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Theorem xpcco2 16874
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 16865 . . 3 (𝑋 × 𝑌) = (Base‘𝑇)
5 eqid 2651 . . 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 5182 . . . 4 ((𝑀𝑋𝑁𝑌) → ⟨𝑀, 𝑁⟩ ∈ (𝑋 × 𝑌))
129, 10, 11syl2anc 694 . . 3 (𝜑 → ⟨𝑀, 𝑁⟩ ∈ (𝑋 × 𝑌))
13 xpcco2.p . . . 4 (𝜑𝑃𝑋)
14 xpcco2.q . . . 4 (𝜑𝑄𝑌)
15 opelxpi 5182 . . . 4 ((𝑃𝑋𝑄𝑌) → ⟨𝑃, 𝑄⟩ ∈ (𝑋 × 𝑌))
1613, 14, 15syl2anc 694 . . 3 (𝜑 → ⟨𝑃, 𝑄⟩ ∈ (𝑋 × 𝑌))
17 xpcco2.r . . . 4 (𝜑𝑅𝑋)
18 xpcco2.s . . . 4 (𝜑𝑆𝑌)
19 opelxpi 5182 . . . 4 ((𝑅𝑋𝑆𝑌) → ⟨𝑅, 𝑆⟩ ∈ (𝑋 × 𝑌))
2017, 18, 19syl2anc 694 . . 3 (𝜑 → ⟨𝑅, 𝑆⟩ ∈ (𝑋 × 𝑌))
21 xpcco2.f . . . . 5 (𝜑𝐹 ∈ (𝑀𝐻𝑃))
22 xpcco2.g . . . . 5 (𝜑𝐺 ∈ (𝑁𝐽𝑄))
23 opelxpi 5182 . . . . 5 ((𝐹 ∈ (𝑀𝐻𝑃) ∧ 𝐺 ∈ (𝑁𝐽𝑄)) → ⟨𝐹, 𝐺⟩ ∈ ((𝑀𝐻𝑃) × (𝑁𝐽𝑄)))
2421, 22, 23syl2anc 694 . . . 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 16873 . . . 4 (𝜑 → (⟨𝑀, 𝑁⟩(Hom ‘𝑇)⟨𝑃, 𝑄⟩) = ((𝑀𝐻𝑃) × (𝑁𝐽𝑄)))
2824, 27eleqtrrd 2733 . . 3 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (⟨𝑀, 𝑁⟩(Hom ‘𝑇)⟨𝑃, 𝑄⟩))
29 xpcco2.k . . . . 5 (𝜑𝐾 ∈ (𝑃𝐻𝑅))
30 xpcco2.l . . . . 5 (𝜑𝐿 ∈ (𝑄𝐽𝑆))
31 opelxpi 5182 . . . . 5 ((𝐾 ∈ (𝑃𝐻𝑅) ∧ 𝐿 ∈ (𝑄𝐽𝑆)) → ⟨𝐾, 𝐿⟩ ∈ ((𝑃𝐻𝑅) × (𝑄𝐽𝑆)))
3229, 30, 31syl2anc 694 . . . 4 (𝜑 → ⟨𝐾, 𝐿⟩ ∈ ((𝑃𝐻𝑅) × (𝑄𝐽𝑆)))
331, 2, 3, 25, 26, 13, 14, 17, 18, 5xpchom2 16873 . . . 4 (𝜑 → (⟨𝑃, 𝑄⟩(Hom ‘𝑇)⟨𝑅, 𝑆⟩) = ((𝑃𝐻𝑅) × (𝑄𝐽𝑆)))
3432, 33eleqtrrd 2733 . . 3 (𝜑 → ⟨𝐾, 𝐿⟩ ∈ (⟨𝑃, 𝑄⟩(Hom ‘𝑇)⟨𝑅, 𝑆⟩))
351, 4, 5, 6, 7, 8, 12, 16, 20, 28, 34xpcco 16870 . 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 7222 . . . . . . 7 ((𝑀𝑋𝑁𝑌) → (1st ‘⟨𝑀, 𝑁⟩) = 𝑀)
379, 10, 36syl2anc 694 . . . . . 6 (𝜑 → (1st ‘⟨𝑀, 𝑁⟩) = 𝑀)
38 op1stg 7222 . . . . . . 7 ((𝑃𝑋𝑄𝑌) → (1st ‘⟨𝑃, 𝑄⟩) = 𝑃)
3913, 14, 38syl2anc 694 . . . . . 6 (𝜑 → (1st ‘⟨𝑃, 𝑄⟩) = 𝑃)
4037, 39opeq12d 4441 . . . . 5 (𝜑 → ⟨(1st ‘⟨𝑀, 𝑁⟩), (1st ‘⟨𝑃, 𝑄⟩)⟩ = ⟨𝑀, 𝑃⟩)
41 op1stg 7222 . . . . . 6 ((𝑅𝑋𝑆𝑌) → (1st ‘⟨𝑅, 𝑆⟩) = 𝑅)
4217, 18, 41syl2anc 694 . . . . 5 (𝜑 → (1st ‘⟨𝑅, 𝑆⟩) = 𝑅)
4340, 42oveq12d 6708 . . . 4 (𝜑 → (⟨(1st ‘⟨𝑀, 𝑁⟩), (1st ‘⟨𝑃, 𝑄⟩)⟩ · (1st ‘⟨𝑅, 𝑆⟩)) = (⟨𝑀, 𝑃· 𝑅))
44 op1stg 7222 . . . . 5 ((𝐾 ∈ (𝑃𝐻𝑅) ∧ 𝐿 ∈ (𝑄𝐽𝑆)) → (1st ‘⟨𝐾, 𝐿⟩) = 𝐾)
4529, 30, 44syl2anc 694 . . . 4 (𝜑 → (1st ‘⟨𝐾, 𝐿⟩) = 𝐾)
46 op1stg 7222 . . . . 5 ((𝐹 ∈ (𝑀𝐻𝑃) ∧ 𝐺 ∈ (𝑁𝐽𝑄)) → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
4721, 22, 46syl2anc 694 . . . 4 (𝜑 → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
4843, 45, 47oveq123d 6711 . . 3 (𝜑 → ((1st ‘⟨𝐾, 𝐿⟩)(⟨(1st ‘⟨𝑀, 𝑁⟩), (1st ‘⟨𝑃, 𝑄⟩)⟩ · (1st ‘⟨𝑅, 𝑆⟩))(1st ‘⟨𝐹, 𝐺⟩)) = (𝐾(⟨𝑀, 𝑃· 𝑅)𝐹))
49 op2ndg 7223 . . . . . . 7 ((𝑀𝑋𝑁𝑌) → (2nd ‘⟨𝑀, 𝑁⟩) = 𝑁)
509, 10, 49syl2anc 694 . . . . . 6 (𝜑 → (2nd ‘⟨𝑀, 𝑁⟩) = 𝑁)
51 op2ndg 7223 . . . . . . 7 ((𝑃𝑋𝑄𝑌) → (2nd ‘⟨𝑃, 𝑄⟩) = 𝑄)
5213, 14, 51syl2anc 694 . . . . . 6 (𝜑 → (2nd ‘⟨𝑃, 𝑄⟩) = 𝑄)
5350, 52opeq12d 4441 . . . . 5 (𝜑 → ⟨(2nd ‘⟨𝑀, 𝑁⟩), (2nd ‘⟨𝑃, 𝑄⟩)⟩ = ⟨𝑁, 𝑄⟩)
54 op2ndg 7223 . . . . . 6 ((𝑅𝑋𝑆𝑌) → (2nd ‘⟨𝑅, 𝑆⟩) = 𝑆)
5517, 18, 54syl2anc 694 . . . . 5 (𝜑 → (2nd ‘⟨𝑅, 𝑆⟩) = 𝑆)
5653, 55oveq12d 6708 . . . 4 (𝜑 → (⟨(2nd ‘⟨𝑀, 𝑁⟩), (2nd ‘⟨𝑃, 𝑄⟩)⟩ (2nd ‘⟨𝑅, 𝑆⟩)) = (⟨𝑁, 𝑄 𝑆))
57 op2ndg 7223 . . . . 5 ((𝐾 ∈ (𝑃𝐻𝑅) ∧ 𝐿 ∈ (𝑄𝐽𝑆)) → (2nd ‘⟨𝐾, 𝐿⟩) = 𝐿)
5829, 30, 57syl2anc 694 . . . 4 (𝜑 → (2nd ‘⟨𝐾, 𝐿⟩) = 𝐿)
59 op2ndg 7223 . . . . 5 ((𝐹 ∈ (𝑀𝐻𝑃) ∧ 𝐺 ∈ (𝑁𝐽𝑄)) → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
6021, 22, 59syl2anc 694 . . . 4 (𝜑 → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
6156, 58, 60oveq123d 6711 . . 3 (𝜑 → ((2nd ‘⟨𝐾, 𝐿⟩)(⟨(2nd ‘⟨𝑀, 𝑁⟩), (2nd ‘⟨𝑃, 𝑄⟩)⟩ (2nd ‘⟨𝑅, 𝑆⟩))(2nd ‘⟨𝐹, 𝐺⟩)) = (𝐿(⟨𝑁, 𝑄 𝑆)𝐺))
6248, 61opeq12d 4441 . 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 2685 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 1523  wcel 2030  cop 4216   × cxp 5141  cfv 5926  (class class class)co 6690  1st c1st 7208  2nd c2nd 7209  Basecbs 15904  Hom chom 15999  compcco 16000   ×c cxpc 16855
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-z 11416  df-dec 11532  df-uz 11726  df-fz 12365  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-hom 16013  df-cco 16014  df-xpc 16859
This theorem is referenced by:  prfcl  16890  evlfcllem  16908  curf1cl  16915  curf2cl  16918  curfcl  16919  uncfcurf  16926  hofcl  16946
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