fucocolem3 - Mathbox for Zhi Wang

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Theorem fucocolem3 50116

Description: Lemma for fucoco 50118. 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 50115	. 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 18011	. . . . . . . . . 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 499	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (𝐷 Func 𝐸))
22	21	func1st2nd 49837	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝐹)(𝐷 Func 𝐸)(2^nd ‘𝐹))
23	22	adantr 485	. . . . . 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 18011	. . . . . . . . . . 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 499	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷))
29	28	func1st2nd 49837	. . . . . . . 8 ⊢ (𝜑 → (1^st ‘𝐺)(𝐶 Func 𝐷)(2^nd ‘𝐺))
30	24, 15, 29	funcf1 17924	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝐺):(Base‘𝐶)⟶(Base‘𝐷))
31	30	ffvelcdmda 7081	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1^st ‘𝐺)‘𝑥) ∈ (Base‘𝐷))
32	27	simprd 500	. . . . . . . . 9 ⊢ (𝜑 → 𝐿 ∈ (𝐶 Func 𝐷))
33	32	func1st2nd 49837	. . . . . . . 8 ⊢ (𝜑 → (1^st ‘𝐿)(𝐶 Func 𝐷)(2^nd ‘𝐿))
34	24, 15, 33	funcf1 17924	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝐿):(Base‘𝐶)⟶(Base‘𝐷))
35	34	ffvelcdmda 7081	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1^st ‘𝐿)‘𝑥) ∈ (Base‘𝐷))
36	25	natrcl 18011	. . . . . . . . . . 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 500	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ (𝐶 Func 𝐷))
39	38	func1st2nd 49837	. . . . . . . 8 ⊢ (𝜑 → (1^st ‘𝑁)(𝐶 Func 𝐷)(2^nd ‘𝑁))
40	24, 15, 39	funcf1 17924	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝑁):(Base‘𝐶)⟶(Base‘𝐷))
41	40	ffvelcdmda 7081	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1^st ‘𝑁)‘𝑥) ∈ (Base‘𝐷))
42	25, 2	nat1st2nd 18012	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ (⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩(𝐶 Nat 𝐷)⟨(1^st ‘𝐿), (2^nd ‘𝐿)⟩))
43	42	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑆 ∈ (⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩(𝐶 Nat 𝐷)⟨(1^st ‘𝐿), (2^nd ‘𝐿)⟩))
44		simpr 489	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
45	25, 43, 24, 16, 44	natcl 18014	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (𝑆‘𝑥) ∈ (((1^st ‘𝐺)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐿)‘𝑥)))
46	25, 4	nat1st2nd 18012	. . . . . . . 8 ⊢ (𝜑 → 𝑉 ∈ (⟨(1^st ‘𝐿), (2^nd ‘𝐿)⟩(𝐶 Nat 𝐷)⟨(1^st ‘𝑁), (2^nd ‘𝑁)⟩))
47	46	adantr 485	. . . . . . 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 18014	. . . . . 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 17929	. . . . 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 7428	. . . 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 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑅 ∈ (𝐹(𝐷 Nat 𝐸)𝐾))
52	2	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑆 ∈ (𝐺(𝐶 Nat 𝐷)𝐿))
53	3	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑈 ∈ (𝐾(𝐷 Nat 𝐸)𝑀))
54	4	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑉 ∈ (𝐿(𝐶 Nat 𝐷)𝑁))
55	21	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐹 ∈ (𝐷 Func 𝐸))
56	38	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑁 ∈ (𝐶 Func 𝐷))
57	18, 1	nat1st2nd 18012	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ (⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩(𝐷 Nat 𝐸)⟨(1^st ‘𝐾), (2^nd ‘𝐾)⟩))
58	57	adantr 485	. . . . . 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 18014	. . . . 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 17926	. . . . . 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 7082	. . . . 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 50114	. . . 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 5205	. 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

Syntax hints: → wi 4 ∧ wa 400 = wceq 1570 ∈ wcel 2143 ⟨cop 4596 class class class wbr 5110 ↦ cmpt 5193 ‘cfv 6538 (class class class)co 7412 1^st c1st 7985 2^nd c2nd 7986 Basecbs 17270 Hom chom 17322 compcco 17323 Func cfunc 17912 Nat cnat 18002 FuncCat cfuc 18003 ×_c cxpc 18225 ∘_F cfuco 50077

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5239 ax-sep 5258 ax-nul 5270 ax-pow 5338 ax-pr 5406 ax-un 7734 ax-cnex 11157 ax-resscn 11158 ax-1cn 11159 ax-icn 11160 ax-addcl 11161 ax-addrcl 11162 ax-mulcl 11163 ax-mulrcl 11164 ax-mulcom 11165 ax-addass 11166 ax-mulass 11167 ax-distr 11168 ax-i2m1 11169 ax-1ne0 11170 ax-1rid 11171 ax-rnegex 11172 ax-rrecex 11173 ax-cnre 11174 ax-pre-lttri 11175 ax-pre-lttrn 11176 ax-pre-ltadd 11177 ax-pre-mulgt0 11178

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3746 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4489 df-pw 4565 df-sn 4591 df-pr 4593 df-tp 4595 df-op 4597 df-uni 4874 df-iun 4959 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7864 df-1st 7987 df-2nd 7988 df-frecs 8279 df-wrecs 8310 df-recs 8359 df-rdg 8398 df-1o 8454 df-er 8695 df-map 8827 df-ixp 8897 df-en 8945 df-dom 8946 df-sdom 8947 df-fin 8948 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11444 df-neg 11445 df-nn 12235 df-2 12304 df-3 12305 df-4 12306 df-5 12307 df-6 12308 df-7 12309 df-8 12310 df-9 12311 df-n0 12506 df-z 12593 df-dec 12713 df-uz 12864 df-fz 13537 df-struct 17208 df-slot 17243 df-ndx 17255 df-base 17271 df-hom 17335 df-cco 17336 df-cat 17725 df-func 17916 df-cofu 17918 df-nat 18004 df-fuc 18005 df-xpc 18229 df-fuco 50078

This theorem is referenced by: fucoco 50118

W3C validator