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Theorem isoco 16209
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 2609 . 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 16203 . 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 16198 1 (𝜑 → (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹) ∈ (𝑋𝐼𝑍))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wcel 1976  cop 4130  cfv 5790  (class class class)co 6527  Basecbs 15644  compcco 15729  Catccat 16097  Invcinv 16177  Isociso 16178
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-1st 7037  df-2nd 7038  df-cat 16101  df-cid 16102  df-sect 16179  df-inv 16180  df-iso 16181
This theorem is referenced by:  cictr  16237
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