Theorem xpcco2 16650

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 16641	. . 3 ⊢ (𝑋 × 𝑌) = (Base‘𝑇)
5		eqid 2610	. . 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		opelxpi 5072	. . . 4 ⊢ ((𝑀 ∈ 𝑋 ∧ 𝑁 ∈ 𝑌) → ⟨𝑀, 𝑁⟩ ∈ (𝑋 × 𝑌))
12	9, 10, 11	syl2anc 691	. . 3 ⊢ (𝜑 → ⟨𝑀, 𝑁⟩ ∈ (𝑋 × 𝑌))
13		xpcco2.p	. . . 4 ⊢ (𝜑 → 𝑃 ∈ 𝑋)
14		xpcco2.q	. . . 4 ⊢ (𝜑 → 𝑄 ∈ 𝑌)
15		opelxpi 5072	. . . 4 ⊢ ((𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌) → ⟨𝑃, 𝑄⟩ ∈ (𝑋 × 𝑌))
16	13, 14, 15	syl2anc 691	. . 3 ⊢ (𝜑 → ⟨𝑃, 𝑄⟩ ∈ (𝑋 × 𝑌))
17		xpcco2.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑋)
18		xpcco2.s	. . . 4 ⊢ (𝜑 → 𝑆 ∈ 𝑌)
19		opelxpi 5072	. . . 4 ⊢ ((𝑅 ∈ 𝑋 ∧ 𝑆 ∈ 𝑌) → ⟨𝑅, 𝑆⟩ ∈ (𝑋 × 𝑌))
20	17, 18, 19	syl2anc 691	. . 3 ⊢ (𝜑 → ⟨𝑅, 𝑆⟩ ∈ (𝑋 × 𝑌))
21		xpcco2.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝑀𝐻𝑃))
22		xpcco2.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (𝑁𝐽𝑄))
23		opelxpi 5072	. . . . 5 ⊢ ((𝐹 ∈ (𝑀𝐻𝑃) ∧ 𝐺 ∈ (𝑁𝐽𝑄)) → ⟨𝐹, 𝐺⟩ ∈ ((𝑀𝐻𝑃) × (𝑁𝐽𝑄)))
24	21, 22, 23	syl2anc 691	. . . 4 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ ((𝑀𝐻𝑃) × (𝑁𝐽𝑄)))
25		xpcco2.h	. . . . 5 ⊢ 𝐻 = (Hom ‘𝐶)
26		xpcco2.j	. . . . 5 ⊢ 𝐽 = (Hom ‘𝐷)
27	1, 2, 3, 25, 26, 9, 10, 13, 14, 5	xpchom2 16649	. . . 4 ⊢ (𝜑 → (⟨𝑀, 𝑁⟩(Hom ‘𝑇)⟨𝑃, 𝑄⟩) = ((𝑀𝐻𝑃) × (𝑁𝐽𝑄)))
28	24, 27	eleqtrrd 2691	. . 3 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (⟨𝑀, 𝑁⟩(Hom ‘𝑇)⟨𝑃, 𝑄⟩))
29		xpcco2.k	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ (𝑃𝐻𝑅))
30		xpcco2.l	. . . . 5 ⊢ (𝜑 → 𝐿 ∈ (𝑄𝐽𝑆))
31		opelxpi 5072	. . . . 5 ⊢ ((𝐾 ∈ (𝑃𝐻𝑅) ∧ 𝐿 ∈ (𝑄𝐽𝑆)) → ⟨𝐾, 𝐿⟩ ∈ ((𝑃𝐻𝑅) × (𝑄𝐽𝑆)))
32	29, 30, 31	syl2anc 691	. . . 4 ⊢ (𝜑 → ⟨𝐾, 𝐿⟩ ∈ ((𝑃𝐻𝑅) × (𝑄𝐽𝑆)))
33	1, 2, 3, 25, 26, 13, 14, 17, 18, 5	xpchom2 16649	. . . 4 ⊢ (𝜑 → (⟨𝑃, 𝑄⟩(Hom ‘𝑇)⟨𝑅, 𝑆⟩) = ((𝑃𝐻𝑅) × (𝑄𝐽𝑆)))
34	32, 33	eleqtrrd 2691	. . 3 ⊢ (𝜑 → ⟨𝐾, 𝐿⟩ ∈ (⟨𝑃, 𝑄⟩(Hom ‘𝑇)⟨𝑅, 𝑆⟩))
35	1, 4, 5, 6, 7, 8, 12, 16, 20, 28, 34	xpcco 16646	. 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⟩))⟩)
36		op1stg 7071	. . . . . . 7 ⊢ ((𝑀 ∈ 𝑋 ∧ 𝑁 ∈ 𝑌) → (1st ‘⟨𝑀, 𝑁⟩) = 𝑀)
37	9, 10, 36	syl2anc 691	. . . . . 6 ⊢ (𝜑 → (1st ‘⟨𝑀, 𝑁⟩) = 𝑀)
38		op1stg 7071	. . . . . . 7 ⊢ ((𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌) → (1st ‘⟨𝑃, 𝑄⟩) = 𝑃)
39	13, 14, 38	syl2anc 691	. . . . . 6 ⊢ (𝜑 → (1st ‘⟨𝑃, 𝑄⟩) = 𝑃)
40	37, 39	opeq12d 4348	. . . . 5 ⊢ (𝜑 → ⟨(1st ‘⟨𝑀, 𝑁⟩), (1st ‘⟨𝑃, 𝑄⟩)⟩ = ⟨𝑀, 𝑃⟩)
41		op1stg 7071	. . . . . 6 ⊢ ((𝑅 ∈ 𝑋 ∧ 𝑆 ∈ 𝑌) → (1st ‘⟨𝑅, 𝑆⟩) = 𝑅)
42	17, 18, 41	syl2anc 691	. . . . 5 ⊢ (𝜑 → (1st ‘⟨𝑅, 𝑆⟩) = 𝑅)
43	40, 42	oveq12d 6567	. . . 4 ⊢ (𝜑 → (⟨(1st ‘⟨𝑀, 𝑁⟩), (1st ‘⟨𝑃, 𝑄⟩)⟩ · (1st ‘⟨𝑅, 𝑆⟩)) = (⟨𝑀, 𝑃⟩ · 𝑅))
44		op1stg 7071	. . . . 5 ⊢ ((𝐾 ∈ (𝑃𝐻𝑅) ∧ 𝐿 ∈ (𝑄𝐽𝑆)) → (1st ‘⟨𝐾, 𝐿⟩) = 𝐾)
45	29, 30, 44	syl2anc 691	. . . 4 ⊢ (𝜑 → (1st ‘⟨𝐾, 𝐿⟩) = 𝐾)
46		op1stg 7071	. . . . 5 ⊢ ((𝐹 ∈ (𝑀𝐻𝑃) ∧ 𝐺 ∈ (𝑁𝐽𝑄)) → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
47	21, 22, 46	syl2anc 691	. . . 4 ⊢ (𝜑 → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
48	43, 45, 47	oveq123d 6570	. . 3 ⊢ (𝜑 → ((1st ‘⟨𝐾, 𝐿⟩)(⟨(1st ‘⟨𝑀, 𝑁⟩), (1st ‘⟨𝑃, 𝑄⟩)⟩ · (1st ‘⟨𝑅, 𝑆⟩))(1st ‘⟨𝐹, 𝐺⟩)) = (𝐾(⟨𝑀, 𝑃⟩ · 𝑅)𝐹))
49		op2ndg 7072	. . . . . . 7 ⊢ ((𝑀 ∈ 𝑋 ∧ 𝑁 ∈ 𝑌) → (2nd ‘⟨𝑀, 𝑁⟩) = 𝑁)
50	9, 10, 49	syl2anc 691	. . . . . 6 ⊢ (𝜑 → (2nd ‘⟨𝑀, 𝑁⟩) = 𝑁)
51		op2ndg 7072	. . . . . . 7 ⊢ ((𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌) → (2nd ‘⟨𝑃, 𝑄⟩) = 𝑄)
52	13, 14, 51	syl2anc 691	. . . . . 6 ⊢ (𝜑 → (2nd ‘⟨𝑃, 𝑄⟩) = 𝑄)
53	50, 52	opeq12d 4348	. . . . 5 ⊢ (𝜑 → ⟨(2nd ‘⟨𝑀, 𝑁⟩), (2nd ‘⟨𝑃, 𝑄⟩)⟩ = ⟨𝑁, 𝑄⟩)
54		op2ndg 7072	. . . . . 6 ⊢ ((𝑅 ∈ 𝑋 ∧ 𝑆 ∈ 𝑌) → (2nd ‘⟨𝑅, 𝑆⟩) = 𝑆)
55	17, 18, 54	syl2anc 691	. . . . 5 ⊢ (𝜑 → (2nd ‘⟨𝑅, 𝑆⟩) = 𝑆)
56	53, 55	oveq12d 6567	. . . 4 ⊢ (𝜑 → (⟨(2nd ‘⟨𝑀, 𝑁⟩), (2nd ‘⟨𝑃, 𝑄⟩)⟩ ∙ (2nd ‘⟨𝑅, 𝑆⟩)) = (⟨𝑁, 𝑄⟩ ∙ 𝑆))
57		op2ndg 7072	. . . . 5 ⊢ ((𝐾 ∈ (𝑃𝐻𝑅) ∧ 𝐿 ∈ (𝑄𝐽𝑆)) → (2nd ‘⟨𝐾, 𝐿⟩) = 𝐿)
58	29, 30, 57	syl2anc 691	. . . 4 ⊢ (𝜑 → (2nd ‘⟨𝐾, 𝐿⟩) = 𝐿)
59		op2ndg 7072	. . . . 5 ⊢ ((𝐹 ∈ (𝑀𝐻𝑃) ∧ 𝐺 ∈ (𝑁𝐽𝑄)) → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
60	21, 22, 59	syl2anc 691	. . . 4 ⊢ (𝜑 → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
61	56, 58, 60	oveq123d 6570	. . 3 ⊢ (𝜑 → ((2nd ‘⟨𝐾, 𝐿⟩)(⟨(2nd ‘⟨𝑀, 𝑁⟩), (2nd ‘⟨𝑃, 𝑄⟩)⟩ ∙ (2nd ‘⟨𝑅, 𝑆⟩))(2nd ‘⟨𝐹, 𝐺⟩)) = (𝐿(⟨𝑁, 𝑄⟩ ∙ 𝑆)𝐺))
62	48, 61	opeq12d 4348	. 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)⟩)
63	35, 62	eqtrd 2644	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 1475   ∈ wcel 1977  ⟨cop 4131   × cxp 5036  ‘cfv 5804  (class class class)co 6549  1^st c1st 7057  2^nd c2nd 7058  Basecbs 15695  Hom chom 15779  compcco 15780   ×_c cxpc 16631

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-fz 12198  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-hom 15793  df-cco 15794  df-xpc 16635

This theorem is referenced by:  prfcl  16666  evlfcllem  16684  curf1cl  16691  curf2cl  16694  curfcl  16695  uncfcurf  16702  hofcl  16722