Theorem fuco22natlem 49318

Description: The composed natural transformation is a natural transformation. Use fuco22nat 49319 instead. (New usage is discouraged.) (Contributed by Zhi Wang, 30-Sep-2025.)

Hypotheses

Ref	Expression
fuco22natlem.o	⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = ⟨𝑂, 𝑃⟩)
fuco22natlem.a	⊢ (𝜑 → 𝐴 ∈ (⟨𝐹, 𝐺⟩(𝐶 Nat 𝐷)⟨𝑀, 𝑁⟩))
fuco22natlem.b	⊢ (𝜑 → 𝐵 ∈ (⟨𝐾, 𝐿⟩(𝐷 Nat 𝐸)⟨𝑅, 𝑆⟩))
fuco22natlem.u	⊢ (𝜑 → 𝑈 = ⟨⟨𝐾, 𝐿⟩, ⟨𝐹, 𝐺⟩⟩)
fuco22natlem.v	⊢ (𝜑 → 𝑉 = ⟨⟨𝑅, 𝑆⟩, ⟨𝑀, 𝑁⟩⟩)

Assertion

Ref	Expression
fuco22natlem	⊢ (𝜑 → (𝐵(𝑈𝑃𝑉)𝐴) ∈ ((𝑂‘𝑈)(𝐶 Nat 𝐸)(𝑂‘𝑉)))

Proof of Theorem fuco22natlem

Dummy variables ℎ 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2729	. . 3 ⊢ (𝐶 Nat 𝐸) = (𝐶 Nat 𝐸)
2		eqid 2729	. . 3 ⊢ (Base‘𝐶) = (Base‘𝐶)
3		eqid 2729	. . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
4		eqid 2729	. . 3 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
5		eqid 2729	. . 3 ⊢ (comp‘𝐸) = (comp‘𝐸)
6		fuco22natlem.o	. . . . . 6 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = ⟨𝑂, 𝑃⟩)
7		eqid 2729	. . . . . . 7 ⊢ (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷)
8		fuco22natlem.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ (⟨𝐹, 𝐺⟩(𝐶 Nat 𝐷)⟨𝑀, 𝑁⟩))
9	7, 8	natrcl2 49197	. . . . . 6 ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)
10		eqid 2729	. . . . . . 7 ⊢ (𝐷 Nat 𝐸) = (𝐷 Nat 𝐸)
11		fuco22natlem.b	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ (⟨𝐾, 𝐿⟩(𝐷 Nat 𝐸)⟨𝑅, 𝑆⟩))
12	10, 11	natrcl2 49197	. . . . . 6 ⊢ (𝜑 → 𝐾(𝐷 Func 𝐸)𝐿)
13		fuco22natlem.u	. . . . . 6 ⊢ (𝜑 → 𝑈 = ⟨⟨𝐾, 𝐿⟩, ⟨𝐹, 𝐺⟩⟩)
14	6, 9, 12, 13, 2	fuco11a 49301	. . . . 5 ⊢ (𝜑 → (𝑂‘𝑈) = ⟨(𝐾 ∘ 𝐹), (𝑧 ∈ (Base‘𝐶), 𝑤 ∈ (Base‘𝐶) ↦ (((𝐹‘𝑧)𝐿(𝐹‘𝑤)) ∘ (𝑧𝐺𝑤)))⟩)
15	6, 9, 12, 13	fuco11cl 49300	. . . . 5 ⊢ (𝜑 → (𝑂‘𝑈) ∈ (𝐶 Func 𝐸))
16	14, 15	eqeltrrd 2829	. . . 4 ⊢ (𝜑 → ⟨(𝐾 ∘ 𝐹), (𝑧 ∈ (Base‘𝐶), 𝑤 ∈ (Base‘𝐶) ↦ (((𝐹‘𝑧)𝐿(𝐹‘𝑤)) ∘ (𝑧𝐺𝑤)))⟩ ∈ (𝐶 Func 𝐸))
17		df-br 5096	. . . 4 ⊢ ((𝐾 ∘ 𝐹)(𝐶 Func 𝐸)(𝑧 ∈ (Base‘𝐶), 𝑤 ∈ (Base‘𝐶) ↦ (((𝐹‘𝑧)𝐿(𝐹‘𝑤)) ∘ (𝑧𝐺𝑤))) ↔ ⟨(𝐾 ∘ 𝐹), (𝑧 ∈ (Base‘𝐶), 𝑤 ∈ (Base‘𝐶) ↦ (((𝐹‘𝑧)𝐿(𝐹‘𝑤)) ∘ (𝑧𝐺𝑤)))⟩ ∈ (𝐶 Func 𝐸))
18	16, 17	sylibr 234	. . 3 ⊢ (𝜑 → (𝐾 ∘ 𝐹)(𝐶 Func 𝐸)(𝑧 ∈ (Base‘𝐶), 𝑤 ∈ (Base‘𝐶) ↦ (((𝐹‘𝑧)𝐿(𝐹‘𝑤)) ∘ (𝑧𝐺𝑤))))
19	7, 8	natrcl3 49198	. . . . . 6 ⊢ (𝜑 → 𝑀(𝐶 Func 𝐷)𝑁)
20	10, 11	natrcl3 49198	. . . . . 6 ⊢ (𝜑 → 𝑅(𝐷 Func 𝐸)𝑆)
21		fuco22natlem.v	. . . . . 6 ⊢ (𝜑 → 𝑉 = ⟨⟨𝑅, 𝑆⟩, ⟨𝑀, 𝑁⟩⟩)
22	6, 19, 20, 21, 2	fuco11a 49301	. . . . 5 ⊢ (𝜑 → (𝑂‘𝑉) = ⟨(𝑅 ∘ 𝑀), (𝑧 ∈ (Base‘𝐶), 𝑤 ∈ (Base‘𝐶) ↦ (((𝑀‘𝑧)𝑆(𝑀‘𝑤)) ∘ (𝑧𝑁𝑤)))⟩)
23	6, 19, 20, 21	fuco11cl 49300	. . . . 5 ⊢ (𝜑 → (𝑂‘𝑉) ∈ (𝐶 Func 𝐸))
24	22, 23	eqeltrrd 2829	. . . 4 ⊢ (𝜑 → ⟨(𝑅 ∘ 𝑀), (𝑧 ∈ (Base‘𝐶), 𝑤 ∈ (Base‘𝐶) ↦ (((𝑀‘𝑧)𝑆(𝑀‘𝑤)) ∘ (𝑧𝑁𝑤)))⟩ ∈ (𝐶 Func 𝐸))
25		df-br 5096	. . . 4 ⊢ ((𝑅 ∘ 𝑀)(𝐶 Func 𝐸)(𝑧 ∈ (Base‘𝐶), 𝑤 ∈ (Base‘𝐶) ↦ (((𝑀‘𝑧)𝑆(𝑀‘𝑤)) ∘ (𝑧𝑁𝑤))) ↔ ⟨(𝑅 ∘ 𝑀), (𝑧 ∈ (Base‘𝐶), 𝑤 ∈ (Base‘𝐶) ↦ (((𝑀‘𝑧)𝑆(𝑀‘𝑤)) ∘ (𝑧𝑁𝑤)))⟩ ∈ (𝐶 Func 𝐸))
26	24, 25	sylibr 234	. . 3 ⊢ (𝜑 → (𝑅 ∘ 𝑀)(𝐶 Func 𝐸)(𝑧 ∈ (Base‘𝐶), 𝑤 ∈ (Base‘𝐶) ↦ (((𝑀‘𝑧)𝑆(𝑀‘𝑤)) ∘ (𝑧𝑁𝑤))))
27	6, 13, 21, 8, 11	fucofn22 49313	. . 3 ⊢ (𝜑 → (𝐵(𝑈𝑃𝑉)𝐴) Fn (Base‘𝐶))
28		eqid 2729	. . . . 5 ⊢ (Base‘𝐸) = (Base‘𝐸)
29	12	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐾(𝐷 Func 𝐸)𝐿)
30	29	funcrcl3 49053	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐸 ∈ Cat)
31		eqid 2729	. . . . . . 7 ⊢ (Base‘𝐷) = (Base‘𝐷)
32	31, 28, 29	funcf1 17791	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐾:(Base‘𝐷)⟶(Base‘𝐸))
33	9	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐹(𝐶 Func 𝐷)𝐺)
34	2, 31, 33	funcf1 17791	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐹:(Base‘𝐶)⟶(Base‘𝐷))
35		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
36	34, 35	ffvelcdmd 7023	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (𝐹‘𝑥) ∈ (Base‘𝐷))
37	32, 36	ffvelcdmd 7023	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (𝐾‘(𝐹‘𝑥)) ∈ (Base‘𝐸))
38	19	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑀(𝐶 Func 𝐷)𝑁)
39	2, 31, 38	funcf1 17791	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑀:(Base‘𝐶)⟶(Base‘𝐷))
40	39, 35	ffvelcdmd 7023	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (𝑀‘𝑥) ∈ (Base‘𝐷))
41	32, 40	ffvelcdmd 7023	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (𝐾‘(𝑀‘𝑥)) ∈ (Base‘𝐸))
42	20	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑅(𝐷 Func 𝐸)𝑆)
43	31, 28, 42	funcf1 17791	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑅:(Base‘𝐷)⟶(Base‘𝐸))
44	43, 40	ffvelcdmd 7023	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (𝑅‘(𝑀‘𝑥)) ∈ (Base‘𝐸))
45		eqid 2729	. . . . . . 7 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
46	31, 45, 4, 29, 36, 40	funcf2 17793	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((𝐹‘𝑥)𝐿(𝑀‘𝑥)):((𝐹‘𝑥)(Hom ‘𝐷)(𝑀‘𝑥))⟶((𝐾‘(𝐹‘𝑥))(Hom ‘𝐸)(𝐾‘(𝑀‘𝑥))))
47	8	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐴 ∈ (⟨𝐹, 𝐺⟩(𝐶 Nat 𝐷)⟨𝑀, 𝑁⟩))
48	7, 47, 2, 45, 35	natcl 17881	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (𝐴‘𝑥) ∈ ((𝐹‘𝑥)(Hom ‘𝐷)(𝑀‘𝑥)))
49	46, 48	ffvelcdmd 7023	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (((𝐹‘𝑥)𝐿(𝑀‘𝑥))‘(𝐴‘𝑥)) ∈ ((𝐾‘(𝐹‘𝑥))(Hom ‘𝐸)(𝐾‘(𝑀‘𝑥))))
50	11	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐵 ∈ (⟨𝐾, 𝐿⟩(𝐷 Nat 𝐸)⟨𝑅, 𝑆⟩))
51	2, 31, 19	funcf1 17791	. . . . . . 7 ⊢ (𝜑 → 𝑀:(Base‘𝐶)⟶(Base‘𝐷))
52	51	ffvelcdmda 7022	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (𝑀‘𝑥) ∈ (Base‘𝐷))
53	10, 50, 31, 4, 52	natcl 17881	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (𝐵‘(𝑀‘𝑥)) ∈ ((𝐾‘(𝑀‘𝑥))(Hom ‘𝐸)(𝑅‘(𝑀‘𝑥))))
54	28, 4, 5, 30, 37, 41, 44, 49, 53	catcocl 17609	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((𝐵‘(𝑀‘𝑥))(⟨(𝐾‘(𝐹‘𝑥)), (𝐾‘(𝑀‘𝑥))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑥)))(((𝐹‘𝑥)𝐿(𝑀‘𝑥))‘(𝐴‘𝑥))) ∈ ((𝐾‘(𝐹‘𝑥))(Hom ‘𝐸)(𝑅‘(𝑀‘𝑥))))
55	6	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = ⟨𝑂, 𝑃⟩)
56	13	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑈 = ⟨⟨𝐾, 𝐿⟩, ⟨𝐹, 𝐺⟩⟩)
57	21	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑉 = ⟨⟨𝑅, 𝑆⟩, ⟨𝑀, 𝑁⟩⟩)
58		eqidd 2730	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (⟨(𝐾‘(𝐹‘𝑥)), (𝐾‘(𝑀‘𝑥))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑥))) = (⟨(𝐾‘(𝐹‘𝑥)), (𝐾‘(𝑀‘𝑥))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑥))))
59	55, 56, 57, 47, 50, 35, 58	fuco23 49314	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((𝐵(𝑈𝑃𝑉)𝐴)‘𝑥) = ((𝐵‘(𝑀‘𝑥))(⟨(𝐾‘(𝐹‘𝑥)), (𝐾‘(𝑀‘𝑥))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑥)))(((𝐹‘𝑥)𝐿(𝑀‘𝑥))‘(𝐴‘𝑥))))
60	34, 35	fvco3d 6927	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((𝐾 ∘ 𝐹)‘𝑥) = (𝐾‘(𝐹‘𝑥)))
61	39, 35	fvco3d 6927	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((𝑅 ∘ 𝑀)‘𝑥) = (𝑅‘(𝑀‘𝑥)))
62	60, 61	oveq12d 7371	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (((𝐾 ∘ 𝐹)‘𝑥)(Hom ‘𝐸)((𝑅 ∘ 𝑀)‘𝑥)) = ((𝐾‘(𝐹‘𝑥))(Hom ‘𝐸)(𝑅‘(𝑀‘𝑥))))
63	54, 59, 62	3eltr4d 2843	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((𝐵(𝑈𝑃𝑉)𝐴)‘𝑥) ∈ (((𝐾 ∘ 𝐹)‘𝑥)(Hom ‘𝐸)((𝑅 ∘ 𝑀)‘𝑥)))
64		simplrl 776	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ ℎ ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝑥 ∈ (Base‘𝐶))
65		simplrr 777	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ ℎ ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝑦 ∈ (Base‘𝐶))
66	8	ad2antrr 726	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ ℎ ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝐴 ∈ (⟨𝐹, 𝐺⟩(𝐶 Nat 𝐷)⟨𝑀, 𝑁⟩))
67		simpr 484	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ ℎ ∈ (𝑥(Hom ‘𝐶)𝑦)) → ℎ ∈ (𝑥(Hom ‘𝐶)𝑦))
68	11	ad2antrr 726	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ ℎ ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝐵 ∈ (⟨𝐾, 𝐿⟩(𝐷 Nat 𝐸)⟨𝑅, 𝑆⟩))
69	6	ad2antrr 726	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ ℎ ∈ (𝑥(Hom ‘𝐶)𝑦)) → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = ⟨𝑂, 𝑃⟩)
70	13	ad2antrr 726	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ ℎ ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝑈 = ⟨⟨𝐾, 𝐿⟩, ⟨𝐹, 𝐺⟩⟩)
71	21	ad2antrr 726	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ ℎ ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝑉 = ⟨⟨𝑅, 𝑆⟩, ⟨𝑀, 𝑁⟩⟩)
72	64, 65, 66, 67, 68, 69, 70, 71	fuco22natlem3 49317	. . . 4 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ ℎ ∈ (𝑥(Hom ‘𝐶)𝑦)) → (((𝐵(𝑈𝑃𝑉)𝐴)‘𝑦)(⟨((𝐾 ∘ 𝐹)‘𝑥), ((𝐾 ∘ 𝐹)‘𝑦)⟩(comp‘𝐸)((𝑅 ∘ 𝑀)‘𝑦))((((𝐹‘𝑥)𝐿(𝐹‘𝑦)) ∘ (𝑥𝐺𝑦))‘ℎ)) = (((((𝑀‘𝑥)𝑆(𝑀‘𝑦)) ∘ (𝑥𝑁𝑦))‘ℎ)(⟨((𝐾 ∘ 𝐹)‘𝑥), ((𝑅 ∘ 𝑀)‘𝑥)⟩(comp‘𝐸)((𝑅 ∘ 𝑀)‘𝑦))((𝐵(𝑈𝑃𝑉)𝐴)‘𝑥)))
73		fveq2 6826	. . . . . . . . . 10 ⊢ (𝑧 = 𝑥 → (𝐹‘𝑧) = (𝐹‘𝑥))
74	73	oveq1d 7368	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → ((𝐹‘𝑧)𝐿(𝐹‘𝑤)) = ((𝐹‘𝑥)𝐿(𝐹‘𝑤)))
75		oveq1 7360	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → (𝑧𝐺𝑤) = (𝑥𝐺𝑤))
76	74, 75	coeq12d 5811	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → (((𝐹‘𝑧)𝐿(𝐹‘𝑤)) ∘ (𝑧𝐺𝑤)) = (((𝐹‘𝑥)𝐿(𝐹‘𝑤)) ∘ (𝑥𝐺𝑤)))
77		fveq2 6826	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → (𝐹‘𝑤) = (𝐹‘𝑦))
78	77	oveq2d 7369	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → ((𝐹‘𝑥)𝐿(𝐹‘𝑤)) = ((𝐹‘𝑥)𝐿(𝐹‘𝑦)))
79		oveq2 7361	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → (𝑥𝐺𝑤) = (𝑥𝐺𝑦))
80	78, 79	coeq12d 5811	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → (((𝐹‘𝑥)𝐿(𝐹‘𝑤)) ∘ (𝑥𝐺𝑤)) = (((𝐹‘𝑥)𝐿(𝐹‘𝑦)) ∘ (𝑥𝐺𝑦)))
81		eqid 2729	. . . . . . . 8 ⊢ (𝑧 ∈ (Base‘𝐶), 𝑤 ∈ (Base‘𝐶) ↦ (((𝐹‘𝑧)𝐿(𝐹‘𝑤)) ∘ (𝑧𝐺𝑤))) = (𝑧 ∈ (Base‘𝐶), 𝑤 ∈ (Base‘𝐶) ↦ (((𝐹‘𝑧)𝐿(𝐹‘𝑤)) ∘ (𝑧𝐺𝑤)))
82		ovex 7386	. . . . . . . . 9 ⊢ ((𝐹‘𝑥)𝐿(𝐹‘𝑦)) ∈ V
83		ovex 7386	. . . . . . . . 9 ⊢ (𝑥𝐺𝑦) ∈ V
84	82, 83	coex 7870	. . . . . . . 8 ⊢ (((𝐹‘𝑥)𝐿(𝐹‘𝑦)) ∘ (𝑥𝐺𝑦)) ∈ V
85	76, 80, 81, 84	ovmpo 7513	. . . . . . 7 ⊢ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (𝑥(𝑧 ∈ (Base‘𝐶), 𝑤 ∈ (Base‘𝐶) ↦ (((𝐹‘𝑧)𝐿(𝐹‘𝑤)) ∘ (𝑧𝐺𝑤)))𝑦) = (((𝐹‘𝑥)𝐿(𝐹‘𝑦)) ∘ (𝑥𝐺𝑦)))
86	85	ad2antlr 727	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ ℎ ∈ (𝑥(Hom ‘𝐶)𝑦)) → (𝑥(𝑧 ∈ (Base‘𝐶), 𝑤 ∈ (Base‘𝐶) ↦ (((𝐹‘𝑧)𝐿(𝐹‘𝑤)) ∘ (𝑧𝐺𝑤)))𝑦) = (((𝐹‘𝑥)𝐿(𝐹‘𝑦)) ∘ (𝑥𝐺𝑦)))
87	86	fveq1d 6828	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ ℎ ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((𝑥(𝑧 ∈ (Base‘𝐶), 𝑤 ∈ (Base‘𝐶) ↦ (((𝐹‘𝑧)𝐿(𝐹‘𝑤)) ∘ (𝑧𝐺𝑤)))𝑦)‘ℎ) = ((((𝐹‘𝑥)𝐿(𝐹‘𝑦)) ∘ (𝑥𝐺𝑦))‘ℎ))
88	87	oveq2d 7369	. . . 4 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ ℎ ∈ (𝑥(Hom ‘𝐶)𝑦)) → (((𝐵(𝑈𝑃𝑉)𝐴)‘𝑦)(⟨((𝐾 ∘ 𝐹)‘𝑥), ((𝐾 ∘ 𝐹)‘𝑦)⟩(comp‘𝐸)((𝑅 ∘ 𝑀)‘𝑦))((𝑥(𝑧 ∈ (Base‘𝐶), 𝑤 ∈ (Base‘𝐶) ↦ (((𝐹‘𝑧)𝐿(𝐹‘𝑤)) ∘ (𝑧𝐺𝑤)))𝑦)‘ℎ)) = (((𝐵(𝑈𝑃𝑉)𝐴)‘𝑦)(⟨((𝐾 ∘ 𝐹)‘𝑥), ((𝐾 ∘ 𝐹)‘𝑦)⟩(comp‘𝐸)((𝑅 ∘ 𝑀)‘𝑦))((((𝐹‘𝑥)𝐿(𝐹‘𝑦)) ∘ (𝑥𝐺𝑦))‘ℎ)))
89		fveq2 6826	. . . . . . . . . 10 ⊢ (𝑧 = 𝑥 → (𝑀‘𝑧) = (𝑀‘𝑥))
90	89	oveq1d 7368	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → ((𝑀‘𝑧)𝑆(𝑀‘𝑤)) = ((𝑀‘𝑥)𝑆(𝑀‘𝑤)))
91		oveq1 7360	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → (𝑧𝑁𝑤) = (𝑥𝑁𝑤))
92	90, 91	coeq12d 5811	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → (((𝑀‘𝑧)𝑆(𝑀‘𝑤)) ∘ (𝑧𝑁𝑤)) = (((𝑀‘𝑥)𝑆(𝑀‘𝑤)) ∘ (𝑥𝑁𝑤)))
93		fveq2 6826	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → (𝑀‘𝑤) = (𝑀‘𝑦))
94	93	oveq2d 7369	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → ((𝑀‘𝑥)𝑆(𝑀‘𝑤)) = ((𝑀‘𝑥)𝑆(𝑀‘𝑦)))
95		oveq2 7361	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → (𝑥𝑁𝑤) = (𝑥𝑁𝑦))
96	94, 95	coeq12d 5811	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → (((𝑀‘𝑥)𝑆(𝑀‘𝑤)) ∘ (𝑥𝑁𝑤)) = (((𝑀‘𝑥)𝑆(𝑀‘𝑦)) ∘ (𝑥𝑁𝑦)))
97		eqid 2729	. . . . . . . 8 ⊢ (𝑧 ∈ (Base‘𝐶), 𝑤 ∈ (Base‘𝐶) ↦ (((𝑀‘𝑧)𝑆(𝑀‘𝑤)) ∘ (𝑧𝑁𝑤))) = (𝑧 ∈ (Base‘𝐶), 𝑤 ∈ (Base‘𝐶) ↦ (((𝑀‘𝑧)𝑆(𝑀‘𝑤)) ∘ (𝑧𝑁𝑤)))
98		ovex 7386	. . . . . . . . 9 ⊢ ((𝑀‘𝑥)𝑆(𝑀‘𝑦)) ∈ V
99		ovex 7386	. . . . . . . . 9 ⊢ (𝑥𝑁𝑦) ∈ V
100	98, 99	coex 7870	. . . . . . . 8 ⊢ (((𝑀‘𝑥)𝑆(𝑀‘𝑦)) ∘ (𝑥𝑁𝑦)) ∈ V
101	92, 96, 97, 100	ovmpo 7513	. . . . . . 7 ⊢ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (𝑥(𝑧 ∈ (Base‘𝐶), 𝑤 ∈ (Base‘𝐶) ↦ (((𝑀‘𝑧)𝑆(𝑀‘𝑤)) ∘ (𝑧𝑁𝑤)))𝑦) = (((𝑀‘𝑥)𝑆(𝑀‘𝑦)) ∘ (𝑥𝑁𝑦)))
102	101	ad2antlr 727	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ ℎ ∈ (𝑥(Hom ‘𝐶)𝑦)) → (𝑥(𝑧 ∈ (Base‘𝐶), 𝑤 ∈ (Base‘𝐶) ↦ (((𝑀‘𝑧)𝑆(𝑀‘𝑤)) ∘ (𝑧𝑁𝑤)))𝑦) = (((𝑀‘𝑥)𝑆(𝑀‘𝑦)) ∘ (𝑥𝑁𝑦)))
103	102	fveq1d 6828	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ ℎ ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((𝑥(𝑧 ∈ (Base‘𝐶), 𝑤 ∈ (Base‘𝐶) ↦ (((𝑀‘𝑧)𝑆(𝑀‘𝑤)) ∘ (𝑧𝑁𝑤)))𝑦)‘ℎ) = ((((𝑀‘𝑥)𝑆(𝑀‘𝑦)) ∘ (𝑥𝑁𝑦))‘ℎ))
104	103	oveq1d 7368	. . . 4 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ ℎ ∈ (𝑥(Hom ‘𝐶)𝑦)) → (((𝑥(𝑧 ∈ (Base‘𝐶), 𝑤 ∈ (Base‘𝐶) ↦ (((𝑀‘𝑧)𝑆(𝑀‘𝑤)) ∘ (𝑧𝑁𝑤)))𝑦)‘ℎ)(⟨((𝐾 ∘ 𝐹)‘𝑥), ((𝑅 ∘ 𝑀)‘𝑥)⟩(comp‘𝐸)((𝑅 ∘ 𝑀)‘𝑦))((𝐵(𝑈𝑃𝑉)𝐴)‘𝑥)) = (((((𝑀‘𝑥)𝑆(𝑀‘𝑦)) ∘ (𝑥𝑁𝑦))‘ℎ)(⟨((𝐾 ∘ 𝐹)‘𝑥), ((𝑅 ∘ 𝑀)‘𝑥)⟩(comp‘𝐸)((𝑅 ∘ 𝑀)‘𝑦))((𝐵(𝑈𝑃𝑉)𝐴)‘𝑥)))
105	72, 88, 104	3eqtr4d 2774	. . 3 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ ℎ ∈ (𝑥(Hom ‘𝐶)𝑦)) → (((𝐵(𝑈𝑃𝑉)𝐴)‘𝑦)(⟨((𝐾 ∘ 𝐹)‘𝑥), ((𝐾 ∘ 𝐹)‘𝑦)⟩(comp‘𝐸)((𝑅 ∘ 𝑀)‘𝑦))((𝑥(𝑧 ∈ (Base‘𝐶), 𝑤 ∈ (Base‘𝐶) ↦ (((𝐹‘𝑧)𝐿(𝐹‘𝑤)) ∘ (𝑧𝐺𝑤)))𝑦)‘ℎ)) = (((𝑥(𝑧 ∈ (Base‘𝐶), 𝑤 ∈ (Base‘𝐶) ↦ (((𝑀‘𝑧)𝑆(𝑀‘𝑤)) ∘ (𝑧𝑁𝑤)))𝑦)‘ℎ)(⟨((𝐾 ∘ 𝐹)‘𝑥), ((𝑅 ∘ 𝑀)‘𝑥)⟩(comp‘𝐸)((𝑅 ∘ 𝑀)‘𝑦))((𝐵(𝑈𝑃𝑉)𝐴)‘𝑥)))
106	1, 2, 3, 4, 5, 18, 26, 27, 63, 105	isnatd 49196	. 2 ⊢ (𝜑 → (𝐵(𝑈𝑃𝑉)𝐴) ∈ (⟨(𝐾 ∘ 𝐹), (𝑧 ∈ (Base‘𝐶), 𝑤 ∈ (Base‘𝐶) ↦ (((𝐹‘𝑧)𝐿(𝐹‘𝑤)) ∘ (𝑧𝐺𝑤)))⟩(𝐶 Nat 𝐸)⟨(𝑅 ∘ 𝑀), (𝑧 ∈ (Base‘𝐶), 𝑤 ∈ (Base‘𝐶) ↦ (((𝑀‘𝑧)𝑆(𝑀‘𝑤)) ∘ (𝑧𝑁𝑤)))⟩))
107	14, 22	oveq12d 7371	. 2 ⊢ (𝜑 → ((𝑂‘𝑈)(𝐶 Nat 𝐸)(𝑂‘𝑉)) = (⟨(𝐾 ∘ 𝐹), (𝑧 ∈ (Base‘𝐶), 𝑤 ∈ (Base‘𝐶) ↦ (((𝐹‘𝑧)𝐿(𝐹‘𝑤)) ∘ (𝑧𝐺𝑤)))⟩(𝐶 Nat 𝐸)⟨(𝑅 ∘ 𝑀), (𝑧 ∈ (Base‘𝐶), 𝑤 ∈ (Base‘𝐶) ↦ (((𝑀‘𝑧)𝑆(𝑀‘𝑤)) ∘ (𝑧𝑁𝑤)))⟩))
108	106, 107	eleqtrrd 2831	1 ⊢ (𝜑 → (𝐵(𝑈𝑃𝑉)𝐴) ∈ ((𝑂‘𝑈)(𝐶 Nat 𝐸)(𝑂‘𝑉)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ⟨cop 4585   class class class wbr 5095   ∘ ccom 5627  ‘cfv 6486  (class class class)co 7353   ∈ cmpo 7355  Basecbs 17138  Hom chom 17190  compcco 17191   Func cfunc 17779   Nat cnat 17869   ∘_F cfuco 49289

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 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  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-ral 3045  df-rex 3054  df-rmo 3345  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7310  df-ov 7356  df-oprab 7357  df-mpo 7358  df-1st 7931  df-2nd 7932  df-map 8762  df-ixp 8832  df-cat 17592  df-cid 17593  df-func 17783  df-cofu 17785  df-nat 17871  df-fuco 49290

This theorem is referenced by:  fuco22nat  49319