fucocolem2 - Mathbox for Zhi Wang

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Theorem fucocolem2 50288

Description: Lemma for fucoco 50291. 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 4844	. . . . . . 7 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ = ⟨⟨𝐹, 𝐺⟩, ⟨𝐾, 𝐿⟩⟩)
4		fucoco.z	. . . . . . 7 ⊢ (𝜑 → 𝑍 = ⟨𝑀, 𝑁⟩)
5	3, 4	oveq12d 7435	. . . . . 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 7438	. . . . 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 2762	. . . . . 6 ⊢ (Base‘𝐷) = (Base‘𝐷)
16		eqid 2762	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
17		eqid 2762	. . . . . 6 ⊢ (comp‘𝐸) = (comp‘𝐸)
18		fucocolem2.od	. . . . . 6 ⊢ ∗ = (comp‘𝐷)
19	9, 10, 11, 12, 13, 14, 15, 16, 17, 18	xpcfucco3 50192	. . . . 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 2797	. . . 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 6886	. . 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 7420	. . 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 2815	. 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 50191	. . . . 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 2863	. . . 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 5702	. . . 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 18	. . 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 5701	. . . 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 18	. . 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 50284	. 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 2762	. . . . . . . . . . 11 ⊢ (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷)
33	32	natrcl 18048	. . . . . . . . . 10 ⊢ (𝑉 ∈ (𝐿(𝐶 Nat 𝐷)𝑁) → (𝐿 ∈ (𝐶 Func 𝐷) ∧ 𝑁 ∈ (𝐶 Func 𝐷)))
34	14, 33	syl 18	. . . . . . . . 9 ⊢ (𝜑 → (𝐿 ∈ (𝐶 Func 𝐷) ∧ 𝑁 ∈ (𝐶 Func 𝐷)))
35	34	simprd 501	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ (𝐶 Func 𝐷))
36	35	func1st2nd 50010	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝑁)(𝐶 Func 𝐷)(2^nd ‘𝑁))
37	16, 15, 36	funcf1 17961	. . . . . 6 ⊢ (𝜑 → (1^st ‘𝑁):(Base‘𝐶)⟶(Base‘𝐷))
38	37	ffvelcdmda 7081	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1^st ‘𝑁)‘𝑥) ∈ (Base‘𝐷))
39		fveq2 6882	. . . . . . . . 9 ⊢ (𝑝 = ((1^st ‘𝑁)‘𝑥) → ((1^st ‘𝐹)‘𝑝) = ((1^st ‘𝐹)‘((1^st ‘𝑁)‘𝑥)))
40		fveq2 6882	. . . . . . . . 9 ⊢ (𝑝 = ((1^st ‘𝑁)‘𝑥) → ((1^st ‘𝐾)‘𝑝) = ((1^st ‘𝐾)‘((1^st ‘𝑁)‘𝑥)))
41	39, 40	opeq12d 4844	. . . . . . . 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 6882	. . . . . . . 8 ⊢ (𝑝 = ((1^st ‘𝑁)‘𝑥) → ((1^st ‘𝑀)‘𝑝) = ((1^st ‘𝑀)‘((1^st ‘𝑁)‘𝑥)))
43	41, 42	oveq12d 7435	. . . . . . 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 6882	. . . . . . 7 ⊢ (𝑝 = ((1^st ‘𝑁)‘𝑥) → (𝑈‘𝑝) = (𝑈‘((1^st ‘𝑁)‘𝑥)))
45		fveq2 6882	. . . . . . 7 ⊢ (𝑝 = ((1^st ‘𝑁)‘𝑥) → (𝑅‘𝑝) = (𝑅‘((1^st ‘𝑁)‘𝑥)))
46	43, 44, 45	oveq123d 7438	. . . . . 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 2762	. . . . . 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 7450	. . . . . 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 6996	. . . . 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 18	. . . 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 6882	. . . . . . . . . 10 ⊢ (𝑝 = 𝑥 → ((1^st ‘𝐺)‘𝑝) = ((1^st ‘𝐺)‘𝑥))
52		fveq2 6882	. . . . . . . . . 10 ⊢ (𝑝 = 𝑥 → ((1^st ‘𝐿)‘𝑝) = ((1^st ‘𝐿)‘𝑥))
53	51, 52	opeq12d 4844	. . . . . . . . 9 ⊢ (𝑝 = 𝑥 → ⟨((1^st ‘𝐺)‘𝑝), ((1^st ‘𝐿)‘𝑝)⟩ = ⟨((1^st ‘𝐺)‘𝑥), ((1^st ‘𝐿)‘𝑥)⟩)
54		fveq2 6882	. . . . . . . . 9 ⊢ (𝑝 = 𝑥 → ((1^st ‘𝑁)‘𝑝) = ((1^st ‘𝑁)‘𝑥))
55	53, 54	oveq12d 7435	. . . . . . . 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 6882	. . . . . . . 8 ⊢ (𝑝 = 𝑥 → (𝑉‘𝑝) = (𝑉‘𝑥))
57		fveq2 6882	. . . . . . . 8 ⊢ (𝑝 = 𝑥 → (𝑆‘𝑝) = (𝑆‘𝑥))
58	55, 56, 57	oveq123d 7438	. . . . . . 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 2762	. . . . . . 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 7450	. . . . . . 7 ⊢ ((𝑉‘𝑝)(⟨((1^st ‘𝐺)‘𝑝), ((1^st ‘𝐿)‘𝑝)⟩ ∗ ((1^st ‘𝑁)‘𝑝))(𝑆‘𝑝)) ∈ V
61	58, 59, 60	fvmpt3i 6996	. . . . . 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 6886	. . . . 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 487	. . . 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 7435	. . 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 5202	. 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 2801	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

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ⟨cop 4593 ↦ cmpt 5190 × cxp 5657 ‘cfv 6537 (class class class)co 7417 1^st c1st 7988 2^nd c2nd 7989 Basecbs 17307 compcco 17360 Func cfunc 17949 Nat cnat 18039 FuncCat cfuc 18040 ×_c cxpc 18262 ∘_F cfuco 50250

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 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7740 ax-cnex 11184 ax-resscn 11185 ax-1cn 11186 ax-icn 11187 ax-addcl 11188 ax-addrcl 11189 ax-mulcl 11190 ax-mulrcl 11191 ax-mulcom 11192 ax-addass 11193 ax-mulass 11194 ax-distr 11195 ax-i2m1 11196 ax-1ne0 11197 ax-1rid 11198 ax-rnegex 11199 ax-rrecex 11200 ax-cnre 11201 ax-pre-lttri 11202 ax-pre-lttrn 11203 ax-pre-ltadd 11204 ax-pre-mulgt0 11205

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 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4871 df-iun 4956 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7374 df-ov 7420 df-oprab 7421 df-mpo 7422 df-om 7867 df-1st 7990 df-2nd 7991 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-er 8700 df-map 8832 df-ixp 8909 df-en 8957 df-dom 8958 df-sdom 8959 df-fin 8960 df-pnf 11273 df-mnf 11274 df-xr 11275 df-ltxr 11276 df-le 11277 df-sub 11471 df-neg 11472 df-nn 12262 df-2 12331 df-3 12332 df-4 12333 df-5 12334 df-6 12335 df-7 12336 df-8 12337 df-9 12338 df-n0 12533 df-z 12620 df-dec 12741 df-uz 12892 df-fz 13566 df-struct 17245 df-slot 17280 df-ndx 17292 df-base 17308 df-hom 17372 df-cco 17373 df-cat 17762 df-func 17953 df-cofu 17955 df-nat 18041 df-fuc 18042 df-xpc 18266 df-fuco 50251

This theorem is used by: fucocolem3 50289

W3C validator