xpcfucco3 - Mathbox for Zhi Wang

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Theorem xpcfucco3 49383

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 49381	. 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 2733	. . . 4 ⊢ (𝐵 FuncCat 𝐶) = (𝐵 FuncCat 𝐶)
9		eqid 2733	. . . 4 ⊢ (𝐵 Nat 𝐶) = (𝐵 Nat 𝐶)
10		xpcfucco3.x	. . . 4 ⊢ 𝑋 = (Base‘𝐵)
11		xpcfucco3.o1	. . . 4 ⊢ · = (comp‘𝐶)
12		eqid 2733	. . . 4 ⊢ (comp‘(𝐵 FuncCat 𝐶)) = (comp‘(𝐵 FuncCat 𝐶))
13	8, 9, 10, 11, 12, 3, 5	fucco 17874	. . 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 2733	. . . 4 ⊢ (𝐷 FuncCat 𝐸) = (𝐷 FuncCat 𝐸)
15		eqid 2733	. . . 4 ⊢ (𝐷 Nat 𝐸) = (𝐷 Nat 𝐸)
16		xpcfucco3.y	. . . 4 ⊢ 𝑌 = (Base‘𝐷)
17		xpcfucco3.o2	. . . 4 ⊢ ∙ = (comp‘𝐸)
18		eqid 2733	. . . 4 ⊢ (comp‘(𝐷 FuncCat 𝐸)) = (comp‘(𝐷 FuncCat 𝐸))
19	14, 15, 16, 17, 18, 4, 6	fucco 17874	. . 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 4832	. 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 2768	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 1541 ∈ wcel 2113 ⟨cop 4581 ↦ cmpt 5174 ‘cfv 6486 (class class class)co 7352 1^st c1st 7925 Basecbs 17122 compcco 17175 Nat cnat 17853 FuncCat cfuc 17854 ×_c cxpc 18076

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5219 ax-sep 5236 ax-nul 5246 ax-pow 5305 ax-pr 5372 ax-un 7674 ax-cnex 11069 ax-resscn 11070 ax-1cn 11071 ax-icn 11072 ax-addcl 11073 ax-addrcl 11074 ax-mulcl 11075 ax-mulrcl 11076 ax-mulcom 11077 ax-addass 11078 ax-mulass 11079 ax-distr 11080 ax-i2m1 11081 ax-1ne0 11082 ax-1rid 11083 ax-rnegex 11084 ax-rrecex 11085 ax-cnre 11086 ax-pre-lttri 11087 ax-pre-lttrn 11088 ax-pre-ltadd 11089 ax-pre-mulgt0 11090

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-tp 4580 df-op 4582 df-uni 4859 df-iun 4943 df-br 5094 df-opab 5156 df-mpt 5175 df-tr 5201 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7309 df-ov 7355 df-oprab 7356 df-mpo 7357 df-om 7803 df-1st 7927 df-2nd 7928 df-frecs 8217 df-wrecs 8248 df-recs 8297 df-rdg 8335 df-1o 8391 df-er 8628 df-ixp 8828 df-en 8876 df-dom 8877 df-sdom 8878 df-fin 8879 df-pnf 11155 df-mnf 11156 df-xr 11157 df-ltxr 11158 df-le 11159 df-sub 11353 df-neg 11354 df-nn 12133 df-2 12195 df-3 12196 df-4 12197 df-5 12198 df-6 12199 df-7 12200 df-8 12201 df-9 12202 df-n0 12389 df-z 12476 df-dec 12595 df-uz 12739 df-fz 13410 df-struct 17060 df-slot 17095 df-ndx 17107 df-base 17123 df-hom 17187 df-cco 17188 df-func 17767 df-nat 17855 df-fuc 17856 df-xpc 18080

This theorem is referenced by: fucocolem2 49479

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