)‘𝐹)) = ( 1𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  cop 4159   class class class wbr 4618  cfv 5852  (class class class)co 6610  Basecbs 15792  Hom chom 15884  compcco 15885  Catccat 16257  Idccid 16258  Sectcsect 16336  Invcinv 16337  Isociso 16338
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-1st 7120  df-2nd 7121  df-cat 16261  df-cid 16262  df-sect 16339  df-inv 16340  df-iso 16341
This theorem is referenced by: (None)
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