fucoco - Mathbox for Zhi Wang

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Theorem fucoco 49346

Description: Composition in the source category is mapped to composition in the target. See also fucoco2 49347. (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 2729	. . . . . . . . 9 ⊢ (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷)
2		fucoco.v	. . . . . . . . 9 ⊢ (𝜑 → 𝑉 ∈ (𝐿(𝐶 Nat 𝐷)𝑁))
3	1, 2	nat1st2nd 17916	. . . . . . . 8 ⊢ (𝜑 → 𝑉 ∈ (⟨(1^st ‘𝐿), (2^nd ‘𝐿)⟩(𝐶 Nat 𝐷)⟨(1^st ‘𝑁), (2^nd ‘𝑁)⟩))
4	3	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘𝐶)) → 𝑉 ∈ (⟨(1^st ‘𝐿), (2^nd ‘𝐿)⟩(𝐶 Nat 𝐷)⟨(1^st ‘𝑁), (2^nd ‘𝑁)⟩))
5		eqid 2729	. . . . . . . . 9 ⊢ (𝐷 Nat 𝐸) = (𝐷 Nat 𝐸)
6		fucoco.r	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ (𝐹(𝐷 Nat 𝐸)𝐾))
7	5, 6	nat1st2nd 17916	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ (⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩(𝐷 Nat 𝐸)⟨(1^st ‘𝐾), (2^nd ‘𝐾)⟩))
8	7	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘𝐶)) → 𝑅 ∈ (⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩(𝐷 Nat 𝐸)⟨(1^st ‘𝐾), (2^nd ‘𝐾)⟩))
9		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘𝐶)) → 𝑝 ∈ (Base‘𝐶))
10		eqid 2729	. . . . . . 7 ⊢ (comp‘𝐸) = (comp‘𝐸)
11	4, 8, 9, 10	fuco23alem 49340	. . . . . 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 7402	. . . . 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 7403	. . . 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 480	. . . . 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 480	. . . . 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 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘𝐶)) → 𝑈 ∈ (𝐾(𝐷 Nat 𝐸)𝑀))
19	2	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘𝐶)) → 𝑉 ∈ (𝐿(𝐶 Nat 𝐷)𝑁))
20	5	natrcl 17915	. . . . . . . 8 ⊢ (𝑅 ∈ (𝐹(𝐷 Nat 𝐸)𝐾) → (𝐹 ∈ (𝐷 Func 𝐸) ∧ 𝐾 ∈ (𝐷 Func 𝐸)))
21	6, 20	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝐹 ∈ (𝐷 Func 𝐸) ∧ 𝐾 ∈ (𝐷 Func 𝐸)))
22	21	simprd 495	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (𝐷 Func 𝐸))
23	22	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘𝐶)) → 𝐾 ∈ (𝐷 Func 𝐸))
24	1	natrcl 17915	. . . . . . . 8 ⊢ (𝑆 ∈ (𝐺(𝐶 Nat 𝐷)𝐿) → (𝐺 ∈ (𝐶 Func 𝐷) ∧ 𝐿 ∈ (𝐶 Func 𝐷)))
25	15, 24	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝐺 ∈ (𝐶 Func 𝐷) ∧ 𝐿 ∈ (𝐶 Func 𝐷)))
26	25	simprd 495	. . . . . 6 ⊢ (𝜑 → 𝐿 ∈ (𝐶 Func 𝐷))
27	26	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘𝐶)) → 𝐿 ∈ (𝐶 Func 𝐷))
28		eqid 2729	. . . . . . 7 ⊢ (Base‘𝐷) = (Base‘𝐷)
29		eqid 2729	. . . . . . 7 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
30		eqid 2729	. . . . . . 7 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
31	22	func1st2nd 49065	. . . . . . . 8 ⊢ (𝜑 → (1^st ‘𝐾)(𝐷 Func 𝐸)(2^nd ‘𝐾))
32	31	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘𝐶)) → (1^st ‘𝐾)(𝐷 Func 𝐸)(2^nd ‘𝐾))
33		eqid 2729	. . . . . . . . 9 ⊢ (Base‘𝐶) = (Base‘𝐶)
34	26	func1st2nd 49065	. . . . . . . . 9 ⊢ (𝜑 → (1^st ‘𝐿)(𝐶 Func 𝐷)(2^nd ‘𝐿))
35	33, 28, 34	funcf1 17828	. . . . . . . 8 ⊢ (𝜑 → (1^st ‘𝐿):(Base‘𝐶)⟶(Base‘𝐷))
36	35	ffvelcdmda 7056	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘𝐶)) → ((1^st ‘𝐿)‘𝑝) ∈ (Base‘𝐷))
37	1	natrcl 17915	. . . . . . . . . . . 12 ⊢ (𝑉 ∈ (𝐿(𝐶 Nat 𝐷)𝑁) → (𝐿 ∈ (𝐶 Func 𝐷) ∧ 𝑁 ∈ (𝐶 Func 𝐷)))
38	2, 37	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐿 ∈ (𝐶 Func 𝐷) ∧ 𝑁 ∈ (𝐶 Func 𝐷)))
39	38	simprd 495	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ (𝐶 Func 𝐷))
40	39	func1st2nd 49065	. . . . . . . . 9 ⊢ (𝜑 → (1^st ‘𝑁)(𝐶 Func 𝐷)(2^nd ‘𝑁))
41	33, 28, 40	funcf1 17828	. . . . . . . 8 ⊢ (𝜑 → (1^st ‘𝑁):(Base‘𝐶)⟶(Base‘𝐷))
42	41	ffvelcdmda 7056	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘𝐶)) → ((1^st ‘𝑁)‘𝑝) ∈ (Base‘𝐷))
43	28, 29, 30, 32, 36, 42	funcf2 17830	. . . . . 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 17918	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘𝐶)) → (𝑉‘𝑝) ∈ (((1^st ‘𝐿)‘𝑝)(Hom ‘𝐷)((1^st ‘𝑁)‘𝑝)))
45	43, 44	ffvelcdmd 7057	. . . . 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 17918	. . . . 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 49342	. . . 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 2767	. . 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 5200	. 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 2729	. . 3 ⊢ (comp‘𝐷) = (comp‘𝐷)
59	6, 15, 17, 2, 50, 51, 52, 53, 54, 55, 56, 57, 58	fucocolem3 49344	. 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 49345	. 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 2774	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 395 = wceq 1540 ∈ wcel 2109 ⟨cop 4595 class class class wbr 5107 ↦ cmpt 5188 ‘cfv 6511 (class class class)co 7387 1^st c1st 7966 2^nd c2nd 7967 Basecbs 17179 Hom chom 17231 compcco 17232 Func cfunc 17816 Nat cnat 17906 FuncCat cfuc 17907 ×_c cxpc 18129 ∘_F cfuco 49305

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4872 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-1st 7968 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-er 8671 df-map 8801 df-ixp 8871 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-7 12254 df-8 12255 df-9 12256 df-n0 12443 df-z 12530 df-dec 12650 df-uz 12794 df-fz 13469 df-struct 17117 df-slot 17152 df-ndx 17164 df-base 17180 df-hom 17244 df-cco 17245 df-cat 17629 df-cid 17630 df-func 17820 df-cofu 17822 df-nat 17908 df-fuc 17909 df-xpc 18133 df-fuco 49306

This theorem is referenced by: fucoco2 49347

W3C validator