xpcfucco3 - Mathbox for Zhi Wang

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Theorem xpcfucco3 50019

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 50017	. 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 2763	. . . 4 ⊢ (𝐵 FuncCat 𝐶) = (𝐵 FuncCat 𝐶)
9		eqid 2763	. . . 4 ⊢ (𝐵 Nat 𝐶) = (𝐵 Nat 𝐶)
10		xpcfucco3.x	. . . 4 ⊢ 𝑋 = (Base‘𝐵)
11		xpcfucco3.o1	. . . 4 ⊢ · = (comp‘𝐶)
12		eqid 2763	. . . 4 ⊢ (comp‘(𝐵 FuncCat 𝐶)) = (comp‘(𝐵 FuncCat 𝐶))
13	8, 9, 10, 11, 12, 3, 5	fucco 18023	. . 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 2763	. . . 4 ⊢ (𝐷 FuncCat 𝐸) = (𝐷 FuncCat 𝐸)
15		eqid 2763	. . . 4 ⊢ (𝐷 Nat 𝐸) = (𝐷 Nat 𝐸)
16		xpcfucco3.y	. . . 4 ⊢ 𝑌 = (Base‘𝐷)
17		xpcfucco3.o2	. . . 4 ⊢ ∙ = (comp‘𝐸)
18		eqid 2763	. . . 4 ⊢ (comp‘(𝐷 FuncCat 𝐸)) = (comp‘(𝐷 FuncCat 𝐸))
19	14, 15, 16, 17, 18, 4, 6	fucco 18023	. . 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 4847	. 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 2798	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 1570 ∈ wcel 2143 ⟨cop 4596 ↦ cmpt 5193 ‘cfv 6538 (class class class)co 7412 1^st c1st 7985 Basecbs 17270 compcco 17323 Nat cnat 18002 FuncCat cfuc 18003 ×_c cxpc 18225

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5239 ax-sep 5258 ax-nul 5270 ax-pow 5338 ax-pr 5406 ax-un 7734 ax-cnex 11157 ax-resscn 11158 ax-1cn 11159 ax-icn 11160 ax-addcl 11161 ax-addrcl 11162 ax-mulcl 11163 ax-mulrcl 11164 ax-mulcom 11165 ax-addass 11166 ax-mulass 11167 ax-distr 11168 ax-i2m1 11169 ax-1ne0 11170 ax-1rid 11171 ax-rnegex 11172 ax-rrecex 11173 ax-cnre 11174 ax-pre-lttri 11175 ax-pre-lttrn 11176 ax-pre-ltadd 11177 ax-pre-mulgt0 11178

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3746 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4489 df-pw 4565 df-sn 4591 df-pr 4593 df-tp 4595 df-op 4597 df-uni 4874 df-iun 4959 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7864 df-1st 7987 df-2nd 7988 df-frecs 8279 df-wrecs 8310 df-recs 8359 df-rdg 8398 df-1o 8454 df-er 8695 df-ixp 8897 df-en 8945 df-dom 8946 df-sdom 8947 df-fin 8948 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11444 df-neg 11445 df-nn 12235 df-2 12304 df-3 12305 df-4 12306 df-5 12307 df-6 12308 df-7 12309 df-8 12310 df-9 12311 df-n0 12506 df-z 12593 df-dec 12713 df-uz 12864 df-fz 13537 df-struct 17208 df-slot 17243 df-ndx 17255 df-base 17271 df-hom 17335 df-cco 17336 df-func 17916 df-nat 18004 df-fuc 18005 df-xpc 18229

This theorem is referenced by: fucocolem2 50115

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