Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  xpcco2nd Structured version   Visualization version   GIF version

Theorem xpcco2nd 16872
 Description: Value of composition in the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypotheses
Ref Expression
xpcco1st.t 𝑇 = (𝐶 ×c 𝐷)
xpcco1st.b 𝐵 = (Base‘𝑇)
xpcco1st.k 𝐾 = (Hom ‘𝑇)
xpcco1st.o 𝑂 = (comp‘𝑇)
xpcco1st.x (𝜑𝑋𝐵)
xpcco1st.y (𝜑𝑌𝐵)
xpcco1st.z (𝜑𝑍𝐵)
xpcco1st.f (𝜑𝐹 ∈ (𝑋𝐾𝑌))
xpcco1st.g (𝜑𝐺 ∈ (𝑌𝐾𝑍))
xpcco2nd.1 · = (comp‘𝐷)
Assertion
Ref Expression
xpcco2nd (𝜑 → (2nd ‘(𝐺(⟨𝑋, 𝑌𝑂𝑍)𝐹)) = ((2nd𝐺)(⟨(2nd𝑋), (2nd𝑌)⟩ · (2nd𝑍))(2nd𝐹)))

Proof of Theorem xpcco2nd
StepHypRef Expression
1 xpcco1st.t . . 3 𝑇 = (𝐶 ×c 𝐷)
2 xpcco1st.b . . 3 𝐵 = (Base‘𝑇)
3 xpcco1st.k . . 3 𝐾 = (Hom ‘𝑇)
4 eqid 2651 . . 3 (comp‘𝐶) = (comp‘𝐶)
5 xpcco2nd.1 . . 3 · = (comp‘𝐷)
6 xpcco1st.o . . 3 𝑂 = (comp‘𝑇)
7 xpcco1st.x . . 3 (𝜑𝑋𝐵)
8 xpcco1st.y . . 3 (𝜑𝑌𝐵)
9 xpcco1st.z . . 3 (𝜑𝑍𝐵)
10 xpcco1st.f . . 3 (𝜑𝐹 ∈ (𝑋𝐾𝑌))
11 xpcco1st.g . . 3 (𝜑𝐺 ∈ (𝑌𝐾𝑍))
121, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11xpcco 16870 . 2 (𝜑 → (𝐺(⟨𝑋, 𝑌𝑂𝑍)𝐹) = ⟨((1st𝐺)(⟨(1st𝑋), (1st𝑌)⟩(comp‘𝐶)(1st𝑍))(1st𝐹)), ((2nd𝐺)(⟨(2nd𝑋), (2nd𝑌)⟩ · (2nd𝑍))(2nd𝐹))⟩)
13 ovex 6718 . . 3 ((1st𝐺)(⟨(1st𝑋), (1st𝑌)⟩(comp‘𝐶)(1st𝑍))(1st𝐹)) ∈ V
14 ovex 6718 . . 3 ((2nd𝐺)(⟨(2nd𝑋), (2nd𝑌)⟩ · (2nd𝑍))(2nd𝐹)) ∈ V
1513, 14op2ndd 7221 . 2 ((𝐺(⟨𝑋, 𝑌𝑂𝑍)𝐹) = ⟨((1st𝐺)(⟨(1st𝑋), (1st𝑌)⟩(comp‘𝐶)(1st𝑍))(1st𝐹)), ((2nd𝐺)(⟨(2nd𝑋), (2nd𝑌)⟩ · (2nd𝑍))(2nd𝐹))⟩ → (2nd ‘(𝐺(⟨𝑋, 𝑌𝑂𝑍)𝐹)) = ((2nd𝐺)(⟨(2nd𝑋), (2nd𝑌)⟩ · (2nd𝑍))(2nd𝐹)))
1612, 15syl 17 1 (𝜑 → (2nd ‘(𝐺(⟨𝑋, 𝑌𝑂𝑍)𝐹)) = ((2nd𝐺)(⟨(2nd𝑋), (2nd𝑌)⟩ · (2nd𝑍))(2nd𝐹)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1523   ∈ wcel 2030  ⟨cop 4216  ‘cfv 5926  (class class class)co 6690  1st c1st 7208  2nd c2nd 7209  Basecbs 15904  Hom chom 15999  compcco 16000   ×c cxpc 16855 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-z 11416  df-dec 11532  df-uz 11726  df-fz 12365  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-hom 16013  df-cco 16014  df-xpc 16859 This theorem is referenced by:  2ndfcl  16885
 Copyright terms: Public domain W3C validator