Theorem fucocolem4 50571

Description: Lemma for fucoco 50572. 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⟩)
fucoco.q	⊢ 𝑄 = (𝐶 FuncCat 𝐸)
fucoco.oq	⊢ ∙ = (comp‘𝑄)

Assertion

Ref

Expression

fucocolem4

⊢ (𝜑 → (((𝑌𝑃𝑍)‘𝐵)(⟨(𝑂‘𝑋), (𝑂‘𝑌)⟩ ∙ (𝑂‘𝑍))((𝑋𝑃𝑌)‘𝐴)) = (𝑥 ∈ (Base‘𝐶) ↦ (((𝑈‘((1st ‘𝑁)‘𝑥))(⟨((1st ‘𝐾)‘((1st ‘𝐿)‘𝑥)), ((1st ‘𝐾)‘((1st ‘𝑁)‘𝑥))⟩(comp‘𝐸)((1st ‘𝑀)‘((1st ‘𝑁)‘𝑥)))((((1st ‘𝐿)‘𝑥)(2nd ‘𝐾)((1st ‘𝑁)‘𝑥))‘(𝑉‘𝑥)))(⟨((1st ‘𝐹)‘((1st ‘𝐺)‘𝑥)), ((1st ‘𝐾)‘((1st ‘𝐿)‘𝑥))⟩(comp‘𝐸)((1st ‘𝑀)‘((1st ‘𝑁)‘𝑥)))((𝑅‘((1st ‘𝐿)‘𝑥))(⟨((1st ‘𝐹)‘((1st ‘𝐺)‘𝑥)), ((1st ‘𝐹)‘((1st ‘𝐿)‘𝑥))⟩(comp‘𝐸)((1st ‘𝐾)‘((1st ‘𝐿)‘𝑥)))((((1st ‘𝐺)‘𝑥)(2nd ‘𝐹)((1st ‘𝐿)‘𝑥))‘(𝑆‘𝑥))))))

Distinct variable groups:   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   𝑥,𝐴   𝑥,𝐵   𝑥,𝑂   𝑥,𝑃   𝑥,𝑌

Allowed substitution hints:   𝑄(𝑥)   ∙ (𝑥)

Proof of Theorem fucocolem4

Step	Hyp	Ref	Expression
1		fucoco.q	. . 3 ⊢ 𝑄 = (𝐶 FuncCat 𝐸)
2		eqid 2760	. . 3 ⊢ (𝐶 Nat 𝐸) = (𝐶 Nat 𝐸)
3		eqid 2760	. . 3 ⊢ (Base‘𝐶) = (Base‘𝐶)
4		eqid 2760	. . 3 ⊢ (comp‘𝐸) = (comp‘𝐸)
5		fucoco.oq	. . 3 ⊢ ∙ = (comp‘𝑄)
6		fucoco.a	. . . . . 6 ⊢ (𝜑 → 𝐴 = ⟨𝑅, 𝑆⟩)
7	6	fveq2d 6889	. . . . 5 ⊢ (𝜑 → ((𝑋𝑃𝑌)‘𝐴) = ((𝑋𝑃𝑌)‘⟨𝑅, 𝑆⟩))
8		df-ov 7423	. . . . 5 ⊢ (𝑅(𝑋𝑃𝑌)𝑆) = ((𝑋𝑃𝑌)‘⟨𝑅, 𝑆⟩)
9	7, 8	eqtr4di 2813	. . . 4 ⊢ (𝜑 → ((𝑋𝑃𝑌)‘𝐴) = (𝑅(𝑋𝑃𝑌)𝑆))
10		fucoco.o	. . . . 5 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = ⟨𝑂, 𝑃⟩)
11		fucoco.s	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ (𝐺(𝐶 Nat 𝐷)𝐿))
12		fucoco.r	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ (𝐹(𝐷 Nat 𝐸)𝐾))
13		fucoco.x	. . . . 5 ⊢ (𝜑 → 𝑋 = ⟨𝐹, 𝐺⟩)
14		fucoco.y	. . . . 5 ⊢ (𝜑 → 𝑌 = ⟨𝐾, 𝐿⟩)
15	10, 11, 12, 13, 14	fuco22nat 50561	. . . 4 ⊢ (𝜑 → (𝑅(𝑋𝑃𝑌)𝑆) ∈ ((𝑂‘𝑋)(𝐶 Nat 𝐸)(𝑂‘𝑌)))
16	9, 15	eqeltrd 2860	. . 3 ⊢ (𝜑 → ((𝑋𝑃𝑌)‘𝐴) ∈ ((𝑂‘𝑋)(𝐶 Nat 𝐸)(𝑂‘𝑌)))
17		fucoco.b	. . . . . 6 ⊢ (𝜑 → 𝐵 = ⟨𝑈, 𝑉⟩)
18	17	fveq2d 6889	. . . . 5 ⊢ (𝜑 → ((𝑌𝑃𝑍)‘𝐵) = ((𝑌𝑃𝑍)‘⟨𝑈, 𝑉⟩))
19		df-ov 7423	. . . . 5 ⊢ (𝑈(𝑌𝑃𝑍)𝑉) = ((𝑌𝑃𝑍)‘⟨𝑈, 𝑉⟩)
20	18, 19	eqtr4di 2813	. . . 4 ⊢ (𝜑 → ((𝑌𝑃𝑍)‘𝐵) = (𝑈(𝑌𝑃𝑍)𝑉))
21		fucoco.v	. . . . 5 ⊢ (𝜑 → 𝑉 ∈ (𝐿(𝐶 Nat 𝐷)𝑁))
22		fucoco.u	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ (𝐾(𝐷 Nat 𝐸)𝑀))
23		fucoco.z	. . . . 5 ⊢ (𝜑 → 𝑍 = ⟨𝑀, 𝑁⟩)
24	10, 21, 22, 14, 23	fuco22nat 50561	. . . 4 ⊢ (𝜑 → (𝑈(𝑌𝑃𝑍)𝑉) ∈ ((𝑂‘𝑌)(𝐶 Nat 𝐸)(𝑂‘𝑍)))
25	20, 24	eqeltrd 2860	. . 3 ⊢ (𝜑 → ((𝑌𝑃𝑍)‘𝐵) ∈ ((𝑂‘𝑌)(𝐶 Nat 𝐸)(𝑂‘𝑍)))
26	1, 2, 3, 4, 5, 16, 25	fucco 18145	. 2 ⊢ (𝜑 → (((𝑌𝑃𝑍)‘𝐵)(⟨(𝑂‘𝑋), (𝑂‘𝑌)⟩ ∙ (𝑂‘𝑍))((𝑋𝑃𝑌)‘𝐴)) = (𝑥 ∈ (Base‘𝐶) ↦ ((((𝑌𝑃𝑍)‘𝐵)‘𝑥)(⟨((1st ‘(𝑂‘𝑋))‘𝑥), ((1st ‘(𝑂‘𝑌))‘𝑥)⟩(comp‘𝐸)((1st ‘(𝑂‘𝑍))‘𝑥))(((𝑋𝑃𝑌)‘𝐴)‘𝑥))))
27		eqid 2760	. . . . . . . . . . . . . 14 ⊢ (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷)
28	27	natrcl 18133	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ (𝐺(𝐶 Nat 𝐷)𝐿) → (𝐺 ∈ (𝐶 Func 𝐷) ∧ 𝐿 ∈ (𝐶 Func 𝐷)))
29	11, 28	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐺 ∈ (𝐶 Func 𝐷) ∧ 𝐿 ∈ (𝐶 Func 𝐷)))
30	29	simpld 500	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷))
31	30	func1st2nd 50283	. . . . . . . . . 10 ⊢ (𝜑 → (1st ‘𝐺)(𝐶 Func 𝐷)(2nd ‘𝐺))
32		eqid 2760	. . . . . . . . . . . . . 14 ⊢ (𝐷 Nat 𝐸) = (𝐷 Nat 𝐸)
33	32	natrcl 18133	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ (𝐹(𝐷 Nat 𝐸)𝐾) → (𝐹 ∈ (𝐷 Func 𝐸) ∧ 𝐾 ∈ (𝐷 Func 𝐸)))
34	12, 33	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹 ∈ (𝐷 Func 𝐸) ∧ 𝐾 ∈ (𝐷 Func 𝐸)))
35	34	simpld 500	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ (𝐷 Func 𝐸))
36	35	func1st2nd 50283	. . . . . . . . . 10 ⊢ (𝜑 → (1st ‘𝐹)(𝐷 Func 𝐸)(2nd ‘𝐹))
37		relfunc 18042	. . . . . . . . . . . . 13 ⊢ Rel (𝐷 Func 𝐸)
38		1st2nd 8050	. . . . . . . . . . . . 13 ⊢ ((Rel (𝐷 Func 𝐸) ∧ 𝐹 ∈ (𝐷 Func 𝐸)) → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
39	37, 35, 38	sylancr 599	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
40		relfunc 18042	. . . . . . . . . . . . 13 ⊢ Rel (𝐶 Func 𝐷)
41		1st2nd 8050	. . . . . . . . . . . . 13 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → 𝐺 = ⟨(1st ‘𝐺), (2nd ‘𝐺)⟩)
42	40, 30, 41	sylancr 599	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺 = ⟨(1st ‘𝐺), (2nd ‘𝐺)⟩)
43	39, 42	opeq12d 4847	. . . . . . . . . . 11 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ = ⟨⟨(1st ‘𝐹), (2nd ‘𝐹)⟩, ⟨(1st ‘𝐺), (2nd ‘𝐺)⟩⟩)
44	13, 43	eqtrd 2795	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 = ⟨⟨(1st ‘𝐹), (2nd ‘𝐹)⟩, ⟨(1st ‘𝐺), (2nd ‘𝐺)⟩⟩)
45	10, 31, 36, 44	fuco111 50545	. . . . . . . . 9 ⊢ (𝜑 → (1st ‘(𝑂‘𝑋)) = ((1st ‘𝐹) ∘ (1st ‘𝐺)))
46	45	fveq1d 6887	. . . . . . . 8 ⊢ (𝜑 → ((1st ‘(𝑂‘𝑋))‘𝑥) = (((1st ‘𝐹) ∘ (1st ‘𝐺))‘𝑥))
47	46	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘(𝑂‘𝑋))‘𝑥) = (((1st ‘𝐹) ∘ (1st ‘𝐺))‘𝑥))
48		eqid 2760	. . . . . . . . . 10 ⊢ (Base‘𝐷) = (Base‘𝐷)
49	3, 48, 31	funcf1 18046	. . . . . . . . 9 ⊢ (𝜑 → (1st ‘𝐺):(Base‘𝐶)⟶(Base‘𝐷))
50	49	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (1st ‘𝐺):(Base‘𝐶)⟶(Base‘𝐷))
51		simpr 490	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
52	50, 51	fvco3d 6986	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (((1st ‘𝐹) ∘ (1st ‘𝐺))‘𝑥) = ((1st ‘𝐹)‘((1st ‘𝐺)‘𝑥)))
53	47, 52	eqtrd 2795	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘(𝑂‘𝑋))‘𝑥) = ((1st ‘𝐹)‘((1st ‘𝐺)‘𝑥)))
54	29	simprd 501	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐿 ∈ (𝐶 Func 𝐷))
55	54	func1st2nd 50283	. . . . . . . . . 10 ⊢ (𝜑 → (1st ‘𝐿)(𝐶 Func 𝐷)(2nd ‘𝐿))
56	34	simprd 501	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐾 ∈ (𝐷 Func 𝐸))
57	56	func1st2nd 50283	. . . . . . . . . 10 ⊢ (𝜑 → (1st ‘𝐾)(𝐷 Func 𝐸)(2nd ‘𝐾))
58		1st2nd 8050	. . . . . . . . . . . . 13 ⊢ ((Rel (𝐷 Func 𝐸) ∧ 𝐾 ∈ (𝐷 Func 𝐸)) → 𝐾 = ⟨(1st ‘𝐾), (2nd ‘𝐾)⟩)
59	37, 56, 58	sylancr 599	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾 = ⟨(1st ‘𝐾), (2nd ‘𝐾)⟩)
60		1st2nd 8050	. . . . . . . . . . . . 13 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐿 ∈ (𝐶 Func 𝐷)) → 𝐿 = ⟨(1st ‘𝐿), (2nd ‘𝐿)⟩)
61	40, 54, 60	sylancr 599	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐿 = ⟨(1st ‘𝐿), (2nd ‘𝐿)⟩)
62	59, 61	opeq12d 4847	. . . . . . . . . . 11 ⊢ (𝜑 → ⟨𝐾, 𝐿⟩ = ⟨⟨(1st ‘𝐾), (2nd ‘𝐾)⟩, ⟨(1st ‘𝐿), (2nd ‘𝐿)⟩⟩)
63	14, 62	eqtrd 2795	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 = ⟨⟨(1st ‘𝐾), (2nd ‘𝐾)⟩, ⟨(1st ‘𝐿), (2nd ‘𝐿)⟩⟩)
64	10, 55, 57, 63	fuco111 50545	. . . . . . . . 9 ⊢ (𝜑 → (1st ‘(𝑂‘𝑌)) = ((1st ‘𝐾) ∘ (1st ‘𝐿)))
65	64	fveq1d 6887	. . . . . . . 8 ⊢ (𝜑 → ((1st ‘(𝑂‘𝑌))‘𝑥) = (((1st ‘𝐾) ∘ (1st ‘𝐿))‘𝑥))
66	65	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘(𝑂‘𝑌))‘𝑥) = (((1st ‘𝐾) ∘ (1st ‘𝐿))‘𝑥))
67	3, 48, 55	funcf1 18046	. . . . . . . . 9 ⊢ (𝜑 → (1st ‘𝐿):(Base‘𝐶)⟶(Base‘𝐷))
68	67	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (1st ‘𝐿):(Base‘𝐶)⟶(Base‘𝐷))
69	68, 51	fvco3d 6986	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (((1st ‘𝐾) ∘ (1st ‘𝐿))‘𝑥) = ((1st ‘𝐾)‘((1st ‘𝐿)‘𝑥)))
70	66, 69	eqtrd 2795	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘(𝑂‘𝑌))‘𝑥) = ((1st ‘𝐾)‘((1st ‘𝐿)‘𝑥)))
71	53, 70	opeq12d 4847	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ⟨((1st ‘(𝑂‘𝑋))‘𝑥), ((1st ‘(𝑂‘𝑌))‘𝑥)⟩ = ⟨((1st ‘𝐹)‘((1st ‘𝐺)‘𝑥)), ((1st ‘𝐾)‘((1st ‘𝐿)‘𝑥))⟩)
72	27	natrcl 18133	. . . . . . . . . . . 12 ⊢ (𝑉 ∈ (𝐿(𝐶 Nat 𝐷)𝑁) → (𝐿 ∈ (𝐶 Func 𝐷) ∧ 𝑁 ∈ (𝐶 Func 𝐷)))
73	21, 72	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐿 ∈ (𝐶 Func 𝐷) ∧ 𝑁 ∈ (𝐶 Func 𝐷)))
74	73	simprd 501	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ (𝐶 Func 𝐷))
75	74	func1st2nd 50283	. . . . . . . . 9 ⊢ (𝜑 → (1st ‘𝑁)(𝐶 Func 𝐷)(2nd ‘𝑁))
76	32	natrcl 18133	. . . . . . . . . . . 12 ⊢ (𝑈 ∈ (𝐾(𝐷 Nat 𝐸)𝑀) → (𝐾 ∈ (𝐷 Func 𝐸) ∧ 𝑀 ∈ (𝐷 Func 𝐸)))
77	22, 76	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐾 ∈ (𝐷 Func 𝐸) ∧ 𝑀 ∈ (𝐷 Func 𝐸)))
78	77	simprd 501	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ (𝐷 Func 𝐸))
79	78	func1st2nd 50283	. . . . . . . . 9 ⊢ (𝜑 → (1st ‘𝑀)(𝐷 Func 𝐸)(2nd ‘𝑀))
80		1st2nd 8050	. . . . . . . . . . . 12 ⊢ ((Rel (𝐷 Func 𝐸) ∧ 𝑀 ∈ (𝐷 Func 𝐸)) → 𝑀 = ⟨(1st ‘𝑀), (2nd ‘𝑀)⟩)
81	37, 78, 80	sylancr 599	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 = ⟨(1st ‘𝑀), (2nd ‘𝑀)⟩)
82		1st2nd 8050	. . . . . . . . . . . 12 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝑁 ∈ (𝐶 Func 𝐷)) → 𝑁 = ⟨(1st ‘𝑁), (2nd ‘𝑁)⟩)
83	40, 74, 82	sylancr 599	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 = ⟨(1st ‘𝑁), (2nd ‘𝑁)⟩)
84	81, 83	opeq12d 4847	. . . . . . . . . 10 ⊢ (𝜑 → ⟨𝑀, 𝑁⟩ = ⟨⟨(1st ‘𝑀), (2nd ‘𝑀)⟩, ⟨(1st ‘𝑁), (2nd ‘𝑁)⟩⟩)
85	23, 84	eqtrd 2795	. . . . . . . . 9 ⊢ (𝜑 → 𝑍 = ⟨⟨(1st ‘𝑀), (2nd ‘𝑀)⟩, ⟨(1st ‘𝑁), (2nd ‘𝑁)⟩⟩)
86	10, 75, 79, 85	fuco111 50545	. . . . . . . 8 ⊢ (𝜑 → (1st ‘(𝑂‘𝑍)) = ((1st ‘𝑀) ∘ (1st ‘𝑁)))
87	86	fveq1d 6887	. . . . . . 7 ⊢ (𝜑 → ((1st ‘(𝑂‘𝑍))‘𝑥) = (((1st ‘𝑀) ∘ (1st ‘𝑁))‘𝑥))
88	87	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘(𝑂‘𝑍))‘𝑥) = (((1st ‘𝑀) ∘ (1st ‘𝑁))‘𝑥))
89	3, 48, 75	funcf1 18046	. . . . . . . 8 ⊢ (𝜑 → (1st ‘𝑁):(Base‘𝐶)⟶(Base‘𝐷))
90	89	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (1st ‘𝑁):(Base‘𝐶)⟶(Base‘𝐷))
91	90, 51	fvco3d 6986	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (((1st ‘𝑀) ∘ (1st ‘𝑁))‘𝑥) = ((1st ‘𝑀)‘((1st ‘𝑁)‘𝑥)))
92	88, 91	eqtrd 2795	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘(𝑂‘𝑍))‘𝑥) = ((1st ‘𝑀)‘((1st ‘𝑁)‘𝑥)))
93	71, 92	oveq12d 7438	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (⟨((1st ‘(𝑂‘𝑋))‘𝑥), ((1st ‘(𝑂‘𝑌))‘𝑥)⟩(comp‘𝐸)((1st ‘(𝑂‘𝑍))‘𝑥)) = (⟨((1st ‘𝐹)‘((1st ‘𝐺)‘𝑥)), ((1st ‘𝐾)‘((1st ‘𝐿)‘𝑥))⟩(comp‘𝐸)((1st ‘𝑀)‘((1st ‘𝑁)‘𝑥))))
94	10, 14, 23, 21, 22	fuco22a 50565	. . . . . 6 ⊢ (𝜑 → (𝑈(𝑌𝑃𝑍)𝑉) = (𝑥 ∈ (Base‘𝐶) ↦ ((𝑈‘((1st ‘𝑁)‘𝑥))(⟨((1st ‘𝐾)‘((1st ‘𝐿)‘𝑥)), ((1st ‘𝐾)‘((1st ‘𝑁)‘𝑥))⟩(comp‘𝐸)((1st ‘𝑀)‘((1st ‘𝑁)‘𝑥)))((((1st ‘𝐿)‘𝑥)(2nd ‘𝐾)((1st ‘𝑁)‘𝑥))‘(𝑉‘𝑥)))))
95	20, 94	eqtrd 2795	. . . . 5 ⊢ (𝜑 → ((𝑌𝑃𝑍)‘𝐵) = (𝑥 ∈ (Base‘𝐶) ↦ ((𝑈‘((1st ‘𝑁)‘𝑥))(⟨((1st ‘𝐾)‘((1st ‘𝐿)‘𝑥)), ((1st ‘𝐾)‘((1st ‘𝑁)‘𝑥))⟩(comp‘𝐸)((1st ‘𝑀)‘((1st ‘𝑁)‘𝑥)))((((1st ‘𝐿)‘𝑥)(2nd ‘𝐾)((1st ‘𝑁)‘𝑥))‘(𝑉‘𝑥)))))
96		ovexd 7455	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((𝑈‘((1st ‘𝑁)‘𝑥))(⟨((1st ‘𝐾)‘((1st ‘𝐿)‘𝑥)), ((1st ‘𝐾)‘((1st ‘𝑁)‘𝑥))⟩(comp‘𝐸)((1st ‘𝑀)‘((1st ‘𝑁)‘𝑥)))((((1st ‘𝐿)‘𝑥)(2nd ‘𝐾)((1st ‘𝑁)‘𝑥))‘(𝑉‘𝑥))) ∈ V)
97	95, 96	fvmpt2d 7007	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (((𝑌𝑃𝑍)‘𝐵)‘𝑥) = ((𝑈‘((1st ‘𝑁)‘𝑥))(⟨((1st ‘𝐾)‘((1st ‘𝐿)‘𝑥)), ((1st ‘𝐾)‘((1st ‘𝑁)‘𝑥))⟩(comp‘𝐸)((1st ‘𝑀)‘((1st ‘𝑁)‘𝑥)))((((1st ‘𝐿)‘𝑥)(2nd ‘𝐾)((1st ‘𝑁)‘𝑥))‘(𝑉‘𝑥))))
98	10, 13, 14, 11, 12	fuco22a 50565	. . . . . 6 ⊢ (𝜑 → (𝑅(𝑋𝑃𝑌)𝑆) = (𝑥 ∈ (Base‘𝐶) ↦ ((𝑅‘((1st ‘𝐿)‘𝑥))(⟨((1st ‘𝐹)‘((1st ‘𝐺)‘𝑥)), ((1st ‘𝐹)‘((1st ‘𝐿)‘𝑥))⟩(comp‘𝐸)((1st ‘𝐾)‘((1st ‘𝐿)‘𝑥)))((((1st ‘𝐺)‘𝑥)(2nd ‘𝐹)((1st ‘𝐿)‘𝑥))‘(𝑆‘𝑥)))))
99	9, 98	eqtrd 2795	. . . . 5 ⊢ (𝜑 → ((𝑋𝑃𝑌)‘𝐴) = (𝑥 ∈ (Base‘𝐶) ↦ ((𝑅‘((1st ‘𝐿)‘𝑥))(⟨((1st ‘𝐹)‘((1st ‘𝐺)‘𝑥)), ((1st ‘𝐹)‘((1st ‘𝐿)‘𝑥))⟩(comp‘𝐸)((1st ‘𝐾)‘((1st ‘𝐿)‘𝑥)))((((1st ‘𝐺)‘𝑥)(2nd ‘𝐹)((1st ‘𝐿)‘𝑥))‘(𝑆‘𝑥)))))
100		ovexd 7455	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((𝑅‘((1st ‘𝐿)‘𝑥))(⟨((1st ‘𝐹)‘((1st ‘𝐺)‘𝑥)), ((1st ‘𝐹)‘((1st ‘𝐿)‘𝑥))⟩(comp‘𝐸)((1st ‘𝐾)‘((1st ‘𝐿)‘𝑥)))((((1st ‘𝐺)‘𝑥)(2nd ‘𝐹)((1st ‘𝐿)‘𝑥))‘(𝑆‘𝑥))) ∈ V)
101	99, 100	fvmpt2d 7007	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (((𝑋𝑃𝑌)‘𝐴)‘𝑥) = ((𝑅‘((1st ‘𝐿)‘𝑥))(⟨((1st ‘𝐹)‘((1st ‘𝐺)‘𝑥)), ((1st ‘𝐹)‘((1st ‘𝐿)‘𝑥))⟩(comp‘𝐸)((1st ‘𝐾)‘((1st ‘𝐿)‘𝑥)))((((1st ‘𝐺)‘𝑥)(2nd ‘𝐹)((1st ‘𝐿)‘𝑥))‘(𝑆‘𝑥))))
102	93, 97, 101	oveq123d 7441	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((((𝑌𝑃𝑍)‘𝐵)‘𝑥)(⟨((1st ‘(𝑂‘𝑋))‘𝑥), ((1st ‘(𝑂‘𝑌))‘𝑥)⟩(comp‘𝐸)((1st ‘(𝑂‘𝑍))‘𝑥))(((𝑋𝑃𝑌)‘𝐴)‘𝑥)) = (((𝑈‘((1st ‘𝑁)‘𝑥))(⟨((1st ‘𝐾)‘((1st ‘𝐿)‘𝑥)), ((1st ‘𝐾)‘((1st ‘𝑁)‘𝑥))⟩(comp‘𝐸)((1st ‘𝑀)‘((1st ‘𝑁)‘𝑥)))((((1st ‘𝐿)‘𝑥)(2nd ‘𝐾)((1st ‘𝑁)‘𝑥))‘(𝑉‘𝑥)))(⟨((1st ‘𝐹)‘((1st ‘𝐺)‘𝑥)), ((1st ‘𝐾)‘((1st ‘𝐿)‘𝑥))⟩(comp‘𝐸)((1st ‘𝑀)‘((1st ‘𝑁)‘𝑥)))((𝑅‘((1st ‘𝐿)‘𝑥))(⟨((1st ‘𝐹)‘((1st ‘𝐺)‘𝑥)), ((1st ‘𝐹)‘((1st ‘𝐿)‘𝑥))⟩(comp‘𝐸)((1st ‘𝐾)‘((1st ‘𝐿)‘𝑥)))((((1st ‘𝐺)‘𝑥)(2nd ‘𝐹)((1st ‘𝐿)‘𝑥))‘(𝑆‘𝑥)))))
103	102	mpteq2dva 5197	. 2 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐶) ↦ ((((𝑌𝑃𝑍)‘𝐵)‘𝑥)(⟨((1st ‘(𝑂‘𝑋))‘𝑥), ((1st ‘(𝑂‘𝑌))‘𝑥)⟩(comp‘𝐸)((1st ‘(𝑂‘𝑍))‘𝑥))(((𝑋𝑃𝑌)‘𝐴)‘𝑥))) = (𝑥 ∈ (Base‘𝐶) ↦ (((𝑈‘((1st ‘𝑁)‘𝑥))(⟨((1st ‘𝐾)‘((1st ‘𝐿)‘𝑥)), ((1st ‘𝐾)‘((1st ‘𝑁)‘𝑥))⟩(comp‘𝐸)((1st ‘𝑀)‘((1st ‘𝑁)‘𝑥)))((((1st ‘𝐿)‘𝑥)(2nd ‘𝐾)((1st ‘𝑁)‘𝑥))‘(𝑉‘𝑥)))(⟨((1st ‘𝐹)‘((1st ‘𝐺)‘𝑥)), ((1st ‘𝐾)‘((1st ‘𝐿)‘𝑥))⟩(comp‘𝐸)((1st ‘𝑀)‘((1st ‘𝑁)‘𝑥)))((𝑅‘((1st ‘𝐿)‘𝑥))(⟨((1st ‘𝐹)‘((1st ‘𝐺)‘𝑥)), ((1st ‘𝐹)‘((1st ‘𝐿)‘𝑥))⟩(comp‘𝐸)((1st ‘𝐾)‘((1st ‘𝐿)‘𝑥)))((((1st ‘𝐺)‘𝑥)(2nd ‘𝐹)((1st ‘𝐿)‘𝑥))‘(𝑆‘𝑥))))))
104	26, 103	eqtrd 2795	1 ⊢ (𝜑 → (((𝑌𝑃𝑍)‘𝐵)(⟨(𝑂‘𝑋), (𝑂‘𝑌)⟩ ∙ (𝑂‘𝑍))((𝑋𝑃𝑌)‘𝐴)) = (𝑥 ∈ (Base‘𝐶) ↦ (((𝑈‘((1st ‘𝑁)‘𝑥))(⟨((1st ‘𝐾)‘((1st ‘𝐿)‘𝑥)), ((1st ‘𝐾)‘((1st ‘𝑁)‘𝑥))⟩(comp‘𝐸)((1st ‘𝑀)‘((1st ‘𝑁)‘𝑥)))((((1st ‘𝐿)‘𝑥)(2nd ‘𝐾)((1st ‘𝑁)‘𝑥))‘(𝑉‘𝑥)))(⟨((1st ‘𝐹)‘((1st ‘𝐺)‘𝑥)), ((1st ‘𝐾)‘((1st ‘𝐿)‘𝑥))⟩(comp‘𝐸)((1st ‘𝑀)‘((1st ‘𝑁)‘𝑥)))((𝑅‘((1st ‘𝐿)‘𝑥))(⟨((1st ‘𝐹)‘((1st ‘𝐺)‘𝑥)), ((1st ‘𝐹)‘((1st ‘𝐿)‘𝑥))⟩(comp‘𝐸)((1st ‘𝐾)‘((1st ‘𝐿)‘𝑥)))((((1st ‘𝐺)‘𝑥)(2nd ‘𝐹)((1st ‘𝐿)‘𝑥))‘(𝑆‘𝑥))))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  Vcvv 3450  ⟨cop 4834   ↦ cmpt 5185   ∘ ccom 5654  Rel wrel 5655  ⟶wf 6534  ‘cfv 6538  (class class class)co 7420  1^st c1st 7999  2^nd c2nd 8000  Basecbs 17392  compcco 17445   Func cfunc 18034   Nat cnat 18124   FuncCat cfuc 18125   ∘_F cfuco 50531

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-cnex 11259  ax-resscn 11260  ax-1cn 11261  ax-icn 11262  ax-addcl 11263  ax-addrcl 11264  ax-mulcl 11265  ax-mulrcl 11266  ax-mulcom 11267  ax-addass 11268  ax-mulass 11269  ax-distr 11270  ax-i2m1 11271  ax-1ne0 11272  ax-1rid 11273  ax-rnegex 11274  ax-rrecex 11275  ax-cnre 11276  ax-pre-lttri 11277  ax-pre-lttrn 11278  ax-pre-ltadd 11279  ax-pre-mulgt0 11280

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  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 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8302  df-wrecs 8333  df-recs 8382  df-rdg 8421  df-1o 8479  df-er 8720  df-map 8852  df-ixp 8929  df-en 8977  df-dom 8978  df-sdom 8979  df-fin 8980  df-pnf 11348  df-mnf 11349  df-xr 11350  df-ltxr 11351  df-le 11352  df-sub 11546  df-neg 11547  df-nn 12339  df-2 12408  df-3 12409  df-4 12410  df-5 12411  df-6 12412  df-7 12413  df-8 12414  df-9 12415  df-n0 12610  df-z 12697  df-dec 12818  df-uz 12969  df-fz 13643  df-struct 17330  df-slot 17365  df-ndx 17377  df-base 17393  df-hom 17457  df-cco 17458  df-cat 17847  df-cid 17848  df-func 18038  df-cofu 18040  df-nat 18126  df-fuc 18127  df-fuco 50532

This theorem is used by:  fucoco  50572