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Theorem isoco 16796
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 2825 . 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 16790 . 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 16785 1 (𝜑 → (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹) ∈ (𝑋𝐼𝑍))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1656  wcel 2164  cop 4405  cfv 6127  (class class class)co 6910  Basecbs 16229  compcco 16324  Catccat 16684  Invcinv 16764  Isociso 16765
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-riota 6871  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-1st 7433  df-2nd 7434  df-cat 16688  df-cid 16689  df-sect 16766  df-inv 16767  df-iso 16768
This theorem is referenced by:  cictr  16824
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