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Theorem isocoinvid 17042
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 17039 . . 3 (𝜑 → ((𝑋𝑁𝑌)‘𝐹)(𝑌𝑁𝑋)𝐹)
9 eqid 2820 . . . . 5 (Sect‘𝐶) = (Sect‘𝐶)
101, 3, 4, 6, 5, 9isinv 17009 . . . 4 (𝜑 → (((𝑋𝑁𝑌)‘𝐹)(𝑌𝑁𝑋)𝐹 ↔ (((𝑋𝑁𝑌)‘𝐹)(𝑌(Sect‘𝐶)𝑋)𝐹𝐹(𝑋(Sect‘𝐶)𝑌)((𝑋𝑁𝑌)‘𝐹))))
11 simpl 485 . . . 4 ((((𝑋𝑁𝑌)‘𝐹)(𝑌(Sect‘𝐶)𝑋)𝐹𝐹(𝑋(Sect‘𝐶)𝑌)((𝑋𝑁𝑌)‘𝐹)) → ((𝑋𝑁𝑌)‘𝐹)(𝑌(Sect‘𝐶)𝑋)𝐹)
1210, 11syl6bi 255 . . 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 2820 . . . 4 (Hom ‘𝐶) = (Hom ‘𝐶)
15 eqid 2820 . . . 4 (comp‘𝐶) = (comp‘𝐶)
16 invcoisoid.1 . . . 4 1 = (Id‘𝐶)
171, 14, 2, 4, 6, 5isohom 17025 . . . . 5 (𝜑 → (𝑌𝐼𝑋) ⊆ (𝑌(Hom ‘𝐶)𝑋))
181, 3, 4, 5, 6, 2invf 17017 . . . . . 6 (𝜑 → (𝑋𝑁𝑌):(𝑋𝐼𝑌)⟶(𝑌𝐼𝑋))
1918, 7ffvelrnd 6828 . . . . 5 (𝜑 → ((𝑋𝑁𝑌)‘𝐹) ∈ (𝑌𝐼𝑋))
2017, 19sseldd 3947 . . . 4 (𝜑 → ((𝑋𝑁𝑌)‘𝐹) ∈ (𝑌(Hom ‘𝐶)𝑋))
211, 14, 2, 4, 5, 6isohom 17025 . . . . 5 (𝜑 → (𝑋𝐼𝑌) ⊆ (𝑋(Hom ‘𝐶)𝑌))
2221, 7sseldd 3947 . . . 4 (𝜑𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
231, 14, 15, 16, 9, 4, 6, 5, 20, 22issect2 17003 . . 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 2826 . . . . 5 (𝜑 → (⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌) = )
2726oveqd 7150 . . . 4 (𝜑 → (𝐹(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)((𝑋𝑁𝑌)‘𝐹)) = (𝐹 ((𝑋𝑁𝑌)‘𝐹)))
2827eqeq1d 2822 . . 3 (𝜑 → ((𝐹(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)((𝑋𝑁𝑌)‘𝐹)) = ( 1𝑌) ↔ (𝐹 ((𝑋𝑁𝑌)‘𝐹)) = ( 1𝑌)))
2923, 28bitrd 281 . 2 (𝜑 → (((𝑋𝑁𝑌)‘𝐹)(𝑌(Sect‘𝐶)𝑋)𝐹 ↔ (𝐹 ((𝑋𝑁𝑌)‘𝐹)) = ( 1𝑌)))
3013, 29mpbid 234 1 (𝜑 → (𝐹 ((𝑋𝑁𝑌)‘𝐹)) = ( 1𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1537  wcel 2114  cop 4549   class class class wbr 5042  cfv 6331  (class class class)co 7133  Basecbs 16462  Hom chom 16555  compcco 16556  Catccat 16914  Idccid 16915  Sectcsect 16993  Invcinv 16994  Isociso 16995
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-reu 3132  df-rmo 3133  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-op 4550  df-uni 4815  df-iun 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5436  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-f1 6336  df-fo 6337  df-f1o 6338  df-fv 6339  df-riota 7091  df-ov 7136  df-oprab 7137  df-mpo 7138  df-1st 7667  df-2nd 7668  df-cat 16918  df-cid 16919  df-sect 16996  df-inv 16997  df-iso 16998
This theorem is referenced by: (None)
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