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Theorem isoco 17047
Description: The composition of two isomorphisms is an isomorphism. Proposition 3.14(2) of [Adamek] p. 29. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
isoco.b 𝐵 = (Base‘𝐶)
isoco.o · = (comp‘𝐶)
isoco.n 𝐼 = (Iso‘𝐶)
isoco.c (𝜑𝐶 ∈ Cat)
isoco.x (𝜑𝑋𝐵)
isoco.y (𝜑𝑌𝐵)
isoco.z (𝜑𝑍𝐵)
isoco.f (𝜑𝐹 ∈ (𝑋𝐼𝑌))
isoco.g (𝜑𝐺 ∈ (𝑌𝐼𝑍))
Assertion
Ref Expression
isoco (𝜑 → (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹) ∈ (𝑋𝐼𝑍))

Proof of Theorem isoco
StepHypRef Expression
1 isoco.b . 2 𝐵 = (Base‘𝐶)
2 eqid 2824 . 2 (Inv‘𝐶) = (Inv‘𝐶)
3 isoco.c . 2 (𝜑𝐶 ∈ Cat)
4 isoco.x . 2 (𝜑𝑋𝐵)
5 isoco.z . 2 (𝜑𝑍𝐵)
6 isoco.n . 2 𝐼 = (Iso‘𝐶)
7 isoco.y . . 3 (𝜑𝑌𝐵)
8 isoco.f . . 3 (𝜑𝐹 ∈ (𝑋𝐼𝑌))
9 isoco.o . . 3 · = (comp‘𝐶)
10 isoco.g . . 3 (𝜑𝐺 ∈ (𝑌𝐼𝑍))
111, 2, 3, 4, 7, 6, 8, 9, 5, 10invco 17041 . 2 (𝜑 → (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹)(𝑋(Inv‘𝐶)𝑍)(((𝑋(Inv‘𝐶)𝑌)‘𝐹)(⟨𝑍, 𝑌· 𝑋)((𝑌(Inv‘𝐶)𝑍)‘𝐺)))
121, 2, 3, 4, 5, 6, 11inviso1 17036 1 (𝜑 → (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹) ∈ (𝑋𝐼𝑍))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2115  cop 4556  cfv 6343  (class class class)co 7149  Basecbs 16483  compcco 16577  Catccat 16935  Invcinv 17015  Isociso 17016
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 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
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 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-1st 7684  df-2nd 7685  df-cat 16939  df-cid 16940  df-sect 17017  df-inv 17018  df-iso 17019
This theorem is referenced by:  cictr  17075
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