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

Theorem xpcco2nd 17551
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 2738 . . 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 17549 . 2 (𝜑 → (𝐺(⟨𝑋, 𝑌𝑂𝑍)𝐹) = ⟨((1st𝐺)(⟨(1st𝑋), (1st𝑌)⟩(comp‘𝐶)(1st𝑍))(1st𝐹)), ((2nd𝐺)(⟨(2nd𝑋), (2nd𝑌)⟩ · (2nd𝑍))(2nd𝐹))⟩)
13 ovex 7203 . . 3 ((1st𝐺)(⟨(1st𝑋), (1st𝑌)⟩(comp‘𝐶)(1st𝑍))(1st𝐹)) ∈ V
14 ovex 7203 . . 3 ((2nd𝐺)(⟨(2nd𝑋), (2nd𝑌)⟩ · (2nd𝑍))(2nd𝐹)) ∈ V
1513, 14op2ndd 7725 . 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 1542  wcel 2114  cop 4522  cfv 6339  (class class class)co 7170  1st c1st 7712  2nd c2nd 7713  Basecbs 16586  Hom chom 16679  compcco 16680   ×c cxpc 17534
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479  ax-cnex 10671  ax-resscn 10672  ax-1cn 10673  ax-icn 10674  ax-addcl 10675  ax-addrcl 10676  ax-mulcl 10677  ax-mulrcl 10678  ax-mulcom 10679  ax-addass 10680  ax-mulass 10681  ax-distr 10682  ax-i2m1 10683  ax-1ne0 10684  ax-1rid 10685  ax-rnegex 10686  ax-rrecex 10687  ax-cnre 10688  ax-pre-lttri 10689  ax-pre-lttrn 10690  ax-pre-ltadd 10691  ax-pre-mulgt0 10692
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-nel 3039  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-riota 7127  df-ov 7173  df-oprab 7174  df-mpo 7175  df-om 7600  df-1st 7714  df-2nd 7715  df-wrecs 7976  df-recs 8037  df-rdg 8075  df-1o 8131  df-er 8320  df-en 8556  df-dom 8557  df-sdom 8558  df-fin 8559  df-pnf 10755  df-mnf 10756  df-xr 10757  df-ltxr 10758  df-le 10759  df-sub 10950  df-neg 10951  df-nn 11717  df-2 11779  df-3 11780  df-4 11781  df-5 11782  df-6 11783  df-7 11784  df-8 11785  df-9 11786  df-n0 11977  df-z 12063  df-dec 12180  df-uz 12325  df-fz 12982  df-struct 16588  df-ndx 16589  df-slot 16590  df-base 16592  df-hom 16692  df-cco 16693  df-xpc 17538
This theorem is referenced by:  2ndfcl  17564
  Copyright terms: Public domain W3C validator