Theorem fucocolem4 49843

Description: Lemma for fucoco 49844. 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 2737	. . 3 ⊢ (𝐶 Nat 𝐸) = (𝐶 Nat 𝐸)
3		eqid 2737	. . 3 ⊢ (Base‘𝐶) = (Base‘𝐶)
4		eqid 2737	. . 3 ⊢ (comp‘𝐸) = (comp‘𝐸)
5		fucoco.oq	. . 3 ⊢ ∙ = (comp‘𝑄)
6		fucoco.a	. . . . . 6 ⊢ (𝜑 → 𝐴 = ⟨𝑅, 𝑆⟩)
7	6	fveq2d 6838	. . . . 5 ⊢ (𝜑 → ((𝑋𝑃𝑌)‘𝐴) = ((𝑋𝑃𝑌)‘⟨𝑅, 𝑆⟩))
8		df-ov 7363	. . . . 5 ⊢ (𝑅(𝑋𝑃𝑌)𝑆) = ((𝑋𝑃𝑌)‘⟨𝑅, 𝑆⟩)
9	7, 8	eqtr4di 2790	. . . 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 49833	. . . 4 ⊢ (𝜑 → (𝑅(𝑋𝑃𝑌)𝑆) ∈ ((𝑂‘𝑋)(𝐶 Nat 𝐸)(𝑂‘𝑌)))
16	9, 15	eqeltrd 2837	. . 3 ⊢ (𝜑 → ((𝑋𝑃𝑌)‘𝐴) ∈ ((𝑂‘𝑋)(𝐶 Nat 𝐸)(𝑂‘𝑌)))
17		fucoco.b	. . . . . 6 ⊢ (𝜑 → 𝐵 = ⟨𝑈, 𝑉⟩)
18	17	fveq2d 6838	. . . . 5 ⊢ (𝜑 → ((𝑌𝑃𝑍)‘𝐵) = ((𝑌𝑃𝑍)‘⟨𝑈, 𝑉⟩))
19		df-ov 7363	. . . . 5 ⊢ (𝑈(𝑌𝑃𝑍)𝑉) = ((𝑌𝑃𝑍)‘⟨𝑈, 𝑉⟩)
20	18, 19	eqtr4di 2790	. . . 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 49833	. . . 4 ⊢ (𝜑 → (𝑈(𝑌𝑃𝑍)𝑉) ∈ ((𝑂‘𝑌)(𝐶 Nat 𝐸)(𝑂‘𝑍)))
25	20, 24	eqeltrd 2837	. . 3 ⊢ (𝜑 → ((𝑌𝑃𝑍)‘𝐵) ∈ ((𝑂‘𝑌)(𝐶 Nat 𝐸)(𝑂‘𝑍)))
26	1, 2, 3, 4, 5, 16, 25	fucco 17923	. 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 2737	. . . . . . . . . . . . . 14 ⊢ (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷)
28	27	natrcl 17911	. . . . . . . . . . . . 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 49563	. . . . . . . . . 10 ⊢ (𝜑 → (1st ‘𝐺)(𝐶 Func 𝐷)(2nd ‘𝐺))
32		eqid 2737	. . . . . . . . . . . . . 14 ⊢ (𝐷 Nat 𝐸) = (𝐷 Nat 𝐸)
33	32	natrcl 17911	. . . . . . . . . . . . 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 49563	. . . . . . . . . 10 ⊢ (𝜑 → (1st ‘𝐹)(𝐷 Func 𝐸)(2nd ‘𝐹))
37		relfunc 17820	. . . . . . . . . . . . 13 ⊢ Rel (𝐷 Func 𝐸)
38		1st2nd 7985	. . . . . . . . . . . . 13 ⊢ ((Rel (𝐷 Func 𝐸) ∧ 𝐹 ∈ (𝐷 Func 𝐸)) → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
39	37, 35, 38	sylancr 588	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
40		relfunc 17820	. . . . . . . . . . . . 13 ⊢ Rel (𝐶 Func 𝐷)
41		1st2nd 7985	. . . . . . . . . . . . 13 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → 𝐺 = ⟨(1st ‘𝐺), (2nd ‘𝐺)⟩)
42	40, 30, 41	sylancr 588	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺 = ⟨(1st ‘𝐺), (2nd ‘𝐺)⟩)
43	39, 42	opeq12d 4825	. . . . . . . . . . 11 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ = ⟨⟨(1st ‘𝐹), (2nd ‘𝐹)⟩, ⟨(1st ‘𝐺), (2nd ‘𝐺)⟩⟩)
44	13, 43	eqtrd 2772	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 = ⟨⟨(1st ‘𝐹), (2nd ‘𝐹)⟩, ⟨(1st ‘𝐺), (2nd ‘𝐺)⟩⟩)
45	10, 31, 36, 44	fuco111 49817	. . . . . . . . 9 ⊢ (𝜑 → (1st ‘(𝑂‘𝑋)) = ((1st ‘𝐹) ∘ (1st ‘𝐺)))
46	45	fveq1d 6836	. . . . . . . 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 2737	. . . . . . . . . 10 ⊢ (Base‘𝐷) = (Base‘𝐷)
49	3, 48, 31	funcf1 17824	. . . . . . . . 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 6934	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (((1st ‘𝐹) ∘ (1st ‘𝐺))‘𝑥) = ((1st ‘𝐹)‘((1st ‘𝐺)‘𝑥)))
53	47, 52	eqtrd 2772	. . . . . 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 49563	. . . . . . . . . 10 ⊢ (𝜑 → (1st ‘𝐿)(𝐶 Func 𝐷)(2nd ‘𝐿))
56	34	simprd 495	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐾 ∈ (𝐷 Func 𝐸))
57	56	func1st2nd 49563	. . . . . . . . . 10 ⊢ (𝜑 → (1st ‘𝐾)(𝐷 Func 𝐸)(2nd ‘𝐾))
58		1st2nd 7985	. . . . . . . . . . . . 13 ⊢ ((Rel (𝐷 Func 𝐸) ∧ 𝐾 ∈ (𝐷 Func 𝐸)) → 𝐾 = ⟨(1st ‘𝐾), (2nd ‘𝐾)⟩)
59	37, 56, 58	sylancr 588	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾 = ⟨(1st ‘𝐾), (2nd ‘𝐾)⟩)
60		1st2nd 7985	. . . . . . . . . . . . 13 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐿 ∈ (𝐶 Func 𝐷)) → 𝐿 = ⟨(1st ‘𝐿), (2nd ‘𝐿)⟩)
61	40, 54, 60	sylancr 588	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐿 = ⟨(1st ‘𝐿), (2nd ‘𝐿)⟩)
62	59, 61	opeq12d 4825	. . . . . . . . . . 11 ⊢ (𝜑 → ⟨𝐾, 𝐿⟩ = ⟨⟨(1st ‘𝐾), (2nd ‘𝐾)⟩, ⟨(1st ‘𝐿), (2nd ‘𝐿)⟩⟩)
63	14, 62	eqtrd 2772	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 = ⟨⟨(1st ‘𝐾), (2nd ‘𝐾)⟩, ⟨(1st ‘𝐿), (2nd ‘𝐿)⟩⟩)
64	10, 55, 57, 63	fuco111 49817	. . . . . . . . 9 ⊢ (𝜑 → (1st ‘(𝑂‘𝑌)) = ((1st ‘𝐾) ∘ (1st ‘𝐿)))
65	64	fveq1d 6836	. . . . . . . 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 17824	. . . . . . . . 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 6934	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (((1st ‘𝐾) ∘ (1st ‘𝐿))‘𝑥) = ((1st ‘𝐾)‘((1st ‘𝐿)‘𝑥)))
70	66, 69	eqtrd 2772	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘(𝑂‘𝑌))‘𝑥) = ((1st ‘𝐾)‘((1st ‘𝐿)‘𝑥)))
71	53, 70	opeq12d 4825	. . . . 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 17911	. . . . . . . . . . . 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 49563	. . . . . . . . 9 ⊢ (𝜑 → (1st ‘𝑁)(𝐶 Func 𝐷)(2nd ‘𝑁))
76	32	natrcl 17911	. . . . . . . . . . . 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 49563	. . . . . . . . 9 ⊢ (𝜑 → (1st ‘𝑀)(𝐷 Func 𝐸)(2nd ‘𝑀))
80		1st2nd 7985	. . . . . . . . . . . 12 ⊢ ((Rel (𝐷 Func 𝐸) ∧ 𝑀 ∈ (𝐷 Func 𝐸)) → 𝑀 = ⟨(1st ‘𝑀), (2nd ‘𝑀)⟩)
81	37, 78, 80	sylancr 588	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 = ⟨(1st ‘𝑀), (2nd ‘𝑀)⟩)
82		1st2nd 7985	. . . . . . . . . . . 12 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝑁 ∈ (𝐶 Func 𝐷)) → 𝑁 = ⟨(1st ‘𝑁), (2nd ‘𝑁)⟩)
83	40, 74, 82	sylancr 588	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 = ⟨(1st ‘𝑁), (2nd ‘𝑁)⟩)
84	81, 83	opeq12d 4825	. . . . . . . . . 10 ⊢ (𝜑 → ⟨𝑀, 𝑁⟩ = ⟨⟨(1st ‘𝑀), (2nd ‘𝑀)⟩, ⟨(1st ‘𝑁), (2nd ‘𝑁)⟩⟩)
85	23, 84	eqtrd 2772	. . . . . . . . 9 ⊢ (𝜑 → 𝑍 = ⟨⟨(1st ‘𝑀), (2nd ‘𝑀)⟩, ⟨(1st ‘𝑁), (2nd ‘𝑁)⟩⟩)
86	10, 75, 79, 85	fuco111 49817	. . . . . . . 8 ⊢ (𝜑 → (1st ‘(𝑂‘𝑍)) = ((1st ‘𝑀) ∘ (1st ‘𝑁)))
87	86	fveq1d 6836	. . . . . . 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 17824	. . . . . . . 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 6934	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (((1st ‘𝑀) ∘ (1st ‘𝑁))‘𝑥) = ((1st ‘𝑀)‘((1st ‘𝑁)‘𝑥)))
92	88, 91	eqtrd 2772	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘(𝑂‘𝑍))‘𝑥) = ((1st ‘𝑀)‘((1st ‘𝑁)‘𝑥)))
93	71, 92	oveq12d 7378	. . . 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 49837	. . . . . 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 2772	. . . . 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 7395	. . . . 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 6955	. . . 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 49837	. . . . . 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 2772	. . . . 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 7395	. . . . 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 6955	. . . 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 7381	. . 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 5179	. 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 2772	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 1542   ∈ wcel 2114  Vcvv 3430  ⟨cop 4574   ↦ cmpt 5167   ∘ ccom 5628  Rel wrel 5629  ⟶wf 6488  ‘cfv 6492  (class class class)co 7360  1^st c1st 7933  2^nd c2nd 7934  Basecbs 17170  compcco 17223   Func cfunc 17812   Nat cnat 17902   FuncCat cfuc 17903   ∘_F cfuco 49803

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  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 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-1o 8398  df-er 8636  df-map 8768  df-ixp 8839  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12166  df-2 12235  df-3 12236  df-4 12237  df-5 12238  df-6 12239  df-7 12240  df-8 12241  df-9 12242  df-n0 12429  df-z 12516  df-dec 12636  df-uz 12780  df-fz 13453  df-struct 17108  df-slot 17143  df-ndx 17155  df-base 17171  df-hom 17235  df-cco 17236  df-cat 17625  df-cid 17626  df-func 17816  df-cofu 17818  df-nat 17904  df-fuc 17905  df-fuco 49804

This theorem is referenced by:  fucoco  49844