fucocolem2 - Mathbox for Zhi Wang

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Theorem fucocolem2 49851

Description: Lemma for fucoco 49854. The composed natural transformations are mapped to composition of 4 natural transformations. (Contributed by Zhi Wang, 2-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
fucocolem2	⊢ (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))))))

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 fucocolem2 Dummy variable 𝑝 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fucoco.x	. . . . . . . 8 ⊢ (𝜑 → 𝑋 = ⟨𝐹, 𝐺⟩)
2		fucoco.y	. . . . . . . 8 ⊢ (𝜑 → 𝑌 = ⟨𝐾, 𝐿⟩)
3	1, 2	opeq12d 4819	. . . . . . 7 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ = ⟨⟨𝐹, 𝐺⟩, ⟨𝐾, 𝐿⟩⟩)
4		fucoco.z	. . . . . . 7 ⊢ (𝜑 → 𝑍 = ⟨𝑀, 𝑁⟩)
5	3, 4	oveq12d 7381	. . . . . 6 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩ · 𝑍) = (⟨⟨𝐹, 𝐺⟩, ⟨𝐾, 𝐿⟩⟩ · ⟨𝑀, 𝑁⟩))
6		fucoco.b	. . . . . 6 ⊢ (𝜑 → 𝐵 = ⟨𝑈, 𝑉⟩)
7		fucoco.a	. . . . . 6 ⊢ (𝜑 → 𝐴 = ⟨𝑅, 𝑆⟩)
8	5, 6, 7	oveq123d 7384	. . . . 5 ⊢ (𝜑 → (𝐵(⟨𝑋, 𝑌⟩ · 𝑍)𝐴) = (⟨𝑈, 𝑉⟩(⟨⟨𝐹, 𝐺⟩, ⟨𝐾, 𝐿⟩⟩ · ⟨𝑀, 𝑁⟩)⟨𝑅, 𝑆⟩))
9		fucocolem2.t	. . . . . 6 ⊢ 𝑇 = ((𝐷 FuncCat 𝐸) ×_c (𝐶 FuncCat 𝐷))
10		fucocolem2.ot	. . . . . 6 ⊢ · = (comp‘𝑇)
11		fucoco.r	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ (𝐹(𝐷 Nat 𝐸)𝐾))
12		fucoco.s	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ (𝐺(𝐶 Nat 𝐷)𝐿))
13		fucoco.u	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ (𝐾(𝐷 Nat 𝐸)𝑀))
14		fucoco.v	. . . . . 6 ⊢ (𝜑 → 𝑉 ∈ (𝐿(𝐶 Nat 𝐷)𝑁))
15		eqid 2740	. . . . . 6 ⊢ (Base‘𝐷) = (Base‘𝐷)
16		eqid 2740	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
17		eqid 2740	. . . . . 6 ⊢ (comp‘𝐸) = (comp‘𝐸)
18		fucocolem2.od	. . . . . 6 ⊢ ∗ = (comp‘𝐷)
19	9, 10, 11, 12, 13, 14, 15, 16, 17, 18	xpcfucco3 49755	. . . . 5 ⊢ (𝜑 → (⟨𝑈, 𝑉⟩(⟨⟨𝐹, 𝐺⟩, ⟨𝐾, 𝐿⟩⟩ · ⟨𝑀, 𝑁⟩)⟨𝑅, 𝑆⟩) = ⟨(𝑝 ∈ (Base‘𝐷) ↦ ((𝑈‘𝑝)(⟨((1^st ‘𝐹)‘𝑝), ((1^st ‘𝐾)‘𝑝)⟩(comp‘𝐸)((1^st ‘𝑀)‘𝑝))(𝑅‘𝑝))), (𝑝 ∈ (Base‘𝐶) ↦ ((𝑉‘𝑝)(⟨((1^st ‘𝐺)‘𝑝), ((1^st ‘𝐿)‘𝑝)⟩ ∗ ((1^st ‘𝑁)‘𝑝))(𝑆‘𝑝)))⟩)
20	8, 19	eqtrd 2775	. . . 4 ⊢ (𝜑 → (𝐵(⟨𝑋, 𝑌⟩ · 𝑍)𝐴) = ⟨(𝑝 ∈ (Base‘𝐷) ↦ ((𝑈‘𝑝)(⟨((1^st ‘𝐹)‘𝑝), ((1^st ‘𝐾)‘𝑝)⟩(comp‘𝐸)((1^st ‘𝑀)‘𝑝))(𝑅‘𝑝))), (𝑝 ∈ (Base‘𝐶) ↦ ((𝑉‘𝑝)(⟨((1^st ‘𝐺)‘𝑝), ((1^st ‘𝐿)‘𝑝)⟩ ∗ ((1^st ‘𝑁)‘𝑝))(𝑆‘𝑝)))⟩)
21	20	fveq2d 6838	. . 3 ⊢ (𝜑 → ((𝑋𝑃𝑍)‘(𝐵(⟨𝑋, 𝑌⟩ · 𝑍)𝐴)) = ((𝑋𝑃𝑍)‘⟨(𝑝 ∈ (Base‘𝐷) ↦ ((𝑈‘𝑝)(⟨((1^st ‘𝐹)‘𝑝), ((1^st ‘𝐾)‘𝑝)⟩(comp‘𝐸)((1^st ‘𝑀)‘𝑝))(𝑅‘𝑝))), (𝑝 ∈ (Base‘𝐶) ↦ ((𝑉‘𝑝)(⟨((1^st ‘𝐺)‘𝑝), ((1^st ‘𝐿)‘𝑝)⟩ ∗ ((1^st ‘𝑁)‘𝑝))(𝑆‘𝑝)))⟩))
22		df-ov 7366	. . 3 ⊢ ((𝑝 ∈ (Base‘𝐷) ↦ ((𝑈‘𝑝)(⟨((1^st ‘𝐹)‘𝑝), ((1^st ‘𝐾)‘𝑝)⟩(comp‘𝐸)((1^st ‘𝑀)‘𝑝))(𝑅‘𝑝)))(𝑋𝑃𝑍)(𝑝 ∈ (Base‘𝐶) ↦ ((𝑉‘𝑝)(⟨((1^st ‘𝐺)‘𝑝), ((1^st ‘𝐿)‘𝑝)⟩ ∗ ((1^st ‘𝑁)‘𝑝))(𝑆‘𝑝)))) = ((𝑋𝑃𝑍)‘⟨(𝑝 ∈ (Base‘𝐷) ↦ ((𝑈‘𝑝)(⟨((1^st ‘𝐹)‘𝑝), ((1^st ‘𝐾)‘𝑝)⟩(comp‘𝐸)((1^st ‘𝑀)‘𝑝))(𝑅‘𝑝))), (𝑝 ∈ (Base‘𝐶) ↦ ((𝑉‘𝑝)(⟨((1^st ‘𝐺)‘𝑝), ((1^st ‘𝐿)‘𝑝)⟩ ∗ ((1^st ‘𝑁)‘𝑝))(𝑆‘𝑝)))⟩)
23	21, 22	eqtr4di 2793	. 2 ⊢ (𝜑 → ((𝑋𝑃𝑍)‘(𝐵(⟨𝑋, 𝑌⟩ · 𝑍)𝐴)) = ((𝑝 ∈ (Base‘𝐷) ↦ ((𝑈‘𝑝)(⟨((1^st ‘𝐹)‘𝑝), ((1^st ‘𝐾)‘𝑝)⟩(comp‘𝐸)((1^st ‘𝑀)‘𝑝))(𝑅‘𝑝)))(𝑋𝑃𝑍)(𝑝 ∈ (Base‘𝐶) ↦ ((𝑉‘𝑝)(⟨((1^st ‘𝐺)‘𝑝), ((1^st ‘𝐿)‘𝑝)⟩ ∗ ((1^st ‘𝑁)‘𝑝))(𝑆‘𝑝)))))
24		fucoco.o	. . 3 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘_F 𝐸) = ⟨𝑂, 𝑃⟩)
25	9, 10, 11, 12, 13, 14	xpcfuccocl 49754	. . . . 5 ⊢ (𝜑 → (⟨𝑈, 𝑉⟩(⟨⟨𝐹, 𝐺⟩, ⟨𝐾, 𝐿⟩⟩ · ⟨𝑀, 𝑁⟩)⟨𝑅, 𝑆⟩) ∈ ((𝐹(𝐷 Nat 𝐸)𝑀) × (𝐺(𝐶 Nat 𝐷)𝑁)))
26	19, 25	eqeltrrd 2841	. . . 4 ⊢ (𝜑 → ⟨(𝑝 ∈ (Base‘𝐷) ↦ ((𝑈‘𝑝)(⟨((1^st ‘𝐹)‘𝑝), ((1^st ‘𝐾)‘𝑝)⟩(comp‘𝐸)((1^st ‘𝑀)‘𝑝))(𝑅‘𝑝))), (𝑝 ∈ (Base‘𝐶) ↦ ((𝑉‘𝑝)(⟨((1^st ‘𝐺)‘𝑝), ((1^st ‘𝐿)‘𝑝)⟩ ∗ ((1^st ‘𝑁)‘𝑝))(𝑆‘𝑝)))⟩ ∈ ((𝐹(𝐷 Nat 𝐸)𝑀) × (𝐺(𝐶 Nat 𝐷)𝑁)))
27		opelxp2 5668	. . . 4 ⊢ (⟨(𝑝 ∈ (Base‘𝐷) ↦ ((𝑈‘𝑝)(⟨((1^st ‘𝐹)‘𝑝), ((1^st ‘𝐾)‘𝑝)⟩(comp‘𝐸)((1^st ‘𝑀)‘𝑝))(𝑅‘𝑝))), (𝑝 ∈ (Base‘𝐶) ↦ ((𝑉‘𝑝)(⟨((1^st ‘𝐺)‘𝑝), ((1^st ‘𝐿)‘𝑝)⟩ ∗ ((1^st ‘𝑁)‘𝑝))(𝑆‘𝑝)))⟩ ∈ ((𝐹(𝐷 Nat 𝐸)𝑀) × (𝐺(𝐶 Nat 𝐷)𝑁)) → (𝑝 ∈ (Base‘𝐶) ↦ ((𝑉‘𝑝)(⟨((1^st ‘𝐺)‘𝑝), ((1^st ‘𝐿)‘𝑝)⟩ ∗ ((1^st ‘𝑁)‘𝑝))(𝑆‘𝑝))) ∈ (𝐺(𝐶 Nat 𝐷)𝑁))
28	26, 27	syl 17	. . 3 ⊢ (𝜑 → (𝑝 ∈ (Base‘𝐶) ↦ ((𝑉‘𝑝)(⟨((1^st ‘𝐺)‘𝑝), ((1^st ‘𝐿)‘𝑝)⟩ ∗ ((1^st ‘𝑁)‘𝑝))(𝑆‘𝑝))) ∈ (𝐺(𝐶 Nat 𝐷)𝑁))
29		opelxp1 5667	. . . 4 ⊢ (⟨(𝑝 ∈ (Base‘𝐷) ↦ ((𝑈‘𝑝)(⟨((1^st ‘𝐹)‘𝑝), ((1^st ‘𝐾)‘𝑝)⟩(comp‘𝐸)((1^st ‘𝑀)‘𝑝))(𝑅‘𝑝))), (𝑝 ∈ (Base‘𝐶) ↦ ((𝑉‘𝑝)(⟨((1^st ‘𝐺)‘𝑝), ((1^st ‘𝐿)‘𝑝)⟩ ∗ ((1^st ‘𝑁)‘𝑝))(𝑆‘𝑝)))⟩ ∈ ((𝐹(𝐷 Nat 𝐸)𝑀) × (𝐺(𝐶 Nat 𝐷)𝑁)) → (𝑝 ∈ (Base‘𝐷) ↦ ((𝑈‘𝑝)(⟨((1^st ‘𝐹)‘𝑝), ((1^st ‘𝐾)‘𝑝)⟩(comp‘𝐸)((1^st ‘𝑀)‘𝑝))(𝑅‘𝑝))) ∈ (𝐹(𝐷 Nat 𝐸)𝑀))
30	26, 29	syl 17	. . 3 ⊢ (𝜑 → (𝑝 ∈ (Base‘𝐷) ↦ ((𝑈‘𝑝)(⟨((1^st ‘𝐹)‘𝑝), ((1^st ‘𝐾)‘𝑝)⟩(comp‘𝐸)((1^st ‘𝑀)‘𝑝))(𝑅‘𝑝))) ∈ (𝐹(𝐷 Nat 𝐸)𝑀))
31	24, 1, 4, 28, 30	fuco22a 49847	. 2 ⊢ (𝜑 → ((𝑝 ∈ (Base‘𝐷) ↦ ((𝑈‘𝑝)(⟨((1^st ‘𝐹)‘𝑝), ((1^st ‘𝐾)‘𝑝)⟩(comp‘𝐸)((1^st ‘𝑀)‘𝑝))(𝑅‘𝑝)))(𝑋𝑃𝑍)(𝑝 ∈ (Base‘𝐶) ↦ ((𝑉‘𝑝)(⟨((1^st ‘𝐺)‘𝑝), ((1^st ‘𝐿)‘𝑝)⟩ ∗ ((1^st ‘𝑁)‘𝑝))(𝑆‘𝑝)))) = (𝑥 ∈ (Base‘𝐶) ↦ (((𝑝 ∈ (Base‘𝐷) ↦ ((𝑈‘𝑝)(⟨((1^st ‘𝐹)‘𝑝), ((1^st ‘𝐾)‘𝑝)⟩(comp‘𝐸)((1^st ‘𝑀)‘𝑝))(𝑅‘𝑝)))‘((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))‘((𝑝 ∈ (Base‘𝐶) ↦ ((𝑉‘𝑝)(⟨((1^st ‘𝐺)‘𝑝), ((1^st ‘𝐿)‘𝑝)⟩ ∗ ((1^st ‘𝑁)‘𝑝))(𝑆‘𝑝)))‘𝑥)))))
32		eqid 2740	. . . . . . . . . . 11 ⊢ (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷)
33	32	natrcl 17918	. . . . . . . . . 10 ⊢ (𝑉 ∈ (𝐿(𝐶 Nat 𝐷)𝑁) → (𝐿 ∈ (𝐶 Func 𝐷) ∧ 𝑁 ∈ (𝐶 Func 𝐷)))
34	14, 33	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝐿 ∈ (𝐶 Func 𝐷) ∧ 𝑁 ∈ (𝐶 Func 𝐷)))
35	34	simprd 496	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ (𝐶 Func 𝐷))
36	35	func1st2nd 49573	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝑁)(𝐶 Func 𝐷)(2^nd ‘𝑁))
37	16, 15, 36	funcf1 17831	. . . . . 6 ⊢ (𝜑 → (1^st ‘𝑁):(Base‘𝐶)⟶(Base‘𝐷))
38	37	ffvelcdmda 7032	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1^st ‘𝑁)‘𝑥) ∈ (Base‘𝐷))
39		fveq2 6834	. . . . . . . . 9 ⊢ (𝑝 = ((1^st ‘𝑁)‘𝑥) → ((1^st ‘𝐹)‘𝑝) = ((1^st ‘𝐹)‘((1^st ‘𝑁)‘𝑥)))
40		fveq2 6834	. . . . . . . . 9 ⊢ (𝑝 = ((1^st ‘𝑁)‘𝑥) → ((1^st ‘𝐾)‘𝑝) = ((1^st ‘𝐾)‘((1^st ‘𝑁)‘𝑥)))
41	39, 40	opeq12d 4819	. . . . . . . 8 ⊢ (𝑝 = ((1^st ‘𝑁)‘𝑥) → ⟨((1^st ‘𝐹)‘𝑝), ((1^st ‘𝐾)‘𝑝)⟩ = ⟨((1^st ‘𝐹)‘((1^st ‘𝑁)‘𝑥)), ((1^st ‘𝐾)‘((1^st ‘𝑁)‘𝑥))⟩)
42		fveq2 6834	. . . . . . . 8 ⊢ (𝑝 = ((1^st ‘𝑁)‘𝑥) → ((1^st ‘𝑀)‘𝑝) = ((1^st ‘𝑀)‘((1^st ‘𝑁)‘𝑥)))
43	41, 42	oveq12d 7381	. . . . . . 7 ⊢ (𝑝 = ((1^st ‘𝑁)‘𝑥) → (⟨((1^st ‘𝐹)‘𝑝), ((1^st ‘𝐾)‘𝑝)⟩(comp‘𝐸)((1^st ‘𝑀)‘𝑝)) = (⟨((1^st ‘𝐹)‘((1^st ‘𝑁)‘𝑥)), ((1^st ‘𝐾)‘((1^st ‘𝑁)‘𝑥))⟩(comp‘𝐸)((1^st ‘𝑀)‘((1^st ‘𝑁)‘𝑥))))
44		fveq2 6834	. . . . . . 7 ⊢ (𝑝 = ((1^st ‘𝑁)‘𝑥) → (𝑈‘𝑝) = (𝑈‘((1^st ‘𝑁)‘𝑥)))
45		fveq2 6834	. . . . . . 7 ⊢ (𝑝 = ((1^st ‘𝑁)‘𝑥) → (𝑅‘𝑝) = (𝑅‘((1^st ‘𝑁)‘𝑥)))
46	43, 44, 45	oveq123d 7384	. . . . . 6 ⊢ (𝑝 = ((1^st ‘𝑁)‘𝑥) → ((𝑈‘𝑝)(⟨((1^st ‘𝐹)‘𝑝), ((1^st ‘𝐾)‘𝑝)⟩(comp‘𝐸)((1^st ‘𝑀)‘𝑝))(𝑅‘𝑝)) = ((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))))
47		eqid 2740	. . . . . 6 ⊢ (𝑝 ∈ (Base‘𝐷) ↦ ((𝑈‘𝑝)(⟨((1^st ‘𝐹)‘𝑝), ((1^st ‘𝐾)‘𝑝)⟩(comp‘𝐸)((1^st ‘𝑀)‘𝑝))(𝑅‘𝑝))) = (𝑝 ∈ (Base‘𝐷) ↦ ((𝑈‘𝑝)(⟨((1^st ‘𝐹)‘𝑝), ((1^st ‘𝐾)‘𝑝)⟩(comp‘𝐸)((1^st ‘𝑀)‘𝑝))(𝑅‘𝑝)))
48		ovex 7396	. . . . . 6 ⊢ ((𝑈‘𝑝)(⟨((1^st ‘𝐹)‘𝑝), ((1^st ‘𝐾)‘𝑝)⟩(comp‘𝐸)((1^st ‘𝑀)‘𝑝))(𝑅‘𝑝)) ∈ V
49	46, 47, 48	fvmpt3i 6948	. . . . 5 ⊢ (((1^st ‘𝑁)‘𝑥) ∈ (Base‘𝐷) → ((𝑝 ∈ (Base‘𝐷) ↦ ((𝑈‘𝑝)(⟨((1^st ‘𝐹)‘𝑝), ((1^st ‘𝐾)‘𝑝)⟩(comp‘𝐸)((1^st ‘𝑀)‘𝑝))(𝑅‘𝑝)))‘((1^st ‘𝑁)‘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))))
50	38, 49	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((𝑝 ∈ (Base‘𝐷) ↦ ((𝑈‘𝑝)(⟨((1^st ‘𝐹)‘𝑝), ((1^st ‘𝐾)‘𝑝)⟩(comp‘𝐸)((1^st ‘𝑀)‘𝑝))(𝑅‘𝑝)))‘((1^st ‘𝑁)‘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))))
51		fveq2 6834	. . . . . . . . . 10 ⊢ (𝑝 = 𝑥 → ((1^st ‘𝐺)‘𝑝) = ((1^st ‘𝐺)‘𝑥))
52		fveq2 6834	. . . . . . . . . 10 ⊢ (𝑝 = 𝑥 → ((1^st ‘𝐿)‘𝑝) = ((1^st ‘𝐿)‘𝑥))
53	51, 52	opeq12d 4819	. . . . . . . . 9 ⊢ (𝑝 = 𝑥 → ⟨((1^st ‘𝐺)‘𝑝), ((1^st ‘𝐿)‘𝑝)⟩ = ⟨((1^st ‘𝐺)‘𝑥), ((1^st ‘𝐿)‘𝑥)⟩)
54		fveq2 6834	. . . . . . . . 9 ⊢ (𝑝 = 𝑥 → ((1^st ‘𝑁)‘𝑝) = ((1^st ‘𝑁)‘𝑥))
55	53, 54	oveq12d 7381	. . . . . . . 8 ⊢ (𝑝 = 𝑥 → (⟨((1^st ‘𝐺)‘𝑝), ((1^st ‘𝐿)‘𝑝)⟩ ∗ ((1^st ‘𝑁)‘𝑝)) = (⟨((1^st ‘𝐺)‘𝑥), ((1^st ‘𝐿)‘𝑥)⟩ ∗ ((1^st ‘𝑁)‘𝑥)))
56		fveq2 6834	. . . . . . . 8 ⊢ (𝑝 = 𝑥 → (𝑉‘𝑝) = (𝑉‘𝑥))
57		fveq2 6834	. . . . . . . 8 ⊢ (𝑝 = 𝑥 → (𝑆‘𝑝) = (𝑆‘𝑥))
58	55, 56, 57	oveq123d 7384	. . . . . . 7 ⊢ (𝑝 = 𝑥 → ((𝑉‘𝑝)(⟨((1^st ‘𝐺)‘𝑝), ((1^st ‘𝐿)‘𝑝)⟩ ∗ ((1^st ‘𝑁)‘𝑝))(𝑆‘𝑝)) = ((𝑉‘𝑥)(⟨((1^st ‘𝐺)‘𝑥), ((1^st ‘𝐿)‘𝑥)⟩ ∗ ((1^st ‘𝑁)‘𝑥))(𝑆‘𝑥)))
59		eqid 2740	. . . . . . 7 ⊢ (𝑝 ∈ (Base‘𝐶) ↦ ((𝑉‘𝑝)(⟨((1^st ‘𝐺)‘𝑝), ((1^st ‘𝐿)‘𝑝)⟩ ∗ ((1^st ‘𝑁)‘𝑝))(𝑆‘𝑝))) = (𝑝 ∈ (Base‘𝐶) ↦ ((𝑉‘𝑝)(⟨((1^st ‘𝐺)‘𝑝), ((1^st ‘𝐿)‘𝑝)⟩ ∗ ((1^st ‘𝑁)‘𝑝))(𝑆‘𝑝)))
60		ovex 7396	. . . . . . 7 ⊢ ((𝑉‘𝑝)(⟨((1^st ‘𝐺)‘𝑝), ((1^st ‘𝐿)‘𝑝)⟩ ∗ ((1^st ‘𝑁)‘𝑝))(𝑆‘𝑝)) ∈ V
61	58, 59, 60	fvmpt3i 6948	. . . . . 6 ⊢ (𝑥 ∈ (Base‘𝐶) → ((𝑝 ∈ (Base‘𝐶) ↦ ((𝑉‘𝑝)(⟨((1^st ‘𝐺)‘𝑝), ((1^st ‘𝐿)‘𝑝)⟩ ∗ ((1^st ‘𝑁)‘𝑝))(𝑆‘𝑝)))‘𝑥) = ((𝑉‘𝑥)(⟨((1^st ‘𝐺)‘𝑥), ((1^st ‘𝐿)‘𝑥)⟩ ∗ ((1^st ‘𝑁)‘𝑥))(𝑆‘𝑥)))
62	61	fveq2d 6838	. . . . 5 ⊢ (𝑥 ∈ (Base‘𝐶) → ((((1^st ‘𝐺)‘𝑥)(2^nd ‘𝐹)((1^st ‘𝑁)‘𝑥))‘((𝑝 ∈ (Base‘𝐶) ↦ ((𝑉‘𝑝)(⟨((1^st ‘𝐺)‘𝑝), ((1^st ‘𝐿)‘𝑝)⟩ ∗ ((1^st ‘𝑁)‘𝑝))(𝑆‘𝑝)))‘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))))
63	62	adantl 482	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((((1^st ‘𝐺)‘𝑥)(2^nd ‘𝐹)((1^st ‘𝑁)‘𝑥))‘((𝑝 ∈ (Base‘𝐶) ↦ ((𝑉‘𝑝)(⟨((1^st ‘𝐺)‘𝑝), ((1^st ‘𝐿)‘𝑝)⟩ ∗ ((1^st ‘𝑁)‘𝑝))(𝑆‘𝑝)))‘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))))
64	50, 63	oveq12d 7381	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (((𝑝 ∈ (Base‘𝐷) ↦ ((𝑈‘𝑝)(⟨((1^st ‘𝐹)‘𝑝), ((1^st ‘𝐾)‘𝑝)⟩(comp‘𝐸)((1^st ‘𝑀)‘𝑝))(𝑅‘𝑝)))‘((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))‘((𝑝 ∈ (Base‘𝐶) ↦ ((𝑉‘𝑝)(⟨((1^st ‘𝐺)‘𝑝), ((1^st ‘𝐿)‘𝑝)⟩ ∗ ((1^st ‘𝑁)‘𝑝))(𝑆‘𝑝)))‘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 ‘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)))))
65	64	mpteq2dva 5172	. 2 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐶) ↦ (((𝑝 ∈ (Base‘𝐷) ↦ ((𝑈‘𝑝)(⟨((1^st ‘𝐹)‘𝑝), ((1^st ‘𝐾)‘𝑝)⟩(comp‘𝐸)((1^st ‘𝑀)‘𝑝))(𝑅‘𝑝)))‘((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))‘((𝑝 ∈ (Base‘𝐶) ↦ ((𝑉‘𝑝)(⟨((1^st ‘𝐺)‘𝑝), ((1^st ‘𝐿)‘𝑝)⟩ ∗ ((1^st ‘𝑁)‘𝑝))(𝑆‘𝑝)))‘𝑥)))) = (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))))))
66	23, 31, 65	3eqtrd 2779	1 ⊢ (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))))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1547 ∈ wcel 2119 ⟨cop 4568 ↦ cmpt 5160 × cxp 5623 ‘cfv 6492 (class class class)co 7363 1^st c1st 7936 2^nd c2nd 7937 Basecbs 17177 compcco 17230 Func cfunc 17819 Nat cnat 17909 FuncCat cfuc 17910 ×_c cxpc 18132 ∘_F cfuco 49813

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2712 ax-rep 5206 ax-sep 5225 ax-nul 5235 ax-pow 5301 ax-pr 5369 ax-un 7685 ax-cnex 11092 ax-resscn 11093 ax-1cn 11094 ax-icn 11095 ax-addcl 11096 ax-addrcl 11097 ax-mulcl 11098 ax-mulrcl 11099 ax-mulcom 11100 ax-addass 11101 ax-mulass 11102 ax-distr 11103 ax-i2m1 11104 ax-1ne0 11105 ax-1rid 11106 ax-rnegex 11107 ax-rrecex 11108 ax-cnre 11109 ax-pre-lttri 11110 ax-pre-lttrn 11111 ax-pre-ltadd 11112 ax-pre-mulgt0 11113

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-nel 3040 df-ral 3055 df-rex 3065 df-reu 3346 df-rab 3393 df-v 3434 df-sbc 3731 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4269 df-if 4462 df-pw 4538 df-sn 4563 df-pr 4565 df-tp 4567 df-op 4569 df-uni 4846 df-iun 4930 df-br 5080 df-opab 5142 df-mpt 5161 df-tr 5187 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7320 df-ov 7366 df-oprab 7367 df-mpo 7368 df-om 7814 df-1st 7938 df-2nd 7939 df-frecs 8228 df-wrecs 8259 df-recs 8308 df-rdg 8346 df-1o 8402 df-er 8640 df-map 8772 df-ixp 8843 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-pnf 11179 df-mnf 11180 df-xr 11181 df-ltxr 11182 df-le 11183 df-sub 11377 df-neg 11378 df-nn 12173 df-2 12242 df-3 12243 df-4 12244 df-5 12245 df-6 12246 df-7 12247 df-8 12248 df-9 12249 df-n0 12436 df-z 12523 df-dec 12643 df-uz 12787 df-fz 13460 df-struct 17115 df-slot 17150 df-ndx 17162 df-base 17178 df-hom 17242 df-cco 17243 df-cat 17632 df-func 17823 df-cofu 17825 df-nat 17911 df-fuc 17912 df-xpc 18136 df-fuco 49814

This theorem is referenced by: fucocolem3 49852

W3C validator