hgt750lema - Mathbox for Thierry Arnoux

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Theorem hgt750lema 34898

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 13973	. . . 4 ⊢ (0..^3) ∈ Fin
2	1	a1i 11	. . 3 ⊢ (𝜑 → (0..^3) ∈ Fin)
3		hgt750leme.n	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ)
4	3	nnnn0d 12528	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
5		3nn0 12485	. . . . . . 7 ⊢ 3 ∈ ℕ₀
6	5	a1i 11	. . . . . 6 ⊢ (𝜑 → 3 ∈ ℕ₀)
7		ssidd 3950	. . . . . 6 ⊢ (𝜑 → ℕ ⊆ ℕ)
8	4, 6, 7	reprfi2 34864	. . . . 5 ⊢ (𝜑 → (ℕ(repr‘3)𝑁) ∈ Fin)
9		ssrab2 4024	. . . . . 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 9198	. . . 4 ⊢ (𝜑 → {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)} ∈ Fin)
12	11	adantr 483	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ (0..^3)) → {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)} ∈ Fin)
13		vmaf 27149	. . . . . 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 3950	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^3)) ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)}) → ℕ ⊆ ℕ)
16	4	nn0zd 12579	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℤ)
17	16	ad2antrr 734	. . . . . . 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 487	. . . . . . . 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 3925	. . . . . . 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 34853	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^3)) ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)}) → 𝑛:(0..^3)⟶ℕ)
22		c0ex 11159	. . . . . . . . 9 ⊢ 0 ∈ V
23	22	tpid1 4717	. . . . . . . 8 ⊢ 0 ∈ {0, 1, 2}
24		fzo0to3tp 13744	. . . . . . . 8 ⊢ (0..^3) = {0, 1, 2}
25	23, 24	eleqtrri 2851	. . . . . . 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	ffvelcdmd 7051	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^3)) ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)}) → (𝑛‘0) ∈ ℕ)
28	14, 27	ffvelcdmd 7051	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^3)) ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)}) → (Λ‘(𝑛‘0)) ∈ ℝ)
29		1ex 11162	. . . . . . . . . 10 ⊢ 1 ∈ V
30	29	tpid2 4719	. . . . . . . . 9 ⊢ 1 ∈ {0, 1, 2}
31	30, 24	eleqtrri 2851	. . . . . . . 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	ffvelcdmd 7051	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^3)) ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)}) → (𝑛‘1) ∈ ℕ)
34	14, 33	ffvelcdmd 7051	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^3)) ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)}) → (Λ‘(𝑛‘1)) ∈ ℝ)
35		2ex 12281	. . . . . . . . . 10 ⊢ 2 ∈ V
36	35	tpid3 4722	. . . . . . . . 9 ⊢ 2 ∈ {0, 1, 2}
37	36, 24	eleqtrri 2851	. . . . . . . 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	ffvelcdmd 7051	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^3)) ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)}) → (𝑛‘2) ∈ ℕ)
40	14, 39	ffvelcdmd 7051	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^3)) ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)}) → (Λ‘(𝑛‘2)) ∈ ℝ)
41	34, 40	remulcld 11198	. . . 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 11198	. . 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 27151	. . . . 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 27151	. . . . . 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 27151	. . . . . 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 11750	. . . 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 11750	. . 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 32970	. 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 2752	. . . 4 ⊢ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)} = {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)}
53		inss2 4180	. . . . . 6 ⊢ (𝑂 ∩ ℙ) ⊆ ℙ
54		prmssnn 16682	. . . . . 6 ⊢ ℙ ⊆ ℕ
55	53, 54	sstri 3936	. . . . 5 ⊢ (𝑂 ∩ ℙ) ⊆ ℕ
56	55	a1i 11	. . . 4 ⊢ (𝜑 → (𝑂 ∩ ℙ) ⊆ ℕ)
57	52, 7, 56, 4, 6	reprdifc 34868	. . 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 15699	. 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 4024	. . . . . . . 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 9198	. . . . . 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 3950	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)}) → ℕ ⊆ ℕ)
64	16	adantr 483	. . . . . . . . . 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 3927	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)}) → 𝑛 ∈ (ℕ(repr‘3)𝑁))
67	63, 64, 65, 66	reprf 34853	. . . . . . . . 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	ffvelcdmd 7051	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)}) → (𝑛‘0) ∈ ℕ)
70	62, 69	ffvelcdmd 7051	. . . . . . 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	ffvelcdmd 7051	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)}) → (𝑛‘1) ∈ ℕ)
73	62, 72	ffvelcdmd 7051	. . . . . . . 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	ffvelcdmd 7051	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)}) → (𝑛‘2) ∈ ℕ)
76	62, 75	ffvelcdmd 7051	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)}) → (Λ‘(𝑛‘2)) ∈ ℝ)
77	73, 76	remulcld 11198	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)}) → ((Λ‘(𝑛‘1)) · (Λ‘(𝑛‘2))) ∈ ℝ)
78	70, 77	remulcld 11198	. . . . . 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 15733	. . . . 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 11196	. . . 4 ⊢ (𝜑 → Σ𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)} ((Λ‘(𝑛‘0)) · ((Λ‘(𝑛‘1)) · (Λ‘(𝑛‘2)))) ∈ ℂ)
81		fsumconst 15789	. . . 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 592	. . 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 6851	. . . . . . . 8 ⊢ (𝑛 = (𝐹‘𝑒) → (𝑛‘0) = ((𝐹‘𝑒)‘0))
84	83	fveq2d 6856	. . . . . . 7 ⊢ (𝑛 = (𝐹‘𝑒) → (Λ‘(𝑛‘0)) = (Λ‘((𝐹‘𝑒)‘0)))
85		fveq1 6851	. . . . . . . . 9 ⊢ (𝑛 = (𝐹‘𝑒) → (𝑛‘1) = ((𝐹‘𝑒)‘1))
86	85	fveq2d 6856	. . . . . . . 8 ⊢ (𝑛 = (𝐹‘𝑒) → (Λ‘(𝑛‘1)) = (Λ‘((𝐹‘𝑒)‘1)))
87		fveq1 6851	. . . . . . . . 9 ⊢ (𝑛 = (𝐹‘𝑒) → (𝑛‘2) = ((𝐹‘𝑒)‘2))
88	87	fveq2d 6856	. . . . . . . 8 ⊢ (𝑛 = (𝐹‘𝑒) → (Λ‘(𝑛‘2)) = (Λ‘((𝐹‘𝑒)‘2)))
89	86, 88	oveq12d 7399	. . . . . . 7 ⊢ (𝑛 = (𝐹‘𝑒) → ((Λ‘(𝑛‘1)) · (Λ‘(𝑛‘2))) = ((Λ‘((𝐹‘𝑒)‘1)) · (Λ‘((𝐹‘𝑒)‘2))))
90	84, 89	oveq12d 7399	. . . . . 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 12283	. . . . . . . . . 10 ⊢ 3 ∈ ℕ
92	91	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 3 ∈ ℕ)
93	92	ralrimivw 3148	. . . . . . . 8 ⊢ (𝜑 → ∀𝑎 ∈ (0..^3)3 ∈ ℕ)
94	93	r19.21bi 3244	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (0..^3)) → 3 ∈ ℕ)
95	16	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (0..^3)) → 𝑁 ∈ ℤ)
96		ssidd 3950	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (0..^3)) → ℕ ⊆ ℕ)
97		simpr 487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (0..^3)) → 𝑎 ∈ (0..^3))
98		fveq1 6851	. . . . . . . . . 10 ⊢ (𝑐 = 𝑑 → (𝑐‘0) = (𝑑‘0))
99	98	eleq1d 2837	. . . . . . . . 9 ⊢ (𝑐 = 𝑑 → ((𝑐‘0) ∈ (𝑂 ∩ ℙ) ↔ (𝑑‘0) ∈ (𝑂 ∩ ℙ)))
100	99	notbid 320	. . . . . . . 8 ⊢ (𝑐 = 𝑑 → (¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ) ↔ ¬ (𝑑‘0) ∈ (𝑂 ∩ ℙ)))
101	100	cbvrabv 3414	. . . . . . 7 ⊢ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)} = {𝑑 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑑‘0) ∈ (𝑂 ∩ ℙ)}
102		fveq1 6851	. . . . . . . . . 10 ⊢ (𝑐 = 𝑑 → (𝑐‘𝑎) = (𝑑‘𝑎))
103	102	eleq1d 2837	. . . . . . . . 9 ⊢ (𝑐 = 𝑑 → ((𝑐‘𝑎) ∈ (𝑂 ∩ ℙ) ↔ (𝑑‘𝑎) ∈ (𝑂 ∩ ℙ)))
104	103	notbid 320	. . . . . . . 8 ⊢ (𝑐 = 𝑑 → (¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ) ↔ ¬ (𝑑‘𝑎) ∈ (𝑂 ∩ ℙ)))
105	104	cbvrabv 3414	. . . . . . 7 ⊢ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)} = {𝑑 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑑‘𝑎) ∈ (𝑂 ∩ ℙ)}
106		eqid 2752	. . . . . . 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 34867	. . . . . 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 2753	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^3)) ∧ 𝑒 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)}) → (𝐹‘𝑒) = (𝐹‘𝑒))
110	78	adantlr 723	. . . . . . 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 11196	. . . . . 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 15722	. . . . 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 6852	. . . . . . . . . 10 ⊢ (𝑒 = 𝑛 → (𝐹‘𝑒) = (𝐹‘𝑛))
114	113	fveq1d 6854	. . . . . . . . 9 ⊢ (𝑒 = 𝑛 → ((𝐹‘𝑒)‘0) = ((𝐹‘𝑛)‘0))
115	114	fveq2d 6856	. . . . . . . 8 ⊢ (𝑒 = 𝑛 → (Λ‘((𝐹‘𝑒)‘0)) = (Λ‘((𝐹‘𝑛)‘0)))
116	113	fveq1d 6854	. . . . . . . . . 10 ⊢ (𝑒 = 𝑛 → ((𝐹‘𝑒)‘1) = ((𝐹‘𝑛)‘1))
117	116	fveq2d 6856	. . . . . . . . 9 ⊢ (𝑒 = 𝑛 → (Λ‘((𝐹‘𝑒)‘1)) = (Λ‘((𝐹‘𝑛)‘1)))
118	113	fveq1d 6854	. . . . . . . . . 10 ⊢ (𝑒 = 𝑛 → ((𝐹‘𝑒)‘2) = ((𝐹‘𝑛)‘2))
119	118	fveq2d 6856	. . . . . . . . 9 ⊢ (𝑒 = 𝑛 → (Λ‘((𝐹‘𝑒)‘2)) = (Λ‘((𝐹‘𝑛)‘2)))
120	117, 119	oveq12d 7399	. . . . . . . 8 ⊢ (𝑒 = 𝑛 → ((Λ‘((𝐹‘𝑒)‘1)) · (Λ‘((𝐹‘𝑒)‘2))) = ((Λ‘((𝐹‘𝑛)‘1)) · (Λ‘((𝐹‘𝑛)‘2))))
121	115, 120	oveq12d 7399	. . . . . . 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 15695	. . . . . 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 7416	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^3)) ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)}) → (0..^3) ∈ V)
125	97	adantr 483	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^3)) ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)}) → 𝑎 ∈ (0..^3))
126	124, 125, 26, 106	pmtridf1o 33224	. . . . . . 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 34895	. . . . . 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 15701	. . . . 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 2792	. . . 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 15701	. . 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 14429	. . . . . . 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 2758	. . . 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 15699	. . . 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 7399	. . 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 2798	. 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 5122	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 398 = wceq 1550 ∈ wcel 2132 {crab 3404 Vcvv 3444 ∖ cdif 3892 ∩ cin 3894 ⊆ wss 3895 ifcif 4470 {cpr 4574 {ctp 4576 ∪ ciun 4939 class class class wbr 5090 ↦ cmpt 5171 I cid 5530 ↾ cres 5638 ∘ ccom 5640 ⟶wf 6502 ‘cfv 6506 (class class class)co 7381 Fincfn 8912 ℂcc 11057 ℝcr 11058 0cc0 11059 1c1 11060 · cmul 11064 ≤ cle 11203 ℕcn 12196 2c2 12258 3c3 12259 ℕ₀cn0 12467 ℤcz 12554 ..^cfzo 13645 ♯chash 14329 Σcsu 15685 ∥ cdvds 16258 ℙcprime 16677 pmTrspcpmtr 19453 Λcvma 27122 reprcrepr 34849

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1805 ax-4 1819 ax-5 1920 ax-6 1977 ax-7 2018 ax-8 2134 ax-9 2142 ax-10 2165 ax-11 2181 ax-12 2202 ax-ext 2724 ax-rep 5217 ax-sep 5236 ax-nul 5246 ax-pow 5312 ax-pr 5380 ax-un 7703 ax-reg 9526 ax-inf2 9582 ax-ac2 10406 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 ax-pre-sup 11137 ax-addf 11138

This theorem depends on definitions: df-bi 209 df-an 399 df-or 857 df-3or 1096 df-3an 1097 df-tru 1553 df-fal 1563 df-ex 1790 df-nf 1794 df-sb 2081 df-mo 2556 df-eu 2586 df-clab 2731 df-cleq 2744 df-clel 2827 df-nfc 2901 df-ne 2948 df-nel 3052 df-ral 3067 df-rex 3077 df-rmo 3357 df-reu 3358 df-rab 3405 df-v 3446 df-sbc 3736 df-csb 3844 df-dif 3898 df-un 3900 df-in 3902 df-ss 3912 df-pss 3915 df-nul 4277 df-if 4471 df-pw 4547 df-sn 4573 df-pr 4575 df-tp 4577 df-op 4579 df-uni 4856 df-int 4896 df-iun 4941 df-iin 4942 df-br 5091 df-opab 5153 df-mpt 5172 df-tr 5198 df-id 5531 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5589 df-se 5590 df-we 5591 df-xp 5642 df-rel 5643 df-cnv 5644 df-co 5645 df-dm 5646 df-rn 5647 df-res 5648 df-ima 5649 df-pred 6273 df-ord 6334 df-on 6335 df-lim 6336 df-suc 6337 df-iota 6462 df-fun 6508 df-fn 6509 df-f 6510 df-f1 6511 df-fo 6512 df-f1o 6513 df-fv 6514 df-isom 6515 df-riota 7338 df-ov 7384 df-oprab 7385 df-mpo 7386 df-of 7645 df-om 7832 df-1st 7955 df-2nd 7956 df-supp 8125 df-frecs 8246 df-wrecs 8277 df-recs 8326 df-rdg 8365 df-1o 8421 df-2o 8422 df-oadd 8425 df-er 8662 df-map 8794 df-pm 8795 df-ixp 8865 df-en 8913 df-dom 8914 df-sdom 8915 df-fin 8916 df-fsupp 9294 df-fi 9343 df-sup 9374 df-inf 9375 df-oi 9444 df-r1 9708 df-rank 9709 df-dju 9845 df-card 9883 df-ac 10058 df-pnf 11204 df-mnf 11205 df-xr 11206 df-ltxr 11207 df-le 11208 df-sub 11402 df-neg 11403 df-div 11831 df-nn 12197 df-2 12266 df-3 12267 df-4 12268 df-5 12269 df-6 12270 df-7 12271 df-8 12272 df-9 12273 df-n0 12468 df-z 12555 df-dec 12675 df-uz 12826 df-q 12936 df-rp 12980 df-xneg 13100 df-xadd 13101 df-xmul 13102 df-ioo 13339 df-ioc 13340 df-ico 13341 df-icc 13342 df-fz 13499 df-fzo 13646 df-fl 13788 df-mod 13866 df-seq 14001 df-exp 14061 df-fac 14273 df-bc 14302 df-hash 14330 df-shft 15066 df-cj 15098 df-re 15099 df-im 15100 df-sqrt 15234 df-abs 15235 df-limsup 15470 df-clim 15487 df-rlim 15488 df-sum 15686 df-prod 15906 df-ef 16069 df-sin 16071 df-cos 16072 df-pi 16074 df-dvds 16259 df-gcd 16501 df-prm 16678 df-pc 16845 df-struct 17155 df-sets 17172 df-slot 17190 df-ndx 17202 df-base 17218 df-ress 17239 df-plusg 17271 df-mulr 17272 df-starv 17273 df-sca 17274 df-vsca 17275 df-ip 17276 df-tset 17277 df-ple 17278 df-ds 17280 df-unif 17281 df-hom 17282 df-cco 17283 df-rest 17423 df-topn 17424 df-0g 17442 df-gsum 17443 df-topgen 17444 df-pt 17445 df-prds 17448 df-xrs 17504 df-qtop 17509 df-imas 17510 df-xps 17512 df-mre 17586 df-mrc 17587 df-acs 17589 df-mgm 18646 df-sgrp 18725 df-mnd 18741 df-submnd 18790 df-mulg 19082 df-cntz 19329 df-pmtr 19454 df-cmn 19794 df-psmet 21385 df-xmet 21386 df-met 21387 df-bl 21388 df-mopn 21389 df-fbas 21390 df-fg 21391 df-cnfld 21394 df-top 22923 df-topon 22940 df-topsp 22962 df-bases 22975 df-cld 23048 df-ntr 23049 df-cls 23050 df-nei 23127 df-lp 23165 df-perf 23166 df-cn 23256 df-cnp 23257 df-haus 23344 df-tx 23591 df-hmeo 23784 df-fil 23875 df-fm 23967 df-flim 23968 df-flf 23969 df-xms 24349 df-ms 24350 df-tms 24351 df-cncf 24909 df-limc 25897 df-dv 25898 df-log 26587 df-vma 27128 df-repr 34850

This theorem is referenced by: hgt750leme 34899

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