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Theorem xpcco1st 17691
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 2737 . . 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 17690 . 2 (𝜑 → (𝐺(⟨𝑋, 𝑌𝑂𝑍)𝐹) = ⟨((1st𝐺)(⟨(1st𝑋), (1st𝑌)⟩ · (1st𝑍))(1st𝐹)), ((2nd𝐺)(⟨(2nd𝑋), (2nd𝑌)⟩(comp‘𝐷)(2nd𝑍))(2nd𝐹))⟩)
13 ovex 7246 . . 3 ((1st𝐺)(⟨(1st𝑋), (1st𝑌)⟩ · (1st𝑍))(1st𝐹)) ∈ V
14 ovex 7246 . . 3 ((2nd𝐺)(⟨(2nd𝑋), (2nd𝑌)⟩(comp‘𝐷)(2nd𝑍))(2nd𝐹)) ∈ V
1513, 14op1std 7771 . 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 1543  wcel 2110  cop 4547  cfv 6380  (class class class)co 7213  1st c1st 7759  2nd c2nd 7760  Basecbs 16760  Hom chom 16813  compcco 16814   ×c cxpc 17675
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-cnex 10785  ax-resscn 10786  ax-1cn 10787  ax-icn 10788  ax-addcl 10789  ax-addrcl 10790  ax-mulcl 10791  ax-mulrcl 10792  ax-mulcom 10793  ax-addass 10794  ax-mulass 10795  ax-distr 10796  ax-i2m1 10797  ax-1ne0 10798  ax-1rid 10799  ax-rnegex 10800  ax-rrecex 10801  ax-cnre 10802  ax-pre-lttri 10803  ax-pre-lttrn 10804  ax-pre-ltadd 10805  ax-pre-mulgt0 10806
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-1st 7761  df-2nd 7762  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-1o 8202  df-er 8391  df-en 8627  df-dom 8628  df-sdom 8629  df-fin 8630  df-pnf 10869  df-mnf 10870  df-xr 10871  df-ltxr 10872  df-le 10873  df-sub 11064  df-neg 11065  df-nn 11831  df-2 11893  df-3 11894  df-4 11895  df-5 11896  df-6 11897  df-7 11898  df-8 11899  df-9 11900  df-n0 12091  df-z 12177  df-dec 12294  df-uz 12439  df-fz 13096  df-struct 16700  df-slot 16735  df-ndx 16745  df-base 16761  df-hom 16826  df-cco 16827  df-xpc 17679
This theorem is referenced by:  1stfcl  17704
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