fucocolem3 - Mathbox for Zhi Wang

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Theorem fucocolem3 50692

Description: Lemma for fucoco 50694. The composed natural transformations are mapped to composition of 4 natural transformations. (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⟩)
fucocolem2.t	⊢ 𝑇 = ((𝐷 FuncCat 𝐸) ×_c (𝐶 FuncCat 𝐷))
fucocolem2.ot	⊢ · = (comp‘𝑇)
fucocolem2.od	⊢ ∗ = (comp‘𝐷)

**Assertion**
Ref	Expression
fucocolem3	⊢ (𝜑 → ((𝑋𝑃𝑍)‘(𝐵(⟨𝑋, 𝑌⟩ · 𝑍)𝐴)) = (𝑥 ∈ (Base‘𝐶) ↦ ((𝑈‘((1^st ‘𝑁)‘𝑥))(⟨((1^st ‘𝐹)‘((1^st ‘𝐺)‘𝑥)), ((1^st ‘𝐾)‘((1^st ‘𝑁)‘𝑥))⟩(comp‘𝐸)((1^st ‘𝑀)‘((1^st ‘𝑁)‘𝑥)))(((𝑅‘((1^st ‘𝑁)‘𝑥))(⟨((1^st ‘𝐹)‘((1^st ‘𝐿)‘𝑥)), ((1^st ‘𝐹)‘((1^st ‘𝑁)‘𝑥))⟩(comp‘𝐸)((1^st ‘𝐾)‘((1^st ‘𝑁)‘𝑥)))((((1^st ‘𝐿)‘𝑥)(2^nd ‘𝐹)((1^st ‘𝑁)‘𝑥))‘(𝑉‘𝑥)))(⟨((1^st ‘𝐹)‘((1^st ‘𝐺)‘𝑥)), ((1^st ‘𝐹)‘((1^st ‘𝐿)‘𝑥))⟩(comp‘𝐸)((1^st ‘𝐾)‘((1^st ‘𝑁)‘𝑥)))((((1^st ‘𝐺)‘𝑥)(2^nd ‘𝐹)((1^st ‘𝐿)‘𝑥))‘(𝑆‘𝑥))))))

Distinct variable groups: 𝑥, ∗ 𝑥,𝐶 𝑥,𝐷 𝑥,𝐸 𝑥,𝐹 𝑥,𝐺 𝑥,𝐾 𝑥,𝐿 𝑥,𝑀 𝑥,𝑁 𝑥,𝑅 𝑥,𝑆 𝑥,𝑈 𝑥,𝑉 𝑥,𝑋 𝑥,𝑍 𝜑,𝑥 Allowed substitution hints: 𝐴(𝑥) 𝐵(𝑥) 𝑃(𝑥) 𝑇(𝑥) · (𝑥) 𝑂(𝑥) 𝑌(𝑥)

**Proof of Theorem fucocolem3**
Step	Hyp	Ref	Expression
1		fucoco.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ (𝐹(𝐷 Nat 𝐸)𝐾))
2		fucoco.s	. . 3 ⊢ (𝜑 → 𝑆 ∈ (𝐺(𝐶 Nat 𝐷)𝐿))
3		fucoco.u	. . 3 ⊢ (𝜑 → 𝑈 ∈ (𝐾(𝐷 Nat 𝐸)𝑀))
4		fucoco.v	. . 3 ⊢ (𝜑 → 𝑉 ∈ (𝐿(𝐶 Nat 𝐷)𝑁))
5		fucoco.o	. . 3 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘_F 𝐸) = ⟨𝑂, 𝑃⟩)
6		fucoco.x	. . 3 ⊢ (𝜑 → 𝑋 = ⟨𝐹, 𝐺⟩)
7		fucoco.y	. . 3 ⊢ (𝜑 → 𝑌 = ⟨𝐾, 𝐿⟩)
8		fucoco.z	. . 3 ⊢ (𝜑 → 𝑍 = ⟨𝑀, 𝑁⟩)
9		fucoco.a	. . 3 ⊢ (𝜑 → 𝐴 = ⟨𝑅, 𝑆⟩)
10		fucoco.b	. . 3 ⊢ (𝜑 → 𝐵 = ⟨𝑈, 𝑉⟩)
11		fucocolem2.t	. . 3 ⊢ 𝑇 = ((𝐷 FuncCat 𝐸) ×_c (𝐶 FuncCat 𝐷))
12		fucocolem2.ot	. . 3 ⊢ · = (comp‘𝑇)
13		fucocolem2.od	. . 3 ⊢ ∗ = (comp‘𝐷)
14	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13	fucocolem2 50691	. 2 ⊢ (𝜑 → ((𝑋𝑃𝑍)‘(𝐵(⟨𝑋, 𝑌⟩ · 𝑍)𝐴)) = (𝑥 ∈ (Base‘𝐶) ↦ (((𝑈‘((1^st ‘𝑁)‘𝑥))(⟨((1^st ‘𝐹)‘((1^st ‘𝑁)‘𝑥)), ((1^st ‘𝐾)‘((1^st ‘𝑁)‘𝑥))⟩(comp‘𝐸)((1^st ‘𝑀)‘((1^st ‘𝑁)‘𝑥)))(𝑅‘((1^st ‘𝑁)‘𝑥)))(⟨((1^st ‘𝐹)‘((1^st ‘𝐺)‘𝑥)), ((1^st ‘𝐹)‘((1^st ‘𝑁)‘𝑥))⟩(comp‘𝐸)((1^st ‘𝑀)‘((1^st ‘𝑁)‘𝑥)))((((1^st ‘𝐺)‘𝑥)(2^nd ‘𝐹)((1^st ‘𝑁)‘𝑥))‘((𝑉‘𝑥)(⟨((1^st ‘𝐺)‘𝑥), ((1^st ‘𝐿)‘𝑥)⟩ ∗ ((1^st ‘𝑁)‘𝑥))(𝑆‘𝑥))))))
15		eqid 2763	. . . . . 6 ⊢ (Base‘𝐷) = (Base‘𝐷)
16		eqid 2763	. . . . . 6 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
17		eqid 2763	. . . . . 6 ⊢ (comp‘𝐸) = (comp‘𝐸)
18		eqid 2763	. . . . . . . . . . 11 ⊢ (𝐷 Nat 𝐸) = (𝐷 Nat 𝐸)
19	18	natrcl 18160	. . . . . . . . . 10 ⊢ (𝑅 ∈ (𝐹(𝐷 Nat 𝐸)𝐾) → (𝐹 ∈ (𝐷 Func 𝐸) ∧ 𝐾 ∈ (𝐷 Func 𝐸)))
20	1, 19	syl 18	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 ∈ (𝐷 Func 𝐸) ∧ 𝐾 ∈ (𝐷 Func 𝐸)))
21	20	simpld 500	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (𝐷 Func 𝐸))
22	21	func1st2nd 50405	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝐹)(𝐷 Func 𝐸)(2^nd ‘𝐹))
23	22	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (1^st ‘𝐹)(𝐷 Func 𝐸)(2^nd ‘𝐹))
24		eqid 2763	. . . . . . . 8 ⊢ (Base‘𝐶) = (Base‘𝐶)
25		eqid 2763	. . . . . . . . . . . 12 ⊢ (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷)
26	25	natrcl 18160	. . . . . . . . . . 11 ⊢ (𝑆 ∈ (𝐺(𝐶 Nat 𝐷)𝐿) → (𝐺 ∈ (𝐶 Func 𝐷) ∧ 𝐿 ∈ (𝐶 Func 𝐷)))
27	2, 26	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (𝐺 ∈ (𝐶 Func 𝐷) ∧ 𝐿 ∈ (𝐶 Func 𝐷)))
28	27	simpld 500	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷))
29	28	func1st2nd 50405	. . . . . . . 8 ⊢ (𝜑 → (1^st ‘𝐺)(𝐶 Func 𝐷)(2^nd ‘𝐺))
30	24, 15, 29	funcf1 18073	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝐺):(Base‘𝐶)⟶(Base‘𝐷))
31	30	ffvelcdmda 7111	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1^st ‘𝐺)‘𝑥) ∈ (Base‘𝐷))
32	27	simprd 501	. . . . . . . . 9 ⊢ (𝜑 → 𝐿 ∈ (𝐶 Func 𝐷))
33	32	func1st2nd 50405	. . . . . . . 8 ⊢ (𝜑 → (1^st ‘𝐿)(𝐶 Func 𝐷)(2^nd ‘𝐿))
34	24, 15, 33	funcf1 18073	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝐿):(Base‘𝐶)⟶(Base‘𝐷))
35	34	ffvelcdmda 7111	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1^st ‘𝐿)‘𝑥) ∈ (Base‘𝐷))
36	25	natrcl 18160	. . . . . . . . . . 11 ⊢ (𝑉 ∈ (𝐿(𝐶 Nat 𝐷)𝑁) → (𝐿 ∈ (𝐶 Func 𝐷) ∧ 𝑁 ∈ (𝐶 Func 𝐷)))
37	4, 36	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (𝐿 ∈ (𝐶 Func 𝐷) ∧ 𝑁 ∈ (𝐶 Func 𝐷)))
38	37	simprd 501	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ (𝐶 Func 𝐷))
39	38	func1st2nd 50405	. . . . . . . 8 ⊢ (𝜑 → (1^st ‘𝑁)(𝐶 Func 𝐷)(2^nd ‘𝑁))
40	24, 15, 39	funcf1 18073	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝑁):(Base‘𝐶)⟶(Base‘𝐷))
41	40	ffvelcdmda 7111	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1^st ‘𝑁)‘𝑥) ∈ (Base‘𝐷))
42	25, 2	nat1st2nd 18161	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ (⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩(𝐶 Nat 𝐷)⟨(1^st ‘𝐿), (2^nd ‘𝐿)⟩))
43	42	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑆 ∈ (⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩(𝐶 Nat 𝐷)⟨(1^st ‘𝐿), (2^nd ‘𝐿)⟩))
44		simpr 490	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
45	25, 43, 24, 16, 44	natcl 18163	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (𝑆‘𝑥) ∈ (((1^st ‘𝐺)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐿)‘𝑥)))
46	25, 4	nat1st2nd 18161	. . . . . . . 8 ⊢ (𝜑 → 𝑉 ∈ (⟨(1^st ‘𝐿), (2^nd ‘𝐿)⟩(𝐶 Nat 𝐷)⟨(1^st ‘𝑁), (2^nd ‘𝑁)⟩))
47	46	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑉 ∈ (⟨(1^st ‘𝐿), (2^nd ‘𝐿)⟩(𝐶 Nat 𝐷)⟨(1^st ‘𝑁), (2^nd ‘𝑁)⟩))
48	25, 47, 24, 16, 44	natcl 18163	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (𝑉‘𝑥) ∈ (((1^st ‘𝐿)‘𝑥)(Hom ‘𝐷)((1^st ‘𝑁)‘𝑥)))
49	15, 16, 13, 17, 23, 31, 35, 41, 45, 48	funcco 18078	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((((1^st ‘𝐺)‘𝑥)(2^nd ‘𝐹)((1^st ‘𝑁)‘𝑥))‘((𝑉‘𝑥)(⟨((1^st ‘𝐺)‘𝑥), ((1^st ‘𝐿)‘𝑥)⟩ ∗ ((1^st ‘𝑁)‘𝑥))(𝑆‘𝑥))) = (((((1^st ‘𝐿)‘𝑥)(2^nd ‘𝐹)((1^st ‘𝑁)‘𝑥))‘(𝑉‘𝑥))(⟨((1^st ‘𝐹)‘((1^st ‘𝐺)‘𝑥)), ((1^st ‘𝐹)‘((1^st ‘𝐿)‘𝑥))⟩(comp‘𝐸)((1^st ‘𝐹)‘((1^st ‘𝑁)‘𝑥)))((((1^st ‘𝐺)‘𝑥)(2^nd ‘𝐹)((1^st ‘𝐿)‘𝑥))‘(𝑆‘𝑥))))
50	49	oveq2d 7462	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (((𝑈‘((1^st ‘𝑁)‘𝑥))(⟨((1^st ‘𝐹)‘((1^st ‘𝑁)‘𝑥)), ((1^st ‘𝐾)‘((1^st ‘𝑁)‘𝑥))⟩(comp‘𝐸)((1^st ‘𝑀)‘((1^st ‘𝑁)‘𝑥)))(𝑅‘((1^st ‘𝑁)‘𝑥)))(⟨((1^st ‘𝐹)‘((1^st ‘𝐺)‘𝑥)), ((1^st ‘𝐹)‘((1^st ‘𝑁)‘𝑥))⟩(comp‘𝐸)((1^st ‘𝑀)‘((1^st ‘𝑁)‘𝑥)))((((1^st ‘𝐺)‘𝑥)(2^nd ‘𝐹)((1^st ‘𝑁)‘𝑥))‘((𝑉‘𝑥)(⟨((1^st ‘𝐺)‘𝑥), ((1^st ‘𝐿)‘𝑥)⟩ ∗ ((1^st ‘𝑁)‘𝑥))(𝑆‘𝑥)))) = (((𝑈‘((1^st ‘𝑁)‘𝑥))(⟨((1^st ‘𝐹)‘((1^st ‘𝑁)‘𝑥)), ((1^st ‘𝐾)‘((1^st ‘𝑁)‘𝑥))⟩(comp‘𝐸)((1^st ‘𝑀)‘((1^st ‘𝑁)‘𝑥)))(𝑅‘((1^st ‘𝑁)‘𝑥)))(⟨((1^st ‘𝐹)‘((1^st ‘𝐺)‘𝑥)), ((1^st ‘𝐹)‘((1^st ‘𝑁)‘𝑥))⟩(comp‘𝐸)((1^st ‘𝑀)‘((1^st ‘𝑁)‘𝑥)))(((((1^st ‘𝐿)‘𝑥)(2^nd ‘𝐹)((1^st ‘𝑁)‘𝑥))‘(𝑉‘𝑥))(⟨((1^st ‘𝐹)‘((1^st ‘𝐺)‘𝑥)), ((1^st ‘𝐹)‘((1^st ‘𝐿)‘𝑥))⟩(comp‘𝐸)((1^st ‘𝐹)‘((1^st ‘𝑁)‘𝑥)))((((1^st ‘𝐺)‘𝑥)(2^nd ‘𝐹)((1^st ‘𝐿)‘𝑥))‘(𝑆‘𝑥)))))
51	1	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑅 ∈ (𝐹(𝐷 Nat 𝐸)𝐾))
52	2	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑆 ∈ (𝐺(𝐶 Nat 𝐷)𝐿))
53	3	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑈 ∈ (𝐾(𝐷 Nat 𝐸)𝑀))
54	4	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑉 ∈ (𝐿(𝐶 Nat 𝐷)𝑁))
55	21	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐹 ∈ (𝐷 Func 𝐸))
56	38	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑁 ∈ (𝐶 Func 𝐷))
57	18, 1	nat1st2nd 18161	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ (⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩(𝐷 Nat 𝐸)⟨(1^st ‘𝐾), (2^nd ‘𝐾)⟩))
58	57	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑅 ∈ (⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩(𝐷 Nat 𝐸)⟨(1^st ‘𝐾), (2^nd ‘𝐾)⟩))
59		eqid 2763	. . . . . 6 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
60	18, 58, 15, 59, 41	natcl 18163	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (𝑅‘((1^st ‘𝑁)‘𝑥)) ∈ (((1^st ‘𝐹)‘((1^st ‘𝑁)‘𝑥))(Hom ‘𝐸)((1^st ‘𝐾)‘((1^st ‘𝑁)‘𝑥))))
61	15, 16, 59, 23, 35, 41	funcf2 18075	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (((1^st ‘𝐿)‘𝑥)(2^nd ‘𝐹)((1^st ‘𝑁)‘𝑥)):(((1^st ‘𝐿)‘𝑥)(Hom ‘𝐷)((1^st ‘𝑁)‘𝑥))⟶(((1^st ‘𝐹)‘((1^st ‘𝐿)‘𝑥))(Hom ‘𝐸)((1^st ‘𝐹)‘((1^st ‘𝑁)‘𝑥))))
62	61, 48	ffvelcdmd 7112	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((((1^st ‘𝐿)‘𝑥)(2^nd ‘𝐹)((1^st ‘𝑁)‘𝑥))‘(𝑉‘𝑥)) ∈ (((1^st ‘𝐹)‘((1^st ‘𝐿)‘𝑥))(Hom ‘𝐸)((1^st ‘𝐹)‘((1^st ‘𝑁)‘𝑥))))
63	51, 52, 53, 54, 44, 55, 56, 60, 62	fucocolem1 50690	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (((𝑈‘((1^st ‘𝑁)‘𝑥))(⟨((1^st ‘𝐹)‘((1^st ‘𝑁)‘𝑥)), ((1^st ‘𝐾)‘((1^st ‘𝑁)‘𝑥))⟩(comp‘𝐸)((1^st ‘𝑀)‘((1^st ‘𝑁)‘𝑥)))(𝑅‘((1^st ‘𝑁)‘𝑥)))(⟨((1^st ‘𝐹)‘((1^st ‘𝐺)‘𝑥)), ((1^st ‘𝐹)‘((1^st ‘𝑁)‘𝑥))⟩(comp‘𝐸)((1^st ‘𝑀)‘((1^st ‘𝑁)‘𝑥)))(((((1^st ‘𝐿)‘𝑥)(2^nd ‘𝐹)((1^st ‘𝑁)‘𝑥))‘(𝑉‘𝑥))(⟨((1^st ‘𝐹)‘((1^st ‘𝐺)‘𝑥)), ((1^st ‘𝐹)‘((1^st ‘𝐿)‘𝑥))⟩(comp‘𝐸)((1^st ‘𝐹)‘((1^st ‘𝑁)‘𝑥)))((((1^st ‘𝐺)‘𝑥)(2^nd ‘𝐹)((1^st ‘𝐿)‘𝑥))‘(𝑆‘𝑥)))) = ((𝑈‘((1^st ‘𝑁)‘𝑥))(⟨((1^st ‘𝐹)‘((1^st ‘𝐺)‘𝑥)), ((1^st ‘𝐾)‘((1^st ‘𝑁)‘𝑥))⟩(comp‘𝐸)((1^st ‘𝑀)‘((1^st ‘𝑁)‘𝑥)))(((𝑅‘((1^st ‘𝑁)‘𝑥))(⟨((1^st ‘𝐹)‘((1^st ‘𝐿)‘𝑥)), ((1^st ‘𝐹)‘((1^st ‘𝑁)‘𝑥))⟩(comp‘𝐸)((1^st ‘𝐾)‘((1^st ‘𝑁)‘𝑥)))((((1^st ‘𝐿)‘𝑥)(2^nd ‘𝐹)((1^st ‘𝑁)‘𝑥))‘(𝑉‘𝑥)))(⟨((1^st ‘𝐹)‘((1^st ‘𝐺)‘𝑥)), ((1^st ‘𝐹)‘((1^st ‘𝐿)‘𝑥))⟩(comp‘𝐸)((1^st ‘𝐾)‘((1^st ‘𝑁)‘𝑥)))((((1^st ‘𝐺)‘𝑥)(2^nd ‘𝐹)((1^st ‘𝐿)‘𝑥))‘(𝑆‘𝑥)))))
64	50, 63	eqtrd 2798	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (((𝑈‘((1^st ‘𝑁)‘𝑥))(⟨((1^st ‘𝐹)‘((1^st ‘𝑁)‘𝑥)), ((1^st ‘𝐾)‘((1^st ‘𝑁)‘𝑥))⟩(comp‘𝐸)((1^st ‘𝑀)‘((1^st ‘𝑁)‘𝑥)))(𝑅‘((1^st ‘𝑁)‘𝑥)))(⟨((1^st ‘𝐹)‘((1^st ‘𝐺)‘𝑥)), ((1^st ‘𝐹)‘((1^st ‘𝑁)‘𝑥))⟩(comp‘𝐸)((1^st ‘𝑀)‘((1^st ‘𝑁)‘𝑥)))((((1^st ‘𝐺)‘𝑥)(2^nd ‘𝐹)((1^st ‘𝑁)‘𝑥))‘((𝑉‘𝑥)(⟨((1^st ‘𝐺)‘𝑥), ((1^st ‘𝐿)‘𝑥)⟩ ∗ ((1^st ‘𝑁)‘𝑥))(𝑆‘𝑥)))) = ((𝑈‘((1^st ‘𝑁)‘𝑥))(⟨((1^st ‘𝐹)‘((1^st ‘𝐺)‘𝑥)), ((1^st ‘𝐾)‘((1^st ‘𝑁)‘𝑥))⟩(comp‘𝐸)((1^st ‘𝑀)‘((1^st ‘𝑁)‘𝑥)))(((𝑅‘((1^st ‘𝑁)‘𝑥))(⟨((1^st ‘𝐹)‘((1^st ‘𝐿)‘𝑥)), ((1^st ‘𝐹)‘((1^st ‘𝑁)‘𝑥))⟩(comp‘𝐸)((1^st ‘𝐾)‘((1^st ‘𝑁)‘𝑥)))((((1^st ‘𝐿)‘𝑥)(2^nd ‘𝐹)((1^st ‘𝑁)‘𝑥))‘(𝑉‘𝑥)))(⟨((1^st ‘𝐹)‘((1^st ‘𝐺)‘𝑥)), ((1^st ‘𝐹)‘((1^st ‘𝐿)‘𝑥))⟩(comp‘𝐸)((1^st ‘𝐾)‘((1^st ‘𝑁)‘𝑥)))((((1^st ‘𝐺)‘𝑥)(2^nd ‘𝐹)((1^st ‘𝐿)‘𝑥))‘(𝑆‘𝑥)))))
65	64	mpteq2dva 5202	. 2 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐶) ↦ (((𝑈‘((1^st ‘𝑁)‘𝑥))(⟨((1^st ‘𝐹)‘((1^st ‘𝑁)‘𝑥)), ((1^st ‘𝐾)‘((1^st ‘𝑁)‘𝑥))⟩(comp‘𝐸)((1^st ‘𝑀)‘((1^st ‘𝑁)‘𝑥)))(𝑅‘((1^st ‘𝑁)‘𝑥)))(⟨((1^st ‘𝐹)‘((1^st ‘𝐺)‘𝑥)), ((1^st ‘𝐹)‘((1^st ‘𝑁)‘𝑥))⟩(comp‘𝐸)((1^st ‘𝑀)‘((1^st ‘𝑁)‘𝑥)))((((1^st ‘𝐺)‘𝑥)(2^nd ‘𝐹)((1^st ‘𝑁)‘𝑥))‘((𝑉‘𝑥)(⟨((1^st ‘𝐺)‘𝑥), ((1^st ‘𝐿)‘𝑥)⟩ ∗ ((1^st ‘𝑁)‘𝑥))(𝑆‘𝑥))))) = (𝑥 ∈ (Base‘𝐶) ↦ ((𝑈‘((1^st ‘𝑁)‘𝑥))(⟨((1^st ‘𝐹)‘((1^st ‘𝐺)‘𝑥)), ((1^st ‘𝐾)‘((1^st ‘𝑁)‘𝑥))⟩(comp‘𝐸)((1^st ‘𝑀)‘((1^st ‘𝑁)‘𝑥)))(((𝑅‘((1^st ‘𝑁)‘𝑥))(⟨((1^st ‘𝐹)‘((1^st ‘𝐿)‘𝑥)), ((1^st ‘𝐹)‘((1^st ‘𝑁)‘𝑥))⟩(comp‘𝐸)((1^st ‘𝐾)‘((1^st ‘𝑁)‘𝑥)))((((1^st ‘𝐿)‘𝑥)(2^nd ‘𝐹)((1^st ‘𝑁)‘𝑥))‘(𝑉‘𝑥)))(⟨((1^st ‘𝐹)‘((1^st ‘𝐺)‘𝑥)), ((1^st ‘𝐹)‘((1^st ‘𝐿)‘𝑥))⟩(comp‘𝐸)((1^st ‘𝐾)‘((1^st ‘𝑁)‘𝑥)))((((1^st ‘𝐺)‘𝑥)(2^nd ‘𝐹)((1^st ‘𝐿)‘𝑥))‘(𝑆‘𝑥))))))
66	14, 65	eqtrd 2798	1 ⊢ (𝜑 → ((𝑋𝑃𝑍)‘(𝐵(⟨𝑋, 𝑌⟩ · 𝑍)𝐴)) = (𝑥 ∈ (Base‘𝐶) ↦ ((𝑈‘((1^st ‘𝑁)‘𝑥))(⟨((1^st ‘𝐹)‘((1^st ‘𝐺)‘𝑥)), ((1^st ‘𝐾)‘((1^st ‘𝑁)‘𝑥))⟩(comp‘𝐸)((1^st ‘𝑀)‘((1^st ‘𝑁)‘𝑥)))(((𝑅‘((1^st ‘𝑁)‘𝑥))(⟨((1^st ‘𝐹)‘((1^st ‘𝐿)‘𝑥)), ((1^st ‘𝐹)‘((1^st ‘𝑁)‘𝑥))⟩(comp‘𝐸)((1^st ‘𝐾)‘((1^st ‘𝑁)‘𝑥)))((((1^st ‘𝐿)‘𝑥)(2^nd ‘𝐹)((1^st ‘𝑁)‘𝑥))‘(𝑉‘𝑥)))(⟨((1^st ‘𝐹)‘((1^st ‘𝐺)‘𝑥)), ((1^st ‘𝐹)‘((1^st ‘𝐿)‘𝑥))⟩(comp‘𝐸)((1^st ‘𝐾)‘((1^st ‘𝑁)‘𝑥)))((((1^st ‘𝐺)‘𝑥)(2^nd ‘𝐹)((1^st ‘𝐿)‘𝑥))‘(𝑆‘𝑥))))))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2147 ⟨cop 4838 class class class wbr 5106 ↦ cmpt 5190 ‘cfv 6561 (class class class)co 7446 1^st c1st 8026 2^nd c2nd 8027 Basecbs 17419 Hom chom 17471 compcco 17472 Func cfunc 18061 Nat cnat 18151 FuncCat cfuc 18152 ×_c cxpc 18374 ∘_F cfuco 50653

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-map 8880 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-cat 17874 df-func 18065 df-cofu 18067 df-nat 18153 df-fuc 18154 df-xpc 18378 df-fuco 50654

This theorem is used by: fucoco 50694

W3C validator