Theorem fucocolem4 49985

Description: Lemma for fucoco 49986. 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 2765	. . 3 ⊢ (𝐶 Nat 𝐸) = (𝐶 Nat 𝐸)
3		eqid 2765	. . 3 ⊢ (Base‘𝐶) = (Base‘𝐶)
4		eqid 2765	. . 3 ⊢ (comp‘𝐸) = (comp‘𝐸)
5		fucoco.oq	. . 3 ⊢ ∙ = (comp‘𝑄)
6		fucoco.a	. . . . . 6 ⊢ (𝜑 → 𝐴 = ⟨𝑅, 𝑆⟩)
7	6	fveq2d 6875	. . . . 5 ⊢ (𝜑 → ((𝑋𝑃𝑌)‘𝐴) = ((𝑋𝑃𝑌)‘⟨𝑅, 𝑆⟩))
8		df-ov 7403	. . . . 5 ⊢ (𝑅(𝑋𝑃𝑌)𝑆) = ((𝑋𝑃𝑌)‘⟨𝑅, 𝑆⟩)
9	7, 8	eqtr4di 2818	. . . 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 49975	. . . 4 ⊢ (𝜑 → (𝑅(𝑋𝑃𝑌)𝑆) ∈ ((𝑂‘𝑋)(𝐶 Nat 𝐸)(𝑂‘𝑌)))
16	9, 15	eqeltrd 2865	. . 3 ⊢ (𝜑 → ((𝑋𝑃𝑌)‘𝐴) ∈ ((𝑂‘𝑋)(𝐶 Nat 𝐸)(𝑂‘𝑌)))
17		fucoco.b	. . . . . 6 ⊢ (𝜑 → 𝐵 = ⟨𝑈, 𝑉⟩)
18	17	fveq2d 6875	. . . . 5 ⊢ (𝜑 → ((𝑌𝑃𝑍)‘𝐵) = ((𝑌𝑃𝑍)‘⟨𝑈, 𝑉⟩))
19		df-ov 7403	. . . . 5 ⊢ (𝑈(𝑌𝑃𝑍)𝑉) = ((𝑌𝑃𝑍)‘⟨𝑈, 𝑉⟩)
20	18, 19	eqtr4di 2818	. . . 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 49975	. . . 4 ⊢ (𝜑 → (𝑈(𝑌𝑃𝑍)𝑉) ∈ ((𝑂‘𝑌)(𝐶 Nat 𝐸)(𝑂‘𝑍)))
25	20, 24	eqeltrd 2865	. . 3 ⊢ (𝜑 → ((𝑌𝑃𝑍)‘𝐵) ∈ ((𝑂‘𝑌)(𝐶 Nat 𝐸)(𝑂‘𝑍)))
26	1, 2, 3, 4, 5, 16, 25	fucco 18012	. 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 2765	. . . . . . . . . . . . . 14 ⊢ (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷)
28	27	natrcl 18000	. . . . . . . . . . . . 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 499	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷))
31	30	func1st2nd 49705	. . . . . . . . . 10 ⊢ (𝜑 → (1st ‘𝐺)(𝐶 Func 𝐷)(2nd ‘𝐺))
32		eqid 2765	. . . . . . . . . . . . . 14 ⊢ (𝐷 Nat 𝐸) = (𝐷 Nat 𝐸)
33	32	natrcl 18000	. . . . . . . . . . . . 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 499	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ (𝐷 Func 𝐸))
36	35	func1st2nd 49705	. . . . . . . . . 10 ⊢ (𝜑 → (1st ‘𝐹)(𝐷 Func 𝐸)(2nd ‘𝐹))
37		relfunc 17909	. . . . . . . . . . . . 13 ⊢ Rel (𝐷 Func 𝐸)
38		1st2nd 8024	. . . . . . . . . . . . 13 ⊢ ((Rel (𝐷 Func 𝐸) ∧ 𝐹 ∈ (𝐷 Func 𝐸)) → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
39	37, 35, 38	sylancr 598	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
40		relfunc 17909	. . . . . . . . . . . . 13 ⊢ Rel (𝐶 Func 𝐷)
41		1st2nd 8024	. . . . . . . . . . . . 13 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → 𝐺 = ⟨(1st ‘𝐺), (2nd ‘𝐺)⟩)
42	40, 30, 41	sylancr 598	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺 = ⟨(1st ‘𝐺), (2nd ‘𝐺)⟩)
43	39, 42	opeq12d 4842	. . . . . . . . . . 11 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ = ⟨⟨(1st ‘𝐹), (2nd ‘𝐹)⟩, ⟨(1st ‘𝐺), (2nd ‘𝐺)⟩⟩)
44	13, 43	eqtrd 2800	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 = ⟨⟨(1st ‘𝐹), (2nd ‘𝐹)⟩, ⟨(1st ‘𝐺), (2nd ‘𝐺)⟩⟩)
45	10, 31, 36, 44	fuco111 49959	. . . . . . . . 9 ⊢ (𝜑 → (1st ‘(𝑂‘𝑋)) = ((1st ‘𝐹) ∘ (1st ‘𝐺)))
46	45	fveq1d 6873	. . . . . . . 8 ⊢ (𝜑 → ((1st ‘(𝑂‘𝑋))‘𝑥) = (((1st ‘𝐹) ∘ (1st ‘𝐺))‘𝑥))
47	46	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘(𝑂‘𝑋))‘𝑥) = (((1st ‘𝐹) ∘ (1st ‘𝐺))‘𝑥))
48		eqid 2765	. . . . . . . . . 10 ⊢ (Base‘𝐷) = (Base‘𝐷)
49	3, 48, 31	funcf1 17913	. . . . . . . . 9 ⊢ (𝜑 → (1st ‘𝐺):(Base‘𝐶)⟶(Base‘𝐷))
50	49	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (1st ‘𝐺):(Base‘𝐶)⟶(Base‘𝐷))
51		simpr 489	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
52	50, 51	fvco3d 6972	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (((1st ‘𝐹) ∘ (1st ‘𝐺))‘𝑥) = ((1st ‘𝐹)‘((1st ‘𝐺)‘𝑥)))
53	47, 52	eqtrd 2800	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘(𝑂‘𝑋))‘𝑥) = ((1st ‘𝐹)‘((1st ‘𝐺)‘𝑥)))
54	29	simprd 500	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐿 ∈ (𝐶 Func 𝐷))
55	54	func1st2nd 49705	. . . . . . . . . 10 ⊢ (𝜑 → (1st ‘𝐿)(𝐶 Func 𝐷)(2nd ‘𝐿))
56	34	simprd 500	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐾 ∈ (𝐷 Func 𝐸))
57	56	func1st2nd 49705	. . . . . . . . . 10 ⊢ (𝜑 → (1st ‘𝐾)(𝐷 Func 𝐸)(2nd ‘𝐾))
58		1st2nd 8024	. . . . . . . . . . . . 13 ⊢ ((Rel (𝐷 Func 𝐸) ∧ 𝐾 ∈ (𝐷 Func 𝐸)) → 𝐾 = ⟨(1st ‘𝐾), (2nd ‘𝐾)⟩)
59	37, 56, 58	sylancr 598	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾 = ⟨(1st ‘𝐾), (2nd ‘𝐾)⟩)
60		1st2nd 8024	. . . . . . . . . . . . 13 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐿 ∈ (𝐶 Func 𝐷)) → 𝐿 = ⟨(1st ‘𝐿), (2nd ‘𝐿)⟩)
61	40, 54, 60	sylancr 598	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐿 = ⟨(1st ‘𝐿), (2nd ‘𝐿)⟩)
62	59, 61	opeq12d 4842	. . . . . . . . . . 11 ⊢ (𝜑 → ⟨𝐾, 𝐿⟩ = ⟨⟨(1st ‘𝐾), (2nd ‘𝐾)⟩, ⟨(1st ‘𝐿), (2nd ‘𝐿)⟩⟩)
63	14, 62	eqtrd 2800	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 = ⟨⟨(1st ‘𝐾), (2nd ‘𝐾)⟩, ⟨(1st ‘𝐿), (2nd ‘𝐿)⟩⟩)
64	10, 55, 57, 63	fuco111 49959	. . . . . . . . 9 ⊢ (𝜑 → (1st ‘(𝑂‘𝑌)) = ((1st ‘𝐾) ∘ (1st ‘𝐿)))
65	64	fveq1d 6873	. . . . . . . 8 ⊢ (𝜑 → ((1st ‘(𝑂‘𝑌))‘𝑥) = (((1st ‘𝐾) ∘ (1st ‘𝐿))‘𝑥))
66	65	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘(𝑂‘𝑌))‘𝑥) = (((1st ‘𝐾) ∘ (1st ‘𝐿))‘𝑥))
67	3, 48, 55	funcf1 17913	. . . . . . . . 9 ⊢ (𝜑 → (1st ‘𝐿):(Base‘𝐶)⟶(Base‘𝐷))
68	67	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (1st ‘𝐿):(Base‘𝐶)⟶(Base‘𝐷))
69	68, 51	fvco3d 6972	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (((1st ‘𝐾) ∘ (1st ‘𝐿))‘𝑥) = ((1st ‘𝐾)‘((1st ‘𝐿)‘𝑥)))
70	66, 69	eqtrd 2800	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘(𝑂‘𝑌))‘𝑥) = ((1st ‘𝐾)‘((1st ‘𝐿)‘𝑥)))
71	53, 70	opeq12d 4842	. . . . 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 18000	. . . . . . . . . . . 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 500	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ (𝐶 Func 𝐷))
75	74	func1st2nd 49705	. . . . . . . . 9 ⊢ (𝜑 → (1st ‘𝑁)(𝐶 Func 𝐷)(2nd ‘𝑁))
76	32	natrcl 18000	. . . . . . . . . . . 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 500	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ (𝐷 Func 𝐸))
79	78	func1st2nd 49705	. . . . . . . . 9 ⊢ (𝜑 → (1st ‘𝑀)(𝐷 Func 𝐸)(2nd ‘𝑀))
80		1st2nd 8024	. . . . . . . . . . . 12 ⊢ ((Rel (𝐷 Func 𝐸) ∧ 𝑀 ∈ (𝐷 Func 𝐸)) → 𝑀 = ⟨(1st ‘𝑀), (2nd ‘𝑀)⟩)
81	37, 78, 80	sylancr 598	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 = ⟨(1st ‘𝑀), (2nd ‘𝑀)⟩)
82		1st2nd 8024	. . . . . . . . . . . 12 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝑁 ∈ (𝐶 Func 𝐷)) → 𝑁 = ⟨(1st ‘𝑁), (2nd ‘𝑁)⟩)
83	40, 74, 82	sylancr 598	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 = ⟨(1st ‘𝑁), (2nd ‘𝑁)⟩)
84	81, 83	opeq12d 4842	. . . . . . . . . 10 ⊢ (𝜑 → ⟨𝑀, 𝑁⟩ = ⟨⟨(1st ‘𝑀), (2nd ‘𝑀)⟩, ⟨(1st ‘𝑁), (2nd ‘𝑁)⟩⟩)
85	23, 84	eqtrd 2800	. . . . . . . . 9 ⊢ (𝜑 → 𝑍 = ⟨⟨(1st ‘𝑀), (2nd ‘𝑀)⟩, ⟨(1st ‘𝑁), (2nd ‘𝑁)⟩⟩)
86	10, 75, 79, 85	fuco111 49959	. . . . . . . 8 ⊢ (𝜑 → (1st ‘(𝑂‘𝑍)) = ((1st ‘𝑀) ∘ (1st ‘𝑁)))
87	86	fveq1d 6873	. . . . . . 7 ⊢ (𝜑 → ((1st ‘(𝑂‘𝑍))‘𝑥) = (((1st ‘𝑀) ∘ (1st ‘𝑁))‘𝑥))
88	87	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘(𝑂‘𝑍))‘𝑥) = (((1st ‘𝑀) ∘ (1st ‘𝑁))‘𝑥))
89	3, 48, 75	funcf1 17913	. . . . . . . 8 ⊢ (𝜑 → (1st ‘𝑁):(Base‘𝐶)⟶(Base‘𝐷))
90	89	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (1st ‘𝑁):(Base‘𝐶)⟶(Base‘𝐷))
91	90, 51	fvco3d 6972	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (((1st ‘𝑀) ∘ (1st ‘𝑁))‘𝑥) = ((1st ‘𝑀)‘((1st ‘𝑁)‘𝑥)))
92	88, 91	eqtrd 2800	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘(𝑂‘𝑍))‘𝑥) = ((1st ‘𝑀)‘((1st ‘𝑁)‘𝑥)))
93	71, 92	oveq12d 7418	. . . 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 49979	. . . . . 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 2800	. . . . 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 7435	. . . . 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 6993	. . . 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 49979	. . . . . 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 2800	. . . . 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 7435	. . . . 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 6993	. . . 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 7421	. . 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 5198	. 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 2800	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))))))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1563   ∈ wcel 2145  Vcvv 3457  ⟨cop 4591   ↦ cmpt 5186   ∘ ccom 5656  Rel wrel 5657  ⟶wf 6521  ‘cfv 6525  (class class class)co 7400  1^st c1st 7972  2^nd c2nd 7973  Basecbs 17259  compcco 17312   Func cfunc 17901   Nat cnat 17991   FuncCat cfuc 17992   ∘_F cfuco 49945

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722  ax-cnex 11144  ax-resscn 11145  ax-1cn 11146  ax-icn 11147  ax-addcl 11148  ax-addrcl 11149  ax-mulcl 11150  ax-mulrcl 11151  ax-mulcom 11152  ax-addass 11153  ax-mulass 11154  ax-distr 11155  ax-i2m1 11156  ax-1ne0 11157  ax-1rid 11158  ax-rnegex 11159  ax-rrecex 11160  ax-cnre 11161  ax-pre-lttri 11162  ax-pre-lttrn 11163  ax-pre-ltadd 11164  ax-pre-mulgt0 11165

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-om 7851  df-1st 7974  df-2nd 7975  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-1o 8441  df-er 8682  df-map 8814  df-ixp 8884  df-en 8932  df-dom 8933  df-sdom 8934  df-fin 8935  df-pnf 11233  df-mnf 11234  df-xr 11235  df-ltxr 11236  df-le 11237  df-sub 11431  df-neg 11432  df-nn 12225  df-2 12294  df-3 12295  df-4 12296  df-5 12297  df-6 12298  df-7 12299  df-8 12300  df-9 12301  df-n0 12496  df-z 12583  df-dec 12703  df-uz 12854  df-fz 13527  df-struct 17197  df-slot 17232  df-ndx 17244  df-base 17260  df-hom 17324  df-cco 17325  df-cat 17714  df-cid 17715  df-func 17905  df-cofu 17907  df-nat 17993  df-fuc 17994  df-fuco 49946

This theorem is referenced by:  fucoco  49986