fucocolem3 - Mathbox for Zhi Wang

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Theorem fucocolem3 50407

Description: Lemma for fucoco 50409. 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 50406	. 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 2761	. . . . . 6 ⊢ (Base‘𝐷) = (Base‘𝐷)
16		eqid 2761	. . . . . 6 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
17		eqid 2761	. . . . . 6 ⊢ (comp‘𝐸) = (comp‘𝐸)
18		eqid 2761	. . . . . . . . . . 11 ⊢ (𝐷 Nat 𝐸) = (𝐷 Nat 𝐸)
19	18	natrcl 18108	. . . . . . . . . 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 50128	. . . . . . 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 2761	. . . . . . . 8 ⊢ (Base‘𝐶) = (Base‘𝐶)
25		eqid 2761	. . . . . . . . . . . 12 ⊢ (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷)
26	25	natrcl 18108	. . . . . . . . . . 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 50128	. . . . . . . 8 ⊢ (𝜑 → (1^st ‘𝐺)(𝐶 Func 𝐷)(2^nd ‘𝐺))
30	24, 15, 29	funcf1 18021	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝐺):(Base‘𝐶)⟶(Base‘𝐷))
31	30	ffvelcdmda 7076	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1^st ‘𝐺)‘𝑥) ∈ (Base‘𝐷))
32	27	simprd 501	. . . . . . . . 9 ⊢ (𝜑 → 𝐿 ∈ (𝐶 Func 𝐷))
33	32	func1st2nd 50128	. . . . . . . 8 ⊢ (𝜑 → (1^st ‘𝐿)(𝐶 Func 𝐷)(2^nd ‘𝐿))
34	24, 15, 33	funcf1 18021	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝐿):(Base‘𝐶)⟶(Base‘𝐷))
35	34	ffvelcdmda 7076	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1^st ‘𝐿)‘𝑥) ∈ (Base‘𝐷))
36	25	natrcl 18108	. . . . . . . . . . 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 50128	. . . . . . . 8 ⊢ (𝜑 → (1^st ‘𝑁)(𝐶 Func 𝐷)(2^nd ‘𝑁))
40	24, 15, 39	funcf1 18021	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝑁):(Base‘𝐶)⟶(Base‘𝐷))
41	40	ffvelcdmda 7076	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1^st ‘𝑁)‘𝑥) ∈ (Base‘𝐷))
42	25, 2	nat1st2nd 18109	. . . . . . . 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 18111	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (𝑆‘𝑥) ∈ (((1^st ‘𝐺)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐿)‘𝑥)))
46	25, 4	nat1st2nd 18109	. . . . . . . 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 18111	. . . . . 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 18026	. . . . 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 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 18109	. . . . . . 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 2761	. . . . . 6 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
60	18, 58, 15, 59, 41	natcl 18111	. . . . 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 18023	. . . . . 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 7077	. . . . 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 50405	. . . 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 2796	. . 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 5198	. 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 2796	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 2145 ⟨cop 4590 class class class wbr 5103 ↦ cmpt 5186 ‘cfv 6531 (class class class)co 7412 1^st c1st 7988 2^nd c2nd 7989 Basecbs 17367 Hom chom 17419 compcco 17420 Func cfunc 18009 Nat cnat 18099 FuncCat cfuc 18100 ×_c cxpc 18322 ∘_F cfuco 50368

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 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7740 ax-cnex 11237 ax-resscn 11238 ax-1cn 11239 ax-icn 11240 ax-addcl 11241 ax-addrcl 11242 ax-mulcl 11243 ax-mulrcl 11244 ax-mulcom 11245 ax-addass 11246 ax-mulass 11247 ax-distr 11248 ax-i2m1 11249 ax-1ne0 11250 ax-1rid 11251 ax-rnegex 11252 ax-rrecex 11253 ax-cnre 11254 ax-pre-lttri 11255 ax-pre-lttrn 11256 ax-pre-ltadd 11257 ax-pre-mulgt0 11258

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 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6297 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6487 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7867 df-1st 7990 df-2nd 7991 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8460 df-er 8701 df-map 8833 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11524 df-neg 11525 df-nn 12317 df-2 12386 df-3 12387 df-4 12388 df-5 12389 df-6 12390 df-7 12391 df-8 12392 df-9 12393 df-n0 12588 df-z 12675 df-dec 12796 df-uz 12947 df-fz 13621 df-struct 17305 df-slot 17340 df-ndx 17352 df-base 17368 df-hom 17432 df-cco 17433 df-cat 17822 df-func 18013 df-cofu 18015 df-nat 18101 df-fuc 18102 df-xpc 18326 df-fuco 50369

This theorem is used by: fucoco 50409

W3C validator