hgt750lema - Mathbox for Thierry Arnoux

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Theorem hgt750lema 32633

Description: An upper bound on the contribution of the non-prime terms in the Statement 7.50 of [Helfgott] p. 69. (Contributed by Thierry Arnoux, 1-Jan-2022.)

**Hypotheses**
Ref	Expression
hgt750leme.o	⊢ 𝑂 = {𝑧 ∈ ℤ ∣ ¬ 2 ∥ 𝑧}
hgt750leme.n	⊢ (𝜑 → 𝑁 ∈ ℕ)
hgt750lemb.2	⊢ (𝜑 → 2 ≤ 𝑁)
hgt750lemb.a	⊢ 𝐴 = {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)}
hgt750lema.f	⊢ 𝐹 = (𝑑 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)} ↦ (𝑑 ∘ if(𝑎 = 0, ( I ↾ (0..^3)), ((pmTrsp‘(0..^3))‘{𝑎, 0}))))

**Assertion**
Ref	Expression
hgt750lema	⊢ (𝜑 → Σ𝑛 ∈ ((ℕ(repr‘3)𝑁) ∖ ((𝑂 ∩ ℙ)(repr‘3)𝑁))((Λ‘(𝑛‘0)) · ((Λ‘(𝑛‘1)) · (Λ‘(𝑛‘2)))) ≤ (3 · Σ𝑛 ∈ 𝐴 ((Λ‘(𝑛‘0)) · ((Λ‘(𝑛‘1)) · (Λ‘(𝑛‘2))))))

Distinct variable groups: 𝑧,𝑂 𝐴,𝑐,𝑑,𝑛 𝑁,𝑐,𝑛 𝜑,𝑐,𝑛 𝑛,𝐹 𝑁,𝑎,𝑑,𝑐,𝑛 𝑂,𝑎,𝑐,𝑑,𝑛 𝜑,𝑎,𝑑 Allowed substitution hints: 𝜑(𝑧) 𝐴(𝑧,𝑎) 𝐹(𝑧,𝑎,𝑐,𝑑) 𝑁(𝑧)

Proof of Theorem hgt750lema Dummy variable 𝑒 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fzofi 13692	. . . 4 ⊢ (0..^3) ∈ Fin
2	1	a1i 11	. . 3 ⊢ (𝜑 → (0..^3) ∈ Fin)
3		hgt750leme.n	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ)
4	3	nnnn0d 12293	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
5		3nn0 12251	. . . . . . 7 ⊢ 3 ∈ ℕ₀
6	5	a1i 11	. . . . . 6 ⊢ (𝜑 → 3 ∈ ℕ₀)
7		ssidd 3949	. . . . . 6 ⊢ (𝜑 → ℕ ⊆ ℕ)
8	4, 6, 7	reprfi2 32599	. . . . 5 ⊢ (𝜑 → (ℕ(repr‘3)𝑁) ∈ Fin)
9		ssrab2 4018	. . . . . 6 ⊢ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)} ⊆ (ℕ(repr‘3)𝑁)
10	9	a1i 11	. . . . 5 ⊢ (𝜑 → {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)} ⊆ (ℕ(repr‘3)𝑁))
11	8, 10	ssfid 9020	. . . 4 ⊢ (𝜑 → {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)} ∈ Fin)
12	11	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ (0..^3)) → {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)} ∈ Fin)
13		vmaf 26266	. . . . . 6 ⊢ Λ:ℕ⟶ℝ
14	13	a1i 11	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^3)) ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)}) → Λ:ℕ⟶ℝ)
15		ssidd 3949	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^3)) ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)}) → ℕ ⊆ ℕ)
16	4	nn0zd 12423	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℤ)
17	16	ad2antrr 723	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^3)) ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)}) → 𝑁 ∈ ℤ)
18	5	a1i 11	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^3)) ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)}) → 3 ∈ ℕ₀)
19		simpr 485	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^3)) ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)}) → 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)})
20	9, 19	sselid 3924	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^3)) ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)}) → 𝑛 ∈ (ℕ(repr‘3)𝑁))
21	15, 17, 18, 20	reprf 32588	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^3)) ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)}) → 𝑛:(0..^3)⟶ℕ)
22		c0ex 10970	. . . . . . . . 9 ⊢ 0 ∈ V
23	22	tpid1 4710	. . . . . . . 8 ⊢ 0 ∈ {0, 1, 2}
24		fzo0to3tp 13471	. . . . . . . 8 ⊢ (0..^3) = {0, 1, 2}
25	23, 24	eleqtrri 2840	. . . . . . 7 ⊢ 0 ∈ (0..^3)
26	25	a1i 11	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^3)) ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)}) → 0 ∈ (0..^3))
27	21, 26	ffvelrnd 6959	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^3)) ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)}) → (𝑛‘0) ∈ ℕ)
28	14, 27	ffvelrnd 6959	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^3)) ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)}) → (Λ‘(𝑛‘0)) ∈ ℝ)
29		1ex 10972	. . . . . . . . . 10 ⊢ 1 ∈ V
30	29	tpid2 4712	. . . . . . . . 9 ⊢ 1 ∈ {0, 1, 2}
31	30, 24	eleqtrri 2840	. . . . . . . 8 ⊢ 1 ∈ (0..^3)
32	31	a1i 11	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^3)) ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)}) → 1 ∈ (0..^3))
33	21, 32	ffvelrnd 6959	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^3)) ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)}) → (𝑛‘1) ∈ ℕ)
34	14, 33	ffvelrnd 6959	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^3)) ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)}) → (Λ‘(𝑛‘1)) ∈ ℝ)
35		2ex 12050	. . . . . . . . . 10 ⊢ 2 ∈ V
36	35	tpid3 4715	. . . . . . . . 9 ⊢ 2 ∈ {0, 1, 2}
37	36, 24	eleqtrri 2840	. . . . . . . 8 ⊢ 2 ∈ (0..^3)
38	37	a1i 11	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^3)) ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)}) → 2 ∈ (0..^3))
39	21, 38	ffvelrnd 6959	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^3)) ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)}) → (𝑛‘2) ∈ ℕ)
40	14, 39	ffvelrnd 6959	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^3)) ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)}) → (Λ‘(𝑛‘2)) ∈ ℝ)
41	34, 40	remulcld 11006	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^3)) ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)}) → ((Λ‘(𝑛‘1)) · (Λ‘(𝑛‘2))) ∈ ℝ)
42	28, 41	remulcld 11006	. . 3 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^3)) ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)}) → ((Λ‘(𝑛‘0)) · ((Λ‘(𝑛‘1)) · (Λ‘(𝑛‘2)))) ∈ ℝ)
43		vmage0 26268	. . . . 5 ⊢ ((𝑛‘0) ∈ ℕ → 0 ≤ (Λ‘(𝑛‘0)))
44	27, 43	syl 17	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^3)) ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)}) → 0 ≤ (Λ‘(𝑛‘0)))
45		vmage0 26268	. . . . . 6 ⊢ ((𝑛‘1) ∈ ℕ → 0 ≤ (Λ‘(𝑛‘1)))
46	33, 45	syl 17	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^3)) ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)}) → 0 ≤ (Λ‘(𝑛‘1)))
47		vmage0 26268	. . . . . 6 ⊢ ((𝑛‘2) ∈ ℕ → 0 ≤ (Λ‘(𝑛‘2)))
48	39, 47	syl 17	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^3)) ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)}) → 0 ≤ (Λ‘(𝑛‘2)))
49	34, 40, 46, 48	mulge0d 11552	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^3)) ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)}) → 0 ≤ ((Λ‘(𝑛‘1)) · (Λ‘(𝑛‘2))))
50	28, 41, 44, 49	mulge0d 11552	. . 3 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^3)) ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)}) → 0 ≤ ((Λ‘(𝑛‘0)) · ((Λ‘(𝑛‘1)) · (Λ‘(𝑛‘2)))))
51	2, 12, 42, 50	fsumiunle 31139	. 2 ⊢ (𝜑 → Σ𝑛 ∈ ∪ 𝑎 ∈ (0..^3){𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)} ((Λ‘(𝑛‘0)) · ((Λ‘(𝑛‘1)) · (Λ‘(𝑛‘2)))) ≤ Σ𝑎 ∈ (0..^3)Σ𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)} ((Λ‘(𝑛‘0)) · ((Λ‘(𝑛‘1)) · (Λ‘(𝑛‘2)))))
52		eqid 2740	. . . 4 ⊢ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)} = {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)}
53		inss2 4169	. . . . . 6 ⊢ (𝑂 ∩ ℙ) ⊆ ℙ
54		prmssnn 16379	. . . . . 6 ⊢ ℙ ⊆ ℕ
55	53, 54	sstri 3935	. . . . 5 ⊢ (𝑂 ∩ ℙ) ⊆ ℕ
56	55	a1i 11	. . . 4 ⊢ (𝜑 → (𝑂 ∩ ℙ) ⊆ ℕ)
57	52, 7, 56, 4, 6	reprdifc 32603	. . 3 ⊢ (𝜑 → ((ℕ(repr‘3)𝑁) ∖ ((𝑂 ∩ ℙ)(repr‘3)𝑁)) = ∪ 𝑎 ∈ (0..^3){𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)})
58	57	sumeq1d 15411	. 2 ⊢ (𝜑 → Σ𝑛 ∈ ((ℕ(repr‘3)𝑁) ∖ ((𝑂 ∩ ℙ)(repr‘3)𝑁))((Λ‘(𝑛‘0)) · ((Λ‘(𝑛‘1)) · (Λ‘(𝑛‘2)))) = Σ𝑛 ∈ ∪ 𝑎 ∈ (0..^3){𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)} ((Λ‘(𝑛‘0)) · ((Λ‘(𝑛‘1)) · (Λ‘(𝑛‘2)))))
59		ssrab2 4018	. . . . . . . 8 ⊢ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)} ⊆ (ℕ(repr‘3)𝑁)
60	59	a1i 11	. . . . . . 7 ⊢ (𝜑 → {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)} ⊆ (ℕ(repr‘3)𝑁))
61	8, 60	ssfid 9020	. . . . . 6 ⊢ (𝜑 → {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)} ∈ Fin)
62	13	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)}) → Λ:ℕ⟶ℝ)
63		ssidd 3949	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)}) → ℕ ⊆ ℕ)
64	16	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)}) → 𝑁 ∈ ℤ)
65	5	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)}) → 3 ∈ ℕ₀)
66	60	sselda 3926	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)}) → 𝑛 ∈ (ℕ(repr‘3)𝑁))
67	63, 64, 65, 66	reprf 32588	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)}) → 𝑛:(0..^3)⟶ℕ)
68	25	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)}) → 0 ∈ (0..^3))
69	67, 68	ffvelrnd 6959	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)}) → (𝑛‘0) ∈ ℕ)
70	62, 69	ffvelrnd 6959	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)}) → (Λ‘(𝑛‘0)) ∈ ℝ)
71	31	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)}) → 1 ∈ (0..^3))
72	67, 71	ffvelrnd 6959	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)}) → (𝑛‘1) ∈ ℕ)
73	62, 72	ffvelrnd 6959	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)}) → (Λ‘(𝑛‘1)) ∈ ℝ)
74	37	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)}) → 2 ∈ (0..^3))
75	67, 74	ffvelrnd 6959	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)}) → (𝑛‘2) ∈ ℕ)
76	62, 75	ffvelrnd 6959	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)}) → (Λ‘(𝑛‘2)) ∈ ℝ)
77	73, 76	remulcld 11006	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)}) → ((Λ‘(𝑛‘1)) · (Λ‘(𝑛‘2))) ∈ ℝ)
78	70, 77	remulcld 11006	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)}) → ((Λ‘(𝑛‘0)) · ((Λ‘(𝑛‘1)) · (Λ‘(𝑛‘2)))) ∈ ℝ)
79	61, 78	fsumrecl 15444	. . . . 5 ⊢ (𝜑 → Σ𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)} ((Λ‘(𝑛‘0)) · ((Λ‘(𝑛‘1)) · (Λ‘(𝑛‘2)))) ∈ ℝ)
80	79	recnd 11004	. . . 4 ⊢ (𝜑 → Σ𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)} ((Λ‘(𝑛‘0)) · ((Λ‘(𝑛‘1)) · (Λ‘(𝑛‘2)))) ∈ ℂ)
81		fsumconst 15500	. . . 4 ⊢ (((0..^3) ∈ Fin ∧ Σ𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)} ((Λ‘(𝑛‘0)) · ((Λ‘(𝑛‘1)) · (Λ‘(𝑛‘2)))) ∈ ℂ) → Σ𝑎 ∈ (0..^3)Σ𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)} ((Λ‘(𝑛‘0)) · ((Λ‘(𝑛‘1)) · (Λ‘(𝑛‘2)))) = ((♯‘(0..^3)) · Σ𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)} ((Λ‘(𝑛‘0)) · ((Λ‘(𝑛‘1)) · (Λ‘(𝑛‘2))))))
82	2, 80, 81	syl2anc 584	. . 3 ⊢ (𝜑 → Σ𝑎 ∈ (0..^3)Σ𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)} ((Λ‘(𝑛‘0)) · ((Λ‘(𝑛‘1)) · (Λ‘(𝑛‘2)))) = ((♯‘(0..^3)) · Σ𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)} ((Λ‘(𝑛‘0)) · ((Λ‘(𝑛‘1)) · (Λ‘(𝑛‘2))))))
83		fveq1 6770	. . . . . . . 8 ⊢ (𝑛 = (𝐹‘𝑒) → (𝑛‘0) = ((𝐹‘𝑒)‘0))
84	83	fveq2d 6775	. . . . . . 7 ⊢ (𝑛 = (𝐹‘𝑒) → (Λ‘(𝑛‘0)) = (Λ‘((𝐹‘𝑒)‘0)))
85		fveq1 6770	. . . . . . . . 9 ⊢ (𝑛 = (𝐹‘𝑒) → (𝑛‘1) = ((𝐹‘𝑒)‘1))
86	85	fveq2d 6775	. . . . . . . 8 ⊢ (𝑛 = (𝐹‘𝑒) → (Λ‘(𝑛‘1)) = (Λ‘((𝐹‘𝑒)‘1)))
87		fveq1 6770	. . . . . . . . 9 ⊢ (𝑛 = (𝐹‘𝑒) → (𝑛‘2) = ((𝐹‘𝑒)‘2))
88	87	fveq2d 6775	. . . . . . . 8 ⊢ (𝑛 = (𝐹‘𝑒) → (Λ‘(𝑛‘2)) = (Λ‘((𝐹‘𝑒)‘2)))
89	86, 88	oveq12d 7289	. . . . . . 7 ⊢ (𝑛 = (𝐹‘𝑒) → ((Λ‘(𝑛‘1)) · (Λ‘(𝑛‘2))) = ((Λ‘((𝐹‘𝑒)‘1)) · (Λ‘((𝐹‘𝑒)‘2))))
90	84, 89	oveq12d 7289	. . . . . 6 ⊢ (𝑛 = (𝐹‘𝑒) → ((Λ‘(𝑛‘0)) · ((Λ‘(𝑛‘1)) · (Λ‘(𝑛‘2)))) = ((Λ‘((𝐹‘𝑒)‘0)) · ((Λ‘((𝐹‘𝑒)‘1)) · (Λ‘((𝐹‘𝑒)‘2)))))
91		3nn 12052	. . . . . . . . . 10 ⊢ 3 ∈ ℕ
92	91	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 3 ∈ ℕ)
93	92	ralrimivw 3111	. . . . . . . 8 ⊢ (𝜑 → ∀𝑎 ∈ (0..^3)3 ∈ ℕ)
94	93	r19.21bi 3135	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (0..^3)) → 3 ∈ ℕ)
95	16	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (0..^3)) → 𝑁 ∈ ℤ)
96		ssidd 3949	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (0..^3)) → ℕ ⊆ ℕ)
97		simpr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (0..^3)) → 𝑎 ∈ (0..^3))
98		fveq1 6770	. . . . . . . . . 10 ⊢ (𝑐 = 𝑑 → (𝑐‘0) = (𝑑‘0))
99	98	eleq1d 2825	. . . . . . . . 9 ⊢ (𝑐 = 𝑑 → ((𝑐‘0) ∈ (𝑂 ∩ ℙ) ↔ (𝑑‘0) ∈ (𝑂 ∩ ℙ)))
100	99	notbid 318	. . . . . . . 8 ⊢ (𝑐 = 𝑑 → (¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ) ↔ ¬ (𝑑‘0) ∈ (𝑂 ∩ ℙ)))
101	100	cbvrabv 3425	. . . . . . 7 ⊢ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)} = {𝑑 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑑‘0) ∈ (𝑂 ∩ ℙ)}
102		fveq1 6770	. . . . . . . . . 10 ⊢ (𝑐 = 𝑑 → (𝑐‘𝑎) = (𝑑‘𝑎))
103	102	eleq1d 2825	. . . . . . . . 9 ⊢ (𝑐 = 𝑑 → ((𝑐‘𝑎) ∈ (𝑂 ∩ ℙ) ↔ (𝑑‘𝑎) ∈ (𝑂 ∩ ℙ)))
104	103	notbid 318	. . . . . . . 8 ⊢ (𝑐 = 𝑑 → (¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ) ↔ ¬ (𝑑‘𝑎) ∈ (𝑂 ∩ ℙ)))
105	104	cbvrabv 3425	. . . . . . 7 ⊢ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)} = {𝑑 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑑‘𝑎) ∈ (𝑂 ∩ ℙ)}
106		eqid 2740	. . . . . . 7 ⊢ if(𝑎 = 0, ( I ↾ (0..^3)), ((pmTrsp‘(0..^3))‘{𝑎, 0})) = if(𝑎 = 0, ( I ↾ (0..^3)), ((pmTrsp‘(0..^3))‘{𝑎, 0}))
107		hgt750lema.f	. . . . . . 7 ⊢ 𝐹 = (𝑑 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)} ↦ (𝑑 ∘ if(𝑎 = 0, ( I ↾ (0..^3)), ((pmTrsp‘(0..^3))‘{𝑎, 0}))))
108	94, 95, 96, 97, 101, 105, 106, 107	reprpmtf1o 32602	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (0..^3)) → 𝐹:{𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)}–1-1-onto→{𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)})
109		eqidd 2741	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^3)) ∧ 𝑒 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)}) → (𝐹‘𝑒) = (𝐹‘𝑒))
110	78	adantlr 712	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^3)) ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)}) → ((Λ‘(𝑛‘0)) · ((Λ‘(𝑛‘1)) · (Λ‘(𝑛‘2)))) ∈ ℝ)
111	110	recnd 11004	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^3)) ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)}) → ((Λ‘(𝑛‘0)) · ((Λ‘(𝑛‘1)) · (Λ‘(𝑛‘2)))) ∈ ℂ)
112	90, 12, 108, 109, 111	fsumf1o 15433	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ (0..^3)) → Σ𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)} ((Λ‘(𝑛‘0)) · ((Λ‘(𝑛‘1)) · (Λ‘(𝑛‘2)))) = Σ𝑒 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)} ((Λ‘((𝐹‘𝑒)‘0)) · ((Λ‘((𝐹‘𝑒)‘1)) · (Λ‘((𝐹‘𝑒)‘2)))))
113		fveq2 6771	. . . . . . . . . 10 ⊢ (𝑒 = 𝑛 → (𝐹‘𝑒) = (𝐹‘𝑛))
114	113	fveq1d 6773	. . . . . . . . 9 ⊢ (𝑒 = 𝑛 → ((𝐹‘𝑒)‘0) = ((𝐹‘𝑛)‘0))
115	114	fveq2d 6775	. . . . . . . 8 ⊢ (𝑒 = 𝑛 → (Λ‘((𝐹‘𝑒)‘0)) = (Λ‘((𝐹‘𝑛)‘0)))
116	113	fveq1d 6773	. . . . . . . . . 10 ⊢ (𝑒 = 𝑛 → ((𝐹‘𝑒)‘1) = ((𝐹‘𝑛)‘1))
117	116	fveq2d 6775	. . . . . . . . 9 ⊢ (𝑒 = 𝑛 → (Λ‘((𝐹‘𝑒)‘1)) = (Λ‘((𝐹‘𝑛)‘1)))
118	113	fveq1d 6773	. . . . . . . . . 10 ⊢ (𝑒 = 𝑛 → ((𝐹‘𝑒)‘2) = ((𝐹‘𝑛)‘2))
119	118	fveq2d 6775	. . . . . . . . 9 ⊢ (𝑒 = 𝑛 → (Λ‘((𝐹‘𝑒)‘2)) = (Λ‘((𝐹‘𝑛)‘2)))
120	117, 119	oveq12d 7289	. . . . . . . 8 ⊢ (𝑒 = 𝑛 → ((Λ‘((𝐹‘𝑒)‘1)) · (Λ‘((𝐹‘𝑒)‘2))) = ((Λ‘((𝐹‘𝑛)‘1)) · (Λ‘((𝐹‘𝑛)‘2))))
121	115, 120	oveq12d 7289	. . . . . . 7 ⊢ (𝑒 = 𝑛 → ((Λ‘((𝐹‘𝑒)‘0)) · ((Λ‘((𝐹‘𝑒)‘1)) · (Λ‘((𝐹‘𝑒)‘2)))) = ((Λ‘((𝐹‘𝑛)‘0)) · ((Λ‘((𝐹‘𝑛)‘1)) · (Λ‘((𝐹‘𝑛)‘2)))))
122	121	cbvsumv 15406	. . . . . 6 ⊢ Σ𝑒 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)} ((Λ‘((𝐹‘𝑒)‘0)) · ((Λ‘((𝐹‘𝑒)‘1)) · (Λ‘((𝐹‘𝑒)‘2)))) = Σ𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)} ((Λ‘((𝐹‘𝑛)‘0)) · ((Λ‘((𝐹‘𝑛)‘1)) · (Λ‘((𝐹‘𝑛)‘2))))
123	122	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ (0..^3)) → Σ𝑒 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)} ((Λ‘((𝐹‘𝑒)‘0)) · ((Λ‘((𝐹‘𝑒)‘1)) · (Λ‘((𝐹‘𝑒)‘2)))) = Σ𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)} ((Λ‘((𝐹‘𝑛)‘0)) · ((Λ‘((𝐹‘𝑛)‘1)) · (Λ‘((𝐹‘𝑛)‘2)))))
124		ovexd 7306	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^3)) ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)}) → (0..^3) ∈ V)
125	97	adantr 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^3)) ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)}) → 𝑎 ∈ (0..^3))
126	124, 125, 26, 106	pmtridf1o 31357	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^3)) ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)}) → if(𝑎 = 0, ( I ↾ (0..^3)), ((pmTrsp‘(0..^3))‘{𝑎, 0})):(0..^3)–1-1-onto→(0..^3))
127	107, 126, 21, 14, 19	hgt750lemg 32630	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^3)) ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)}) → ((Λ‘((𝐹‘𝑛)‘0)) · ((Λ‘((𝐹‘𝑛)‘1)) · (Λ‘((𝐹‘𝑛)‘2)))) = ((Λ‘(𝑛‘0)) · ((Λ‘(𝑛‘1)) · (Λ‘(𝑛‘2)))))
128	127	sumeq2dv 15413	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ (0..^3)) → Σ𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)} ((Λ‘((𝐹‘𝑛)‘0)) · ((Λ‘((𝐹‘𝑛)‘1)) · (Λ‘((𝐹‘𝑛)‘2)))) = Σ𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)} ((Λ‘(𝑛‘0)) · ((Λ‘(𝑛‘1)) · (Λ‘(𝑛‘2)))))
129	112, 123, 128	3eqtrrd 2785	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (0..^3)) → Σ𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)} ((Λ‘(𝑛‘0)) · ((Λ‘(𝑛‘1)) · (Λ‘(𝑛‘2)))) = Σ𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)} ((Λ‘(𝑛‘0)) · ((Λ‘(𝑛‘1)) · (Λ‘(𝑛‘2)))))
130	129	sumeq2dv 15413	. . 3 ⊢ (𝜑 → Σ𝑎 ∈ (0..^3)Σ𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)} ((Λ‘(𝑛‘0)) · ((Λ‘(𝑛‘1)) · (Λ‘(𝑛‘2)))) = Σ𝑎 ∈ (0..^3)Σ𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)} ((Λ‘(𝑛‘0)) · ((Λ‘(𝑛‘1)) · (Λ‘(𝑛‘2)))))
131		hashfzo0 14143	. . . . . . 7 ⊢ (3 ∈ ℕ₀ → (♯‘(0..^3)) = 3)
132	5, 131	ax-mp 5	. . . . . 6 ⊢ (♯‘(0..^3)) = 3
133	132	a1i 11	. . . . 5 ⊢ (𝜑 → (♯‘(0..^3)) = 3)
134	133	eqcomd 2746	. . . 4 ⊢ (𝜑 → 3 = (♯‘(0..^3)))
135		hgt750lemb.a	. . . . . 6 ⊢ 𝐴 = {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)}
136	135	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐴 = {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)})
137	136	sumeq1d 15411	. . . 4 ⊢ (𝜑 → Σ𝑛 ∈ 𝐴 ((Λ‘(𝑛‘0)) · ((Λ‘(𝑛‘1)) · (Λ‘(𝑛‘2)))) = Σ𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)} ((Λ‘(𝑛‘0)) · ((Λ‘(𝑛‘1)) · (Λ‘(𝑛‘2)))))
138	134, 137	oveq12d 7289	. . 3 ⊢ (𝜑 → (3 · Σ𝑛 ∈ 𝐴 ((Λ‘(𝑛‘0)) · ((Λ‘(𝑛‘1)) · (Λ‘(𝑛‘2))))) = ((♯‘(0..^3)) · Σ𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)} ((Λ‘(𝑛‘0)) · ((Λ‘(𝑛‘1)) · (Λ‘(𝑛‘2))))))
139	82, 130, 138	3eqtr4rd 2791	. 2 ⊢ (𝜑 → (3 · Σ𝑛 ∈ 𝐴 ((Λ‘(𝑛‘0)) · ((Λ‘(𝑛‘1)) · (Λ‘(𝑛‘2))))) = Σ𝑎 ∈ (0..^3)Σ𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)} ((Λ‘(𝑛‘0)) · ((Λ‘(𝑛‘1)) · (Λ‘(𝑛‘2)))))
140	51, 58, 139	3brtr4d 5111	1 ⊢ (𝜑 → Σ𝑛 ∈ ((ℕ(repr‘3)𝑁) ∖ ((𝑂 ∩ ℙ)(repr‘3)𝑁))((Λ‘(𝑛‘0)) · ((Λ‘(𝑛‘1)) · (Λ‘(𝑛‘2)))) ≤ (3 · Σ𝑛 ∈ 𝐴 ((Λ‘(𝑛‘0)) · ((Λ‘(𝑛‘1)) · (Λ‘(𝑛‘2))))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1542 ∈ wcel 2110 {crab 3070 Vcvv 3431 ∖ cdif 3889 ∩ cin 3891 ⊆ wss 3892 ifcif 4465 {cpr 4569 {ctp 4571 ∪ ciun 4930 class class class wbr 5079 ↦ cmpt 5162 I cid 5489 ↾ cres 5592 ∘ ccom 5594 ⟶wf 6428 ‘cfv 6432 (class class class)co 7271 Fincfn 8716 ℂcc 10870 ℝcr 10871 0cc0 10872 1c1 10873 · cmul 10877 ≤ cle 11011 ℕcn 11973 2c2 12028 3c3 12029 ℕ₀cn0 12233 ℤcz 12319 ..^cfzo 13381 ♯chash 14042 Σcsu 15395 ∥ cdvds 15961 ℙcprime 16374 pmTrspcpmtr 19047 Λcvma 26239 reprcrepr 32584

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-reg 9329 ax-inf2 9377 ax-ac2 10220 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949 ax-pre-sup 10950 ax-addf 10951 ax-mulf 10952

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-iin 4933 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-se 5546 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-isom 6441 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-of 7527 df-om 7707 df-1st 7824 df-2nd 7825 df-supp 7969 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-1o 8288 df-2o 8289 df-oadd 8292 df-er 8481 df-map 8600 df-pm 8601 df-ixp 8669 df-en 8717 df-dom 8718 df-sdom 8719 df-fin 8720 df-fsupp 9107 df-fi 9148 df-sup 9179 df-inf 9180 df-oi 9247 df-r1 9523 df-rank 9524 df-dju 9660 df-card 9698 df-ac 9873 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12437 df-uz 12582 df-q 12688 df-rp 12730 df-xneg 12847 df-xadd 12848 df-xmul 12849 df-ioo 13082 df-ioc 13083 df-ico 13084 df-icc 13085 df-fz 13239 df-fzo 13382 df-fl 13510 df-mod 13588 df-seq 13720 df-exp 13781 df-fac 13986 df-bc 14015 df-hash 14043 df-shft 14776 df-cj 14808 df-re 14809 df-im 14810 df-sqrt 14944 df-abs 14945 df-limsup 15178 df-clim 15195 df-rlim 15196 df-sum 15396 df-prod 15614 df-ef 15775 df-sin 15777 df-cos 15778 df-pi 15780 df-dvds 15962 df-gcd 16200 df-prm 16375 df-pc 16536 df-struct 16846 df-sets 16863 df-slot 16881 df-ndx 16893 df-base 16911 df-ress 16940 df-plusg 16973 df-mulr 16974 df-starv 16975 df-sca 16976 df-vsca 16977 df-ip 16978 df-tset 16979 df-ple 16980 df-ds 16982 df-unif 16983 df-hom 16984 df-cco 16985 df-rest 17131 df-topn 17132 df-0g 17150 df-gsum 17151 df-topgen 17152 df-pt 17153 df-prds 17156 df-xrs 17211 df-qtop 17216 df-imas 17217 df-xps 17219 df-mre 17293 df-mrc 17294 df-acs 17296 df-mgm 18324 df-sgrp 18373 df-mnd 18384 df-submnd 18429 df-mulg 18699 df-cntz 18921 df-pmtr 19048 df-cmn 19386 df-psmet 20587 df-xmet 20588 df-met 20589 df-bl 20590 df-mopn 20591 df-fbas 20592 df-fg 20593 df-cnfld 20596 df-top 22041 df-topon 22058 df-topsp 22080 df-bases 22094 df-cld 22168 df-ntr 22169 df-cls 22170 df-nei 22247 df-lp 22285 df-perf 22286 df-cn 22376 df-cnp 22377 df-haus 22464 df-tx 22711 df-hmeo 22904 df-fil 22995 df-fm 23087 df-flim 23088 df-flf 23089 df-xms 23471 df-ms 23472 df-tms 23473 df-cncf 24039 df-limc 25028 df-dv 25029 df-log 25710 df-vma 26245 df-repr 32585

This theorem is referenced by: hgt750leme 32634

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