Theorem xpcco2 17820

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	⊢ (𝜑 → 𝑀 ∈ 𝑋)
xpcco2.n	⊢ (𝜑 → 𝑁 ∈ 𝑌)
xpcco2.p	⊢ (𝜑 → 𝑃 ∈ 𝑋)
xpcco2.q	⊢ (𝜑 → 𝑄 ∈ 𝑌)
xpcco2.o1	⊢ · = (comp‘𝐶)
xpcco2.o2	⊢ ∙ = (comp‘𝐷)
xpcco2.o	⊢ 𝑂 = (comp‘𝑇)
xpcco2.r	⊢ (𝜑 → 𝑅 ∈ 𝑋)
xpcco2.s	⊢ (𝜑 → 𝑆 ∈ 𝑌)
xpcco2.f	⊢ (𝜑 → 𝐹 ∈ (𝑀𝐻𝑃))
xpcco2.g	⊢ (𝜑 → 𝐺 ∈ (𝑁𝐽𝑄))
xpcco2.k	⊢ (𝜑 → 𝐾 ∈ (𝑃𝐻𝑅))
xpcco2.l	⊢ (𝜑 → 𝐿 ∈ (𝑄𝐽𝑆))

Assertion

Ref	Expression
xpcco2	⊢ (𝜑 → (⟨𝐾, 𝐿⟩(⟨⟨𝑀, 𝑁⟩, ⟨𝑃, 𝑄⟩⟩𝑂⟨𝑅, 𝑆⟩)⟨𝐹, 𝐺⟩) = ⟨(𝐾(⟨𝑀, 𝑃⟩ · 𝑅)𝐹), (𝐿(⟨𝑁, 𝑄⟩ ∙ 𝑆)𝐺)⟩)

Proof of Theorem xpcco2

Step	Hyp	Ref	Expression
1		xpcco2.t	. . 3 ⊢ 𝑇 = (𝐶 ×c 𝐷)
2		xpcco2.x	. . . 4 ⊢ 𝑋 = (Base‘𝐶)
3		xpcco2.y	. . . 4 ⊢ 𝑌 = (Base‘𝐷)
4	1, 2, 3	xpcbas 17811	. . 3 ⊢ (𝑋 × 𝑌) = (Base‘𝑇)
5		eqid 2738	. . 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 ⊢ (𝜑 → 𝑀 ∈ 𝑋)
10		xpcco2.n	. . . 4 ⊢ (𝜑 → 𝑁 ∈ 𝑌)
11	9, 10	opelxpd 5618	. . 3 ⊢ (𝜑 → ⟨𝑀, 𝑁⟩ ∈ (𝑋 × 𝑌))
12		xpcco2.p	. . . 4 ⊢ (𝜑 → 𝑃 ∈ 𝑋)
13		xpcco2.q	. . . 4 ⊢ (𝜑 → 𝑄 ∈ 𝑌)
14	12, 13	opelxpd 5618	. . 3 ⊢ (𝜑 → ⟨𝑃, 𝑄⟩ ∈ (𝑋 × 𝑌))
15		xpcco2.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑋)
16		xpcco2.s	. . . 4 ⊢ (𝜑 → 𝑆 ∈ 𝑌)
17	15, 16	opelxpd 5618	. . 3 ⊢ (𝜑 → ⟨𝑅, 𝑆⟩ ∈ (𝑋 × 𝑌))
18		xpcco2.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝑀𝐻𝑃))
19		xpcco2.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (𝑁𝐽𝑄))
20	18, 19	opelxpd 5618	. . . 4 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ ((𝑀𝐻𝑃) × (𝑁𝐽𝑄)))
21		xpcco2.h	. . . . 5 ⊢ 𝐻 = (Hom ‘𝐶)
22		xpcco2.j	. . . . 5 ⊢ 𝐽 = (Hom ‘𝐷)
23	1, 2, 3, 21, 22, 9, 10, 12, 13, 5	xpchom2 17819	. . . 4 ⊢ (𝜑 → (⟨𝑀, 𝑁⟩(Hom ‘𝑇)⟨𝑃, 𝑄⟩) = ((𝑀𝐻𝑃) × (𝑁𝐽𝑄)))
24	20, 23	eleqtrrd 2842	. . 3 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (⟨𝑀, 𝑁⟩(Hom ‘𝑇)⟨𝑃, 𝑄⟩))
25		xpcco2.k	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ (𝑃𝐻𝑅))
26		xpcco2.l	. . . . 5 ⊢ (𝜑 → 𝐿 ∈ (𝑄𝐽𝑆))
27	25, 26	opelxpd 5618	. . . 4 ⊢ (𝜑 → ⟨𝐾, 𝐿⟩ ∈ ((𝑃𝐻𝑅) × (𝑄𝐽𝑆)))
28	1, 2, 3, 21, 22, 12, 13, 15, 16, 5	xpchom2 17819	. . . 4 ⊢ (𝜑 → (⟨𝑃, 𝑄⟩(Hom ‘𝑇)⟨𝑅, 𝑆⟩) = ((𝑃𝐻𝑅) × (𝑄𝐽𝑆)))
29	27, 28	eleqtrrd 2842	. . 3 ⊢ (𝜑 → ⟨𝐾, 𝐿⟩ ∈ (⟨𝑃, 𝑄⟩(Hom ‘𝑇)⟨𝑅, 𝑆⟩))
30	1, 4, 5, 6, 7, 8, 11, 14, 17, 24, 29	xpcco 17816	. 2 ⊢ (𝜑 → (⟨𝐾, 𝐿⟩(⟨⟨𝑀, 𝑁⟩, ⟨𝑃, 𝑄⟩⟩𝑂⟨𝑅, 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 7816	. . . . . . 7 ⊢ ((𝑀 ∈ 𝑋 ∧ 𝑁 ∈ 𝑌) → (1st ‘⟨𝑀, 𝑁⟩) = 𝑀)
32	9, 10, 31	syl2anc 583	. . . . . 6 ⊢ (𝜑 → (1st ‘⟨𝑀, 𝑁⟩) = 𝑀)
33		op1stg 7816	. . . . . . 7 ⊢ ((𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌) → (1st ‘⟨𝑃, 𝑄⟩) = 𝑃)
34	12, 13, 33	syl2anc 583	. . . . . 6 ⊢ (𝜑 → (1st ‘⟨𝑃, 𝑄⟩) = 𝑃)
35	32, 34	opeq12d 4809	. . . . 5 ⊢ (𝜑 → ⟨(1st ‘⟨𝑀, 𝑁⟩), (1st ‘⟨𝑃, 𝑄⟩)⟩ = ⟨𝑀, 𝑃⟩)
36		op1stg 7816	. . . . . 6 ⊢ ((𝑅 ∈ 𝑋 ∧ 𝑆 ∈ 𝑌) → (1st ‘⟨𝑅, 𝑆⟩) = 𝑅)
37	15, 16, 36	syl2anc 583	. . . . 5 ⊢ (𝜑 → (1st ‘⟨𝑅, 𝑆⟩) = 𝑅)
38	35, 37	oveq12d 7273	. . . 4 ⊢ (𝜑 → (⟨(1st ‘⟨𝑀, 𝑁⟩), (1st ‘⟨𝑃, 𝑄⟩)⟩ · (1st ‘⟨𝑅, 𝑆⟩)) = (⟨𝑀, 𝑃⟩ · 𝑅))
39		op1stg 7816	. . . . 5 ⊢ ((𝐾 ∈ (𝑃𝐻𝑅) ∧ 𝐿 ∈ (𝑄𝐽𝑆)) → (1st ‘⟨𝐾, 𝐿⟩) = 𝐾)
40	25, 26, 39	syl2anc 583	. . . 4 ⊢ (𝜑 → (1st ‘⟨𝐾, 𝐿⟩) = 𝐾)
41		op1stg 7816	. . . . 5 ⊢ ((𝐹 ∈ (𝑀𝐻𝑃) ∧ 𝐺 ∈ (𝑁𝐽𝑄)) → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
42	18, 19, 41	syl2anc 583	. . . 4 ⊢ (𝜑 → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
43	38, 40, 42	oveq123d 7276	. . 3 ⊢ (𝜑 → ((1st ‘⟨𝐾, 𝐿⟩)(⟨(1st ‘⟨𝑀, 𝑁⟩), (1st ‘⟨𝑃, 𝑄⟩)⟩ · (1st ‘⟨𝑅, 𝑆⟩))(1st ‘⟨𝐹, 𝐺⟩)) = (𝐾(⟨𝑀, 𝑃⟩ · 𝑅)𝐹))
44		op2ndg 7817	. . . . . . 7 ⊢ ((𝑀 ∈ 𝑋 ∧ 𝑁 ∈ 𝑌) → (2nd ‘⟨𝑀, 𝑁⟩) = 𝑁)
45	9, 10, 44	syl2anc 583	. . . . . 6 ⊢ (𝜑 → (2nd ‘⟨𝑀, 𝑁⟩) = 𝑁)
46		op2ndg 7817	. . . . . . 7 ⊢ ((𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌) → (2nd ‘⟨𝑃, 𝑄⟩) = 𝑄)
47	12, 13, 46	syl2anc 583	. . . . . 6 ⊢ (𝜑 → (2nd ‘⟨𝑃, 𝑄⟩) = 𝑄)
48	45, 47	opeq12d 4809	. . . . 5 ⊢ (𝜑 → ⟨(2nd ‘⟨𝑀, 𝑁⟩), (2nd ‘⟨𝑃, 𝑄⟩)⟩ = ⟨𝑁, 𝑄⟩)
49		op2ndg 7817	. . . . . 6 ⊢ ((𝑅 ∈ 𝑋 ∧ 𝑆 ∈ 𝑌) → (2nd ‘⟨𝑅, 𝑆⟩) = 𝑆)
50	15, 16, 49	syl2anc 583	. . . . 5 ⊢ (𝜑 → (2nd ‘⟨𝑅, 𝑆⟩) = 𝑆)
51	48, 50	oveq12d 7273	. . . 4 ⊢ (𝜑 → (⟨(2nd ‘⟨𝑀, 𝑁⟩), (2nd ‘⟨𝑃, 𝑄⟩)⟩ ∙ (2nd ‘⟨𝑅, 𝑆⟩)) = (⟨𝑁, 𝑄⟩ ∙ 𝑆))
52		op2ndg 7817	. . . . 5 ⊢ ((𝐾 ∈ (𝑃𝐻𝑅) ∧ 𝐿 ∈ (𝑄𝐽𝑆)) → (2nd ‘⟨𝐾, 𝐿⟩) = 𝐿)
53	25, 26, 52	syl2anc 583	. . . 4 ⊢ (𝜑 → (2nd ‘⟨𝐾, 𝐿⟩) = 𝐿)
54		op2ndg 7817	. . . . 5 ⊢ ((𝐹 ∈ (𝑀𝐻𝑃) ∧ 𝐺 ∈ (𝑁𝐽𝑄)) → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
55	18, 19, 54	syl2anc 583	. . . 4 ⊢ (𝜑 → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
56	51, 53, 55	oveq123d 7276	. . 3 ⊢ (𝜑 → ((2nd ‘⟨𝐾, 𝐿⟩)(⟨(2nd ‘⟨𝑀, 𝑁⟩), (2nd ‘⟨𝑃, 𝑄⟩)⟩ ∙ (2nd ‘⟨𝑅, 𝑆⟩))(2nd ‘⟨𝐹, 𝐺⟩)) = (𝐿(⟨𝑁, 𝑄⟩ ∙ 𝑆)𝐺))
57	43, 56	opeq12d 4809	. 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)⟩)
58	30, 57	eqtrd 2778	1 ⊢ (𝜑 → (⟨𝐾, 𝐿⟩(⟨⟨𝑀, 𝑁⟩, ⟨𝑃, 𝑄⟩⟩𝑂⟨𝑅, 𝑆⟩)⟨𝐹, 𝐺⟩) = ⟨(𝐾(⟨𝑀, 𝑃⟩ · 𝑅)𝐹), (𝐿(⟨𝑁, 𝑄⟩ ∙ 𝑆)𝐺)⟩)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2108  ⟨cop 4564   × cxp 5578  ‘cfv 6418  (class class class)co 7255  1^st c1st 7802  2^nd c2nd 7803  Basecbs 16840  Hom chom 16899  compcco 16900   ×_c cxpc 17801

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-2 11966  df-3 11967  df-4 11968  df-5 11969  df-6 11970  df-7 11971  df-8 11972  df-9 11973  df-n0 12164  df-z 12250  df-dec 12367  df-uz 12512  df-fz 13169  df-struct 16776  df-slot 16811  df-ndx 16823  df-base 16841  df-hom 16912  df-cco 16913  df-xpc 17805

This theorem is referenced by:  prfcl  17836  evlfcllem  17855  curf1cl  17862  curf2cl  17865  curfcl  17866  uncfcurf  17873  hofcl  17893