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Theorem xpcco1st 17282
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))
xpcco1st.1 · = (comp‘𝐶)
Assertion
Ref Expression
xpcco1st (𝜑 → (1st ‘(𝐺(⟨𝑋, 𝑌𝑂𝑍)𝐹)) = ((1st𝐺)(⟨(1st𝑋), (1st𝑌)⟩ · (1st𝑍))(1st𝐹)))

Proof of Theorem xpcco1st
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 xpcco1st.1 . . 3 · = (comp‘𝐶)
5 eqid 2772 . . 3 (comp‘𝐷) = (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 17281 . 2 (𝜑 → (𝐺(⟨𝑋, 𝑌𝑂𝑍)𝐹) = ⟨((1st𝐺)(⟨(1st𝑋), (1st𝑌)⟩ · (1st𝑍))(1st𝐹)), ((2nd𝐺)(⟨(2nd𝑋), (2nd𝑌)⟩(comp‘𝐷)(2nd𝑍))(2nd𝐹))⟩)
13 ovex 7002 . . 3 ((1st𝐺)(⟨(1st𝑋), (1st𝑌)⟩ · (1st𝑍))(1st𝐹)) ∈ V
14 ovex 7002 . . 3 ((2nd𝐺)(⟨(2nd𝑋), (2nd𝑌)⟩(comp‘𝐷)(2nd𝑍))(2nd𝐹)) ∈ V
1513, 14op1std 7504 . 2 ((𝐺(⟨𝑋, 𝑌𝑂𝑍)𝐹) = ⟨((1st𝐺)(⟨(1st𝑋), (1st𝑌)⟩ · (1st𝑍))(1st𝐹)), ((2nd𝐺)(⟨(2nd𝑋), (2nd𝑌)⟩(comp‘𝐷)(2nd𝑍))(2nd𝐹))⟩ → (1st ‘(𝐺(⟨𝑋, 𝑌𝑂𝑍)𝐹)) = ((1st𝐺)(⟨(1st𝑋), (1st𝑌)⟩ · (1st𝑍))(1st𝐹)))
1612, 15syl 17 1 (𝜑 → (1st ‘(𝐺(⟨𝑋, 𝑌𝑂𝑍)𝐹)) = ((1st𝐺)(⟨(1st𝑋), (1st𝑌)⟩ · (1st𝑍))(1st𝐹)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1507  wcel 2048  cop 4441  cfv 6182  (class class class)co 6970  1st c1st 7492  2nd c2nd 7493  Basecbs 16329  Hom chom 16422  compcco 16423   ×c cxpc 17266
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-rep 5043  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273  ax-cnex 10383  ax-resscn 10384  ax-1cn 10385  ax-icn 10386  ax-addcl 10387  ax-addrcl 10388  ax-mulcl 10389  ax-mulrcl 10390  ax-mulcom 10391  ax-addass 10392  ax-mulass 10393  ax-distr 10394  ax-i2m1 10395  ax-1ne0 10396  ax-1rid 10397  ax-rnegex 10398  ax-rrecex 10399  ax-cnre 10400  ax-pre-lttri 10401  ax-pre-lttrn 10402  ax-pre-ltadd 10403  ax-pre-mulgt0 10404
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-fal 1520  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-nel 3068  df-ral 3087  df-rex 3088  df-reu 3089  df-rab 3091  df-v 3411  df-sbc 3678  df-csb 3783  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-pss 3841  df-nul 4174  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4707  df-int 4744  df-iun 4788  df-br 4924  df-opab 4986  df-mpt 5003  df-tr 5025  df-id 5305  df-eprel 5310  df-po 5319  df-so 5320  df-fr 5359  df-we 5361  df-xp 5406  df-rel 5407  df-cnv 5408  df-co 5409  df-dm 5410  df-rn 5411  df-res 5412  df-ima 5413  df-pred 5980  df-ord 6026  df-on 6027  df-lim 6028  df-suc 6029  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-riota 6931  df-ov 6973  df-oprab 6974  df-mpo 6975  df-om 7391  df-1st 7494  df-2nd 7495  df-wrecs 7743  df-recs 7805  df-rdg 7843  df-1o 7897  df-oadd 7901  df-er 8081  df-en 8299  df-dom 8300  df-sdom 8301  df-fin 8302  df-pnf 10468  df-mnf 10469  df-xr 10470  df-ltxr 10471  df-le 10472  df-sub 10664  df-neg 10665  df-nn 11432  df-2 11496  df-3 11497  df-4 11498  df-5 11499  df-6 11500  df-7 11501  df-8 11502  df-9 11503  df-n0 11701  df-z 11787  df-dec 11905  df-uz 12052  df-fz 12702  df-struct 16331  df-ndx 16332  df-slot 16333  df-base 16335  df-hom 16435  df-cco 16436  df-xpc 17270
This theorem is referenced by:  1stfcl  17295
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