Theorem xpcco2 18268

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 18259	. . 3 ⊢ (𝑋 × 𝑌) = (Base‘𝑇)
5		eqid 2766	. . 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 5705	. . 3 ⊢ (𝜑 → ⟨𝑀, 𝑁⟩ ∈ (𝑋 × 𝑌))
12		xpcco2.p	. . . 4 ⊢ (𝜑 → 𝑃 ∈ 𝑋)
13		xpcco2.q	. . . 4 ⊢ (𝜑 → 𝑄 ∈ 𝑌)
14	12, 13	opelxpd 5705	. . 3 ⊢ (𝜑 → ⟨𝑃, 𝑄⟩ ∈ (𝑋 × 𝑌))
15		xpcco2.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑋)
16		xpcco2.s	. . . 4 ⊢ (𝜑 → 𝑆 ∈ 𝑌)
17	15, 16	opelxpd 5705	. . 3 ⊢ (𝜑 → ⟨𝑅, 𝑆⟩ ∈ (𝑋 × 𝑌))
18		xpcco2.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝑀𝐻𝑃))
19		xpcco2.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (𝑁𝐽𝑄))
20	18, 19	opelxpd 5705	. . . 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 18267	. . . 4 ⊢ (𝜑 → (⟨𝑀, 𝑁⟩(Hom ‘𝑇)⟨𝑃, 𝑄⟩) = ((𝑀𝐻𝑃) × (𝑁𝐽𝑄)))
24	20, 23	eleqtrrd 2869	. . 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 5705	. . . 4 ⊢ (𝜑 → ⟨𝐾, 𝐿⟩ ∈ ((𝑃𝐻𝑅) × (𝑄𝐽𝑆)))
28	1, 2, 3, 21, 22, 12, 13, 15, 16, 5	xpchom2 18267	. . . 4 ⊢ (𝜑 → (⟨𝑃, 𝑄⟩(Hom ‘𝑇)⟨𝑅, 𝑆⟩) = ((𝑃𝐻𝑅) × (𝑄𝐽𝑆)))
29	27, 28	eleqtrrd 2869	. . 3 ⊢ (𝜑 → ⟨𝐾, 𝐿⟩ ∈ (⟨𝑃, 𝑄⟩(Hom ‘𝑇)⟨𝑅, 𝑆⟩))
30	1, 4, 5, 6, 7, 8, 11, 14, 17, 24, 29	xpcco 18264	. 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 8007	. . . . . . 7 ⊢ ((𝑀 ∈ 𝑋 ∧ 𝑁 ∈ 𝑌) → (1st ‘⟨𝑀, 𝑁⟩) = 𝑀)
32	9, 10, 31	syl2anc 596	. . . . . 6 ⊢ (𝜑 → (1st ‘⟨𝑀, 𝑁⟩) = 𝑀)
33		op1stg 8007	. . . . . . 7 ⊢ ((𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌) → (1st ‘⟨𝑃, 𝑄⟩) = 𝑃)
34	12, 13, 33	syl2anc 596	. . . . . 6 ⊢ (𝜑 → (1st ‘⟨𝑃, 𝑄⟩) = 𝑃)
35	32, 34	opeq12d 4851	. . . . 5 ⊢ (𝜑 → ⟨(1st ‘⟨𝑀, 𝑁⟩), (1st ‘⟨𝑃, 𝑄⟩)⟩ = ⟨𝑀, 𝑃⟩)
36		op1stg 8007	. . . . . 6 ⊢ ((𝑅 ∈ 𝑋 ∧ 𝑆 ∈ 𝑌) → (1st ‘⟨𝑅, 𝑆⟩) = 𝑅)
37	15, 16, 36	syl2anc 596	. . . . 5 ⊢ (𝜑 → (1st ‘⟨𝑅, 𝑆⟩) = 𝑅)
38	35, 37	oveq12d 7441	. . . 4 ⊢ (𝜑 → (⟨(1st ‘⟨𝑀, 𝑁⟩), (1st ‘⟨𝑃, 𝑄⟩)⟩ · (1st ‘⟨𝑅, 𝑆⟩)) = (⟨𝑀, 𝑃⟩ · 𝑅))
39		op1stg 8007	. . . . 5 ⊢ ((𝐾 ∈ (𝑃𝐻𝑅) ∧ 𝐿 ∈ (𝑄𝐽𝑆)) → (1st ‘⟨𝐾, 𝐿⟩) = 𝐾)
40	25, 26, 39	syl2anc 596	. . . 4 ⊢ (𝜑 → (1st ‘⟨𝐾, 𝐿⟩) = 𝐾)
41		op1stg 8007	. . . . 5 ⊢ ((𝐹 ∈ (𝑀𝐻𝑃) ∧ 𝐺 ∈ (𝑁𝐽𝑄)) → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
42	18, 19, 41	syl2anc 596	. . . 4 ⊢ (𝜑 → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
43	38, 40, 42	oveq123d 7444	. . 3 ⊢ (𝜑 → ((1st ‘⟨𝐾, 𝐿⟩)(⟨(1st ‘⟨𝑀, 𝑁⟩), (1st ‘⟨𝑃, 𝑄⟩)⟩ · (1st ‘⟨𝑅, 𝑆⟩))(1st ‘⟨𝐹, 𝐺⟩)) = (𝐾(⟨𝑀, 𝑃⟩ · 𝑅)𝐹))
44		op2ndg 8008	. . . . . . 7 ⊢ ((𝑀 ∈ 𝑋 ∧ 𝑁 ∈ 𝑌) → (2nd ‘⟨𝑀, 𝑁⟩) = 𝑁)
45	9, 10, 44	syl2anc 596	. . . . . 6 ⊢ (𝜑 → (2nd ‘⟨𝑀, 𝑁⟩) = 𝑁)
46		op2ndg 8008	. . . . . . 7 ⊢ ((𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌) → (2nd ‘⟨𝑃, 𝑄⟩) = 𝑄)
47	12, 13, 46	syl2anc 596	. . . . . 6 ⊢ (𝜑 → (2nd ‘⟨𝑃, 𝑄⟩) = 𝑄)
48	45, 47	opeq12d 4851	. . . . 5 ⊢ (𝜑 → ⟨(2nd ‘⟨𝑀, 𝑁⟩), (2nd ‘⟨𝑃, 𝑄⟩)⟩ = ⟨𝑁, 𝑄⟩)
49		op2ndg 8008	. . . . . 6 ⊢ ((𝑅 ∈ 𝑋 ∧ 𝑆 ∈ 𝑌) → (2nd ‘⟨𝑅, 𝑆⟩) = 𝑆)
50	15, 16, 49	syl2anc 596	. . . . 5 ⊢ (𝜑 → (2nd ‘⟨𝑅, 𝑆⟩) = 𝑆)
51	48, 50	oveq12d 7441	. . . 4 ⊢ (𝜑 → (⟨(2nd ‘⟨𝑀, 𝑁⟩), (2nd ‘⟨𝑃, 𝑄⟩)⟩ ∙ (2nd ‘⟨𝑅, 𝑆⟩)) = (⟨𝑁, 𝑄⟩ ∙ 𝑆))
52		op2ndg 8008	. . . . 5 ⊢ ((𝐾 ∈ (𝑃𝐻𝑅) ∧ 𝐿 ∈ (𝑄𝐽𝑆)) → (2nd ‘⟨𝐾, 𝐿⟩) = 𝐿)
53	25, 26, 52	syl2anc 596	. . . 4 ⊢ (𝜑 → (2nd ‘⟨𝐾, 𝐿⟩) = 𝐿)
54		op2ndg 8008	. . . . 5 ⊢ ((𝐹 ∈ (𝑀𝐻𝑃) ∧ 𝐺 ∈ (𝑁𝐽𝑄)) → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
55	18, 19, 54	syl2anc 596	. . . 4 ⊢ (𝜑 → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
56	51, 53, 55	oveq123d 7444	. . 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 4851	. 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 2801	1 ⊢ (𝜑 → (⟨𝐾, 𝐿⟩(⟨⟨𝑀, 𝑁⟩, ⟨𝑃, 𝑄⟩⟩𝑂⟨𝑅, 𝑆⟩)⟨𝐹, 𝐺⟩) = ⟨(𝐾(⟨𝑀, 𝑃⟩ · 𝑅)𝐹), (𝐿(⟨𝑁, 𝑄⟩ ∙ 𝑆)𝐺)⟩)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2146  ⟨cop 4600   × cxp 5664  ‘cfv 6543  (class class class)co 7423  1^st c1st 7993  2^nd c2nd 7994  Basecbs 17294  Hom chom 17346  compcco 17347   ×_c cxpc 18249

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745  ax-cnex 11174  ax-resscn 11175  ax-1cn 11176  ax-icn 11177  ax-addcl 11178  ax-addrcl 11179  ax-mulcl 11180  ax-mulrcl 11181  ax-mulcom 11182  ax-addass 11183  ax-mulass 11184  ax-distr 11185  ax-i2m1 11186  ax-1ne0 11187  ax-1rid 11188  ax-rnegex 11189  ax-rrecex 11190  ax-cnre 11191  ax-pre-lttri 11192  ax-pre-lttrn 11193  ax-pre-ltadd 11194  ax-pre-mulgt0 11195

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-nel 3068  df-ral 3083  df-rex 3093  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-tp 4599  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-tr 5224  df-id 5561  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-we 5621  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6309  df-ord 6370  df-on 6371  df-lim 6372  df-suc 6373  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7380  df-ov 7426  df-oprab 7427  df-mpo 7428  df-om 7872  df-1st 7995  df-2nd 7996  df-frecs 8287  df-wrecs 8318  df-recs 8367  df-rdg 8406  df-1o 8462  df-er 8703  df-en 8953  df-dom 8954  df-sdom 8955  df-fin 8956  df-pnf 11263  df-mnf 11264  df-xr 11265  df-ltxr 11266  df-le 11267  df-sub 11461  df-neg 11462  df-nn 12252  df-2 12321  df-3 12322  df-4 12323  df-5 12324  df-6 12325  df-7 12326  df-8 12327  df-9 12328  df-n0 12523  df-z 12610  df-dec 12730  df-uz 12881  df-fz 13554  df-struct 17232  df-slot 17267  df-ndx 17279  df-base 17295  df-hom 17359  df-cco 17360  df-xpc 18253

This theorem is used by:  prfcl  18284  evlfcllem  18302  curf1cl  18309  curf2cl  18312  curfcl  18313  uncfcurf  18320  hofcl  18340  xpcfucco2  50075