hgt750lema - Mathbox for Thierry Arnoux

	Mathbox for Thierry Arnoux		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > hgt750lema		Structured version Visualization version GIF version

Theorem hgt750lema 34635

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 13990	. . . 4 ⊢ (0..^3) ∈ Fin
2	1	a1i 11	. . 3 ⊢ (𝜑 → (0..^3) ∈ Fin)
3		hgt750leme.n	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ)
4	3	nnnn0d 12560	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
5		3nn0 12517	. . . . . . 7 ⊢ 3 ∈ ℕ₀
6	5	a1i 11	. . . . . 6 ⊢ (𝜑 → 3 ∈ ℕ₀)
7		ssidd 3982	. . . . . 6 ⊢ (𝜑 → ℕ ⊆ ℕ)
8	4, 6, 7	reprfi2 34601	. . . . 5 ⊢ (𝜑 → (ℕ(repr‘3)𝑁) ∈ Fin)
9		ssrab2 4055	. . . . . 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 9271	. . . 4 ⊢ (𝜑 → {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)} ∈ Fin)
12	11	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ (0..^3)) → {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)} ∈ Fin)
13		vmaf 27079	. . . . . 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 3982	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^3)) ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)}) → ℕ ⊆ ℕ)
16	4	nn0zd 12612	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℤ)
17	16	ad2antrr 726	. . . . . . 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 484	. . . . . . . 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 3956	. . . . . . 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 34590	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^3)) ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)}) → 𝑛:(0..^3)⟶ℕ)
22		c0ex 11227	. . . . . . . . 9 ⊢ 0 ∈ V
23	22	tpid1 4744	. . . . . . . 8 ⊢ 0 ∈ {0, 1, 2}
24		fzo0to3tp 13766	. . . . . . . 8 ⊢ (0..^3) = {0, 1, 2}
25	23, 24	eleqtrri 2833	. . . . . . 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 7074	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^3)) ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)}) → (𝑛‘0) ∈ ℕ)
28	14, 27	ffvelcdmd 7074	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^3)) ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)}) → (Λ‘(𝑛‘0)) ∈ ℝ)
29		1ex 11229	. . . . . . . . . 10 ⊢ 1 ∈ V
30	29	tpid2 4746	. . . . . . . . 9 ⊢ 1 ∈ {0, 1, 2}
31	30, 24	eleqtrri 2833	. . . . . . . 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 7074	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^3)) ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)}) → (𝑛‘1) ∈ ℕ)
34	14, 33	ffvelcdmd 7074	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^3)) ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)}) → (Λ‘(𝑛‘1)) ∈ ℝ)
35		2ex 12315	. . . . . . . . . 10 ⊢ 2 ∈ V
36	35	tpid3 4749	. . . . . . . . 9 ⊢ 2 ∈ {0, 1, 2}
37	36, 24	eleqtrri 2833	. . . . . . . 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 7074	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^3)) ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)}) → (𝑛‘2) ∈ ℕ)
40	14, 39	ffvelcdmd 7074	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^3)) ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)}) → (Λ‘(𝑛‘2)) ∈ ℝ)
41	34, 40	remulcld 11263	. . . 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 11263	. . 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 27081	. . . . 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 27081	. . . . . 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 27081	. . . . . 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 11812	. . . 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 11812	. . 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 32754	. 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 2735	. . . 4 ⊢ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)} = {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)}
53		inss2 4213	. . . . . 6 ⊢ (𝑂 ∩ ℙ) ⊆ ℙ
54		prmssnn 16693	. . . . . 6 ⊢ ℙ ⊆ ℕ
55	53, 54	sstri 3968	. . . . 5 ⊢ (𝑂 ∩ ℙ) ⊆ ℕ
56	55	a1i 11	. . . 4 ⊢ (𝜑 → (𝑂 ∩ ℙ) ⊆ ℕ)
57	52, 7, 56, 4, 6	reprdifc 34605	. . 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 15714	. 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 4055	. . . . . . . 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 9271	. . . . . 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 3982	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)}) → ℕ ⊆ ℕ)
64	16	adantr 480	. . . . . . . . . 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 3958	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)}) → 𝑛 ∈ (ℕ(repr‘3)𝑁))
67	63, 64, 65, 66	reprf 34590	. . . . . . . . 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 7074	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)}) → (𝑛‘0) ∈ ℕ)
70	62, 69	ffvelcdmd 7074	. . . . . . 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 7074	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)}) → (𝑛‘1) ∈ ℕ)
73	62, 72	ffvelcdmd 7074	. . . . . . . 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 7074	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)}) → (𝑛‘2) ∈ ℕ)
76	62, 75	ffvelcdmd 7074	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)}) → (Λ‘(𝑛‘2)) ∈ ℝ)
77	73, 76	remulcld 11263	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)}) → ((Λ‘(𝑛‘1)) · (Λ‘(𝑛‘2))) ∈ ℝ)
78	70, 77	remulcld 11263	. . . . . 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 15748	. . . . 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 11261	. . . 4 ⊢ (𝜑 → Σ𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)} ((Λ‘(𝑛‘0)) · ((Λ‘(𝑛‘1)) · (Λ‘(𝑛‘2)))) ∈ ℂ)
81		fsumconst 15804	. . . 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 6874	. . . . . . . 8 ⊢ (𝑛 = (𝐹‘𝑒) → (𝑛‘0) = ((𝐹‘𝑒)‘0))
84	83	fveq2d 6879	. . . . . . 7 ⊢ (𝑛 = (𝐹‘𝑒) → (Λ‘(𝑛‘0)) = (Λ‘((𝐹‘𝑒)‘0)))
85		fveq1 6874	. . . . . . . . 9 ⊢ (𝑛 = (𝐹‘𝑒) → (𝑛‘1) = ((𝐹‘𝑒)‘1))
86	85	fveq2d 6879	. . . . . . . 8 ⊢ (𝑛 = (𝐹‘𝑒) → (Λ‘(𝑛‘1)) = (Λ‘((𝐹‘𝑒)‘1)))
87		fveq1 6874	. . . . . . . . 9 ⊢ (𝑛 = (𝐹‘𝑒) → (𝑛‘2) = ((𝐹‘𝑒)‘2))
88	87	fveq2d 6879	. . . . . . . 8 ⊢ (𝑛 = (𝐹‘𝑒) → (Λ‘(𝑛‘2)) = (Λ‘((𝐹‘𝑒)‘2)))
89	86, 88	oveq12d 7421	. . . . . . 7 ⊢ (𝑛 = (𝐹‘𝑒) → ((Λ‘(𝑛‘1)) · (Λ‘(𝑛‘2))) = ((Λ‘((𝐹‘𝑒)‘1)) · (Λ‘((𝐹‘𝑒)‘2))))
90	84, 89	oveq12d 7421	. . . . . 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 12317	. . . . . . . . . 10 ⊢ 3 ∈ ℕ
92	91	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 3 ∈ ℕ)
93	92	ralrimivw 3136	. . . . . . . 8 ⊢ (𝜑 → ∀𝑎 ∈ (0..^3)3 ∈ ℕ)
94	93	r19.21bi 3234	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (0..^3)) → 3 ∈ ℕ)
95	16	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (0..^3)) → 𝑁 ∈ ℤ)
96		ssidd 3982	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (0..^3)) → ℕ ⊆ ℕ)
97		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (0..^3)) → 𝑎 ∈ (0..^3))
98		fveq1 6874	. . . . . . . . . 10 ⊢ (𝑐 = 𝑑 → (𝑐‘0) = (𝑑‘0))
99	98	eleq1d 2819	. . . . . . . . 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 3426	. . . . . . 7 ⊢ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)} = {𝑑 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑑‘0) ∈ (𝑂 ∩ ℙ)}
102		fveq1 6874	. . . . . . . . . 10 ⊢ (𝑐 = 𝑑 → (𝑐‘𝑎) = (𝑑‘𝑎))
103	102	eleq1d 2819	. . . . . . . . 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 3426	. . . . . . 7 ⊢ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)} = {𝑑 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑑‘𝑎) ∈ (𝑂 ∩ ℙ)}
106		eqid 2735	. . . . . . 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 34604	. . . . . 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 2736	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^3)) ∧ 𝑒 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)}) → (𝐹‘𝑒) = (𝐹‘𝑒))
110	78	adantlr 715	. . . . . . 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 11261	. . . . . 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 15737	. . . . 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 6875	. . . . . . . . . 10 ⊢ (𝑒 = 𝑛 → (𝐹‘𝑒) = (𝐹‘𝑛))
114	113	fveq1d 6877	. . . . . . . . 9 ⊢ (𝑒 = 𝑛 → ((𝐹‘𝑒)‘0) = ((𝐹‘𝑛)‘0))
115	114	fveq2d 6879	. . . . . . . 8 ⊢ (𝑒 = 𝑛 → (Λ‘((𝐹‘𝑒)‘0)) = (Λ‘((𝐹‘𝑛)‘0)))
116	113	fveq1d 6877	. . . . . . . . . 10 ⊢ (𝑒 = 𝑛 → ((𝐹‘𝑒)‘1) = ((𝐹‘𝑛)‘1))
117	116	fveq2d 6879	. . . . . . . . 9 ⊢ (𝑒 = 𝑛 → (Λ‘((𝐹‘𝑒)‘1)) = (Λ‘((𝐹‘𝑛)‘1)))
118	113	fveq1d 6877	. . . . . . . . . 10 ⊢ (𝑒 = 𝑛 → ((𝐹‘𝑒)‘2) = ((𝐹‘𝑛)‘2))
119	118	fveq2d 6879	. . . . . . . . 9 ⊢ (𝑒 = 𝑛 → (Λ‘((𝐹‘𝑒)‘2)) = (Λ‘((𝐹‘𝑛)‘2)))
120	117, 119	oveq12d 7421	. . . . . . . 8 ⊢ (𝑒 = 𝑛 → ((Λ‘((𝐹‘𝑒)‘1)) · (Λ‘((𝐹‘𝑒)‘2))) = ((Λ‘((𝐹‘𝑛)‘1)) · (Λ‘((𝐹‘𝑛)‘2))))
121	115, 120	oveq12d 7421	. . . . . . 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 15710	. . . . . 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 7438	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^3)) ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)}) → (0..^3) ∈ V)
125	97	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ (0..^3)) ∧ 𝑛 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)}) → 𝑎 ∈ (0..^3))
126	124, 125, 26, 106	pmtridf1o 33051	. . . . . . 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 34632	. . . . . 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 15716	. . . . 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 2775	. . . 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 15716	. . 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 14446	. . . . . . 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 2741	. . . 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 15714	. . . 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 7421	. . 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 2781	. 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 5151	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 395 = wceq 1540 ∈ wcel 2108 {crab 3415 Vcvv 3459 ∖ cdif 3923 ∩ cin 3925 ⊆ wss 3926 ifcif 4500 {cpr 4603 {ctp 4605 ∪ ciun 4967 class class class wbr 5119 ↦ cmpt 5201 I cid 5547 ↾ cres 5656 ∘ ccom 5658 ⟶wf 6526 ‘cfv 6530 (class class class)co 7403 Fincfn 8957 ℂcc 11125 ℝcr 11126 0cc0 11127 1c1 11128 · cmul 11132 ≤ cle 11268 ℕcn 12238 2c2 12293 3c3 12294 ℕ₀cn0 12499 ℤcz 12586 ..^cfzo 13669 ♯chash 14346 Σcsu 15700 ∥ cdvds 16270 ℙcprime 16688 pmTrspcpmtr 19420 Λcvma 27052 reprcrepr 34586

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7727 ax-reg 9604 ax-inf2 9653 ax-ac2 10475 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204 ax-pre-sup 11205 ax-addf 11206

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 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-tp 4606 df-op 4608 df-uni 4884 df-int 4923 df-iun 4969 df-iin 4970 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-se 5607 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6483 df-fun 6532 df-fn 6533 df-f 6534 df-f1 6535 df-fo 6536 df-f1o 6537 df-fv 6538 df-isom 6539 df-riota 7360 df-ov 7406 df-oprab 7407 df-mpo 7408 df-of 7669 df-om 7860 df-1st 7986 df-2nd 7987 df-supp 8158 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-2o 8479 df-oadd 8482 df-er 8717 df-map 8840 df-pm 8841 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fsupp 9372 df-fi 9421 df-sup 9452 df-inf 9453 df-oi 9522 df-r1 9776 df-rank 9777 df-dju 9913 df-card 9951 df-ac 10128 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11466 df-neg 11467 df-div 11893 df-nn 12239 df-2 12301 df-3 12302 df-4 12303 df-5 12304 df-6 12305 df-7 12306 df-8 12307 df-9 12308 df-n0 12500 df-z 12587 df-dec 12707 df-uz 12851 df-q 12963 df-rp 13007 df-xneg 13126 df-xadd 13127 df-xmul 13128 df-ioo 13364 df-ioc 13365 df-ico 13366 df-icc 13367 df-fz 13523 df-fzo 13670 df-fl 13807 df-mod 13885 df-seq 14018 df-exp 14078 df-fac 14290 df-bc 14319 df-hash 14347 df-shft 15084 df-cj 15116 df-re 15117 df-im 15118 df-sqrt 15252 df-abs 15253 df-limsup 15485 df-clim 15502 df-rlim 15503 df-sum 15701 df-prod 15918 df-ef 16081 df-sin 16083 df-cos 16084 df-pi 16086 df-dvds 16271 df-gcd 16512 df-prm 16689 df-pc 16855 df-struct 17164 df-sets 17181 df-slot 17199 df-ndx 17211 df-base 17227 df-ress 17250 df-plusg 17282 df-mulr 17283 df-starv 17284 df-sca 17285 df-vsca 17286 df-ip 17287 df-tset 17288 df-ple 17289 df-ds 17291 df-unif 17292 df-hom 17293 df-cco 17294 df-rest 17434 df-topn 17435 df-0g 17453 df-gsum 17454 df-topgen 17455 df-pt 17456 df-prds 17459 df-xrs 17514 df-qtop 17519 df-imas 17520 df-xps 17522 df-mre 17596 df-mrc 17597 df-acs 17599 df-mgm 18616 df-sgrp 18695 df-mnd 18711 df-submnd 18760 df-mulg 19049 df-cntz 19298 df-pmtr 19421 df-cmn 19761 df-psmet 21305 df-xmet 21306 df-met 21307 df-bl 21308 df-mopn 21309 df-fbas 21310 df-fg 21311 df-cnfld 21314 df-top 22830 df-topon 22847 df-topsp 22869 df-bases 22882 df-cld 22955 df-ntr 22956 df-cls 22957 df-nei 23034 df-lp 23072 df-perf 23073 df-cn 23163 df-cnp 23164 df-haus 23251 df-tx 23498 df-hmeo 23691 df-fil 23782 df-fm 23874 df-flim 23875 df-flf 23876 df-xms 24257 df-ms 24258 df-tms 24259 df-cncf 24820 df-limc 25817 df-dv 25818 df-log 26515 df-vma 27058 df-repr 34587

This theorem is referenced by: hgt750leme 34636

W3C validator