Theorem fucocolem4 49345

Description: Lemma for fucoco 49346. 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 2729	. . 3 ⊢ (𝐶 Nat 𝐸) = (𝐶 Nat 𝐸)
3		eqid 2729	. . 3 ⊢ (Base‘𝐶) = (Base‘𝐶)
4		eqid 2729	. . 3 ⊢ (comp‘𝐸) = (comp‘𝐸)
5		fucoco.oq	. . 3 ⊢ ∙ = (comp‘𝑄)
6		fucoco.a	. . . . . 6 ⊢ (𝜑 → 𝐴 = ⟨𝑅, 𝑆⟩)
7	6	fveq2d 6862	. . . . 5 ⊢ (𝜑 → ((𝑋𝑃𝑌)‘𝐴) = ((𝑋𝑃𝑌)‘⟨𝑅, 𝑆⟩))
8		df-ov 7390	. . . . 5 ⊢ (𝑅(𝑋𝑃𝑌)𝑆) = ((𝑋𝑃𝑌)‘⟨𝑅, 𝑆⟩)
9	7, 8	eqtr4di 2782	. . . 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 49335	. . . 4 ⊢ (𝜑 → (𝑅(𝑋𝑃𝑌)𝑆) ∈ ((𝑂‘𝑋)(𝐶 Nat 𝐸)(𝑂‘𝑌)))
16	9, 15	eqeltrd 2828	. . 3 ⊢ (𝜑 → ((𝑋𝑃𝑌)‘𝐴) ∈ ((𝑂‘𝑋)(𝐶 Nat 𝐸)(𝑂‘𝑌)))
17		fucoco.b	. . . . . 6 ⊢ (𝜑 → 𝐵 = ⟨𝑈, 𝑉⟩)
18	17	fveq2d 6862	. . . . 5 ⊢ (𝜑 → ((𝑌𝑃𝑍)‘𝐵) = ((𝑌𝑃𝑍)‘⟨𝑈, 𝑉⟩))
19		df-ov 7390	. . . . 5 ⊢ (𝑈(𝑌𝑃𝑍)𝑉) = ((𝑌𝑃𝑍)‘⟨𝑈, 𝑉⟩)
20	18, 19	eqtr4di 2782	. . . 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 49335	. . . 4 ⊢ (𝜑 → (𝑈(𝑌𝑃𝑍)𝑉) ∈ ((𝑂‘𝑌)(𝐶 Nat 𝐸)(𝑂‘𝑍)))
25	20, 24	eqeltrd 2828	. . 3 ⊢ (𝜑 → ((𝑌𝑃𝑍)‘𝐵) ∈ ((𝑂‘𝑌)(𝐶 Nat 𝐸)(𝑂‘𝑍)))
26	1, 2, 3, 4, 5, 16, 25	fucco 17927	. 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 2729	. . . . . . . . . . . . . 14 ⊢ (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷)
28	27	natrcl 17915	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ (𝐺(𝐶 Nat 𝐷)𝐿) → (𝐺 ∈ (𝐶 Func 𝐷) ∧ 𝐿 ∈ (𝐶 Func 𝐷)))
29	11, 28	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐺 ∈ (𝐶 Func 𝐷) ∧ 𝐿 ∈ (𝐶 Func 𝐷)))
30	29	simpld 494	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷))
31	30	func1st2nd 49065	. . . . . . . . . 10 ⊢ (𝜑 → (1st ‘𝐺)(𝐶 Func 𝐷)(2nd ‘𝐺))
32		eqid 2729	. . . . . . . . . . . . . 14 ⊢ (𝐷 Nat 𝐸) = (𝐷 Nat 𝐸)
33	32	natrcl 17915	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ (𝐹(𝐷 Nat 𝐸)𝐾) → (𝐹 ∈ (𝐷 Func 𝐸) ∧ 𝐾 ∈ (𝐷 Func 𝐸)))
34	12, 33	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹 ∈ (𝐷 Func 𝐸) ∧ 𝐾 ∈ (𝐷 Func 𝐸)))
35	34	simpld 494	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ (𝐷 Func 𝐸))
36	35	func1st2nd 49065	. . . . . . . . . 10 ⊢ (𝜑 → (1st ‘𝐹)(𝐷 Func 𝐸)(2nd ‘𝐹))
37		relfunc 17824	. . . . . . . . . . . . 13 ⊢ Rel (𝐷 Func 𝐸)
38		1st2nd 8018	. . . . . . . . . . . . 13 ⊢ ((Rel (𝐷 Func 𝐸) ∧ 𝐹 ∈ (𝐷 Func 𝐸)) → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
39	37, 35, 38	sylancr 587	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
40		relfunc 17824	. . . . . . . . . . . . 13 ⊢ Rel (𝐶 Func 𝐷)
41		1st2nd 8018	. . . . . . . . . . . . 13 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → 𝐺 = ⟨(1st ‘𝐺), (2nd ‘𝐺)⟩)
42	40, 30, 41	sylancr 587	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺 = ⟨(1st ‘𝐺), (2nd ‘𝐺)⟩)
43	39, 42	opeq12d 4845	. . . . . . . . . . 11 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ = ⟨⟨(1st ‘𝐹), (2nd ‘𝐹)⟩, ⟨(1st ‘𝐺), (2nd ‘𝐺)⟩⟩)
44	13, 43	eqtrd 2764	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 = ⟨⟨(1st ‘𝐹), (2nd ‘𝐹)⟩, ⟨(1st ‘𝐺), (2nd ‘𝐺)⟩⟩)
45	10, 31, 36, 44	fuco111 49319	. . . . . . . . 9 ⊢ (𝜑 → (1st ‘(𝑂‘𝑋)) = ((1st ‘𝐹) ∘ (1st ‘𝐺)))
46	45	fveq1d 6860	. . . . . . . 8 ⊢ (𝜑 → ((1st ‘(𝑂‘𝑋))‘𝑥) = (((1st ‘𝐹) ∘ (1st ‘𝐺))‘𝑥))
47	46	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘(𝑂‘𝑋))‘𝑥) = (((1st ‘𝐹) ∘ (1st ‘𝐺))‘𝑥))
48		eqid 2729	. . . . . . . . . 10 ⊢ (Base‘𝐷) = (Base‘𝐷)
49	3, 48, 31	funcf1 17828	. . . . . . . . 9 ⊢ (𝜑 → (1st ‘𝐺):(Base‘𝐶)⟶(Base‘𝐷))
50	49	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (1st ‘𝐺):(Base‘𝐶)⟶(Base‘𝐷))
51		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
52	50, 51	fvco3d 6961	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (((1st ‘𝐹) ∘ (1st ‘𝐺))‘𝑥) = ((1st ‘𝐹)‘((1st ‘𝐺)‘𝑥)))
53	47, 52	eqtrd 2764	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘(𝑂‘𝑋))‘𝑥) = ((1st ‘𝐹)‘((1st ‘𝐺)‘𝑥)))
54	29	simprd 495	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐿 ∈ (𝐶 Func 𝐷))
55	54	func1st2nd 49065	. . . . . . . . . 10 ⊢ (𝜑 → (1st ‘𝐿)(𝐶 Func 𝐷)(2nd ‘𝐿))
56	34	simprd 495	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐾 ∈ (𝐷 Func 𝐸))
57	56	func1st2nd 49065	. . . . . . . . . 10 ⊢ (𝜑 → (1st ‘𝐾)(𝐷 Func 𝐸)(2nd ‘𝐾))
58		1st2nd 8018	. . . . . . . . . . . . 13 ⊢ ((Rel (𝐷 Func 𝐸) ∧ 𝐾 ∈ (𝐷 Func 𝐸)) → 𝐾 = ⟨(1st ‘𝐾), (2nd ‘𝐾)⟩)
59	37, 56, 58	sylancr 587	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾 = ⟨(1st ‘𝐾), (2nd ‘𝐾)⟩)
60		1st2nd 8018	. . . . . . . . . . . . 13 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐿 ∈ (𝐶 Func 𝐷)) → 𝐿 = ⟨(1st ‘𝐿), (2nd ‘𝐿)⟩)
61	40, 54, 60	sylancr 587	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐿 = ⟨(1st ‘𝐿), (2nd ‘𝐿)⟩)
62	59, 61	opeq12d 4845	. . . . . . . . . . 11 ⊢ (𝜑 → ⟨𝐾, 𝐿⟩ = ⟨⟨(1st ‘𝐾), (2nd ‘𝐾)⟩, ⟨(1st ‘𝐿), (2nd ‘𝐿)⟩⟩)
63	14, 62	eqtrd 2764	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 = ⟨⟨(1st ‘𝐾), (2nd ‘𝐾)⟩, ⟨(1st ‘𝐿), (2nd ‘𝐿)⟩⟩)
64	10, 55, 57, 63	fuco111 49319	. . . . . . . . 9 ⊢ (𝜑 → (1st ‘(𝑂‘𝑌)) = ((1st ‘𝐾) ∘ (1st ‘𝐿)))
65	64	fveq1d 6860	. . . . . . . 8 ⊢ (𝜑 → ((1st ‘(𝑂‘𝑌))‘𝑥) = (((1st ‘𝐾) ∘ (1st ‘𝐿))‘𝑥))
66	65	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘(𝑂‘𝑌))‘𝑥) = (((1st ‘𝐾) ∘ (1st ‘𝐿))‘𝑥))
67	3, 48, 55	funcf1 17828	. . . . . . . . 9 ⊢ (𝜑 → (1st ‘𝐿):(Base‘𝐶)⟶(Base‘𝐷))
68	67	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (1st ‘𝐿):(Base‘𝐶)⟶(Base‘𝐷))
69	68, 51	fvco3d 6961	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (((1st ‘𝐾) ∘ (1st ‘𝐿))‘𝑥) = ((1st ‘𝐾)‘((1st ‘𝐿)‘𝑥)))
70	66, 69	eqtrd 2764	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘(𝑂‘𝑌))‘𝑥) = ((1st ‘𝐾)‘((1st ‘𝐿)‘𝑥)))
71	53, 70	opeq12d 4845	. . . . 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 17915	. . . . . . . . . . . 12 ⊢ (𝑉 ∈ (𝐿(𝐶 Nat 𝐷)𝑁) → (𝐿 ∈ (𝐶 Func 𝐷) ∧ 𝑁 ∈ (𝐶 Func 𝐷)))
73	21, 72	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐿 ∈ (𝐶 Func 𝐷) ∧ 𝑁 ∈ (𝐶 Func 𝐷)))
74	73	simprd 495	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ (𝐶 Func 𝐷))
75	74	func1st2nd 49065	. . . . . . . . 9 ⊢ (𝜑 → (1st ‘𝑁)(𝐶 Func 𝐷)(2nd ‘𝑁))
76	32	natrcl 17915	. . . . . . . . . . . 12 ⊢ (𝑈 ∈ (𝐾(𝐷 Nat 𝐸)𝑀) → (𝐾 ∈ (𝐷 Func 𝐸) ∧ 𝑀 ∈ (𝐷 Func 𝐸)))
77	22, 76	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐾 ∈ (𝐷 Func 𝐸) ∧ 𝑀 ∈ (𝐷 Func 𝐸)))
78	77	simprd 495	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ (𝐷 Func 𝐸))
79	78	func1st2nd 49065	. . . . . . . . 9 ⊢ (𝜑 → (1st ‘𝑀)(𝐷 Func 𝐸)(2nd ‘𝑀))
80		1st2nd 8018	. . . . . . . . . . . 12 ⊢ ((Rel (𝐷 Func 𝐸) ∧ 𝑀 ∈ (𝐷 Func 𝐸)) → 𝑀 = ⟨(1st ‘𝑀), (2nd ‘𝑀)⟩)
81	37, 78, 80	sylancr 587	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 = ⟨(1st ‘𝑀), (2nd ‘𝑀)⟩)
82		1st2nd 8018	. . . . . . . . . . . 12 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝑁 ∈ (𝐶 Func 𝐷)) → 𝑁 = ⟨(1st ‘𝑁), (2nd ‘𝑁)⟩)
83	40, 74, 82	sylancr 587	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 = ⟨(1st ‘𝑁), (2nd ‘𝑁)⟩)
84	81, 83	opeq12d 4845	. . . . . . . . . 10 ⊢ (𝜑 → ⟨𝑀, 𝑁⟩ = ⟨⟨(1st ‘𝑀), (2nd ‘𝑀)⟩, ⟨(1st ‘𝑁), (2nd ‘𝑁)⟩⟩)
85	23, 84	eqtrd 2764	. . . . . . . . 9 ⊢ (𝜑 → 𝑍 = ⟨⟨(1st ‘𝑀), (2nd ‘𝑀)⟩, ⟨(1st ‘𝑁), (2nd ‘𝑁)⟩⟩)
86	10, 75, 79, 85	fuco111 49319	. . . . . . . 8 ⊢ (𝜑 → (1st ‘(𝑂‘𝑍)) = ((1st ‘𝑀) ∘ (1st ‘𝑁)))
87	86	fveq1d 6860	. . . . . . 7 ⊢ (𝜑 → ((1st ‘(𝑂‘𝑍))‘𝑥) = (((1st ‘𝑀) ∘ (1st ‘𝑁))‘𝑥))
88	87	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘(𝑂‘𝑍))‘𝑥) = (((1st ‘𝑀) ∘ (1st ‘𝑁))‘𝑥))
89	3, 48, 75	funcf1 17828	. . . . . . . 8 ⊢ (𝜑 → (1st ‘𝑁):(Base‘𝐶)⟶(Base‘𝐷))
90	89	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (1st ‘𝑁):(Base‘𝐶)⟶(Base‘𝐷))
91	90, 51	fvco3d 6961	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (((1st ‘𝑀) ∘ (1st ‘𝑁))‘𝑥) = ((1st ‘𝑀)‘((1st ‘𝑁)‘𝑥)))
92	88, 91	eqtrd 2764	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘(𝑂‘𝑍))‘𝑥) = ((1st ‘𝑀)‘((1st ‘𝑁)‘𝑥)))
93	71, 92	oveq12d 7405	. . . 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 49339	. . . . . 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 2764	. . . . 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 7422	. . . . 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 6981	. . . 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 49339	. . . . . 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 2764	. . . . 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 7422	. . . . 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 6981	. . . 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 7408	. . 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 5200	. 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 2764	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 395   = wceq 1540   ∈ wcel 2109  Vcvv 3447  ⟨cop 4595   ↦ cmpt 5188   ∘ ccom 5642  Rel wrel 5643  ⟶wf 6507  ‘cfv 6511  (class class class)co 7387  1^st c1st 7966  2^nd c2nd 7967  Basecbs 17179  compcco 17232   Func cfunc 17816   Nat cnat 17906   FuncCat cfuc 17907   ∘_F cfuco 49305

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-er 8671  df-map 8801  df-ixp 8871  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-nn 12187  df-2 12249  df-3 12250  df-4 12251  df-5 12252  df-6 12253  df-7 12254  df-8 12255  df-9 12256  df-n0 12443  df-z 12530  df-dec 12650  df-uz 12794  df-fz 13469  df-struct 17117  df-slot 17152  df-ndx 17164  df-base 17180  df-hom 17244  df-cco 17245  df-cat 17629  df-cid 17630  df-func 17820  df-cofu 17822  df-nat 17908  df-fuc 17909  df-fuco 49306

This theorem is referenced by:  fucoco  49346