xpcfucco3 - Mathbox for Zhi Wang

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Theorem xpcfucco3 49755

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 49753	. 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 2740	. . . 4 ⊢ (𝐵 FuncCat 𝐶) = (𝐵 FuncCat 𝐶)
9		eqid 2740	. . . 4 ⊢ (𝐵 Nat 𝐶) = (𝐵 Nat 𝐶)
10		xpcfucco3.x	. . . 4 ⊢ 𝑋 = (Base‘𝐵)
11		xpcfucco3.o1	. . . 4 ⊢ · = (comp‘𝐶)
12		eqid 2740	. . . 4 ⊢ (comp‘(𝐵 FuncCat 𝐶)) = (comp‘(𝐵 FuncCat 𝐶))
13	8, 9, 10, 11, 12, 3, 5	fucco 17930	. . 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 2740	. . . 4 ⊢ (𝐷 FuncCat 𝐸) = (𝐷 FuncCat 𝐸)
15		eqid 2740	. . . 4 ⊢ (𝐷 Nat 𝐸) = (𝐷 Nat 𝐸)
16		xpcfucco3.y	. . . 4 ⊢ 𝑌 = (Base‘𝐷)
17		xpcfucco3.o2	. . . 4 ⊢ ∙ = (comp‘𝐸)
18		eqid 2740	. . . 4 ⊢ (comp‘(𝐷 FuncCat 𝐸)) = (comp‘(𝐷 FuncCat 𝐸))
19	14, 15, 16, 17, 18, 4, 6	fucco 17930	. . 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 4819	. 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 2775	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 1547 ∈ wcel 2119 ⟨cop 4568 ↦ cmpt 5160 ‘cfv 6492 (class class class)co 7363 1^st c1st 7936 Basecbs 17177 compcco 17230 Nat cnat 17909 FuncCat cfuc 17910 ×_c cxpc 18132

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2712 ax-rep 5206 ax-sep 5225 ax-nul 5235 ax-pow 5301 ax-pr 5369 ax-un 7685 ax-cnex 11092 ax-resscn 11093 ax-1cn 11094 ax-icn 11095 ax-addcl 11096 ax-addrcl 11097 ax-mulcl 11098 ax-mulrcl 11099 ax-mulcom 11100 ax-addass 11101 ax-mulass 11102 ax-distr 11103 ax-i2m1 11104 ax-1ne0 11105 ax-1rid 11106 ax-rnegex 11107 ax-rrecex 11108 ax-cnre 11109 ax-pre-lttri 11110 ax-pre-lttrn 11111 ax-pre-ltadd 11112 ax-pre-mulgt0 11113

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-nel 3040 df-ral 3055 df-rex 3065 df-reu 3346 df-rab 3393 df-v 3434 df-sbc 3731 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4269 df-if 4462 df-pw 4538 df-sn 4563 df-pr 4565 df-tp 4567 df-op 4569 df-uni 4846 df-iun 4930 df-br 5080 df-opab 5142 df-mpt 5161 df-tr 5187 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7320 df-ov 7366 df-oprab 7367 df-mpo 7368 df-om 7814 df-1st 7938 df-2nd 7939 df-frecs 8228 df-wrecs 8259 df-recs 8308 df-rdg 8346 df-1o 8402 df-er 8640 df-ixp 8843 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-pnf 11179 df-mnf 11180 df-xr 11181 df-ltxr 11182 df-le 11183 df-sub 11377 df-neg 11378 df-nn 12173 df-2 12242 df-3 12243 df-4 12244 df-5 12245 df-6 12246 df-7 12247 df-8 12248 df-9 12249 df-n0 12436 df-z 12523 df-dec 12643 df-uz 12787 df-fz 13460 df-struct 17115 df-slot 17150 df-ndx 17162 df-base 17178 df-hom 17242 df-cco 17243 df-func 17823 df-nat 17911 df-fuc 17912 df-xpc 18136

This theorem is referenced by: fucocolem2 49851

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