Theorem fucocolem4 49251

Description: Lemma for fucoco 49252. 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 2730	. . 3 ⊢ (𝐶 Nat 𝐸) = (𝐶 Nat 𝐸)
3		eqid 2730	. . 3 ⊢ (Base‘𝐶) = (Base‘𝐶)
4		eqid 2730	. . 3 ⊢ (comp‘𝐸) = (comp‘𝐸)
5		fucoco.oq	. . 3 ⊢ ∙ = (comp‘𝑄)
6		fucoco.a	. . . . . 6 ⊢ (𝜑 → 𝐴 = ⟨𝑅, 𝑆⟩)
7	6	fveq2d 6869	. . . . 5 ⊢ (𝜑 → ((𝑋𝑃𝑌)‘𝐴) = ((𝑋𝑃𝑌)‘⟨𝑅, 𝑆⟩))
8		df-ov 7397	. . . . 5 ⊢ (𝑅(𝑋𝑃𝑌)𝑆) = ((𝑋𝑃𝑌)‘⟨𝑅, 𝑆⟩)
9	7, 8	eqtr4di 2783	. . . 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 49241	. . . 4 ⊢ (𝜑 → (𝑅(𝑋𝑃𝑌)𝑆) ∈ ((𝑂‘𝑋)(𝐶 Nat 𝐸)(𝑂‘𝑌)))
16	9, 15	eqeltrd 2829	. . 3 ⊢ (𝜑 → ((𝑋𝑃𝑌)‘𝐴) ∈ ((𝑂‘𝑋)(𝐶 Nat 𝐸)(𝑂‘𝑌)))
17		fucoco.b	. . . . . 6 ⊢ (𝜑 → 𝐵 = ⟨𝑈, 𝑉⟩)
18	17	fveq2d 6869	. . . . 5 ⊢ (𝜑 → ((𝑌𝑃𝑍)‘𝐵) = ((𝑌𝑃𝑍)‘⟨𝑈, 𝑉⟩))
19		df-ov 7397	. . . . 5 ⊢ (𝑈(𝑌𝑃𝑍)𝑉) = ((𝑌𝑃𝑍)‘⟨𝑈, 𝑉⟩)
20	18, 19	eqtr4di 2783	. . . 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 49241	. . . 4 ⊢ (𝜑 → (𝑈(𝑌𝑃𝑍)𝑉) ∈ ((𝑂‘𝑌)(𝐶 Nat 𝐸)(𝑂‘𝑍)))
25	20, 24	eqeltrd 2829	. . 3 ⊢ (𝜑 → ((𝑌𝑃𝑍)‘𝐵) ∈ ((𝑂‘𝑌)(𝐶 Nat 𝐸)(𝑂‘𝑍)))
26	1, 2, 3, 4, 5, 16, 25	fucco 17933	. 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 2730	. . . . . . . . . . . . . 14 ⊢ (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷)
28	27	natrcl 17921	. . . . . . . . . . . . 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 48993	. . . . . . . . . 10 ⊢ (𝜑 → (1st ‘𝐺)(𝐶 Func 𝐷)(2nd ‘𝐺))
32		eqid 2730	. . . . . . . . . . . . . 14 ⊢ (𝐷 Nat 𝐸) = (𝐷 Nat 𝐸)
33	32	natrcl 17921	. . . . . . . . . . . . 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 48993	. . . . . . . . . 10 ⊢ (𝜑 → (1st ‘𝐹)(𝐷 Func 𝐸)(2nd ‘𝐹))
37		relfunc 17830	. . . . . . . . . . . . 13 ⊢ Rel (𝐷 Func 𝐸)
38		1st2nd 8027	. . . . . . . . . . . . 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 17830	. . . . . . . . . . . . 13 ⊢ Rel (𝐶 Func 𝐷)
41		1st2nd 8027	. . . . . . . . . . . . 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 4853	. . . . . . . . . . 11 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ = ⟨⟨(1st ‘𝐹), (2nd ‘𝐹)⟩, ⟨(1st ‘𝐺), (2nd ‘𝐺)⟩⟩)
44	13, 43	eqtrd 2765	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 = ⟨⟨(1st ‘𝐹), (2nd ‘𝐹)⟩, ⟨(1st ‘𝐺), (2nd ‘𝐺)⟩⟩)
45	10, 31, 36, 44	fuco111 49225	. . . . . . . . 9 ⊢ (𝜑 → (1st ‘(𝑂‘𝑋)) = ((1st ‘𝐹) ∘ (1st ‘𝐺)))
46	45	fveq1d 6867	. . . . . . . 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 2730	. . . . . . . . . 10 ⊢ (Base‘𝐷) = (Base‘𝐷)
49	3, 48, 31	funcf1 17834	. . . . . . . . 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 6968	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (((1st ‘𝐹) ∘ (1st ‘𝐺))‘𝑥) = ((1st ‘𝐹)‘((1st ‘𝐺)‘𝑥)))
53	47, 52	eqtrd 2765	. . . . . 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 48993	. . . . . . . . . 10 ⊢ (𝜑 → (1st ‘𝐿)(𝐶 Func 𝐷)(2nd ‘𝐿))
56	34	simprd 495	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐾 ∈ (𝐷 Func 𝐸))
57	56	func1st2nd 48993	. . . . . . . . . 10 ⊢ (𝜑 → (1st ‘𝐾)(𝐷 Func 𝐸)(2nd ‘𝐾))
58		1st2nd 8027	. . . . . . . . . . . . 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 8027	. . . . . . . . . . . . 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 4853	. . . . . . . . . . 11 ⊢ (𝜑 → ⟨𝐾, 𝐿⟩ = ⟨⟨(1st ‘𝐾), (2nd ‘𝐾)⟩, ⟨(1st ‘𝐿), (2nd ‘𝐿)⟩⟩)
63	14, 62	eqtrd 2765	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 = ⟨⟨(1st ‘𝐾), (2nd ‘𝐾)⟩, ⟨(1st ‘𝐿), (2nd ‘𝐿)⟩⟩)
64	10, 55, 57, 63	fuco111 49225	. . . . . . . . 9 ⊢ (𝜑 → (1st ‘(𝑂‘𝑌)) = ((1st ‘𝐾) ∘ (1st ‘𝐿)))
65	64	fveq1d 6867	. . . . . . . 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 17834	. . . . . . . . 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 6968	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (((1st ‘𝐾) ∘ (1st ‘𝐿))‘𝑥) = ((1st ‘𝐾)‘((1st ‘𝐿)‘𝑥)))
70	66, 69	eqtrd 2765	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘(𝑂‘𝑌))‘𝑥) = ((1st ‘𝐾)‘((1st ‘𝐿)‘𝑥)))
71	53, 70	opeq12d 4853	. . . . 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 17921	. . . . . . . . . . . 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 48993	. . . . . . . . 9 ⊢ (𝜑 → (1st ‘𝑁)(𝐶 Func 𝐷)(2nd ‘𝑁))
76	32	natrcl 17921	. . . . . . . . . . . 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 48993	. . . . . . . . 9 ⊢ (𝜑 → (1st ‘𝑀)(𝐷 Func 𝐸)(2nd ‘𝑀))
80		1st2nd 8027	. . . . . . . . . . . 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 8027	. . . . . . . . . . . 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 4853	. . . . . . . . . 10 ⊢ (𝜑 → ⟨𝑀, 𝑁⟩ = ⟨⟨(1st ‘𝑀), (2nd ‘𝑀)⟩, ⟨(1st ‘𝑁), (2nd ‘𝑁)⟩⟩)
85	23, 84	eqtrd 2765	. . . . . . . . 9 ⊢ (𝜑 → 𝑍 = ⟨⟨(1st ‘𝑀), (2nd ‘𝑀)⟩, ⟨(1st ‘𝑁), (2nd ‘𝑁)⟩⟩)
86	10, 75, 79, 85	fuco111 49225	. . . . . . . 8 ⊢ (𝜑 → (1st ‘(𝑂‘𝑍)) = ((1st ‘𝑀) ∘ (1st ‘𝑁)))
87	86	fveq1d 6867	. . . . . . 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 17834	. . . . . . . 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 6968	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (((1st ‘𝑀) ∘ (1st ‘𝑁))‘𝑥) = ((1st ‘𝑀)‘((1st ‘𝑁)‘𝑥)))
92	88, 91	eqtrd 2765	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘(𝑂‘𝑍))‘𝑥) = ((1st ‘𝑀)‘((1st ‘𝑁)‘𝑥)))
93	71, 92	oveq12d 7412	. . . 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 49245	. . . . . 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 2765	. . . . 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 7429	. . . . 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 6988	. . . 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 49245	. . . . . 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 2765	. . . . 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 7429	. . . . 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 6988	. . . 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 7415	. . 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 5208	. 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 2765	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 3455  ⟨cop 4603   ↦ cmpt 5196   ∘ ccom 5650  Rel wrel 5651  ⟶wf 6515  ‘cfv 6519  (class class class)co 7394  1^st c1st 7975  2^nd c2nd 7976  Basecbs 17185  compcco 17238   Func cfunc 17822   Nat cnat 17912   FuncCat cfuc 17913   ∘_F cfuco 49211

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 2702  ax-rep 5242  ax-sep 5259  ax-nul 5269  ax-pow 5328  ax-pr 5395  ax-un 7718  ax-cnex 11142  ax-resscn 11143  ax-1cn 11144  ax-icn 11145  ax-addcl 11146  ax-addrcl 11147  ax-mulcl 11148  ax-mulrcl 11149  ax-mulcom 11150  ax-addass 11151  ax-mulass 11152  ax-distr 11153  ax-i2m1 11154  ax-1ne0 11155  ax-1rid 11156  ax-rnegex 11157  ax-rrecex 11158  ax-cnre 11159  ax-pre-lttri 11160  ax-pre-lttrn 11161  ax-pre-ltadd 11162  ax-pre-mulgt0 11163

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2880  df-ne 2928  df-nel 3032  df-ral 3047  df-rex 3056  df-rmo 3357  df-reu 3358  df-rab 3412  df-v 3457  df-sbc 3762  df-csb 3871  df-dif 3925  df-un 3927  df-in 3929  df-ss 3939  df-pss 3942  df-nul 4305  df-if 4497  df-pw 4573  df-sn 4598  df-pr 4600  df-tp 4602  df-op 4604  df-uni 4880  df-iun 4965  df-br 5116  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5541  df-eprel 5546  df-po 5554  df-so 5555  df-fr 5599  df-we 5601  df-xp 5652  df-rel 5653  df-cnv 5654  df-co 5655  df-dm 5656  df-rn 5657  df-res 5658  df-ima 5659  df-pred 6282  df-ord 6343  df-on 6344  df-lim 6345  df-suc 6346  df-iota 6472  df-fun 6521  df-fn 6522  df-f 6523  df-f1 6524  df-fo 6525  df-f1o 6526  df-fv 6527  df-riota 7351  df-ov 7397  df-oprab 7398  df-mpo 7399  df-om 7851  df-1st 7977  df-2nd 7978  df-frecs 8269  df-wrecs 8300  df-recs 8349  df-rdg 8387  df-1o 8443  df-er 8682  df-map 8805  df-ixp 8875  df-en 8923  df-dom 8924  df-sdom 8925  df-fin 8926  df-pnf 11228  df-mnf 11229  df-xr 11230  df-ltxr 11231  df-le 11232  df-sub 11425  df-neg 11426  df-nn 12198  df-2 12260  df-3 12261  df-4 12262  df-5 12263  df-6 12264  df-7 12265  df-8 12266  df-9 12267  df-n0 12459  df-z 12546  df-dec 12666  df-uz 12810  df-fz 13482  df-struct 17123  df-slot 17158  df-ndx 17170  df-base 17186  df-hom 17250  df-cco 17251  df-cat 17635  df-cid 17636  df-func 17826  df-cofu 17828  df-nat 17914  df-fuc 17915  df-fuco 49212

This theorem is referenced by:  fucoco  49252