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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 (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 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 (𝜑𝑀𝑋)
10 xpcco2.n . . . 4 (𝜑𝑁𝑌)
119, 10opelxpd 5724 . . 3 (𝜑 → ⟨𝑀, 𝑁⟩ ∈ (𝑋 × 𝑌))
12 xpcco2.p . . . 4 (𝜑𝑃𝑋)
13 xpcco2.q . . . 4 (𝜑𝑄𝑌)
1412, 13opelxpd 5724 . . 3 (𝜑 → ⟨𝑃, 𝑄⟩ ∈ (𝑋 × 𝑌))
15 xpcco2.r . . . 4 (𝜑𝑅𝑋)
16 xpcco2.s . . . 4 (𝜑𝑆𝑌)
1715, 16opelxpd 5724 . . 3 (𝜑 → ⟨𝑅, 𝑆⟩ ∈ (𝑋 × 𝑌))
18 xpcco2.f . . . . 5 (𝜑𝐹 ∈ (𝑀𝐻𝑃))
19 xpcco2.g . . . . 5 (𝜑𝐺 ∈ (𝑁𝐽𝑄))
2018, 19opelxpd 5724 . . . 4 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ ((𝑀𝐻𝑃) × (𝑁𝐽𝑄)))
21 xpcco2.h . . . . 5 𝐻 = (Hom ‘𝐶)
22 xpcco2.j . . . . 5 𝐽 = (Hom ‘𝐷)
231, 2, 3, 21, 22, 9, 10, 12, 13, 5xpchom2 18231 . . . 4 (𝜑 → (⟨𝑀, 𝑁⟩(Hom ‘𝑇)⟨𝑃, 𝑄⟩) = ((𝑀𝐻𝑃) × (𝑁𝐽𝑄)))
2420, 23eleqtrrd 2844 . . 3 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (⟨𝑀, 𝑁⟩(Hom ‘𝑇)⟨𝑃, 𝑄⟩))
25 xpcco2.k . . . . 5 (𝜑𝐾 ∈ (𝑃𝐻𝑅))
26 xpcco2.l . . . . 5 (𝜑𝐿 ∈ (𝑄𝐽𝑆))
2725, 26opelxpd 5724 . . . 4 (𝜑 → ⟨𝐾, 𝐿⟩ ∈ ((𝑃𝐻𝑅) × (𝑄𝐽𝑆)))
281, 2, 3, 21, 22, 12, 13, 15, 16, 5xpchom2 18231 . . . 4 (𝜑 → (⟨𝑃, 𝑄⟩(Hom ‘𝑇)⟨𝑅, 𝑆⟩) = ((𝑃𝐻𝑅) × (𝑄𝐽𝑆)))
2927, 28eleqtrrd 2844 . . 3 (𝜑 → ⟨𝐾, 𝐿⟩ ∈ (⟨𝑃, 𝑄⟩(Hom ‘𝑇)⟨𝑅, 𝑆⟩))
301, 4, 5, 6, 7, 8, 11, 14, 17, 24, 29xpcco 18228 . 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⟩))⟩)
31 op1stg 8026 . . . . . . 7 ((𝑀𝑋𝑁𝑌) → (1st ‘⟨𝑀, 𝑁⟩) = 𝑀)
329, 10, 31syl2anc 584 . . . . . 6 (𝜑 → (1st ‘⟨𝑀, 𝑁⟩) = 𝑀)
33 op1stg 8026 . . . . . . 7 ((𝑃𝑋𝑄𝑌) → (1st ‘⟨𝑃, 𝑄⟩) = 𝑃)
3412, 13, 33syl2anc 584 . . . . . 6 (𝜑 → (1st ‘⟨𝑃, 𝑄⟩) = 𝑃)
3532, 34opeq12d 4881 . . . . 5 (𝜑 → ⟨(1st ‘⟨𝑀, 𝑁⟩), (1st ‘⟨𝑃, 𝑄⟩)⟩ = ⟨𝑀, 𝑃⟩)
36 op1stg 8026 . . . . . 6 ((𝑅𝑋𝑆𝑌) → (1st ‘⟨𝑅, 𝑆⟩) = 𝑅)
3715, 16, 36syl2anc 584 . . . . 5 (𝜑 → (1st ‘⟨𝑅, 𝑆⟩) = 𝑅)
3835, 37oveq12d 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)
4025, 26, 39syl2anc 584 . . . 4 (𝜑 → (1st ‘⟨𝐾, 𝐿⟩) = 𝐾)
41 op1stg 8026 . . . . 5 ((𝐹 ∈ (𝑀𝐻𝑃) ∧ 𝐺 ∈ (𝑁𝐽𝑄)) → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
4218, 19, 41syl2anc 584 . . . 4 (𝜑 → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
4338, 40, 42oveq123d 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 ((𝑀𝑋𝑁𝑌) → (2nd ‘⟨𝑀, 𝑁⟩) = 𝑁)
459, 10, 44syl2anc 584 . . . . . 6 (𝜑 → (2nd ‘⟨𝑀, 𝑁⟩) = 𝑁)
46 op2ndg 8027 . . . . . . 7 ((𝑃𝑋𝑄𝑌) → (2nd ‘⟨𝑃, 𝑄⟩) = 𝑄)
4712, 13, 46syl2anc 584 . . . . . 6 (𝜑 → (2nd ‘⟨𝑃, 𝑄⟩) = 𝑄)
4845, 47opeq12d 4881 . . . . 5 (𝜑 → ⟨(2nd ‘⟨𝑀, 𝑁⟩), (2nd ‘⟨𝑃, 𝑄⟩)⟩ = ⟨𝑁, 𝑄⟩)
49 op2ndg 8027 . . . . . 6 ((𝑅𝑋𝑆𝑌) → (2nd ‘⟨𝑅, 𝑆⟩) = 𝑆)
5015, 16, 49syl2anc 584 . . . . 5 (𝜑 → (2nd ‘⟨𝑅, 𝑆⟩) = 𝑆)
5148, 50oveq12d 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)
5325, 26, 52syl2anc 584 . . . 4 (𝜑 → (2nd ‘⟨𝐾, 𝐿⟩) = 𝐿)
54 op2ndg 8027 . . . . 5 ((𝐹 ∈ (𝑀𝐻𝑃) ∧ 𝐺 ∈ (𝑁𝐽𝑄)) → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
5518, 19, 54syl2anc 584 . . . 4 (𝜑 → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
5651, 53, 55oveq123d 7452 . . 3 (𝜑 → ((2nd ‘⟨𝐾, 𝐿⟩)(⟨(2nd ‘⟨𝑀, 𝑁⟩), (2nd ‘⟨𝑃, 𝑄⟩)⟩ (2nd ‘⟨𝑅, 𝑆⟩))(2nd ‘⟨𝐹, 𝐺⟩)) = (𝐿(⟨𝑁, 𝑄 𝑆)𝐺))
5743, 56opeq12d 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)⟩)
5830, 57eqtrd 2777 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 1540  wcel 2108  cop 4632   × cxp 5683  cfv 6561  (class class class)co 7431  1st c1st 8012  2nd 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
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