fucoco - Mathbox for Zhi Wang

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Theorem fucoco 49986

Description: Composition in the source category is mapped to composition in the target. See also fucoco2 49987. (Contributed by Zhi Wang, 3-Oct-2025.)

**Hypotheses**
Ref	Expression
fucoco.r	⊢ (𝜑 → 𝑅 ∈ (𝐹(𝐷 Nat 𝐸)𝐾))
fucoco.s	⊢ (𝜑 → 𝑆 ∈ (𝐺(𝐶 Nat 𝐷)𝐿))
fucoco.u	⊢ (𝜑 → 𝑈 ∈ (𝐾(𝐷 Nat 𝐸)𝑀))
fucoco.v	⊢ (𝜑 → 𝑉 ∈ (𝐿(𝐶 Nat 𝐷)𝑁))
fucoco.o	⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘_F 𝐸) = ⟨𝑂, 𝑃⟩)
fucoco.x	⊢ (𝜑 → 𝑋 = ⟨𝐹, 𝐺⟩)
fucoco.y	⊢ (𝜑 → 𝑌 = ⟨𝐾, 𝐿⟩)
fucoco.z	⊢ (𝜑 → 𝑍 = ⟨𝑀, 𝑁⟩)
fucoco.a	⊢ (𝜑 → 𝐴 = ⟨𝑅, 𝑆⟩)
fucoco.b	⊢ (𝜑 → 𝐵 = ⟨𝑈, 𝑉⟩)
fucoco.q	⊢ 𝑄 = (𝐶 FuncCat 𝐸)
fucoco.oq	⊢ ∙ = (comp‘𝑄)
fucoco.t	⊢ 𝑇 = ((𝐷 FuncCat 𝐸) ×_c (𝐶 FuncCat 𝐷))
fucoco.ot	⊢ · = (comp‘𝑇)

**Assertion**
Ref	Expression
fucoco	⊢ (𝜑 → ((𝑋𝑃𝑍)‘(𝐵(⟨𝑋, 𝑌⟩ · 𝑍)𝐴)) = (((𝑌𝑃𝑍)‘𝐵)(⟨(𝑂‘𝑋), (𝑂‘𝑌)⟩ ∙ (𝑂‘𝑍))((𝑋𝑃𝑌)‘𝐴)))

Proof of Theorem fucoco Dummy variable 𝑝 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2765	. . . . . . . . 9 ⊢ (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷)
2		fucoco.v	. . . . . . . . 9 ⊢ (𝜑 → 𝑉 ∈ (𝐿(𝐶 Nat 𝐷)𝑁))
3	1, 2	nat1st2nd 18001	. . . . . . . 8 ⊢ (𝜑 → 𝑉 ∈ (⟨(1^st ‘𝐿), (2^nd ‘𝐿)⟩(𝐶 Nat 𝐷)⟨(1^st ‘𝑁), (2^nd ‘𝑁)⟩))
4	3	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘𝐶)) → 𝑉 ∈ (⟨(1^st ‘𝐿), (2^nd ‘𝐿)⟩(𝐶 Nat 𝐷)⟨(1^st ‘𝑁), (2^nd ‘𝑁)⟩))
5		eqid 2765	. . . . . . . . 9 ⊢ (𝐷 Nat 𝐸) = (𝐷 Nat 𝐸)
6		fucoco.r	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ (𝐹(𝐷 Nat 𝐸)𝐾))
7	5, 6	nat1st2nd 18001	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ (⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩(𝐷 Nat 𝐸)⟨(1^st ‘𝐾), (2^nd ‘𝐾)⟩))
8	7	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘𝐶)) → 𝑅 ∈ (⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩(𝐷 Nat 𝐸)⟨(1^st ‘𝐾), (2^nd ‘𝐾)⟩))
9		simpr 489	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘𝐶)) → 𝑝 ∈ (Base‘𝐶))
10		eqid 2765	. . . . . . 7 ⊢ (comp‘𝐸) = (comp‘𝐸)
11	4, 8, 9, 10	fuco23alem 49980	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘𝐶)) → ((𝑅‘((1^st ‘𝑁)‘𝑝))(⟨((1^st ‘𝐹)‘((1^st ‘𝐿)‘𝑝)), ((1^st ‘𝐹)‘((1^st ‘𝑁)‘𝑝))⟩(comp‘𝐸)((1^st ‘𝐾)‘((1^st ‘𝑁)‘𝑝)))((((1^st ‘𝐿)‘𝑝)(2^nd ‘𝐹)((1^st ‘𝑁)‘𝑝))‘(𝑉‘𝑝))) = (((((1^st ‘𝐿)‘𝑝)(2^nd ‘𝐾)((1^st ‘𝑁)‘𝑝))‘(𝑉‘𝑝))(⟨((1^st ‘𝐹)‘((1^st ‘𝐿)‘𝑝)), ((1^st ‘𝐾)‘((1^st ‘𝐿)‘𝑝))⟩(comp‘𝐸)((1^st ‘𝐾)‘((1^st ‘𝑁)‘𝑝)))(𝑅‘((1^st ‘𝐿)‘𝑝))))
12	11	oveq1d 7415	. . . . 5 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘𝐶)) → (((𝑅‘((1^st ‘𝑁)‘𝑝))(⟨((1^st ‘𝐹)‘((1^st ‘𝐿)‘𝑝)), ((1^st ‘𝐹)‘((1^st ‘𝑁)‘𝑝))⟩(comp‘𝐸)((1^st ‘𝐾)‘((1^st ‘𝑁)‘𝑝)))((((1^st ‘𝐿)‘𝑝)(2^nd ‘𝐹)((1^st ‘𝑁)‘𝑝))‘(𝑉‘𝑝)))(⟨((1^st ‘𝐹)‘((1^st ‘𝐺)‘𝑝)), ((1^st ‘𝐹)‘((1^st ‘𝐿)‘𝑝))⟩(comp‘𝐸)((1^st ‘𝐾)‘((1^st ‘𝑁)‘𝑝)))((((1^st ‘𝐺)‘𝑝)(2^nd ‘𝐹)((1^st ‘𝐿)‘𝑝))‘(𝑆‘𝑝))) = ((((((1^st ‘𝐿)‘𝑝)(2^nd ‘𝐾)((1^st ‘𝑁)‘𝑝))‘(𝑉‘𝑝))(⟨((1^st ‘𝐹)‘((1^st ‘𝐿)‘𝑝)), ((1^st ‘𝐾)‘((1^st ‘𝐿)‘𝑝))⟩(comp‘𝐸)((1^st ‘𝐾)‘((1^st ‘𝑁)‘𝑝)))(𝑅‘((1^st ‘𝐿)‘𝑝)))(⟨((1^st ‘𝐹)‘((1^st ‘𝐺)‘𝑝)), ((1^st ‘𝐹)‘((1^st ‘𝐿)‘𝑝))⟩(comp‘𝐸)((1^st ‘𝐾)‘((1^st ‘𝑁)‘𝑝)))((((1^st ‘𝐺)‘𝑝)(2^nd ‘𝐹)((1^st ‘𝐿)‘𝑝))‘(𝑆‘𝑝))))
13	12	oveq2d 7416	. . . 4 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘𝐶)) → ((𝑈‘((1^st ‘𝑁)‘𝑝))(⟨((1^st ‘𝐹)‘((1^st ‘𝐺)‘𝑝)), ((1^st ‘𝐾)‘((1^st ‘𝑁)‘𝑝))⟩(comp‘𝐸)((1^st ‘𝑀)‘((1^st ‘𝑁)‘𝑝)))(((𝑅‘((1^st ‘𝑁)‘𝑝))(⟨((1^st ‘𝐹)‘((1^st ‘𝐿)‘𝑝)), ((1^st ‘𝐹)‘((1^st ‘𝑁)‘𝑝))⟩(comp‘𝐸)((1^st ‘𝐾)‘((1^st ‘𝑁)‘𝑝)))((((1^st ‘𝐿)‘𝑝)(2^nd ‘𝐹)((1^st ‘𝑁)‘𝑝))‘(𝑉‘𝑝)))(⟨((1^st ‘𝐹)‘((1^st ‘𝐺)‘𝑝)), ((1^st ‘𝐹)‘((1^st ‘𝐿)‘𝑝))⟩(comp‘𝐸)((1^st ‘𝐾)‘((1^st ‘𝑁)‘𝑝)))((((1^st ‘𝐺)‘𝑝)(2^nd ‘𝐹)((1^st ‘𝐿)‘𝑝))‘(𝑆‘𝑝)))) = ((𝑈‘((1^st ‘𝑁)‘𝑝))(⟨((1^st ‘𝐹)‘((1^st ‘𝐺)‘𝑝)), ((1^st ‘𝐾)‘((1^st ‘𝑁)‘𝑝))⟩(comp‘𝐸)((1^st ‘𝑀)‘((1^st ‘𝑁)‘𝑝)))((((((1^st ‘𝐿)‘𝑝)(2^nd ‘𝐾)((1^st ‘𝑁)‘𝑝))‘(𝑉‘𝑝))(⟨((1^st ‘𝐹)‘((1^st ‘𝐿)‘𝑝)), ((1^st ‘𝐾)‘((1^st ‘𝐿)‘𝑝))⟩(comp‘𝐸)((1^st ‘𝐾)‘((1^st ‘𝑁)‘𝑝)))(𝑅‘((1^st ‘𝐿)‘𝑝)))(⟨((1^st ‘𝐹)‘((1^st ‘𝐺)‘𝑝)), ((1^st ‘𝐹)‘((1^st ‘𝐿)‘𝑝))⟩(comp‘𝐸)((1^st ‘𝐾)‘((1^st ‘𝑁)‘𝑝)))((((1^st ‘𝐺)‘𝑝)(2^nd ‘𝐹)((1^st ‘𝐿)‘𝑝))‘(𝑆‘𝑝)))))
14	6	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘𝐶)) → 𝑅 ∈ (𝐹(𝐷 Nat 𝐸)𝐾))
15		fucoco.s	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ (𝐺(𝐶 Nat 𝐷)𝐿))
16	15	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘𝐶)) → 𝑆 ∈ (𝐺(𝐶 Nat 𝐷)𝐿))
17		fucoco.u	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ (𝐾(𝐷 Nat 𝐸)𝑀))
18	17	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘𝐶)) → 𝑈 ∈ (𝐾(𝐷 Nat 𝐸)𝑀))
19	2	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘𝐶)) → 𝑉 ∈ (𝐿(𝐶 Nat 𝐷)𝑁))
20	5	natrcl 18000	. . . . . . . 8 ⊢ (𝑅 ∈ (𝐹(𝐷 Nat 𝐸)𝐾) → (𝐹 ∈ (𝐷 Func 𝐸) ∧ 𝐾 ∈ (𝐷 Func 𝐸)))
21	6, 20	syl 18	. . . . . . 7 ⊢ (𝜑 → (𝐹 ∈ (𝐷 Func 𝐸) ∧ 𝐾 ∈ (𝐷 Func 𝐸)))
22	21	simprd 500	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (𝐷 Func 𝐸))
23	22	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘𝐶)) → 𝐾 ∈ (𝐷 Func 𝐸))
24	1	natrcl 18000	. . . . . . . 8 ⊢ (𝑆 ∈ (𝐺(𝐶 Nat 𝐷)𝐿) → (𝐺 ∈ (𝐶 Func 𝐷) ∧ 𝐿 ∈ (𝐶 Func 𝐷)))
25	15, 24	syl 18	. . . . . . 7 ⊢ (𝜑 → (𝐺 ∈ (𝐶 Func 𝐷) ∧ 𝐿 ∈ (𝐶 Func 𝐷)))
26	25	simprd 500	. . . . . 6 ⊢ (𝜑 → 𝐿 ∈ (𝐶 Func 𝐷))
27	26	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘𝐶)) → 𝐿 ∈ (𝐶 Func 𝐷))
28		eqid 2765	. . . . . . 7 ⊢ (Base‘𝐷) = (Base‘𝐷)
29		eqid 2765	. . . . . . 7 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
30		eqid 2765	. . . . . . 7 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
31	22	func1st2nd 49705	. . . . . . . 8 ⊢ (𝜑 → (1^st ‘𝐾)(𝐷 Func 𝐸)(2^nd ‘𝐾))
32	31	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘𝐶)) → (1^st ‘𝐾)(𝐷 Func 𝐸)(2^nd ‘𝐾))
33		eqid 2765	. . . . . . . . 9 ⊢ (Base‘𝐶) = (Base‘𝐶)
34	26	func1st2nd 49705	. . . . . . . . 9 ⊢ (𝜑 → (1^st ‘𝐿)(𝐶 Func 𝐷)(2^nd ‘𝐿))
35	33, 28, 34	funcf1 17913	. . . . . . . 8 ⊢ (𝜑 → (1^st ‘𝐿):(Base‘𝐶)⟶(Base‘𝐷))
36	35	ffvelcdmda 7069	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘𝐶)) → ((1^st ‘𝐿)‘𝑝) ∈ (Base‘𝐷))
37	1	natrcl 18000	. . . . . . . . . . . 12 ⊢ (𝑉 ∈ (𝐿(𝐶 Nat 𝐷)𝑁) → (𝐿 ∈ (𝐶 Func 𝐷) ∧ 𝑁 ∈ (𝐶 Func 𝐷)))
38	2, 37	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐿 ∈ (𝐶 Func 𝐷) ∧ 𝑁 ∈ (𝐶 Func 𝐷)))
39	38	simprd 500	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ (𝐶 Func 𝐷))
40	39	func1st2nd 49705	. . . . . . . . 9 ⊢ (𝜑 → (1^st ‘𝑁)(𝐶 Func 𝐷)(2^nd ‘𝑁))
41	33, 28, 40	funcf1 17913	. . . . . . . 8 ⊢ (𝜑 → (1^st ‘𝑁):(Base‘𝐶)⟶(Base‘𝐷))
42	41	ffvelcdmda 7069	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘𝐶)) → ((1^st ‘𝑁)‘𝑝) ∈ (Base‘𝐷))
43	28, 29, 30, 32, 36, 42	funcf2 17915	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘𝐶)) → (((1^st ‘𝐿)‘𝑝)(2^nd ‘𝐾)((1^st ‘𝑁)‘𝑝)):(((1^st ‘𝐿)‘𝑝)(Hom ‘𝐷)((1^st ‘𝑁)‘𝑝))⟶(((1^st ‘𝐾)‘((1^st ‘𝐿)‘𝑝))(Hom ‘𝐸)((1^st ‘𝐾)‘((1^st ‘𝑁)‘𝑝))))
44	1, 4, 33, 29, 9	natcl 18003	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘𝐶)) → (𝑉‘𝑝) ∈ (((1^st ‘𝐿)‘𝑝)(Hom ‘𝐷)((1^st ‘𝑁)‘𝑝)))
45	43, 44	ffvelcdmd 7070	. . . . 5 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘𝐶)) → ((((1^st ‘𝐿)‘𝑝)(2^nd ‘𝐾)((1^st ‘𝑁)‘𝑝))‘(𝑉‘𝑝)) ∈ (((1^st ‘𝐾)‘((1^st ‘𝐿)‘𝑝))(Hom ‘𝐸)((1^st ‘𝐾)‘((1^st ‘𝑁)‘𝑝))))
46	5, 8, 28, 30, 36	natcl 18003	. . . . 5 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘𝐶)) → (𝑅‘((1^st ‘𝐿)‘𝑝)) ∈ (((1^st ‘𝐹)‘((1^st ‘𝐿)‘𝑝))(Hom ‘𝐸)((1^st ‘𝐾)‘((1^st ‘𝐿)‘𝑝))))
47	14, 16, 18, 19, 9, 23, 27, 45, 46	fucocolem1 49982	. . . 4 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘𝐶)) → (((𝑈‘((1^st ‘𝑁)‘𝑝))(⟨((1^st ‘𝐾)‘((1^st ‘𝐿)‘𝑝)), ((1^st ‘𝐾)‘((1^st ‘𝑁)‘𝑝))⟩(comp‘𝐸)((1^st ‘𝑀)‘((1^st ‘𝑁)‘𝑝)))((((1^st ‘𝐿)‘𝑝)(2^nd ‘𝐾)((1^st ‘𝑁)‘𝑝))‘(𝑉‘𝑝)))(⟨((1^st ‘𝐹)‘((1^st ‘𝐺)‘𝑝)), ((1^st ‘𝐾)‘((1^st ‘𝐿)‘𝑝))⟩(comp‘𝐸)((1^st ‘𝑀)‘((1^st ‘𝑁)‘𝑝)))((𝑅‘((1^st ‘𝐿)‘𝑝))(⟨((1^st ‘𝐹)‘((1^st ‘𝐺)‘𝑝)), ((1^st ‘𝐹)‘((1^st ‘𝐿)‘𝑝))⟩(comp‘𝐸)((1^st ‘𝐾)‘((1^st ‘𝐿)‘𝑝)))((((1^st ‘𝐺)‘𝑝)(2^nd ‘𝐹)((1^st ‘𝐿)‘𝑝))‘(𝑆‘𝑝)))) = ((𝑈‘((1^st ‘𝑁)‘𝑝))(⟨((1^st ‘𝐹)‘((1^st ‘𝐺)‘𝑝)), ((1^st ‘𝐾)‘((1^st ‘𝑁)‘𝑝))⟩(comp‘𝐸)((1^st ‘𝑀)‘((1^st ‘𝑁)‘𝑝)))((((((1^st ‘𝐿)‘𝑝)(2^nd ‘𝐾)((1^st ‘𝑁)‘𝑝))‘(𝑉‘𝑝))(⟨((1^st ‘𝐹)‘((1^st ‘𝐿)‘𝑝)), ((1^st ‘𝐾)‘((1^st ‘𝐿)‘𝑝))⟩(comp‘𝐸)((1^st ‘𝐾)‘((1^st ‘𝑁)‘𝑝)))(𝑅‘((1^st ‘𝐿)‘𝑝)))(⟨((1^st ‘𝐹)‘((1^st ‘𝐺)‘𝑝)), ((1^st ‘𝐹)‘((1^st ‘𝐿)‘𝑝))⟩(comp‘𝐸)((1^st ‘𝐾)‘((1^st ‘𝑁)‘𝑝)))((((1^st ‘𝐺)‘𝑝)(2^nd ‘𝐹)((1^st ‘𝐿)‘𝑝))‘(𝑆‘𝑝)))))
48	13, 47	eqtr4d 2803	. . 3 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘𝐶)) → ((𝑈‘((1^st ‘𝑁)‘𝑝))(⟨((1^st ‘𝐹)‘((1^st ‘𝐺)‘𝑝)), ((1^st ‘𝐾)‘((1^st ‘𝑁)‘𝑝))⟩(comp‘𝐸)((1^st ‘𝑀)‘((1^st ‘𝑁)‘𝑝)))(((𝑅‘((1^st ‘𝑁)‘𝑝))(⟨((1^st ‘𝐹)‘((1^st ‘𝐿)‘𝑝)), ((1^st ‘𝐹)‘((1^st ‘𝑁)‘𝑝))⟩(comp‘𝐸)((1^st ‘𝐾)‘((1^st ‘𝑁)‘𝑝)))((((1^st ‘𝐿)‘𝑝)(2^nd ‘𝐹)((1^st ‘𝑁)‘𝑝))‘(𝑉‘𝑝)))(⟨((1^st ‘𝐹)‘((1^st ‘𝐺)‘𝑝)), ((1^st ‘𝐹)‘((1^st ‘𝐿)‘𝑝))⟩(comp‘𝐸)((1^st ‘𝐾)‘((1^st ‘𝑁)‘𝑝)))((((1^st ‘𝐺)‘𝑝)(2^nd ‘𝐹)((1^st ‘𝐿)‘𝑝))‘(𝑆‘𝑝)))) = (((𝑈‘((1^st ‘𝑁)‘𝑝))(⟨((1^st ‘𝐾)‘((1^st ‘𝐿)‘𝑝)), ((1^st ‘𝐾)‘((1^st ‘𝑁)‘𝑝))⟩(comp‘𝐸)((1^st ‘𝑀)‘((1^st ‘𝑁)‘𝑝)))((((1^st ‘𝐿)‘𝑝)(2^nd ‘𝐾)((1^st ‘𝑁)‘𝑝))‘(𝑉‘𝑝)))(⟨((1^st ‘𝐹)‘((1^st ‘𝐺)‘𝑝)), ((1^st ‘𝐾)‘((1^st ‘𝐿)‘𝑝))⟩(comp‘𝐸)((1^st ‘𝑀)‘((1^st ‘𝑁)‘𝑝)))((𝑅‘((1^st ‘𝐿)‘𝑝))(⟨((1^st ‘𝐹)‘((1^st ‘𝐺)‘𝑝)), ((1^st ‘𝐹)‘((1^st ‘𝐿)‘𝑝))⟩(comp‘𝐸)((1^st ‘𝐾)‘((1^st ‘𝐿)‘𝑝)))((((1^st ‘𝐺)‘𝑝)(2^nd ‘𝐹)((1^st ‘𝐿)‘𝑝))‘(𝑆‘𝑝)))))
49	48	mpteq2dva 5198	. 2 ⊢ (𝜑 → (𝑝 ∈ (Base‘𝐶) ↦ ((𝑈‘((1^st ‘𝑁)‘𝑝))(⟨((1^st ‘𝐹)‘((1^st ‘𝐺)‘𝑝)), ((1^st ‘𝐾)‘((1^st ‘𝑁)‘𝑝))⟩(comp‘𝐸)((1^st ‘𝑀)‘((1^st ‘𝑁)‘𝑝)))(((𝑅‘((1^st ‘𝑁)‘𝑝))(⟨((1^st ‘𝐹)‘((1^st ‘𝐿)‘𝑝)), ((1^st ‘𝐹)‘((1^st ‘𝑁)‘𝑝))⟩(comp‘𝐸)((1^st ‘𝐾)‘((1^st ‘𝑁)‘𝑝)))((((1^st ‘𝐿)‘𝑝)(2^nd ‘𝐹)((1^st ‘𝑁)‘𝑝))‘(𝑉‘𝑝)))(⟨((1^st ‘𝐹)‘((1^st ‘𝐺)‘𝑝)), ((1^st ‘𝐹)‘((1^st ‘𝐿)‘𝑝))⟩(comp‘𝐸)((1^st ‘𝐾)‘((1^st ‘𝑁)‘𝑝)))((((1^st ‘𝐺)‘𝑝)(2^nd ‘𝐹)((1^st ‘𝐿)‘𝑝))‘(𝑆‘𝑝))))) = (𝑝 ∈ (Base‘𝐶) ↦ (((𝑈‘((1^st ‘𝑁)‘𝑝))(⟨((1^st ‘𝐾)‘((1^st ‘𝐿)‘𝑝)), ((1^st ‘𝐾)‘((1^st ‘𝑁)‘𝑝))⟩(comp‘𝐸)((1^st ‘𝑀)‘((1^st ‘𝑁)‘𝑝)))((((1^st ‘𝐿)‘𝑝)(2^nd ‘𝐾)((1^st ‘𝑁)‘𝑝))‘(𝑉‘𝑝)))(⟨((1^st ‘𝐹)‘((1^st ‘𝐺)‘𝑝)), ((1^st ‘𝐾)‘((1^st ‘𝐿)‘𝑝))⟩(comp‘𝐸)((1^st ‘𝑀)‘((1^st ‘𝑁)‘𝑝)))((𝑅‘((1^st ‘𝐿)‘𝑝))(⟨((1^st ‘𝐹)‘((1^st ‘𝐺)‘𝑝)), ((1^st ‘𝐹)‘((1^st ‘𝐿)‘𝑝))⟩(comp‘𝐸)((1^st ‘𝐾)‘((1^st ‘𝐿)‘𝑝)))((((1^st ‘𝐺)‘𝑝)(2^nd ‘𝐹)((1^st ‘𝐿)‘𝑝))‘(𝑆‘𝑝))))))
50		fucoco.o	. . 3 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘_F 𝐸) = ⟨𝑂, 𝑃⟩)
51		fucoco.x	. . 3 ⊢ (𝜑 → 𝑋 = ⟨𝐹, 𝐺⟩)
52		fucoco.y	. . 3 ⊢ (𝜑 → 𝑌 = ⟨𝐾, 𝐿⟩)
53		fucoco.z	. . 3 ⊢ (𝜑 → 𝑍 = ⟨𝑀, 𝑁⟩)
54		fucoco.a	. . 3 ⊢ (𝜑 → 𝐴 = ⟨𝑅, 𝑆⟩)
55		fucoco.b	. . 3 ⊢ (𝜑 → 𝐵 = ⟨𝑈, 𝑉⟩)
56		fucoco.t	. . 3 ⊢ 𝑇 = ((𝐷 FuncCat 𝐸) ×_c (𝐶 FuncCat 𝐷))
57		fucoco.ot	. . 3 ⊢ · = (comp‘𝑇)
58		eqid 2765	. . 3 ⊢ (comp‘𝐷) = (comp‘𝐷)
59	6, 15, 17, 2, 50, 51, 52, 53, 54, 55, 56, 57, 58	fucocolem3 49984	. 2 ⊢ (𝜑 → ((𝑋𝑃𝑍)‘(𝐵(⟨𝑋, 𝑌⟩ · 𝑍)𝐴)) = (𝑝 ∈ (Base‘𝐶) ↦ ((𝑈‘((1^st ‘𝑁)‘𝑝))(⟨((1^st ‘𝐹)‘((1^st ‘𝐺)‘𝑝)), ((1^st ‘𝐾)‘((1^st ‘𝑁)‘𝑝))⟩(comp‘𝐸)((1^st ‘𝑀)‘((1^st ‘𝑁)‘𝑝)))(((𝑅‘((1^st ‘𝑁)‘𝑝))(⟨((1^st ‘𝐹)‘((1^st ‘𝐿)‘𝑝)), ((1^st ‘𝐹)‘((1^st ‘𝑁)‘𝑝))⟩(comp‘𝐸)((1^st ‘𝐾)‘((1^st ‘𝑁)‘𝑝)))((((1^st ‘𝐿)‘𝑝)(2^nd ‘𝐹)((1^st ‘𝑁)‘𝑝))‘(𝑉‘𝑝)))(⟨((1^st ‘𝐹)‘((1^st ‘𝐺)‘𝑝)), ((1^st ‘𝐹)‘((1^st ‘𝐿)‘𝑝))⟩(comp‘𝐸)((1^st ‘𝐾)‘((1^st ‘𝑁)‘𝑝)))((((1^st ‘𝐺)‘𝑝)(2^nd ‘𝐹)((1^st ‘𝐿)‘𝑝))‘(𝑆‘𝑝))))))
60		fucoco.q	. . 3 ⊢ 𝑄 = (𝐶 FuncCat 𝐸)
61		fucoco.oq	. . 3 ⊢ ∙ = (comp‘𝑄)
62	6, 15, 17, 2, 50, 51, 52, 53, 54, 55, 60, 61	fucocolem4 49985	. 2 ⊢ (𝜑 → (((𝑌𝑃𝑍)‘𝐵)(⟨(𝑂‘𝑋), (𝑂‘𝑌)⟩ ∙ (𝑂‘𝑍))((𝑋𝑃𝑌)‘𝐴)) = (𝑝 ∈ (Base‘𝐶) ↦ (((𝑈‘((1^st ‘𝑁)‘𝑝))(⟨((1^st ‘𝐾)‘((1^st ‘𝐿)‘𝑝)), ((1^st ‘𝐾)‘((1^st ‘𝑁)‘𝑝))⟩(comp‘𝐸)((1^st ‘𝑀)‘((1^st ‘𝑁)‘𝑝)))((((1^st ‘𝐿)‘𝑝)(2^nd ‘𝐾)((1^st ‘𝑁)‘𝑝))‘(𝑉‘𝑝)))(⟨((1^st ‘𝐹)‘((1^st ‘𝐺)‘𝑝)), ((1^st ‘𝐾)‘((1^st ‘𝐿)‘𝑝))⟩(comp‘𝐸)((1^st ‘𝑀)‘((1^st ‘𝑁)‘𝑝)))((𝑅‘((1^st ‘𝐿)‘𝑝))(⟨((1^st ‘𝐹)‘((1^st ‘𝐺)‘𝑝)), ((1^st ‘𝐹)‘((1^st ‘𝐿)‘𝑝))⟩(comp‘𝐸)((1^st ‘𝐾)‘((1^st ‘𝐿)‘𝑝)))((((1^st ‘𝐺)‘𝑝)(2^nd ‘𝐹)((1^st ‘𝐿)‘𝑝))‘(𝑆‘𝑝))))))
63	49, 59, 62	3eqtr4d 2810	1 ⊢ (𝜑 → ((𝑋𝑃𝑍)‘(𝐵(⟨𝑋, 𝑌⟩ · 𝑍)𝐴)) = (((𝑌𝑃𝑍)‘𝐵)(⟨(𝑂‘𝑋), (𝑂‘𝑌)⟩ ∙ (𝑂‘𝑍))((𝑋𝑃𝑌)‘𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 = wceq 1563 ∈ wcel 2145 ⟨cop 4591 class class class wbr 5105 ↦ cmpt 5186 ‘cfv 6525 (class class class)co 7400 1^st c1st 7972 2^nd c2nd 7973 Basecbs 17259 Hom chom 17311 compcco 17312 Func cfunc 17901 Nat cnat 17991 FuncCat cfuc 17992 ×_c cxpc 18214 ∘_F cfuco 49945

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-rep 5232 ax-sep 5251 ax-nul 5261 ax-pow 5327 ax-pr 5395 ax-un 7722 ax-cnex 11144 ax-resscn 11145 ax-1cn 11146 ax-icn 11147 ax-addcl 11148 ax-addrcl 11149 ax-mulcl 11150 ax-mulrcl 11151 ax-mulcom 11152 ax-addass 11153 ax-mulass 11154 ax-distr 11155 ax-i2m1 11156 ax-1ne0 11157 ax-1rid 11158 ax-rnegex 11159 ax-rrecex 11160 ax-cnre 11161 ax-pre-lttri 11162 ax-pre-lttrn 11163 ax-pre-ltadd 11164 ax-pre-mulgt0 11165

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3370 df-reu 3371 df-rab 3418 df-v 3459 df-sbc 3748 df-csb 3856 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-pss 3927 df-nul 4289 df-if 4484 df-pw 4560 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4869 df-iun 4954 df-br 5106 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5547 df-eprel 5552 df-po 5560 df-so 5561 df-fr 5605 df-we 5607 df-xp 5658 df-rel 5659 df-cnv 5660 df-co 5661 df-dm 5662 df-rn 5663 df-res 5664 df-ima 5665 df-pred 6292 df-ord 6353 df-on 6354 df-lim 6355 df-suc 6356 df-iota 6481 df-fun 6527 df-fn 6528 df-f 6529 df-f1 6530 df-fo 6531 df-f1o 6532 df-fv 6533 df-riota 7357 df-ov 7403 df-oprab 7404 df-mpo 7405 df-om 7851 df-1st 7974 df-2nd 7975 df-frecs 8266 df-wrecs 8297 df-recs 8346 df-rdg 8385 df-1o 8441 df-er 8682 df-map 8814 df-ixp 8884 df-en 8932 df-dom 8933 df-sdom 8934 df-fin 8935 df-pnf 11233 df-mnf 11234 df-xr 11235 df-ltxr 11236 df-le 11237 df-sub 11431 df-neg 11432 df-nn 12225 df-2 12294 df-3 12295 df-4 12296 df-5 12297 df-6 12298 df-7 12299 df-8 12300 df-9 12301 df-n0 12496 df-z 12583 df-dec 12703 df-uz 12854 df-fz 13527 df-struct 17197 df-slot 17232 df-ndx 17244 df-base 17260 df-hom 17324 df-cco 17325 df-cat 17714 df-cid 17715 df-func 17905 df-cofu 17907 df-nat 17993 df-fuc 17994 df-xpc 18218 df-fuco 49946

This theorem is referenced by: fucoco2 49987

W3C validator