xpcfucco3 - Mathbox for Zhi Wang

	Mathbox for Zhi Wang		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > xpcfucco3		Structured version Visualization version GIF version

Theorem xpcfucco3 50595

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 50593	. 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 18172	. . 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 18172	. . 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 4851	. 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

This proof depends on syntax axioms: → wi 4 = wceq 1570 ∈ wcel 2147 ⟨cop 4838 ↦ cmpt 5190 ‘cfv 6561 (class class class)co 7446 1^st c1st 8026 Basecbs 17419 compcco 17472 Nat cnat 18151 FuncCat cfuc 18152 ×_c cxpc 18374

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5236 ax-sep 5253 ax-nul 5264 ax-pow 5332 ax-pr 5396 ax-un 7778 ax-cnex 11286 ax-resscn 11287 ax-1cn 11288 ax-icn 11289 ax-addcl 11290 ax-addrcl 11291 ax-mulcl 11292 ax-mulrcl 11293 ax-mulcom 11294 ax-addass 11295 ax-mulass 11296 ax-distr 11297 ax-i2m1 11298 ax-1ne0 11299 ax-1rid 11300 ax-rnegex 11301 ax-rrecex 11302 ax-cnre 11303 ax-pre-lttri 11304 ax-pre-lttrn 11305 ax-pre-ltadd 11306 ax-pre-mulgt0 11307

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4839 df-uni 4871 df-iun 4956 df-br 5107 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5551 df-eprel 5556 df-po 5564 df-so 5565 df-fr 5609 df-we 5611 df-xp 5662 df-rel 5663 df-cnv 5664 df-co 5665 df-dm 5666 df-rn 5667 df-res 5668 df-ima 5669 df-pred 6325 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6515 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7403 df-ov 7449 df-oprab 7450 df-mpo 7451 df-om 7905 df-1st 8028 df-2nd 8029 df-frecs 8330 df-wrecs 8361 df-recs 8410 df-rdg 8449 df-1o 8507 df-er 8748 df-ixp 8957 df-en 9005 df-dom 9006 df-sdom 9007 df-fin 9008 df-pnf 11375 df-mnf 11376 df-xr 11377 df-ltxr 11378 df-le 11379 df-sub 11573 df-neg 11574 df-nn 12366 df-2 12435 df-3 12436 df-4 12437 df-5 12438 df-6 12439 df-7 12440 df-8 12441 df-9 12442 df-n0 12637 df-z 12724 df-dec 12845 df-uz 12996 df-fz 13670 df-struct 17357 df-slot 17392 df-ndx 17404 df-base 17420 df-hom 17484 df-cco 17485 df-func 18065 df-nat 18153 df-fuc 18154 df-xpc 18378

This theorem is used by: fucocolem2 50691

W3C validator