hgt750lema - Mathbox for Thierry Arnoux

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Theorem hgt750lema 30863

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 12813	. . . 4 ⊢ (0..^3) ∈ Fin
2	1	a1i 11	. . 3 ⊢ (𝜑 → (0..^3) ∈ Fin)
3		hgt750leme.n	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ)
4	3	nnnn0d 11389	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
5		3nn0 11348	. . . . . . 7 ⊢ 3 ∈ ℕ₀
6	5	a1i 11	. . . . . 6 ⊢ (𝜑 → 3 ∈ ℕ₀)
7		ssid 3657	. . . . . . 7 ⊢ ℕ ⊆ ℕ
8	7	a1i 11	. . . . . 6 ⊢ (𝜑 → ℕ ⊆ ℕ)
9	4, 6, 8	reprfi2 30829	. . . . 5 ⊢ (𝜑 → (ℕ(repr‘3)𝑁) ∈ Fin)
10		ssrab2 3720	. . . . . 6 ⊢ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)} ⊆ (ℕ(repr‘3)𝑁)
11	10	a1i 11	. . . . 5 ⊢ (𝜑 → {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)} ⊆ (ℕ(repr‘3)𝑁))
12	9, 11	ssfid 8224	. . . 4 ⊢ (𝜑 → {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)} ∈ Fin)
13	12	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ (0..^3)) → {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)} ∈ Fin)
14		vmaf 24890	. . . . . 6 ⊢ Λ:ℕ⟶ℝ
15	14	a1i 11	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^3)) ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)}) → Λ:ℕ⟶ℝ)
16	7	a1i 11	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^3)) ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)}) → ℕ ⊆ ℕ)
17	4	nn0zd 11518	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℤ)
18	17	ad2antrr 762	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^3)) ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)}) → 𝑁 ∈ ℤ)
19	5	a1i 11	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^3)) ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)}) → 3 ∈ ℕ₀)
20		simpr 476	. . . . . . . 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)})
21	10, 20	sseldi 3634	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^3)) ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)}) → 𝑛 ∈ (ℕ(repr‘3)𝑁))
22	16, 18, 19, 21	reprf 30818	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^3)) ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)}) → 𝑛:(0..^3)⟶ℕ)
23		c0ex 10072	. . . . . . . . 9 ⊢ 0 ∈ V
24	23	tpid1 4335	. . . . . . . 8 ⊢ 0 ∈ {0, 1, 2}
25		fzo0to3tp 12594	. . . . . . . 8 ⊢ (0..^3) = {0, 1, 2}
26	24, 25	eleqtrri 2729	. . . . . . 7 ⊢ 0 ∈ (0..^3)
27	26	a1i 11	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^3)) ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)}) → 0 ∈ (0..^3))
28	22, 27	ffvelrnd 6400	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^3)) ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)}) → (𝑛‘0) ∈ ℕ)
29	15, 28	ffvelrnd 6400	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^3)) ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)}) → (Λ‘(𝑛‘0)) ∈ ℝ)
30		1ex 10073	. . . . . . . . . 10 ⊢ 1 ∈ V
31	30	tpid2 4336	. . . . . . . . 9 ⊢ 1 ∈ {0, 1, 2}
32	31, 25	eleqtrri 2729	. . . . . . . 8 ⊢ 1 ∈ (0..^3)
33	32	a1i 11	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^3)) ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)}) → 1 ∈ (0..^3))
34	22, 33	ffvelrnd 6400	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^3)) ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)}) → (𝑛‘1) ∈ ℕ)
35	15, 34	ffvelrnd 6400	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^3)) ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)}) → (Λ‘(𝑛‘1)) ∈ ℝ)
36		2ex 11130	. . . . . . . . . 10 ⊢ 2 ∈ V
37	36	tpid3 4338	. . . . . . . . 9 ⊢ 2 ∈ {0, 1, 2}
38	37, 25	eleqtrri 2729	. . . . . . . 8 ⊢ 2 ∈ (0..^3)
39	38	a1i 11	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^3)) ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)}) → 2 ∈ (0..^3))
40	22, 39	ffvelrnd 6400	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^3)) ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)}) → (𝑛‘2) ∈ ℕ)
41	15, 40	ffvelrnd 6400	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^3)) ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)}) → (Λ‘(𝑛‘2)) ∈ ℝ)
42	35, 41	remulcld 10108	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^3)) ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)}) → ((Λ‘(𝑛‘1)) · (Λ‘(𝑛‘2))) ∈ ℝ)
43	29, 42	remulcld 10108	. . 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)
44		vmage0 24892	. . . . 5 ⊢ ((𝑛‘0) ∈ ℕ → 0 ≤ (Λ‘(𝑛‘0)))
45	28, 44	syl 17	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^3)) ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)}) → 0 ≤ (Λ‘(𝑛‘0)))
46		vmage0 24892	. . . . . 6 ⊢ ((𝑛‘1) ∈ ℕ → 0 ≤ (Λ‘(𝑛‘1)))
47	34, 46	syl 17	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^3)) ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)}) → 0 ≤ (Λ‘(𝑛‘1)))
48		vmage0 24892	. . . . . 6 ⊢ ((𝑛‘2) ∈ ℕ → 0 ≤ (Λ‘(𝑛‘2)))
49	40, 48	syl 17	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^3)) ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)}) → 0 ≤ (Λ‘(𝑛‘2)))
50	35, 41, 47, 49	mulge0d 10642	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^3)) ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)}) → 0 ≤ ((Λ‘(𝑛‘1)) · (Λ‘(𝑛‘2))))
51	29, 42, 45, 50	mulge0d 10642	. . 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)))))
52	2, 13, 43, 51	fsumiunle 29703	. 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)))))
53		eqid 2651	. . . 4 ⊢ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)} = {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)}
54		inss2 3867	. . . . . 6 ⊢ (𝑂 ∩ ℙ) ⊆ ℙ
55		prmssnn 15437	. . . . . 6 ⊢ ℙ ⊆ ℕ
56	54, 55	sstri 3645	. . . . 5 ⊢ (𝑂 ∩ ℙ) ⊆ ℕ
57	56	a1i 11	. . . 4 ⊢ (𝜑 → (𝑂 ∩ ℙ) ⊆ ℕ)
58	53, 8, 57, 4, 6	reprdifc 30833	. . 3 ⊢ (𝜑 → ((ℕ(repr‘3)𝑁) ∖ ((𝑂 ∩ ℙ)(repr‘3)𝑁)) = ∪ 𝑎 ∈ (0..^3){𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)})
59	58	sumeq1d 14475	. 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)))))
60		ssrab2 3720	. . . . . . . 8 ⊢ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)} ⊆ (ℕ(repr‘3)𝑁)
61	60	a1i 11	. . . . . . 7 ⊢ (𝜑 → {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)} ⊆ (ℕ(repr‘3)𝑁))
62	9, 61	ssfid 8224	. . . . . 6 ⊢ (𝜑 → {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)} ∈ Fin)
63	14	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)}) → Λ:ℕ⟶ℝ)
64	7	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)}) → ℕ ⊆ ℕ)
65	17	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)}) → 𝑁 ∈ ℤ)
66	5	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)}) → 3 ∈ ℕ₀)
67	61	sselda 3636	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)}) → 𝑛 ∈ (ℕ(repr‘3)𝑁))
68	64, 65, 66, 67	reprf 30818	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)}) → 𝑛:(0..^3)⟶ℕ)
69	26	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)}) → 0 ∈ (0..^3))
70	68, 69	ffvelrnd 6400	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)}) → (𝑛‘0) ∈ ℕ)
71	63, 70	ffvelrnd 6400	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)}) → (Λ‘(𝑛‘0)) ∈ ℝ)
72	32	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)}) → 1 ∈ (0..^3))
73	68, 72	ffvelrnd 6400	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)}) → (𝑛‘1) ∈ ℕ)
74	63, 73	ffvelrnd 6400	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)}) → (Λ‘(𝑛‘1)) ∈ ℝ)
75	38	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)}) → 2 ∈ (0..^3))
76	68, 75	ffvelrnd 6400	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)}) → (𝑛‘2) ∈ ℕ)
77	63, 76	ffvelrnd 6400	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)}) → (Λ‘(𝑛‘2)) ∈ ℝ)
78	74, 77	remulcld 10108	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)}) → ((Λ‘(𝑛‘1)) · (Λ‘(𝑛‘2))) ∈ ℝ)
79	71, 78	remulcld 10108	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)}) → ((Λ‘(𝑛‘0)) · ((Λ‘(𝑛‘1)) · (Λ‘(𝑛‘2)))) ∈ ℝ)
80	62, 79	fsumrecl 14509	. . . . 5 ⊢ (𝜑 → Σ𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)} ((Λ‘(𝑛‘0)) · ((Λ‘(𝑛‘1)) · (Λ‘(𝑛‘2)))) ∈ ℝ)
81	80	recnd 10106	. . . 4 ⊢ (𝜑 → Σ𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)} ((Λ‘(𝑛‘0)) · ((Λ‘(𝑛‘1)) · (Λ‘(𝑛‘2)))) ∈ ℂ)
82		fsumconst 14566	. . . 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))))))
83	2, 81, 82	syl2anc 694	. . 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))))))
84		fveq1 6228	. . . . . . . 8 ⊢ (𝑛 = (𝐹‘𝑒) → (𝑛‘0) = ((𝐹‘𝑒)‘0))
85	84	fveq2d 6233	. . . . . . 7 ⊢ (𝑛 = (𝐹‘𝑒) → (Λ‘(𝑛‘0)) = (Λ‘((𝐹‘𝑒)‘0)))
86		fveq1 6228	. . . . . . . . 9 ⊢ (𝑛 = (𝐹‘𝑒) → (𝑛‘1) = ((𝐹‘𝑒)‘1))
87	86	fveq2d 6233	. . . . . . . 8 ⊢ (𝑛 = (𝐹‘𝑒) → (Λ‘(𝑛‘1)) = (Λ‘((𝐹‘𝑒)‘1)))
88		fveq1 6228	. . . . . . . . 9 ⊢ (𝑛 = (𝐹‘𝑒) → (𝑛‘2) = ((𝐹‘𝑒)‘2))
89	88	fveq2d 6233	. . . . . . . 8 ⊢ (𝑛 = (𝐹‘𝑒) → (Λ‘(𝑛‘2)) = (Λ‘((𝐹‘𝑒)‘2)))
90	87, 89	oveq12d 6708	. . . . . . 7 ⊢ (𝑛 = (𝐹‘𝑒) → ((Λ‘(𝑛‘1)) · (Λ‘(𝑛‘2))) = ((Λ‘((𝐹‘𝑒)‘1)) · (Λ‘((𝐹‘𝑒)‘2))))
91	85, 90	oveq12d 6708	. . . . . 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)))))
92		3nn 11224	. . . . . . . . . 10 ⊢ 3 ∈ ℕ
93	92	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 3 ∈ ℕ)
94	93	ralrimivw 2996	. . . . . . . 8 ⊢ (𝜑 → ∀𝑎 ∈ (0..^3)3 ∈ ℕ)
95	94	r19.21bi 2961	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (0..^3)) → 3 ∈ ℕ)
96	17	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (0..^3)) → 𝑁 ∈ ℤ)
97	7	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (0..^3)) → ℕ ⊆ ℕ)
98		simpr 476	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (0..^3)) → 𝑎 ∈ (0..^3))
99		fveq1 6228	. . . . . . . . . 10 ⊢ (𝑐 = 𝑑 → (𝑐‘0) = (𝑑‘0))
100	99	eleq1d 2715	. . . . . . . . 9 ⊢ (𝑐 = 𝑑 → ((𝑐‘0) ∈ (𝑂 ∩ ℙ) ↔ (𝑑‘0) ∈ (𝑂 ∩ ℙ)))
101	100	notbid 307	. . . . . . . 8 ⊢ (𝑐 = 𝑑 → (¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ) ↔ ¬ (𝑑‘0) ∈ (𝑂 ∩ ℙ)))
102	101	cbvrabv 3230	. . . . . . 7 ⊢ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)} = {𝑑 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑑‘0) ∈ (𝑂 ∩ ℙ)}
103		fveq1 6228	. . . . . . . . . 10 ⊢ (𝑐 = 𝑑 → (𝑐‘𝑎) = (𝑑‘𝑎))
104	103	eleq1d 2715	. . . . . . . . 9 ⊢ (𝑐 = 𝑑 → ((𝑐‘𝑎) ∈ (𝑂 ∩ ℙ) ↔ (𝑑‘𝑎) ∈ (𝑂 ∩ ℙ)))
105	104	notbid 307	. . . . . . . 8 ⊢ (𝑐 = 𝑑 → (¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ) ↔ ¬ (𝑑‘𝑎) ∈ (𝑂 ∩ ℙ)))
106	105	cbvrabv 3230	. . . . . . 7 ⊢ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)} = {𝑑 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑑‘𝑎) ∈ (𝑂 ∩ ℙ)}
107		eqid 2651	. . . . . . 7 ⊢ if(𝑎 = 0, ( I ↾ (0..^3)), ((pmTrsp‘(0..^3))‘{𝑎, 0})) = if(𝑎 = 0, ( I ↾ (0..^3)), ((pmTrsp‘(0..^3))‘{𝑎, 0}))
108		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}))))
109	95, 96, 97, 98, 102, 106, 107, 108	reprpmtf1o 30832	. . . . . 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)})
110		eqidd 2652	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^3)) ∧ 𝑒 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)}) → (𝐹‘𝑒) = (𝐹‘𝑒))
111	79	adantlr 751	. . . . . . 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)
112	111	recnd 10106	. . . . . 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)
113	91, 13, 109, 110, 112	fsumf1o 14498	. . . . 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)))))
114		fveq2 6229	. . . . . . . . . 10 ⊢ (𝑒 = 𝑛 → (𝐹‘𝑒) = (𝐹‘𝑛))
115	114	fveq1d 6231	. . . . . . . . 9 ⊢ (𝑒 = 𝑛 → ((𝐹‘𝑒)‘0) = ((𝐹‘𝑛)‘0))
116	115	fveq2d 6233	. . . . . . . 8 ⊢ (𝑒 = 𝑛 → (Λ‘((𝐹‘𝑒)‘0)) = (Λ‘((𝐹‘𝑛)‘0)))
117	114	fveq1d 6231	. . . . . . . . . 10 ⊢ (𝑒 = 𝑛 → ((𝐹‘𝑒)‘1) = ((𝐹‘𝑛)‘1))
118	117	fveq2d 6233	. . . . . . . . 9 ⊢ (𝑒 = 𝑛 → (Λ‘((𝐹‘𝑒)‘1)) = (Λ‘((𝐹‘𝑛)‘1)))
119	114	fveq1d 6231	. . . . . . . . . 10 ⊢ (𝑒 = 𝑛 → ((𝐹‘𝑒)‘2) = ((𝐹‘𝑛)‘2))
120	119	fveq2d 6233	. . . . . . . . 9 ⊢ (𝑒 = 𝑛 → (Λ‘((𝐹‘𝑒)‘2)) = (Λ‘((𝐹‘𝑛)‘2)))
121	118, 120	oveq12d 6708	. . . . . . . 8 ⊢ (𝑒 = 𝑛 → ((Λ‘((𝐹‘𝑒)‘1)) · (Λ‘((𝐹‘𝑒)‘2))) = ((Λ‘((𝐹‘𝑛)‘1)) · (Λ‘((𝐹‘𝑛)‘2))))
122	116, 121	oveq12d 6708	. . . . . . 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)))))
123	122	cbvsumv 14470	. . . . . 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))))
124	123	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)))))
125		ovexd 6720	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^3)) ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)}) → (0..^3) ∈ V)
126	98	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^3)) ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)}) → 𝑎 ∈ (0..^3))
127	125, 126, 27, 107	pmtridf1o 29984	. . . . . . 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))
128	108, 127, 22, 15, 20	hgt750lemg 30860	. . . . . 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)))))
129	128	sumeq2dv 14477	. . . . 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)))))
130	113, 124, 129	3eqtrrd 2690	. . . 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)))))
131	130	sumeq2dv 14477	. . 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)))))
132		hashfzo0 13255	. . . . . . 7 ⊢ (3 ∈ ℕ₀ → (#‘(0..^3)) = 3)
133	5, 132	ax-mp 5	. . . . . 6 ⊢ (#‘(0..^3)) = 3
134	133	a1i 11	. . . . 5 ⊢ (𝜑 → (#‘(0..^3)) = 3)
135	134	eqcomd 2657	. . . 4 ⊢ (𝜑 → 3 = (#‘(0..^3)))
136		hgt750lemb.a	. . . . . 6 ⊢ 𝐴 = {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)}
137	136	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐴 = {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)})
138	137	sumeq1d 14475	. . . 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)))))
139	135, 138	oveq12d 6708	. . 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))))))
140	83, 131, 139	3eqtr4rd 2696	. 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)))))
141	52, 59, 140	3brtr4d 4717	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 383 = wceq 1523 ∈ wcel 2030 {crab 2945 Vcvv 3231 ∖ cdif 3604 ∩ cin 3606 ⊆ wss 3607 ifcif 4119 {cpr 4212 {ctp 4214 ∪ ciun 4552 class class class wbr 4685 ↦ cmpt 4762 I cid 5052 ↾ cres 5145 ∘ ccom 5147 ⟶wf 5922 ‘cfv 5926 (class class class)co 6690 Fincfn 7997 ℂcc 9972 ℝcr 9973 0cc0 9974 1c1 9975 · cmul 9979 ≤ cle 10113 ℕcn 11058 2c2 11108 3c3 11109 ℕ₀cn0 11330 ℤcz 11415 ..^cfzo 12504 #chash 13157 Σcsu 14460 ∥ cdvds 15027 ℙcprime 15432 pmTrspcpmtr 17907 Λcvma 24863 reprcrepr 30814

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-reg 8538 ax-inf2 8576 ax-ac2 9323 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 ax-pre-sup 10052 ax-addf 10053 ax-mulf 10054

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-fal 1529 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-iin 4555 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-se 5103 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-isom 5935 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-of 6939 df-om 7108 df-1st 7210 df-2nd 7211 df-supp 7341 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-2o 7606 df-oadd 7609 df-er 7787 df-map 7901 df-pm 7902 df-ixp 7951 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-fsupp 8317 df-fi 8358 df-sup 8389 df-inf 8390 df-oi 8456 df-r1 8665 df-rank 8666 df-card 8803 df-ac 8977 df-cda 9028 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-div 10723 df-nn 11059 df-2 11117 df-3 11118 df-4 11119 df-5 11120 df-6 11121 df-7 11122 df-8 11123 df-9 11124 df-n0 11331 df-z 11416 df-dec 11532 df-uz 11726 df-q 11827 df-rp 11871 df-xneg 11984 df-xadd 11985 df-xmul 11986 df-ioo 12217 df-ioc 12218 df-ico 12219 df-icc 12220 df-fz 12365 df-fzo 12505 df-fl 12633 df-mod 12709 df-seq 12842 df-exp 12901 df-fac 13101 df-bc 13130 df-hash 13158 df-shft 13851 df-cj 13883 df-re 13884 df-im 13885 df-sqrt 14019 df-abs 14020 df-limsup 14246 df-clim 14263 df-rlim 14264 df-sum 14461 df-prod 14680 df-ef 14842 df-sin 14844 df-cos 14845 df-pi 14847 df-dvds 15028 df-gcd 15264 df-prm 15433 df-pc 15589 df-struct 15906 df-ndx 15907 df-slot 15908 df-base 15910 df-sets 15911 df-ress 15912 df-plusg 16001 df-mulr 16002 df-starv 16003 df-sca 16004 df-vsca 16005 df-ip 16006 df-tset 16007 df-ple 16008 df-ds 16011 df-unif 16012 df-hom 16013 df-cco 16014 df-rest 16130 df-topn 16131 df-0g 16149 df-gsum 16150 df-topgen 16151 df-pt 16152 df-prds 16155 df-xrs 16209 df-qtop 16214 df-imas 16215 df-xps 16217 df-mre 16293 df-mrc 16294 df-acs 16296 df-mgm 17289 df-sgrp 17331 df-mnd 17342 df-submnd 17383 df-mulg 17588 df-cntz 17796 df-pmtr 17908 df-cmn 18241 df-psmet 19786 df-xmet 19787 df-met 19788 df-bl 19789 df-mopn 19790 df-fbas 19791 df-fg 19792 df-cnfld 19795 df-top 20747 df-topon 20764 df-topsp 20785 df-bases 20798 df-cld 20871 df-ntr 20872 df-cls 20873 df-nei 20950 df-lp 20988 df-perf 20989 df-cn 21079 df-cnp 21080 df-haus 21167 df-tx 21413 df-hmeo 21606 df-fil 21697 df-fm 21789 df-flim 21790 df-flf 21791 df-xms 22172 df-ms 22173 df-tms 22174 df-cncf 22728 df-limc 23675 df-dv 23676 df-log 24348 df-vma 24869 df-repr 30815

This theorem is referenced by: hgt750leme 30864

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