fucoco - Mathbox for Zhi Wang

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Theorem fucoco 49012

Description: Composition in the source category is mapped to composition in the target. See also fucoco2 49013. (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 2734	. . . . . . . . 9 ⊢ (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷)
2		fucoco.v	. . . . . . . . 9 ⊢ (𝜑 → 𝑉 ∈ (𝐿(𝐶 Nat 𝐷)𝑁))
3	1, 2	nat1st2nd 17971	. . . . . . . 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 2734	. . . . . . . . 9 ⊢ (𝐷 Nat 𝐸) = (𝐷 Nat 𝐸)
6		fucoco.r	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ (𝐹(𝐷 Nat 𝐸)𝐾))
7	5, 6	nat1st2nd 17971	. . . . . . . 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 2734	. . . . . . 7 ⊢ (comp‘𝐸) = (comp‘𝐸)
11	4, 8, 9, 10	fuco23alem 49006	. . . . . 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 7429	. . . . 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 7430	. . . 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 17970	. . . . . . . 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 17970	. . . . . . . 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 2734	. . . . . . 7 ⊢ (Base‘𝐷) = (Base‘𝐷)
29		eqid 2734	. . . . . . 7 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
30		eqid 2734	. . . . . . 7 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
31		relfunc 17879	. . . . . . . . 9 ⊢ Rel (𝐷 Func 𝐸)
32		1st2ndbr 8050	. . . . . . . . 9 ⊢ ((Rel (𝐷 Func 𝐸) ∧ 𝐾 ∈ (𝐷 Func 𝐸)) → (1^st ‘𝐾)(𝐷 Func 𝐸)(2^nd ‘𝐾))
33	31, 22, 32	sylancr 587	. . . . . . . 8 ⊢ (𝜑 → (1^st ‘𝐾)(𝐷 Func 𝐸)(2^nd ‘𝐾))
34	33	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘𝐶)) → (1^st ‘𝐾)(𝐷 Func 𝐸)(2^nd ‘𝐾))
35		eqid 2734	. . . . . . . . 9 ⊢ (Base‘𝐶) = (Base‘𝐶)
36		relfunc 17879	. . . . . . . . . 10 ⊢ Rel (𝐶 Func 𝐷)
37		1st2ndbr 8050	. . . . . . . . . 10 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐿 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝐿)(𝐶 Func 𝐷)(2^nd ‘𝐿))
38	36, 26, 37	sylancr 587	. . . . . . . . 9 ⊢ (𝜑 → (1^st ‘𝐿)(𝐶 Func 𝐷)(2^nd ‘𝐿))
39	35, 28, 38	funcf1 17883	. . . . . . . 8 ⊢ (𝜑 → (1^st ‘𝐿):(Base‘𝐶)⟶(Base‘𝐷))
40	39	ffvelcdmda 7085	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘𝐶)) → ((1^st ‘𝐿)‘𝑝) ∈ (Base‘𝐷))
41	1	natrcl 17970	. . . . . . . . . . . 12 ⊢ (𝑉 ∈ (𝐿(𝐶 Nat 𝐷)𝑁) → (𝐿 ∈ (𝐶 Func 𝐷) ∧ 𝑁 ∈ (𝐶 Func 𝐷)))
42	2, 41	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐿 ∈ (𝐶 Func 𝐷) ∧ 𝑁 ∈ (𝐶 Func 𝐷)))
43	42	simprd 495	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ (𝐶 Func 𝐷))
44		1st2ndbr 8050	. . . . . . . . . 10 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝑁 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝑁)(𝐶 Func 𝐷)(2^nd ‘𝑁))
45	36, 43, 44	sylancr 587	. . . . . . . . 9 ⊢ (𝜑 → (1^st ‘𝑁)(𝐶 Func 𝐷)(2^nd ‘𝑁))
46	35, 28, 45	funcf1 17883	. . . . . . . 8 ⊢ (𝜑 → (1^st ‘𝑁):(Base‘𝐶)⟶(Base‘𝐷))
47	46	ffvelcdmda 7085	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘𝐶)) → ((1^st ‘𝑁)‘𝑝) ∈ (Base‘𝐷))
48	28, 29, 30, 34, 40, 47	funcf2 17885	. . . . . 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))))
49	1, 4, 35, 29, 9	natcl 17973	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘𝐶)) → (𝑉‘𝑝) ∈ (((1^st ‘𝐿)‘𝑝)(Hom ‘𝐷)((1^st ‘𝑁)‘𝑝)))
50	48, 49	ffvelcdmd 7086	. . . . 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))))
51	5, 8, 28, 30, 40	natcl 17973	. . . . 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))))
52	14, 16, 18, 19, 9, 23, 27, 50, 51	fucocolem1 49008	. . . 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)))))
53	13, 52	eqtr4d 2772	. . 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)))))
54	53	mpteq2dva 5224	. 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))))))
55		fucoco.o	. . 3 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘_F 𝐸) = ⟨𝑂, 𝑃⟩)
56		fucoco.x	. . 3 ⊢ (𝜑 → 𝑋 = ⟨𝐹, 𝐺⟩)
57		fucoco.y	. . 3 ⊢ (𝜑 → 𝑌 = ⟨𝐾, 𝐿⟩)
58		fucoco.z	. . 3 ⊢ (𝜑 → 𝑍 = ⟨𝑀, 𝑁⟩)
59		fucoco.a	. . 3 ⊢ (𝜑 → 𝐴 = ⟨𝑅, 𝑆⟩)
60		fucoco.b	. . 3 ⊢ (𝜑 → 𝐵 = ⟨𝑈, 𝑉⟩)
61		fucoco.t	. . 3 ⊢ 𝑇 = ((𝐷 FuncCat 𝐸) ×_c (𝐶 FuncCat 𝐷))
62		fucoco.ot	. . 3 ⊢ · = (comp‘𝑇)
63		eqid 2734	. . 3 ⊢ (comp‘𝐷) = (comp‘𝐷)
64	6, 15, 17, 2, 55, 56, 57, 58, 59, 60, 61, 62, 63	fucocolem3 49010	. 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))))))
65		fucoco.q	. . 3 ⊢ 𝑄 = (𝐶 FuncCat 𝐸)
66		fucoco.oq	. . 3 ⊢ ∙ = (comp‘𝑄)
67	6, 15, 17, 2, 55, 56, 57, 58, 59, 60, 65, 66	fucocolem4 49011	. 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))))))
68	54, 64, 67	3eqtr4d 2779	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 1539 ∈ wcel 2107 ⟨cop 4614 class class class wbr 5125 ↦ cmpt 5207 Rel wrel 5672 ‘cfv 6542 (class class class)co 7414 1^st c1st 7995 2^nd c2nd 7996 Basecbs 17230 Hom chom 17285 compcco 17286 Func cfunc 17871 Nat cnat 17961 FuncCat cfuc 17962 ×_c cxpc 18184 ∘_F cfuco 48971

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2706 ax-rep 5261 ax-sep 5278 ax-nul 5288 ax-pow 5347 ax-pr 5414 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2808 df-nfc 2884 df-ne 2932 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3364 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3773 df-csb 3882 df-dif 3936 df-un 3938 df-in 3940 df-ss 3950 df-pss 3953 df-nul 4316 df-if 4508 df-pw 4584 df-sn 4609 df-pr 4611 df-tp 4613 df-op 4615 df-uni 4890 df-iun 4975 df-br 5126 df-opab 5188 df-mpt 5208 df-tr 5242 df-id 5560 df-eprel 5566 df-po 5574 df-so 5575 df-fr 5619 df-we 5621 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6303 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7371 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7871 df-1st 7997 df-2nd 7998 df-frecs 8289 df-wrecs 8320 df-recs 8394 df-rdg 8433 df-1o 8489 df-er 8728 df-map 8851 df-ixp 8921 df-en 8969 df-dom 8970 df-sdom 8971 df-fin 8972 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11477 df-neg 11478 df-nn 12250 df-2 12312 df-3 12313 df-4 12314 df-5 12315 df-6 12316 df-7 12317 df-8 12318 df-9 12319 df-n0 12511 df-z 12598 df-dec 12718 df-uz 12862 df-fz 13531 df-struct 17167 df-slot 17202 df-ndx 17214 df-base 17231 df-hom 17298 df-cco 17299 df-cat 17683 df-cid 17684 df-func 17875 df-cofu 17877 df-nat 17963 df-fuc 17964 df-xpc 18188 df-fuco 48972

This theorem is referenced by: fucoco2 49013

W3C validator