Theorem xpcco2 18232

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 18223	. . 3 ⊢ (𝑋 × 𝑌) = (Base‘𝑇)
5		eqid 2737	. . 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 5724	. . 3 ⊢ (𝜑 → ⟨𝑀, 𝑁⟩ ∈ (𝑋 × 𝑌))
12		xpcco2.p	. . . 4 ⊢ (𝜑 → 𝑃 ∈ 𝑋)
13		xpcco2.q	. . . 4 ⊢ (𝜑 → 𝑄 ∈ 𝑌)
14	12, 13	opelxpd 5724	. . 3 ⊢ (𝜑 → ⟨𝑃, 𝑄⟩ ∈ (𝑋 × 𝑌))
15		xpcco2.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑋)
16		xpcco2.s	. . . 4 ⊢ (𝜑 → 𝑆 ∈ 𝑌)
17	15, 16	opelxpd 5724	. . 3 ⊢ (𝜑 → ⟨𝑅, 𝑆⟩ ∈ (𝑋 × 𝑌))
18		xpcco2.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝑀𝐻𝑃))
19		xpcco2.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (𝑁𝐽𝑄))
20	18, 19	opelxpd 5724	. . . 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 18231	. . . 4 ⊢ (𝜑 → (⟨𝑀, 𝑁⟩(Hom ‘𝑇)⟨𝑃, 𝑄⟩) = ((𝑀𝐻𝑃) × (𝑁𝐽𝑄)))
24	20, 23	eleqtrrd 2844	. . 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 5724	. . . 4 ⊢ (𝜑 → ⟨𝐾, 𝐿⟩ ∈ ((𝑃𝐻𝑅) × (𝑄𝐽𝑆)))
28	1, 2, 3, 21, 22, 12, 13, 15, 16, 5	xpchom2 18231	. . . 4 ⊢ (𝜑 → (⟨𝑃, 𝑄⟩(Hom ‘𝑇)⟨𝑅, 𝑆⟩) = ((𝑃𝐻𝑅) × (𝑄𝐽𝑆)))
29	27, 28	eleqtrrd 2844	. . 3 ⊢ (𝜑 → ⟨𝐾, 𝐿⟩ ∈ (⟨𝑃, 𝑄⟩(Hom ‘𝑇)⟨𝑅, 𝑆⟩))
30	1, 4, 5, 6, 7, 8, 11, 14, 17, 24, 29	xpcco 18228	. 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 8026	. . . . . . 7 ⊢ ((𝑀 ∈ 𝑋 ∧ 𝑁 ∈ 𝑌) → (1st ‘⟨𝑀, 𝑁⟩) = 𝑀)
32	9, 10, 31	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (1st ‘⟨𝑀, 𝑁⟩) = 𝑀)
33		op1stg 8026	. . . . . . 7 ⊢ ((𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌) → (1st ‘⟨𝑃, 𝑄⟩) = 𝑃)
34	12, 13, 33	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (1st ‘⟨𝑃, 𝑄⟩) = 𝑃)
35	32, 34	opeq12d 4881	. . . . 5 ⊢ (𝜑 → ⟨(1st ‘⟨𝑀, 𝑁⟩), (1st ‘⟨𝑃, 𝑄⟩)⟩ = ⟨𝑀, 𝑃⟩)
36		op1stg 8026	. . . . . 6 ⊢ ((𝑅 ∈ 𝑋 ∧ 𝑆 ∈ 𝑌) → (1st ‘⟨𝑅, 𝑆⟩) = 𝑅)
37	15, 16, 36	syl2anc 584	. . . . 5 ⊢ (𝜑 → (1st ‘⟨𝑅, 𝑆⟩) = 𝑅)
38	35, 37	oveq12d 7449	. . . 4 ⊢ (𝜑 → (⟨(1st ‘⟨𝑀, 𝑁⟩), (1st ‘⟨𝑃, 𝑄⟩)⟩ · (1st ‘⟨𝑅, 𝑆⟩)) = (⟨𝑀, 𝑃⟩ · 𝑅))
39		op1stg 8026	. . . . 5 ⊢ ((𝐾 ∈ (𝑃𝐻𝑅) ∧ 𝐿 ∈ (𝑄𝐽𝑆)) → (1st ‘⟨𝐾, 𝐿⟩) = 𝐾)
40	25, 26, 39	syl2anc 584	. . . 4 ⊢ (𝜑 → (1st ‘⟨𝐾, 𝐿⟩) = 𝐾)
41		op1stg 8026	. . . . 5 ⊢ ((𝐹 ∈ (𝑀𝐻𝑃) ∧ 𝐺 ∈ (𝑁𝐽𝑄)) → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
42	18, 19, 41	syl2anc 584	. . . 4 ⊢ (𝜑 → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
43	38, 40, 42	oveq123d 7452	. . 3 ⊢ (𝜑 → ((1st ‘⟨𝐾, 𝐿⟩)(⟨(1st ‘⟨𝑀, 𝑁⟩), (1st ‘⟨𝑃, 𝑄⟩)⟩ · (1st ‘⟨𝑅, 𝑆⟩))(1st ‘⟨𝐹, 𝐺⟩)) = (𝐾(⟨𝑀, 𝑃⟩ · 𝑅)𝐹))
44		op2ndg 8027	. . . . . . 7 ⊢ ((𝑀 ∈ 𝑋 ∧ 𝑁 ∈ 𝑌) → (2nd ‘⟨𝑀, 𝑁⟩) = 𝑁)
45	9, 10, 44	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (2nd ‘⟨𝑀, 𝑁⟩) = 𝑁)
46		op2ndg 8027	. . . . . . 7 ⊢ ((𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌) → (2nd ‘⟨𝑃, 𝑄⟩) = 𝑄)
47	12, 13, 46	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (2nd ‘⟨𝑃, 𝑄⟩) = 𝑄)
48	45, 47	opeq12d 4881	. . . . 5 ⊢ (𝜑 → ⟨(2nd ‘⟨𝑀, 𝑁⟩), (2nd ‘⟨𝑃, 𝑄⟩)⟩ = ⟨𝑁, 𝑄⟩)
49		op2ndg 8027	. . . . . 6 ⊢ ((𝑅 ∈ 𝑋 ∧ 𝑆 ∈ 𝑌) → (2nd ‘⟨𝑅, 𝑆⟩) = 𝑆)
50	15, 16, 49	syl2anc 584	. . . . 5 ⊢ (𝜑 → (2nd ‘⟨𝑅, 𝑆⟩) = 𝑆)
51	48, 50	oveq12d 7449	. . . 4 ⊢ (𝜑 → (⟨(2nd ‘⟨𝑀, 𝑁⟩), (2nd ‘⟨𝑃, 𝑄⟩)⟩ ∙ (2nd ‘⟨𝑅, 𝑆⟩)) = (⟨𝑁, 𝑄⟩ ∙ 𝑆))
52		op2ndg 8027	. . . . 5 ⊢ ((𝐾 ∈ (𝑃𝐻𝑅) ∧ 𝐿 ∈ (𝑄𝐽𝑆)) → (2nd ‘⟨𝐾, 𝐿⟩) = 𝐿)
53	25, 26, 52	syl2anc 584	. . . 4 ⊢ (𝜑 → (2nd ‘⟨𝐾, 𝐿⟩) = 𝐿)
54		op2ndg 8027	. . . . 5 ⊢ ((𝐹 ∈ (𝑀𝐻𝑃) ∧ 𝐺 ∈ (𝑁𝐽𝑄)) → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
55	18, 19, 54	syl2anc 584	. . . 4 ⊢ (𝜑 → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
56	51, 53, 55	oveq123d 7452	. . 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 4881	. 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 2777	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 1540   ∈ wcel 2108  ⟨cop 4632   × cxp 5683  ‘cfv 6561  (class class class)co 7431  1^st c1st 8012  2^nd c2nd 8013  Basecbs 17247  Hom chom 17308  compcco 17309   ×_c cxpc 18213

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-fz 13548  df-struct 17184  df-slot 17219  df-ndx 17231  df-base 17248  df-hom 17321  df-cco 17322  df-xpc 18217

This theorem is referenced by:  prfcl  18248  evlfcllem  18266  curf1cl  18273  curf2cl  18276  curfcl  18277  uncfcurf  18284  hofcl  18304  xpcfucco2  48962