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Theorem invcoisoid 17040
Description: The inverse of an isomorphism composed with the isomorphism is the identity. (Contributed by AV, 5-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)
invcoisoid.o = (⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)
Assertion
Ref Expression
invcoisoid (𝜑 → (((𝑋𝑁𝑌)‘𝐹) 𝐹) = ( 1𝑋))

Proof of Theorem invcoisoid
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, 7invisoinvr 17039 . . 3 (𝜑𝐹(𝑋𝑁𝑌)((𝑋𝑁𝑌)‘𝐹))
9 eqid 2821 . . . . 5 (Sect‘𝐶) = (Sect‘𝐶)
101, 3, 4, 5, 6, 9isinv 17008 . . . 4 (𝜑 → (𝐹(𝑋𝑁𝑌)((𝑋𝑁𝑌)‘𝐹) ↔ (𝐹(𝑋(Sect‘𝐶)𝑌)((𝑋𝑁𝑌)‘𝐹) ∧ ((𝑋𝑁𝑌)‘𝐹)(𝑌(Sect‘𝐶)𝑋)𝐹)))
11 simpl 486 . . . 4 ((𝐹(𝑋(Sect‘𝐶)𝑌)((𝑋𝑁𝑌)‘𝐹) ∧ ((𝑋𝑁𝑌)‘𝐹)(𝑌(Sect‘𝐶)𝑋)𝐹) → 𝐹(𝑋(Sect‘𝐶)𝑌)((𝑋𝑁𝑌)‘𝐹))
1210, 11syl6bi 256 . . 3 (𝜑 → (𝐹(𝑋𝑁𝑌)((𝑋𝑁𝑌)‘𝐹) → 𝐹(𝑋(Sect‘𝐶)𝑌)((𝑋𝑁𝑌)‘𝐹)))
138, 12mpd 15 . 2 (𝜑𝐹(𝑋(Sect‘𝐶)𝑌)((𝑋𝑁𝑌)‘𝐹))
14 eqid 2821 . . . 4 (Hom ‘𝐶) = (Hom ‘𝐶)
15 eqid 2821 . . . 4 (comp‘𝐶) = (comp‘𝐶)
16 invcoisoid.1 . . . 4 1 = (Id‘𝐶)
171, 14, 2, 4, 5, 6isohom 17024 . . . . 5 (𝜑 → (𝑋𝐼𝑌) ⊆ (𝑋(Hom ‘𝐶)𝑌))
1817, 7sseldd 3944 . . . 4 (𝜑𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
191, 14, 2, 4, 6, 5isohom 17024 . . . . 5 (𝜑 → (𝑌𝐼𝑋) ⊆ (𝑌(Hom ‘𝐶)𝑋))
201, 3, 4, 5, 6, 2invf 17016 . . . . . 6 (𝜑 → (𝑋𝑁𝑌):(𝑋𝐼𝑌)⟶(𝑌𝐼𝑋))
2120, 7ffvelrnd 6825 . . . . 5 (𝜑 → ((𝑋𝑁𝑌)‘𝐹) ∈ (𝑌𝐼𝑋))
2219, 21sseldd 3944 . . . 4 (𝜑 → ((𝑋𝑁𝑌)‘𝐹) ∈ (𝑌(Hom ‘𝐶)𝑋))
231, 14, 15, 16, 9, 4, 5, 6, 18, 22issect2 17002 . . 3 (𝜑 → (𝐹(𝑋(Sect‘𝐶)𝑌)((𝑋𝑁𝑌)‘𝐹) ↔ (((𝑋𝑁𝑌)‘𝐹)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ( 1𝑋)))
24 invcoisoid.o . . . . . . 7 = (⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)
2524a1i 11 . . . . . 6 (𝜑 = (⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋))
2625eqcomd 2827 . . . . 5 (𝜑 → (⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋) = )
2726oveqd 7147 . . . 4 (𝜑 → (((𝑋𝑁𝑌)‘𝐹)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = (((𝑋𝑁𝑌)‘𝐹) 𝐹))
2827eqeq1d 2823 . . 3 (𝜑 → ((((𝑋𝑁𝑌)‘𝐹)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ( 1𝑋) ↔ (((𝑋𝑁𝑌)‘𝐹) 𝐹) = ( 1𝑋)))
2923, 28bitrd 282 . 2 (𝜑 → (𝐹(𝑋(Sect‘𝐶)𝑌)((𝑋𝑁𝑌)‘𝐹) ↔ (((𝑋𝑁𝑌)‘𝐹) 𝐹) = ( 1𝑋)))
3013, 29mpbid 235 1 (𝜑 → (((𝑋𝑁𝑌)‘𝐹) 𝐹) = ( 1𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  cop 4546   class class class wbr 5039  cfv 6328  (class class class)co 7130  Basecbs 16461  Hom chom 16554  compcco 16555  Catccat 16913  Idccid 16914  Sectcsect 16992  Invcinv 16993  Isociso 16994
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-reu 3133  df-rmo 3134  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7088  df-ov 7133  df-oprab 7134  df-mpo 7135  df-1st 7664  df-2nd 7665  df-cat 16917  df-cid 16918  df-sect 16995  df-inv 16996  df-iso 16997
This theorem is referenced by:  rcaninv  17042
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