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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 (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 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 (𝜑𝑀𝑋)
10 xpcco2.n . . . 4 (𝜑𝑁𝑌)
11 opelxpi 5072 . . . 4 ((𝑀𝑋𝑁𝑌) → ⟨𝑀, 𝑁⟩ ∈ (𝑋 × 𝑌))
129, 10, 11syl2anc 691 . . 3 (𝜑 → ⟨𝑀, 𝑁⟩ ∈ (𝑋 × 𝑌))
13 xpcco2.p . . . 4 (𝜑𝑃𝑋)
14 xpcco2.q . . . 4 (𝜑𝑄𝑌)
15 opelxpi 5072 . . . 4 ((𝑃𝑋𝑄𝑌) → ⟨𝑃, 𝑄⟩ ∈ (𝑋 × 𝑌))
1613, 14, 15syl2anc 691 . . 3 (𝜑 → ⟨𝑃, 𝑄⟩ ∈ (𝑋 × 𝑌))
17 xpcco2.r . . . 4 (𝜑𝑅𝑋)
18 xpcco2.s . . . 4 (𝜑𝑆𝑌)
19 opelxpi 5072 . . . 4 ((𝑅𝑋𝑆𝑌) → ⟨𝑅, 𝑆⟩ ∈ (𝑋 × 𝑌))
2017, 18, 19syl2anc 691 . . 3 (𝜑 → ⟨𝑅, 𝑆⟩ ∈ (𝑋 × 𝑌))
21 xpcco2.f . . . . 5 (𝜑𝐹 ∈ (𝑀𝐻𝑃))
22 xpcco2.g . . . . 5 (𝜑𝐺 ∈ (𝑁𝐽𝑄))
23 opelxpi 5072 . . . . 5 ((𝐹 ∈ (𝑀𝐻𝑃) ∧ 𝐺 ∈ (𝑁𝐽𝑄)) → ⟨𝐹, 𝐺⟩ ∈ ((𝑀𝐻𝑃) × (𝑁𝐽𝑄)))
2421, 22, 23syl2anc 691 . . . 4 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ ((𝑀𝐻𝑃) × (𝑁𝐽𝑄)))
25 xpcco2.h . . . . 5 𝐻 = (Hom ‘𝐶)
26 xpcco2.j . . . . 5 𝐽 = (Hom ‘𝐷)
271, 2, 3, 25, 26, 9, 10, 13, 14, 5xpchom2 16649 . . . 4 (𝜑 → (⟨𝑀, 𝑁⟩(Hom ‘𝑇)⟨𝑃, 𝑄⟩) = ((𝑀𝐻𝑃) × (𝑁𝐽𝑄)))
2824, 27eleqtrrd 2691 . . 3 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (⟨𝑀, 𝑁⟩(Hom ‘𝑇)⟨𝑃, 𝑄⟩))
29 xpcco2.k . . . . 5 (𝜑𝐾 ∈ (𝑃𝐻𝑅))
30 xpcco2.l . . . . 5 (𝜑𝐿 ∈ (𝑄𝐽𝑆))
31 opelxpi 5072 . . . . 5 ((𝐾 ∈ (𝑃𝐻𝑅) ∧ 𝐿 ∈ (𝑄𝐽𝑆)) → ⟨𝐾, 𝐿⟩ ∈ ((𝑃𝐻𝑅) × (𝑄𝐽𝑆)))
3229, 30, 31syl2anc 691 . . . 4 (𝜑 → ⟨𝐾, 𝐿⟩ ∈ ((𝑃𝐻𝑅) × (𝑄𝐽𝑆)))
331, 2, 3, 25, 26, 13, 14, 17, 18, 5xpchom2 16649 . . . 4 (𝜑 → (⟨𝑃, 𝑄⟩(Hom ‘𝑇)⟨𝑅, 𝑆⟩) = ((𝑃𝐻𝑅) × (𝑄𝐽𝑆)))
3432, 33eleqtrrd 2691 . . 3 (𝜑 → ⟨𝐾, 𝐿⟩ ∈ (⟨𝑃, 𝑄⟩(Hom ‘𝑇)⟨𝑅, 𝑆⟩))
351, 4, 5, 6, 7, 8, 12, 16, 20, 28, 34xpcco 16646 . 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⟩))⟩)
36 op1stg 7071 . . . . . . 7 ((𝑀𝑋𝑁𝑌) → (1st ‘⟨𝑀, 𝑁⟩) = 𝑀)
379, 10, 36syl2anc 691 . . . . . 6 (𝜑 → (1st ‘⟨𝑀, 𝑁⟩) = 𝑀)
38 op1stg 7071 . . . . . . 7 ((𝑃𝑋𝑄𝑌) → (1st ‘⟨𝑃, 𝑄⟩) = 𝑃)
3913, 14, 38syl2anc 691 . . . . . 6 (𝜑 → (1st ‘⟨𝑃, 𝑄⟩) = 𝑃)
4037, 39opeq12d 4348 . . . . 5 (𝜑 → ⟨(1st ‘⟨𝑀, 𝑁⟩), (1st ‘⟨𝑃, 𝑄⟩)⟩ = ⟨𝑀, 𝑃⟩)
41 op1stg 7071 . . . . . 6 ((𝑅𝑋𝑆𝑌) → (1st ‘⟨𝑅, 𝑆⟩) = 𝑅)
4217, 18, 41syl2anc 691 . . . . 5 (𝜑 → (1st ‘⟨𝑅, 𝑆⟩) = 𝑅)
4340, 42oveq12d 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)
4529, 30, 44syl2anc 691 . . . 4 (𝜑 → (1st ‘⟨𝐾, 𝐿⟩) = 𝐾)
46 op1stg 7071 . . . . 5 ((𝐹 ∈ (𝑀𝐻𝑃) ∧ 𝐺 ∈ (𝑁𝐽𝑄)) → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
4721, 22, 46syl2anc 691 . . . 4 (𝜑 → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
4843, 45, 47oveq123d 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 ((𝑀𝑋𝑁𝑌) → (2nd ‘⟨𝑀, 𝑁⟩) = 𝑁)
509, 10, 49syl2anc 691 . . . . . 6 (𝜑 → (2nd ‘⟨𝑀, 𝑁⟩) = 𝑁)
51 op2ndg 7072 . . . . . . 7 ((𝑃𝑋𝑄𝑌) → (2nd ‘⟨𝑃, 𝑄⟩) = 𝑄)
5213, 14, 51syl2anc 691 . . . . . 6 (𝜑 → (2nd ‘⟨𝑃, 𝑄⟩) = 𝑄)
5350, 52opeq12d 4348 . . . . 5 (𝜑 → ⟨(2nd ‘⟨𝑀, 𝑁⟩), (2nd ‘⟨𝑃, 𝑄⟩)⟩ = ⟨𝑁, 𝑄⟩)
54 op2ndg 7072 . . . . . 6 ((𝑅𝑋𝑆𝑌) → (2nd ‘⟨𝑅, 𝑆⟩) = 𝑆)
5517, 18, 54syl2anc 691 . . . . 5 (𝜑 → (2nd ‘⟨𝑅, 𝑆⟩) = 𝑆)
5653, 55oveq12d 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)
5829, 30, 57syl2anc 691 . . . 4 (𝜑 → (2nd ‘⟨𝐾, 𝐿⟩) = 𝐿)
59 op2ndg 7072 . . . . 5 ((𝐹 ∈ (𝑀𝐻𝑃) ∧ 𝐺 ∈ (𝑁𝐽𝑄)) → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
6021, 22, 59syl2anc 691 . . . 4 (𝜑 → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
6156, 58, 60oveq123d 6570 . . 3 (𝜑 → ((2nd ‘⟨𝐾, 𝐿⟩)(⟨(2nd ‘⟨𝑀, 𝑁⟩), (2nd ‘⟨𝑃, 𝑄⟩)⟩ (2nd ‘⟨𝑅, 𝑆⟩))(2nd ‘⟨𝐹, 𝐺⟩)) = (𝐿(⟨𝑁, 𝑄 𝑆)𝐺))
6248, 61opeq12d 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)⟩)
6335, 62eqtrd 2644 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 1475   ∈ wcel 1977  ⟨cop 4131   × cxp 5036  ‘cfv 5804  (class class class)co 6549  1st c1st 7057  2nd 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
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