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Theorem xpcco2 17213
Description: Value of composition in the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypotheses
Ref Expression
xpcco2.t 𝑇 = (𝐶 ×c 𝐷)
xpcco2.x 𝑋 = (Base‘𝐶)
xpcco2.y 𝑌 = (Base‘𝐷)
xpcco2.h 𝐻 = (Hom ‘𝐶)
xpcco2.j 𝐽 = (Hom ‘𝐷)
xpcco2.m (𝜑𝑀𝑋)
xpcco2.n (𝜑𝑁𝑌)
xpcco2.p (𝜑𝑃𝑋)
xpcco2.q (𝜑𝑄𝑌)
xpcco2.o1 · = (comp‘𝐶)
xpcco2.o2 = (comp‘𝐷)
xpcco2.o 𝑂 = (comp‘𝑇)
xpcco2.r (𝜑𝑅𝑋)
xpcco2.s (𝜑𝑆𝑌)
xpcco2.f (𝜑𝐹 ∈ (𝑀𝐻𝑃))
xpcco2.g (𝜑𝐺 ∈ (𝑁𝐽𝑄))
xpcco2.k (𝜑𝐾 ∈ (𝑃𝐻𝑅))
xpcco2.l (𝜑𝐿 ∈ (𝑄𝐽𝑆))
Assertion
Ref Expression
xpcco2 (𝜑 → (⟨𝐾, 𝐿⟩(⟨⟨𝑀, 𝑁⟩, ⟨𝑃, 𝑄⟩⟩𝑂𝑅, 𝑆⟩)⟨𝐹, 𝐺⟩) = ⟨(𝐾(⟨𝑀, 𝑃· 𝑅)𝐹), (𝐿(⟨𝑁, 𝑄 𝑆)𝐺)⟩)

Proof of Theorem xpcco2
StepHypRef Expression
1 xpcco2.t . . 3 𝑇 = (𝐶 ×c 𝐷)
2 xpcco2.x . . . 4 𝑋 = (Base‘𝐶)
3 xpcco2.y . . . 4 𝑌 = (Base‘𝐷)
41, 2, 3xpcbas 17204 . . 3 (𝑋 × 𝑌) = (Base‘𝑇)
5 eqid 2778 . . 3 (Hom ‘𝑇) = (Hom ‘𝑇)
6 xpcco2.o1 . . 3 · = (comp‘𝐶)
7 xpcco2.o2 . . 3 = (comp‘𝐷)
8 xpcco2.o . . 3 𝑂 = (comp‘𝑇)
9 xpcco2.m . . . 4 (𝜑𝑀𝑋)
10 xpcco2.n . . . 4 (𝜑𝑁𝑌)
119, 10opelxpd 5393 . . 3 (𝜑 → ⟨𝑀, 𝑁⟩ ∈ (𝑋 × 𝑌))
12 xpcco2.p . . . 4 (𝜑𝑃𝑋)
13 xpcco2.q . . . 4 (𝜑𝑄𝑌)
1412, 13opelxpd 5393 . . 3 (𝜑 → ⟨𝑃, 𝑄⟩ ∈ (𝑋 × 𝑌))
15 xpcco2.r . . . 4 (𝜑𝑅𝑋)
16 xpcco2.s . . . 4 (𝜑𝑆𝑌)
1715, 16opelxpd 5393 . . 3 (𝜑 → ⟨𝑅, 𝑆⟩ ∈ (𝑋 × 𝑌))
18 xpcco2.f . . . . 5 (𝜑𝐹 ∈ (𝑀𝐻𝑃))
19 xpcco2.g . . . . 5 (𝜑𝐺 ∈ (𝑁𝐽𝑄))
2018, 19opelxpd 5393 . . . 4 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ ((𝑀𝐻𝑃) × (𝑁𝐽𝑄)))
21 xpcco2.h . . . . 5 𝐻 = (Hom ‘𝐶)
22 xpcco2.j . . . . 5 𝐽 = (Hom ‘𝐷)
231, 2, 3, 21, 22, 9, 10, 12, 13, 5xpchom2 17212 . . . 4 (𝜑 → (⟨𝑀, 𝑁⟩(Hom ‘𝑇)⟨𝑃, 𝑄⟩) = ((𝑀𝐻𝑃) × (𝑁𝐽𝑄)))
2420, 23eleqtrrd 2862 . . 3 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (⟨𝑀, 𝑁⟩(Hom ‘𝑇)⟨𝑃, 𝑄⟩))
25 xpcco2.k . . . . 5 (𝜑𝐾 ∈ (𝑃𝐻𝑅))
26 xpcco2.l . . . . 5 (𝜑𝐿 ∈ (𝑄𝐽𝑆))
2725, 26opelxpd 5393 . . . 4 (𝜑 → ⟨𝐾, 𝐿⟩ ∈ ((𝑃𝐻𝑅) × (𝑄𝐽𝑆)))
281, 2, 3, 21, 22, 12, 13, 15, 16, 5xpchom2 17212 . . . 4 (𝜑 → (⟨𝑃, 𝑄⟩(Hom ‘𝑇)⟨𝑅, 𝑆⟩) = ((𝑃𝐻𝑅) × (𝑄𝐽𝑆)))
2927, 28eleqtrrd 2862 . . 3 (𝜑 → ⟨𝐾, 𝐿⟩ ∈ (⟨𝑃, 𝑄⟩(Hom ‘𝑇)⟨𝑅, 𝑆⟩))
301, 4, 5, 6, 7, 8, 11, 14, 17, 24, 29xpcco 17209 . 2 (𝜑 → (⟨𝐾, 𝐿⟩(⟨⟨𝑀, 𝑁⟩, ⟨𝑃, 𝑄⟩⟩𝑂𝑅, 𝑆⟩)⟨𝐹, 𝐺⟩) = ⟨((1st ‘⟨𝐾, 𝐿⟩)(⟨(1st ‘⟨𝑀, 𝑁⟩), (1st ‘⟨𝑃, 𝑄⟩)⟩ · (1st ‘⟨𝑅, 𝑆⟩))(1st ‘⟨𝐹, 𝐺⟩)), ((2nd ‘⟨𝐾, 𝐿⟩)(⟨(2nd ‘⟨𝑀, 𝑁⟩), (2nd ‘⟨𝑃, 𝑄⟩)⟩ (2nd ‘⟨𝑅, 𝑆⟩))(2nd ‘⟨𝐹, 𝐺⟩))⟩)
31 op1stg 7457 . . . . . . 7 ((𝑀𝑋𝑁𝑌) → (1st ‘⟨𝑀, 𝑁⟩) = 𝑀)
329, 10, 31syl2anc 579 . . . . . 6 (𝜑 → (1st ‘⟨𝑀, 𝑁⟩) = 𝑀)
33 op1stg 7457 . . . . . . 7 ((𝑃𝑋𝑄𝑌) → (1st ‘⟨𝑃, 𝑄⟩) = 𝑃)
3412, 13, 33syl2anc 579 . . . . . 6 (𝜑 → (1st ‘⟨𝑃, 𝑄⟩) = 𝑃)
3532, 34opeq12d 4644 . . . . 5 (𝜑 → ⟨(1st ‘⟨𝑀, 𝑁⟩), (1st ‘⟨𝑃, 𝑄⟩)⟩ = ⟨𝑀, 𝑃⟩)
36 op1stg 7457 . . . . . 6 ((𝑅𝑋𝑆𝑌) → (1st ‘⟨𝑅, 𝑆⟩) = 𝑅)
3715, 16, 36syl2anc 579 . . . . 5 (𝜑 → (1st ‘⟨𝑅, 𝑆⟩) = 𝑅)
3835, 37oveq12d 6940 . . . 4 (𝜑 → (⟨(1st ‘⟨𝑀, 𝑁⟩), (1st ‘⟨𝑃, 𝑄⟩)⟩ · (1st ‘⟨𝑅, 𝑆⟩)) = (⟨𝑀, 𝑃· 𝑅))
39 op1stg 7457 . . . . 5 ((𝐾 ∈ (𝑃𝐻𝑅) ∧ 𝐿 ∈ (𝑄𝐽𝑆)) → (1st ‘⟨𝐾, 𝐿⟩) = 𝐾)
4025, 26, 39syl2anc 579 . . . 4 (𝜑 → (1st ‘⟨𝐾, 𝐿⟩) = 𝐾)
41 op1stg 7457 . . . . 5 ((𝐹 ∈ (𝑀𝐻𝑃) ∧ 𝐺 ∈ (𝑁𝐽𝑄)) → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
4218, 19, 41syl2anc 579 . . . 4 (𝜑 → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
4338, 40, 42oveq123d 6943 . . 3 (𝜑 → ((1st ‘⟨𝐾, 𝐿⟩)(⟨(1st ‘⟨𝑀, 𝑁⟩), (1st ‘⟨𝑃, 𝑄⟩)⟩ · (1st ‘⟨𝑅, 𝑆⟩))(1st ‘⟨𝐹, 𝐺⟩)) = (𝐾(⟨𝑀, 𝑃· 𝑅)𝐹))
44 op2ndg 7458 . . . . . . 7 ((𝑀𝑋𝑁𝑌) → (2nd ‘⟨𝑀, 𝑁⟩) = 𝑁)
459, 10, 44syl2anc 579 . . . . . 6 (𝜑 → (2nd ‘⟨𝑀, 𝑁⟩) = 𝑁)
46 op2ndg 7458 . . . . . . 7 ((𝑃𝑋𝑄𝑌) → (2nd ‘⟨𝑃, 𝑄⟩) = 𝑄)
4712, 13, 46syl2anc 579 . . . . . 6 (𝜑 → (2nd ‘⟨𝑃, 𝑄⟩) = 𝑄)
4845, 47opeq12d 4644 . . . . 5 (𝜑 → ⟨(2nd ‘⟨𝑀, 𝑁⟩), (2nd ‘⟨𝑃, 𝑄⟩)⟩ = ⟨𝑁, 𝑄⟩)
49 op2ndg 7458 . . . . . 6 ((𝑅𝑋𝑆𝑌) → (2nd ‘⟨𝑅, 𝑆⟩) = 𝑆)
5015, 16, 49syl2anc 579 . . . . 5 (𝜑 → (2nd ‘⟨𝑅, 𝑆⟩) = 𝑆)
5148, 50oveq12d 6940 . . . 4 (𝜑 → (⟨(2nd ‘⟨𝑀, 𝑁⟩), (2nd ‘⟨𝑃, 𝑄⟩)⟩ (2nd ‘⟨𝑅, 𝑆⟩)) = (⟨𝑁, 𝑄 𝑆))
52 op2ndg 7458 . . . . 5 ((𝐾 ∈ (𝑃𝐻𝑅) ∧ 𝐿 ∈ (𝑄𝐽𝑆)) → (2nd ‘⟨𝐾, 𝐿⟩) = 𝐿)
5325, 26, 52syl2anc 579 . . . 4 (𝜑 → (2nd ‘⟨𝐾, 𝐿⟩) = 𝐿)
54 op2ndg 7458 . . . . 5 ((𝐹 ∈ (𝑀𝐻𝑃) ∧ 𝐺 ∈ (𝑁𝐽𝑄)) → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
5518, 19, 54syl2anc 579 . . . 4 (𝜑 → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
5651, 53, 55oveq123d 6943 . . 3 (𝜑 → ((2nd ‘⟨𝐾, 𝐿⟩)(⟨(2nd ‘⟨𝑀, 𝑁⟩), (2nd ‘⟨𝑃, 𝑄⟩)⟩ (2nd ‘⟨𝑅, 𝑆⟩))(2nd ‘⟨𝐹, 𝐺⟩)) = (𝐿(⟨𝑁, 𝑄 𝑆)𝐺))
5743, 56opeq12d 4644 . 2 (𝜑 → ⟨((1st ‘⟨𝐾, 𝐿⟩)(⟨(1st ‘⟨𝑀, 𝑁⟩), (1st ‘⟨𝑃, 𝑄⟩)⟩ · (1st ‘⟨𝑅, 𝑆⟩))(1st ‘⟨𝐹, 𝐺⟩)), ((2nd ‘⟨𝐾, 𝐿⟩)(⟨(2nd ‘⟨𝑀, 𝑁⟩), (2nd ‘⟨𝑃, 𝑄⟩)⟩ (2nd ‘⟨𝑅, 𝑆⟩))(2nd ‘⟨𝐹, 𝐺⟩))⟩ = ⟨(𝐾(⟨𝑀, 𝑃· 𝑅)𝐹), (𝐿(⟨𝑁, 𝑄 𝑆)𝐺)⟩)
5830, 57eqtrd 2814 1 (𝜑 → (⟨𝐾, 𝐿⟩(⟨⟨𝑀, 𝑁⟩, ⟨𝑃, 𝑄⟩⟩𝑂𝑅, 𝑆⟩)⟨𝐹, 𝐺⟩) = ⟨(𝐾(⟨𝑀, 𝑃· 𝑅)𝐹), (𝐿(⟨𝑁, 𝑄 𝑆)𝐺)⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1601  wcel 2107  cop 4404   × cxp 5353  cfv 6135  (class class class)co 6922  1st c1st 7443  2nd c2nd 7444  Basecbs 16255  Hom chom 16349  compcco 16350   ×c cxpc 17194
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-cnex 10328  ax-resscn 10329  ax-1cn 10330  ax-icn 10331  ax-addcl 10332  ax-addrcl 10333  ax-mulcl 10334  ax-mulrcl 10335  ax-mulcom 10336  ax-addass 10337  ax-mulass 10338  ax-distr 10339  ax-i2m1 10340  ax-1ne0 10341  ax-1rid 10342  ax-rnegex 10343  ax-rrecex 10344  ax-cnre 10345  ax-pre-lttri 10346  ax-pre-lttrn 10347  ax-pre-ltadd 10348  ax-pre-mulgt0 10349
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-fal 1615  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-int 4711  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-om 7344  df-1st 7445  df-2nd 7446  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-1o 7843  df-oadd 7847  df-er 8026  df-en 8242  df-dom 8243  df-sdom 8244  df-fin 8245  df-pnf 10413  df-mnf 10414  df-xr 10415  df-ltxr 10416  df-le 10417  df-sub 10608  df-neg 10609  df-nn 11375  df-2 11438  df-3 11439  df-4 11440  df-5 11441  df-6 11442  df-7 11443  df-8 11444  df-9 11445  df-n0 11643  df-z 11729  df-dec 11846  df-uz 11993  df-fz 12644  df-struct 16257  df-ndx 16258  df-slot 16259  df-base 16261  df-hom 16362  df-cco 16363  df-xpc 17198
This theorem is referenced by:  prfcl  17229  evlfcllem  17247  curf1cl  17254  curf2cl  17257  curfcl  17258  uncfcurf  17265  hofcl  17285
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