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

Theorem isoco 16260
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 2610 . 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 16254 . 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 16249 1 (𝜑 → (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹) ∈ (𝑋𝐼𝑍))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1475  wcel 1977  cop 4131  cfv 5804  (class class class)co 6549  Basecbs 15695  compcco 15780  Catccat 16148  Invcinv 16228  Isociso 16229
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-cat 16152  df-cid 16153  df-sect 16230  df-inv 16231  df-iso 16232
This theorem is referenced by:  cictr  16288
  Copyright terms: Public domain W3C validator