MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  invisoinvl Structured version   Visualization version   GIF version

Theorem invisoinvl 17052
Description: The inverse of an isomorphism 𝐹 (which is unique because of invf 17030 and is therefore denoted by ((𝑋𝑁𝑌)‘𝐹), see also remark 3.12 in [Adamek] p. 28) is invers to the isomorphism. (Contributed by AV, 9-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))
Assertion
Ref Expression
invisoinvl (𝜑 → ((𝑋𝑁𝑌)‘𝐹)(𝑌𝑁𝑋)𝐹)

Proof of Theorem invisoinvl
StepHypRef Expression
1 invisoinv.b . . . 4 𝐵 = (Base‘𝐶)
2 invisoinv.n . . . 4 𝑁 = (Inv‘𝐶)
3 invisoinv.c . . . 4 (𝜑𝐶 ∈ Cat)
4 invisoinv.x . . . 4 (𝜑𝑋𝐵)
5 invisoinv.y . . . 4 (𝜑𝑌𝐵)
6 invisoinv.i . . . 4 𝐼 = (Iso‘𝐶)
7 invisoinv.f . . . 4 (𝜑𝐹 ∈ (𝑋𝐼𝑌))
8 eqid 2798 . . . 4 (comp‘𝐶) = (comp‘𝐶)
9 eqid 2798 . . . . . 6 (Id‘𝐶) = (Id‘𝐶)
101, 9, 3, 5idiso 17050 . . . . 5 (𝜑 → ((Id‘𝐶)‘𝑌) ∈ (𝑌(Iso‘𝐶)𝑌))
116a1i 11 . . . . . 6 (𝜑𝐼 = (Iso‘𝐶))
1211oveqd 7152 . . . . 5 (𝜑 → (𝑌𝐼𝑌) = (𝑌(Iso‘𝐶)𝑌))
1310, 12eleqtrrd 2893 . . . 4 (𝜑 → ((Id‘𝐶)‘𝑌) ∈ (𝑌𝐼𝑌))
141, 2, 3, 4, 5, 6, 7, 8, 5, 13invco 17033 . . 3 (𝜑 → (((Id‘𝐶)‘𝑌)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)𝐹)(𝑋𝑁𝑌)(((𝑋𝑁𝑌)‘𝐹)(⟨𝑌, 𝑌⟩(comp‘𝐶)𝑋)((𝑌𝑁𝑌)‘((Id‘𝐶)‘𝑌))))
15 eqid 2798 . . . 4 (Hom ‘𝐶) = (Hom ‘𝐶)
161, 15, 6, 3, 4, 5isohom 17038 . . . . 5 (𝜑 → (𝑋𝐼𝑌) ⊆ (𝑋(Hom ‘𝐶)𝑌))
1716, 7sseldd 3916 . . . 4 (𝜑𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
181, 15, 9, 3, 4, 8, 5, 17catlid 16946 . . 3 (𝜑 → (((Id‘𝐶)‘𝑌)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)𝐹) = 𝐹)
192a1i 11 . . . . . . . 8 (𝜑𝑁 = (Inv‘𝐶))
2019oveqd 7152 . . . . . . 7 (𝜑 → (𝑌𝑁𝑌) = (𝑌(Inv‘𝐶)𝑌))
2120fveq1d 6647 . . . . . 6 (𝜑 → ((𝑌𝑁𝑌)‘((Id‘𝐶)‘𝑌)) = ((𝑌(Inv‘𝐶)𝑌)‘((Id‘𝐶)‘𝑌)))
221, 9, 3, 5idinv 17051 . . . . . 6 (𝜑 → ((𝑌(Inv‘𝐶)𝑌)‘((Id‘𝐶)‘𝑌)) = ((Id‘𝐶)‘𝑌))
2321, 22eqtrd 2833 . . . . 5 (𝜑 → ((𝑌𝑁𝑌)‘((Id‘𝐶)‘𝑌)) = ((Id‘𝐶)‘𝑌))
2423oveq2d 7151 . . . 4 (𝜑 → (((𝑋𝑁𝑌)‘𝐹)(⟨𝑌, 𝑌⟩(comp‘𝐶)𝑋)((𝑌𝑁𝑌)‘((Id‘𝐶)‘𝑌))) = (((𝑋𝑁𝑌)‘𝐹)(⟨𝑌, 𝑌⟩(comp‘𝐶)𝑋)((Id‘𝐶)‘𝑌)))
251, 15, 6, 3, 5, 4isohom 17038 . . . . . 6 (𝜑 → (𝑌𝐼𝑋) ⊆ (𝑌(Hom ‘𝐶)𝑋))
261, 2, 3, 4, 5, 6invf 17030 . . . . . . 7 (𝜑 → (𝑋𝑁𝑌):(𝑋𝐼𝑌)⟶(𝑌𝐼𝑋))
2726, 7ffvelrnd 6829 . . . . . 6 (𝜑 → ((𝑋𝑁𝑌)‘𝐹) ∈ (𝑌𝐼𝑋))
2825, 27sseldd 3916 . . . . 5 (𝜑 → ((𝑋𝑁𝑌)‘𝐹) ∈ (𝑌(Hom ‘𝐶)𝑋))
291, 15, 9, 3, 5, 8, 4, 28catrid 16947 . . . 4 (𝜑 → (((𝑋𝑁𝑌)‘𝐹)(⟨𝑌, 𝑌⟩(comp‘𝐶)𝑋)((Id‘𝐶)‘𝑌)) = ((𝑋𝑁𝑌)‘𝐹))
3024, 29eqtrd 2833 . . 3 (𝜑 → (((𝑋𝑁𝑌)‘𝐹)(⟨𝑌, 𝑌⟩(comp‘𝐶)𝑋)((𝑌𝑁𝑌)‘((Id‘𝐶)‘𝑌))) = ((𝑋𝑁𝑌)‘𝐹))
3114, 18, 303brtr3d 5061 . 2 (𝜑𝐹(𝑋𝑁𝑌)((𝑋𝑁𝑌)‘𝐹))
321, 2, 3, 5, 4invsym 17024 . 2 (𝜑 → (((𝑋𝑁𝑌)‘𝐹)(𝑌𝑁𝑋)𝐹𝐹(𝑋𝑁𝑌)((𝑋𝑁𝑌)‘𝐹)))
3331, 32mpbird 260 1 (𝜑 → ((𝑋𝑁𝑌)‘𝐹)(𝑌𝑁𝑋)𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2111  cop 4531   class class class wbr 5030  cfv 6324  (class class class)co 7135  Basecbs 16475  Hom chom 16568  compcco 16569  Catccat 16927  Idccid 16928  Invcinv 17007  Isociso 17008
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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441
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 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-1st 7671  df-2nd 7672  df-cat 16931  df-cid 16932  df-sect 17009  df-inv 17010  df-iso 17011
This theorem is referenced by:  invisoinvr  17053  isocoinvid  17055
  Copyright terms: Public domain W3C validator