xpcfucco3 - Mathbox for Zhi Wang

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Theorem xpcfucco3 48878

Description: Value of composition in the binary product of categories of functors; expressed explicitly. (Contributed by Zhi Wang, 1-Oct-2025.)

**Hypotheses**
Ref	Expression
xpcfuchom2.t	⊢ 𝑇 = ((𝐵 FuncCat 𝐶) ×_c (𝐷 FuncCat 𝐸))
xpcfucco2.o	⊢ 𝑂 = (comp‘𝑇)
xpcfucco2.f	⊢ (𝜑 → 𝐹 ∈ (𝑀(𝐵 Nat 𝐶)𝑃))
xpcfucco2.g	⊢ (𝜑 → 𝐺 ∈ (𝑁(𝐷 Nat 𝐸)𝑄))
xpcfucco2.k	⊢ (𝜑 → 𝐾 ∈ (𝑃(𝐵 Nat 𝐶)𝑅))
xpcfucco2.l	⊢ (𝜑 → 𝐿 ∈ (𝑄(𝐷 Nat 𝐸)𝑆))
xpcfucco3.x	⊢ 𝑋 = (Base‘𝐵)
xpcfucco3.y	⊢ 𝑌 = (Base‘𝐷)
xpcfucco3.o1	⊢ · = (comp‘𝐶)
xpcfucco3.o2	⊢ ∙ = (comp‘𝐸)

**Assertion**
Ref	Expression
xpcfucco3	⊢ (𝜑 → (⟨𝐾, 𝐿⟩(⟨⟨𝑀, 𝑁⟩, ⟨𝑃, 𝑄⟩⟩𝑂⟨𝑅, 𝑆⟩)⟨𝐹, 𝐺⟩) = ⟨(𝑥 ∈ 𝑋 ↦ ((𝐾‘𝑥)(⟨((1^st ‘𝑀)‘𝑥), ((1^st ‘𝑃)‘𝑥)⟩ · ((1^st ‘𝑅)‘𝑥))(𝐹‘𝑥))), (𝑦 ∈ 𝑌 ↦ ((𝐿‘𝑦)(⟨((1^st ‘𝑁)‘𝑦), ((1^st ‘𝑄)‘𝑦)⟩ ∙ ((1^st ‘𝑆)‘𝑦))(𝐺‘𝑦)))⟩)

Distinct variable groups: 𝑥, · 𝑦, ∙ 𝑥,𝐵 𝑥,𝐶 𝑦,𝐷 𝑦,𝐸 𝑥,𝐹 𝑦,𝐺 𝑥,𝐾 𝑦,𝐿 𝑥,𝑀 𝑦,𝑁 𝑥,𝑃 𝑦,𝑄 𝑥,𝑅 𝑦,𝑆 𝑥,𝑋 𝑦,𝑌 𝜑,𝑥 𝜑,𝑦 Allowed substitution hints: 𝐵(𝑦) 𝐶(𝑦) 𝐷(𝑥) 𝑃(𝑦) 𝑄(𝑥) 𝑅(𝑦) 𝑆(𝑥) ∙ (𝑥) 𝑇(𝑥,𝑦) · (𝑦) 𝐸(𝑥) 𝐹(𝑦) 𝐺(𝑥) 𝐾(𝑦) 𝐿(𝑥) 𝑀(𝑦) 𝑁(𝑥) 𝑂(𝑥,𝑦) 𝑋(𝑦) 𝑌(𝑥)

**Proof of Theorem xpcfucco3**
Step	Hyp	Ref	Expression
1		xpcfuchom2.t	. . 3 ⊢ 𝑇 = ((𝐵 FuncCat 𝐶) ×_c (𝐷 FuncCat 𝐸))
2		xpcfucco2.o	. . 3 ⊢ 𝑂 = (comp‘𝑇)
3		xpcfucco2.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝑀(𝐵 Nat 𝐶)𝑃))
4		xpcfucco2.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ (𝑁(𝐷 Nat 𝐸)𝑄))
5		xpcfucco2.k	. . 3 ⊢ (𝜑 → 𝐾 ∈ (𝑃(𝐵 Nat 𝐶)𝑅))
6		xpcfucco2.l	. . 3 ⊢ (𝜑 → 𝐿 ∈ (𝑄(𝐷 Nat 𝐸)𝑆))
7	1, 2, 3, 4, 5, 6	xpcfucco2 48876	. 2 ⊢ (𝜑 → (⟨𝐾, 𝐿⟩(⟨⟨𝑀, 𝑁⟩, ⟨𝑃, 𝑄⟩⟩𝑂⟨𝑅, 𝑆⟩)⟨𝐹, 𝐺⟩) = ⟨(𝐾(⟨𝑀, 𝑃⟩(comp‘(𝐵 FuncCat 𝐶))𝑅)𝐹), (𝐿(⟨𝑁, 𝑄⟩(comp‘(𝐷 FuncCat 𝐸))𝑆)𝐺)⟩)
8		eqid 2737	. . . 4 ⊢ (𝐵 FuncCat 𝐶) = (𝐵 FuncCat 𝐶)
9		eqid 2737	. . . 4 ⊢ (𝐵 Nat 𝐶) = (𝐵 Nat 𝐶)
10		xpcfucco3.x	. . . 4 ⊢ 𝑋 = (Base‘𝐵)
11		xpcfucco3.o1	. . . 4 ⊢ · = (comp‘𝐶)
12		eqid 2737	. . . 4 ⊢ (comp‘(𝐵 FuncCat 𝐶)) = (comp‘(𝐵 FuncCat 𝐶))
13	8, 9, 10, 11, 12, 3, 5	fucco 18028	. . 3 ⊢ (𝜑 → (𝐾(⟨𝑀, 𝑃⟩(comp‘(𝐵 FuncCat 𝐶))𝑅)𝐹) = (𝑥 ∈ 𝑋 ↦ ((𝐾‘𝑥)(⟨((1^st ‘𝑀)‘𝑥), ((1^st ‘𝑃)‘𝑥)⟩ · ((1^st ‘𝑅)‘𝑥))(𝐹‘𝑥))))
14		eqid 2737	. . . 4 ⊢ (𝐷 FuncCat 𝐸) = (𝐷 FuncCat 𝐸)
15		eqid 2737	. . . 4 ⊢ (𝐷 Nat 𝐸) = (𝐷 Nat 𝐸)
16		xpcfucco3.y	. . . 4 ⊢ 𝑌 = (Base‘𝐷)
17		xpcfucco3.o2	. . . 4 ⊢ ∙ = (comp‘𝐸)
18		eqid 2737	. . . 4 ⊢ (comp‘(𝐷 FuncCat 𝐸)) = (comp‘(𝐷 FuncCat 𝐸))
19	14, 15, 16, 17, 18, 4, 6	fucco 18028	. . 3 ⊢ (𝜑 → (𝐿(⟨𝑁, 𝑄⟩(comp‘(𝐷 FuncCat 𝐸))𝑆)𝐺) = (𝑦 ∈ 𝑌 ↦ ((𝐿‘𝑦)(⟨((1^st ‘𝑁)‘𝑦), ((1^st ‘𝑄)‘𝑦)⟩ ∙ ((1^st ‘𝑆)‘𝑦))(𝐺‘𝑦))))
20	13, 19	opeq12d 4889	. 2 ⊢ (𝜑 → ⟨(𝐾(⟨𝑀, 𝑃⟩(comp‘(𝐵 FuncCat 𝐶))𝑅)𝐹), (𝐿(⟨𝑁, 𝑄⟩(comp‘(𝐷 FuncCat 𝐸))𝑆)𝐺)⟩ = ⟨(𝑥 ∈ 𝑋 ↦ ((𝐾‘𝑥)(⟨((1^st ‘𝑀)‘𝑥), ((1^st ‘𝑃)‘𝑥)⟩ · ((1^st ‘𝑅)‘𝑥))(𝐹‘𝑥))), (𝑦 ∈ 𝑌 ↦ ((𝐿‘𝑦)(⟨((1^st ‘𝑁)‘𝑦), ((1^st ‘𝑄)‘𝑦)⟩ ∙ ((1^st ‘𝑆)‘𝑦))(𝐺‘𝑦)))⟩)
21	7, 20	eqtrd 2777	1 ⊢ (𝜑 → (⟨𝐾, 𝐿⟩(⟨⟨𝑀, 𝑁⟩, ⟨𝑃, 𝑄⟩⟩𝑂⟨𝑅, 𝑆⟩)⟨𝐹, 𝐺⟩) = ⟨(𝑥 ∈ 𝑋 ↦ ((𝐾‘𝑥)(⟨((1^st ‘𝑀)‘𝑥), ((1^st ‘𝑃)‘𝑥)⟩ · ((1^st ‘𝑅)‘𝑥))(𝐹‘𝑥))), (𝑦 ∈ 𝑌 ↦ ((𝐿‘𝑦)(⟨((1^st ‘𝑁)‘𝑦), ((1^st ‘𝑄)‘𝑦)⟩ ∙ ((1^st ‘𝑆)‘𝑦))(𝐺‘𝑦)))⟩)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2108 ⟨cop 4640 ↦ cmpt 5234 ‘cfv 6569 (class class class)co 7438 1^st c1st 8020 Basecbs 17254 compcco 17319 Nat cnat 18005 FuncCat cfuc 18006 ×_c cxpc 18233

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 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 5288 ax-sep 5305 ax-nul 5315 ax-pow 5374 ax-pr 5441 ax-un 7761 ax-cnex 11218 ax-resscn 11219 ax-1cn 11220 ax-icn 11221 ax-addcl 11222 ax-addrcl 11223 ax-mulcl 11224 ax-mulrcl 11225 ax-mulcom 11226 ax-addass 11227 ax-mulass 11228 ax-distr 11229 ax-i2m1 11230 ax-1ne0 11231 ax-1rid 11232 ax-rnegex 11233 ax-rrecex 11234 ax-cnre 11235 ax-pre-lttri 11236 ax-pre-lttrn 11237 ax-pre-ltadd 11238 ax-pre-mulgt0 11239

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 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 3483 df-sbc 3795 df-csb 3912 df-dif 3969 df-un 3971 df-in 3973 df-ss 3983 df-pss 3986 df-nul 4343 df-if 4535 df-pw 4610 df-sn 4635 df-pr 4637 df-tp 4639 df-op 4641 df-uni 4916 df-iun 5001 df-br 5152 df-opab 5214 df-mpt 5235 df-tr 5269 df-id 5587 df-eprel 5593 df-po 5601 df-so 5602 df-fr 5645 df-we 5647 df-xp 5699 df-rel 5700 df-cnv 5701 df-co 5702 df-dm 5703 df-rn 5704 df-res 5705 df-ima 5706 df-pred 6329 df-ord 6395 df-on 6396 df-lim 6397 df-suc 6398 df-iota 6522 df-fun 6571 df-fn 6572 df-f 6573 df-f1 6574 df-fo 6575 df-f1o 6576 df-fv 6577 df-riota 7395 df-ov 7441 df-oprab 7442 df-mpo 7443 df-om 7895 df-1st 8022 df-2nd 8023 df-frecs 8314 df-wrecs 8345 df-recs 8419 df-rdg 8458 df-1o 8514 df-er 8753 df-ixp 8946 df-en 8994 df-dom 8995 df-sdom 8996 df-fin 8997 df-pnf 11304 df-mnf 11305 df-xr 11306 df-ltxr 11307 df-le 11308 df-sub 11501 df-neg 11502 df-nn 12274 df-2 12336 df-3 12337 df-4 12338 df-5 12339 df-6 12340 df-7 12341 df-8 12342 df-9 12343 df-n0 12534 df-z 12621 df-dec 12741 df-uz 12886 df-fz 13554 df-struct 17190 df-slot 17225 df-ndx 17237 df-base 17255 df-hom 17331 df-cco 17332 df-func 17918 df-nat 18007 df-fuc 18008 df-xpc 18237

This theorem is referenced by: fucocolem2 48921

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