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Theorem xpccofval 18118

Description: Value of composition in the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.) (Proof shortened by AV, 2-Mar-2024.)

**Hypotheses**
Ref	Expression
xpccofval.t	⊢ 𝑇 = (𝐶 ×_c 𝐷)
xpccofval.b	⊢ 𝐵 = (Base‘𝑇)
xpccofval.k	⊢ 𝐾 = (Hom ‘𝑇)
xpccofval.o1	⊢ · = (comp‘𝐶)
xpccofval.o2	⊢ ∙ = (comp‘𝐷)
xpccofval.o	⊢ 𝑂 = (comp‘𝑇)

**Assertion**
Ref	Expression
xpccofval	⊢ 𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2^nd ‘𝑥)𝐾𝑦), 𝑓 ∈ (𝐾‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩ · (1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩ ∙ (2^nd ‘𝑦))(2^nd ‘𝑓))⟩))

Distinct variable groups: 𝑓,𝑔,𝑥,𝑦,𝐵 𝐶,𝑓,𝑔,𝑥,𝑦 𝐷,𝑓,𝑔,𝑥,𝑦 · ,𝑓,𝑔,𝑥,𝑦 ∙ ,𝑓,𝑔,𝑥,𝑦 𝑓,𝐾,𝑔,𝑥,𝑦 𝑥,𝑂,𝑦 Allowed substitution hints: 𝑇(𝑥,𝑦,𝑓,𝑔) 𝑂(𝑓,𝑔)

Proof of Theorem xpccofval Dummy variables 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xpccofval.t	. . . 4 ⊢ 𝑇 = (𝐶 ×_c 𝐷)
2		eqid 2732	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
3		eqid 2732	. . . 4 ⊢ (Base‘𝐷) = (Base‘𝐷)
4		eqid 2732	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
5		eqid 2732	. . . 4 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
6		xpccofval.o1	. . . 4 ⊢ · = (comp‘𝐶)
7		xpccofval.o2	. . . 4 ⊢ ∙ = (comp‘𝐷)
8		simpl 483	. . . 4 ⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝐶 ∈ V)
9		simpr 485	. . . 4 ⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝐷 ∈ V)
10		xpccofval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑇)
11	1, 2, 3	xpcbas 18114	. . . . . 6 ⊢ ((Base‘𝐶) × (Base‘𝐷)) = (Base‘𝑇)
12	10, 11	eqtr4i 2763	. . . . 5 ⊢ 𝐵 = ((Base‘𝐶) × (Base‘𝐷))
13	12	a1i 11	. . . 4 ⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝐵 = ((Base‘𝐶) × (Base‘𝐷)))
14		xpccofval.k	. . . . . 6 ⊢ 𝐾 = (Hom ‘𝑇)
15	1, 10, 4, 5, 14	xpchomfval 18115	. . . . 5 ⊢ 𝐾 = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (((1^st ‘𝑢)(Hom ‘𝐶)(1^st ‘𝑣)) × ((2^nd ‘𝑢)(Hom ‘𝐷)(2^nd ‘𝑣))))
16	15	a1i 11	. . . 4 ⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝐾 = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (((1^st ‘𝑢)(Hom ‘𝐶)(1^st ‘𝑣)) × ((2^nd ‘𝑢)(Hom ‘𝐷)(2^nd ‘𝑣)))))
17		eqidd 2733	. . . 4 ⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2^nd ‘𝑥)𝐾𝑦), 𝑓 ∈ (𝐾‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩ · (1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩ ∙ (2^nd ‘𝑦))(2^nd ‘𝑓))⟩)) = (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2^nd ‘𝑥)𝐾𝑦), 𝑓 ∈ (𝐾‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩ · (1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩ ∙ (2^nd ‘𝑦))(2^nd ‘𝑓))⟩)))
18	1, 2, 3, 4, 5, 6, 7, 8, 9, 13, 16, 17	xpcval 18113	. . 3 ⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝑇 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐾⟩, ⟨(comp‘ndx), (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2^nd ‘𝑥)𝐾𝑦), 𝑓 ∈ (𝐾‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩ · (1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩ ∙ (2^nd ‘𝑦))(2^nd ‘𝑓))⟩))⟩})
19		catstr 17893	. . 3 ⊢ {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐾⟩, ⟨(comp‘ndx), (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2^nd ‘𝑥)𝐾𝑦), 𝑓 ∈ (𝐾‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩ · (1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩ ∙ (2^nd ‘𝑦))(2^nd ‘𝑓))⟩))⟩} Struct ⟨1, ;15⟩
20		ccoid 17343	. . 3 ⊢ comp = Slot (comp‘ndx)
21		snsstp3 4815	. . 3 ⊢ {⟨(comp‘ndx), (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2^nd ‘𝑥)𝐾𝑦), 𝑓 ∈ (𝐾‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩ · (1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩ ∙ (2^nd ‘𝑦))(2^nd ‘𝑓))⟩))⟩} ⊆ {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐾⟩, ⟨(comp‘ndx), (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2^nd ‘𝑥)𝐾𝑦), 𝑓 ∈ (𝐾‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩ · (1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩ ∙ (2^nd ‘𝑦))(2^nd ‘𝑓))⟩))⟩}
22	10	fvexi 6893	. . . . . 6 ⊢ 𝐵 ∈ V
23	22, 22	xpex 7724	. . . . 5 ⊢ (𝐵 × 𝐵) ∈ V
24	23, 22	mpoex 8050	. . . 4 ⊢ (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2^nd ‘𝑥)𝐾𝑦), 𝑓 ∈ (𝐾‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩ · (1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩ ∙ (2^nd ‘𝑦))(2^nd ‘𝑓))⟩)) ∈ V
25	24	a1i 11	. . 3 ⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2^nd ‘𝑥)𝐾𝑦), 𝑓 ∈ (𝐾‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩ · (1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩ ∙ (2^nd ‘𝑦))(2^nd ‘𝑓))⟩)) ∈ V)
26		xpccofval.o	. . 3 ⊢ 𝑂 = (comp‘𝑇)
27	18, 19, 20, 21, 25, 26	strfv3 17122	. 2 ⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2^nd ‘𝑥)𝐾𝑦), 𝑓 ∈ (𝐾‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩ · (1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩ ∙ (2^nd ‘𝑦))(2^nd ‘𝑓))⟩)))
28		fnxpc 18112	. . . . . . . 8 ⊢ ×_c Fn (V × V)
29	28	fndmi 6643	. . . . . . 7 ⊢ dom ×_c = (V × V)
30	29	ndmov 7575	. . . . . 6 ⊢ (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝐶 ×_c 𝐷) = ∅)
31	1, 30	eqtrid 2784	. . . . 5 ⊢ (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝑇 = ∅)
32	31	fveq2d 6883	. . . 4 ⊢ (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (comp‘𝑇) = (comp‘∅))
33	20	str0 17106	. . . 4 ⊢ ∅ = (comp‘∅)
34	32, 26, 33	3eqtr4g 2797	. . 3 ⊢ (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝑂 = ∅)
35	31	fveq2d 6883	. . . . . 6 ⊢ (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (Base‘𝑇) = (Base‘∅))
36		base0 17133	. . . . . 6 ⊢ ∅ = (Base‘∅)
37	35, 10, 36	3eqtr4g 2797	. . . . 5 ⊢ (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝐵 = ∅)
38	37	olcd 872	. . . 4 ⊢ (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → ((𝐵 × 𝐵) = ∅ ∨ 𝐵 = ∅))
39		0mpo0 7477	. . . 4 ⊢ (((𝐵 × 𝐵) = ∅ ∨ 𝐵 = ∅) → (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2^nd ‘𝑥)𝐾𝑦), 𝑓 ∈ (𝐾‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩ · (1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩ ∙ (2^nd ‘𝑦))(2^nd ‘𝑓))⟩)) = ∅)
40	38, 39	syl 17	. . 3 ⊢ (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2^nd ‘𝑥)𝐾𝑦), 𝑓 ∈ (𝐾‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩ · (1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩ ∙ (2^nd ‘𝑦))(2^nd ‘𝑓))⟩)) = ∅)
41	34, 40	eqtr4d 2775	. 2 ⊢ (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2^nd ‘𝑥)𝐾𝑦), 𝑓 ∈ (𝐾‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩ · (1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩ ∙ (2^nd ‘𝑦))(2^nd ‘𝑓))⟩)))
42	27, 41	pm2.61i 182	1 ⊢ 𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2^nd ‘𝑥)𝐾𝑦), 𝑓 ∈ (𝐾‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩ · (1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩ ∙ (2^nd ‘𝑦))(2^nd ‘𝑓))⟩))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ∧ wa 396 ∨ wo 845 = wceq 1541 ∈ wcel 2106 Vcvv 3474 ∅c0 4319 {ctp 4627 ⟨cop 4629 × cxp 5668 ‘cfv 6533 (class class class)co 7394 ∈ cmpo 7396 1^st c1st 7957 2^nd c2nd 7958 1c1 11095 5c5 12254 cdc 12661 ndxcnx 17110 Basecbs 17128 Hom chom 17192 compcco 17193 ×_c cxpc 18104

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5357 ax-pr 5421 ax-un 7709 ax-cnex 11150 ax-resscn 11151 ax-1cn 11152 ax-icn 11153 ax-addcl 11154 ax-addrcl 11155 ax-mulcl 11156 ax-mulrcl 11157 ax-mulcom 11158 ax-addass 11159 ax-mulass 11160 ax-distr 11161 ax-i2m1 11162 ax-1ne0 11163 ax-1rid 11164 ax-rnegex 11165 ax-rrecex 11166 ax-cnre 11167 ax-pre-lttri 11168 ax-pre-lttrn 11169 ax-pre-ltadd 11170 ax-pre-mulgt0 11171

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3775 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5568 df-eprel 5574 df-po 5582 df-so 5583 df-fr 5625 df-we 5627 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7350 df-ov 7397 df-oprab 7398 df-mpo 7399 df-om 7840 df-1st 7959 df-2nd 7960 df-frecs 8250 df-wrecs 8281 df-recs 8355 df-rdg 8394 df-1o 8450 df-er 8688 df-en 8925 df-dom 8926 df-sdom 8927 df-fin 8928 df-pnf 11234 df-mnf 11235 df-xr 11236 df-ltxr 11237 df-le 11238 df-sub 11430 df-neg 11431 df-nn 12197 df-2 12259 df-3 12260 df-4 12261 df-5 12262 df-6 12263 df-7 12264 df-8 12265 df-9 12266 df-n0 12457 df-z 12543 df-dec 12662 df-uz 12807 df-fz 13469 df-struct 17064 df-slot 17099 df-ndx 17111 df-base 17129 df-hom 17205 df-cco 17206 df-xpc 18108

This theorem is referenced by: xpcco 18119

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