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Theorem invcoisoid 17746
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 17745 . . 3 (𝜑𝐹(𝑋𝑁𝑌)((𝑋𝑁𝑌)‘𝐹))
9 eqid 2731 . . . . 5 (Sect‘𝐶) = (Sect‘𝐶)
101, 3, 4, 5, 6, 9isinv 17714 . . . 4 (𝜑 → (𝐹(𝑋𝑁𝑌)((𝑋𝑁𝑌)‘𝐹) ↔ (𝐹(𝑋(Sect‘𝐶)𝑌)((𝑋𝑁𝑌)‘𝐹) ∧ ((𝑋𝑁𝑌)‘𝐹)(𝑌(Sect‘𝐶)𝑋)𝐹)))
11 simpl 482 . . . 4 ((𝐹(𝑋(Sect‘𝐶)𝑌)((𝑋𝑁𝑌)‘𝐹) ∧ ((𝑋𝑁𝑌)‘𝐹)(𝑌(Sect‘𝐶)𝑋)𝐹) → 𝐹(𝑋(Sect‘𝐶)𝑌)((𝑋𝑁𝑌)‘𝐹))
1210, 11syl6bi 253 . . 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 2731 . . . 4 (Hom ‘𝐶) = (Hom ‘𝐶)
15 eqid 2731 . . . 4 (comp‘𝐶) = (comp‘𝐶)
16 invcoisoid.1 . . . 4 1 = (Id‘𝐶)
171, 14, 2, 4, 5, 6isohom 17730 . . . . 5 (𝜑 → (𝑋𝐼𝑌) ⊆ (𝑋(Hom ‘𝐶)𝑌))
1817, 7sseldd 3983 . . . 4 (𝜑𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
191, 14, 2, 4, 6, 5isohom 17730 . . . . 5 (𝜑 → (𝑌𝐼𝑋) ⊆ (𝑌(Hom ‘𝐶)𝑋))
201, 3, 4, 5, 6, 2invf 17722 . . . . . 6 (𝜑 → (𝑋𝑁𝑌):(𝑋𝐼𝑌)⟶(𝑌𝐼𝑋))
2120, 7ffvelcdmd 7087 . . . . 5 (𝜑 → ((𝑋𝑁𝑌)‘𝐹) ∈ (𝑌𝐼𝑋))
2219, 21sseldd 3983 . . . 4 (𝜑 → ((𝑋𝑁𝑌)‘𝐹) ∈ (𝑌(Hom ‘𝐶)𝑋))
231, 14, 15, 16, 9, 4, 5, 6, 18, 22issect2 17708 . . 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 2737 . . . . 5 (𝜑 → (⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋) = )
2726oveqd 7429 . . . 4 (𝜑 → (((𝑋𝑁𝑌)‘𝐹)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = (((𝑋𝑁𝑌)‘𝐹) 𝐹))
2827eqeq1d 2733 . . 3 (𝜑 → ((((𝑋𝑁𝑌)‘𝐹)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ( 1𝑋) ↔ (((𝑋𝑁𝑌)‘𝐹) 𝐹) = ( 1𝑋)))
2923, 28bitrd 279 . 2 (𝜑 → (𝐹(𝑋(Sect‘𝐶)𝑌)((𝑋𝑁𝑌)‘𝐹) ↔ (((𝑋𝑁𝑌)‘𝐹) 𝐹) = ( 1𝑋)))
3013, 29mpbid 231 1 (𝜑 → (((𝑋𝑁𝑌)‘𝐹) 𝐹) = ( 1𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2105  cop 4634   class class class wbr 5148  cfv 6543  (class class class)co 7412  Basecbs 17151  Hom chom 17215  compcco 17216  Catccat 17615  Idccid 17616  Sectcsect 17698  Invcinv 17699  Isociso 17700
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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7979  df-2nd 7980  df-cat 17619  df-cid 17620  df-sect 17701  df-inv 17702  df-iso 17703
This theorem is referenced by:  rcaninv  17748
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