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Theorem xpcco2 17432
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 17423 . . 3 (𝑋 × 𝑌) = (Base‘𝑇)
5 eqid 2801 . . 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 5561 . . 3 (𝜑 → ⟨𝑀, 𝑁⟩ ∈ (𝑋 × 𝑌))
12 xpcco2.p . . . 4 (𝜑𝑃𝑋)
13 xpcco2.q . . . 4 (𝜑𝑄𝑌)
1412, 13opelxpd 5561 . . 3 (𝜑 → ⟨𝑃, 𝑄⟩ ∈ (𝑋 × 𝑌))
15 xpcco2.r . . . 4 (𝜑𝑅𝑋)
16 xpcco2.s . . . 4 (𝜑𝑆𝑌)
1715, 16opelxpd 5561 . . 3 (𝜑 → ⟨𝑅, 𝑆⟩ ∈ (𝑋 × 𝑌))
18 xpcco2.f . . . . 5 (𝜑𝐹 ∈ (𝑀𝐻𝑃))
19 xpcco2.g . . . . 5 (𝜑𝐺 ∈ (𝑁𝐽𝑄))
2018, 19opelxpd 5561 . . . 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 17431 . . . 4 (𝜑 → (⟨𝑀, 𝑁⟩(Hom ‘𝑇)⟨𝑃, 𝑄⟩) = ((𝑀𝐻𝑃) × (𝑁𝐽𝑄)))
2420, 23eleqtrrd 2896 . . 3 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (⟨𝑀, 𝑁⟩(Hom ‘𝑇)⟨𝑃, 𝑄⟩))
25 xpcco2.k . . . . 5 (𝜑𝐾 ∈ (𝑃𝐻𝑅))
26 xpcco2.l . . . . 5 (𝜑𝐿 ∈ (𝑄𝐽𝑆))
2725, 26opelxpd 5561 . . . 4 (𝜑 → ⟨𝐾, 𝐿⟩ ∈ ((𝑃𝐻𝑅) × (𝑄𝐽𝑆)))
281, 2, 3, 21, 22, 12, 13, 15, 16, 5xpchom2 17431 . . . 4 (𝜑 → (⟨𝑃, 𝑄⟩(Hom ‘𝑇)⟨𝑅, 𝑆⟩) = ((𝑃𝐻𝑅) × (𝑄𝐽𝑆)))
2927, 28eleqtrrd 2896 . . 3 (𝜑 → ⟨𝐾, 𝐿⟩ ∈ (⟨𝑃, 𝑄⟩(Hom ‘𝑇)⟨𝑅, 𝑆⟩))
301, 4, 5, 6, 7, 8, 11, 14, 17, 24, 29xpcco 17428 . 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 7687 . . . . . . 7 ((𝑀𝑋𝑁𝑌) → (1st ‘⟨𝑀, 𝑁⟩) = 𝑀)
329, 10, 31syl2anc 587 . . . . . 6 (𝜑 → (1st ‘⟨𝑀, 𝑁⟩) = 𝑀)
33 op1stg 7687 . . . . . . 7 ((𝑃𝑋𝑄𝑌) → (1st ‘⟨𝑃, 𝑄⟩) = 𝑃)
3412, 13, 33syl2anc 587 . . . . . 6 (𝜑 → (1st ‘⟨𝑃, 𝑄⟩) = 𝑃)
3532, 34opeq12d 4776 . . . . 5 (𝜑 → ⟨(1st ‘⟨𝑀, 𝑁⟩), (1st ‘⟨𝑃, 𝑄⟩)⟩ = ⟨𝑀, 𝑃⟩)
36 op1stg 7687 . . . . . 6 ((𝑅𝑋𝑆𝑌) → (1st ‘⟨𝑅, 𝑆⟩) = 𝑅)
3715, 16, 36syl2anc 587 . . . . 5 (𝜑 → (1st ‘⟨𝑅, 𝑆⟩) = 𝑅)
3835, 37oveq12d 7157 . . . 4 (𝜑 → (⟨(1st ‘⟨𝑀, 𝑁⟩), (1st ‘⟨𝑃, 𝑄⟩)⟩ · (1st ‘⟨𝑅, 𝑆⟩)) = (⟨𝑀, 𝑃· 𝑅))
39 op1stg 7687 . . . . 5 ((𝐾 ∈ (𝑃𝐻𝑅) ∧ 𝐿 ∈ (𝑄𝐽𝑆)) → (1st ‘⟨𝐾, 𝐿⟩) = 𝐾)
4025, 26, 39syl2anc 587 . . . 4 (𝜑 → (1st ‘⟨𝐾, 𝐿⟩) = 𝐾)
41 op1stg 7687 . . . . 5 ((𝐹 ∈ (𝑀𝐻𝑃) ∧ 𝐺 ∈ (𝑁𝐽𝑄)) → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
4218, 19, 41syl2anc 587 . . . 4 (𝜑 → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
4338, 40, 42oveq123d 7160 . . 3 (𝜑 → ((1st ‘⟨𝐾, 𝐿⟩)(⟨(1st ‘⟨𝑀, 𝑁⟩), (1st ‘⟨𝑃, 𝑄⟩)⟩ · (1st ‘⟨𝑅, 𝑆⟩))(1st ‘⟨𝐹, 𝐺⟩)) = (𝐾(⟨𝑀, 𝑃· 𝑅)𝐹))
44 op2ndg 7688 . . . . . . 7 ((𝑀𝑋𝑁𝑌) → (2nd ‘⟨𝑀, 𝑁⟩) = 𝑁)
459, 10, 44syl2anc 587 . . . . . 6 (𝜑 → (2nd ‘⟨𝑀, 𝑁⟩) = 𝑁)
46 op2ndg 7688 . . . . . . 7 ((𝑃𝑋𝑄𝑌) → (2nd ‘⟨𝑃, 𝑄⟩) = 𝑄)
4712, 13, 46syl2anc 587 . . . . . 6 (𝜑 → (2nd ‘⟨𝑃, 𝑄⟩) = 𝑄)
4845, 47opeq12d 4776 . . . . 5 (𝜑 → ⟨(2nd ‘⟨𝑀, 𝑁⟩), (2nd ‘⟨𝑃, 𝑄⟩)⟩ = ⟨𝑁, 𝑄⟩)
49 op2ndg 7688 . . . . . 6 ((𝑅𝑋𝑆𝑌) → (2nd ‘⟨𝑅, 𝑆⟩) = 𝑆)
5015, 16, 49syl2anc 587 . . . . 5 (𝜑 → (2nd ‘⟨𝑅, 𝑆⟩) = 𝑆)
5148, 50oveq12d 7157 . . . 4 (𝜑 → (⟨(2nd ‘⟨𝑀, 𝑁⟩), (2nd ‘⟨𝑃, 𝑄⟩)⟩ (2nd ‘⟨𝑅, 𝑆⟩)) = (⟨𝑁, 𝑄 𝑆))
52 op2ndg 7688 . . . . 5 ((𝐾 ∈ (𝑃𝐻𝑅) ∧ 𝐿 ∈ (𝑄𝐽𝑆)) → (2nd ‘⟨𝐾, 𝐿⟩) = 𝐿)
5325, 26, 52syl2anc 587 . . . 4 (𝜑 → (2nd ‘⟨𝐾, 𝐿⟩) = 𝐿)
54 op2ndg 7688 . . . . 5 ((𝐹 ∈ (𝑀𝐻𝑃) ∧ 𝐺 ∈ (𝑁𝐽𝑄)) → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
5518, 19, 54syl2anc 587 . . . 4 (𝜑 → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
5651, 53, 55oveq123d 7160 . . 3 (𝜑 → ((2nd ‘⟨𝐾, 𝐿⟩)(⟨(2nd ‘⟨𝑀, 𝑁⟩), (2nd ‘⟨𝑃, 𝑄⟩)⟩ (2nd ‘⟨𝑅, 𝑆⟩))(2nd ‘⟨𝐹, 𝐺⟩)) = (𝐿(⟨𝑁, 𝑄 𝑆)𝐺))
5743, 56opeq12d 4776 . 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 2836 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 1538  wcel 2112  cop 4534   × cxp 5521  cfv 6328  (class class class)co 7139  1st c1st 7673  2nd c2nd 7674  Basecbs 16478  Hom chom 16571  compcco 16572   ×c cxpc 17413
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-oadd 8093  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-nn 11630  df-2 11692  df-3 11693  df-4 11694  df-5 11695  df-6 11696  df-7 11697  df-8 11698  df-9 11699  df-n0 11890  df-z 11974  df-dec 12091  df-uz 12236  df-fz 12890  df-struct 16480  df-ndx 16481  df-slot 16482  df-base 16484  df-hom 16584  df-cco 16585  df-xpc 17417
This theorem is referenced by:  prfcl  17448  evlfcllem  17466  curf1cl  17473  curf2cl  17476  curfcl  17477  uncfcurf  17484  hofcl  17504
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