nnsum3primesgbe - Mathbox for Alexander van der Vekens

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Theorem nnsum3primesgbe 46759

Description: Any even Goldbach number is the sum of at most 3 (actually 2) primes. (Contributed by AV, 2-Aug-2020.)

**Assertion**
Ref	Expression
nnsum3primesgbe	⊢ (𝑁 ∈ GoldbachEven → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑_m (1...𝑑))(𝑑 ≤ 3 ∧ 𝑁 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘)))

Distinct variable group: 𝑁,𝑑,𝑓,𝑘

Proof of Theorem nnsum3primesgbe Dummy variables 𝑝 𝑞 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isgbe 46718	. 2 ⊢ (𝑁 ∈ GoldbachEven ↔ (𝑁 ∈ Even ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑁 = (𝑝 + 𝑞))))
2		2nn 12290	. . . . . . . 8 ⊢ 2 ∈ ℕ
3	2	a1i 11	. . . . . . 7 ⊢ (((𝑝 ∈ ℙ ∧ 𝑞 ∈ ℙ) ∧ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑁 = (𝑝 + 𝑞))) → 2 ∈ ℕ)
4		oveq2 7420	. . . . . . . . . . 11 ⊢ (𝑑 = 2 → (1...𝑑) = (1...2))
5		df-2 12280	. . . . . . . . . . . . 13 ⊢ 2 = (1 + 1)
6	5	oveq2i 7423	. . . . . . . . . . . 12 ⊢ (1...2) = (1...(1 + 1))
7		1z 12597	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℤ
8		fzpr 13561	. . . . . . . . . . . . 13 ⊢ (1 ∈ ℤ → (1...(1 + 1)) = {1, (1 + 1)})
9	7, 8	ax-mp 5	. . . . . . . . . . . 12 ⊢ (1...(1 + 1)) = {1, (1 + 1)}
10		1p1e2 12342	. . . . . . . . . . . . 13 ⊢ (1 + 1) = 2
11	10	preq2i 4741	. . . . . . . . . . . 12 ⊢ {1, (1 + 1)} = {1, 2}
12	6, 9, 11	3eqtri 2763	. . . . . . . . . . 11 ⊢ (1...2) = {1, 2}
13	4, 12	eqtrdi 2787	. . . . . . . . . 10 ⊢ (𝑑 = 2 → (1...𝑑) = {1, 2})
14	13	oveq2d 7428	. . . . . . . . 9 ⊢ (𝑑 = 2 → (ℙ ↑_m (1...𝑑)) = (ℙ ↑_m {1, 2}))
15		breq1 5151	. . . . . . . . . 10 ⊢ (𝑑 = 2 → (𝑑 ≤ 3 ↔ 2 ≤ 3))
16	13	sumeq1d 15652	. . . . . . . . . . 11 ⊢ (𝑑 = 2 → Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘) = Σ𝑘 ∈ {1, 2} (𝑓‘𝑘))
17	16	eqeq2d 2742	. . . . . . . . . 10 ⊢ (𝑑 = 2 → (𝑁 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘) ↔ 𝑁 = Σ𝑘 ∈ {1, 2} (𝑓‘𝑘)))
18	15, 17	anbi12d 630	. . . . . . . . 9 ⊢ (𝑑 = 2 → ((𝑑 ≤ 3 ∧ 𝑁 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘)) ↔ (2 ≤ 3 ∧ 𝑁 = Σ𝑘 ∈ {1, 2} (𝑓‘𝑘))))
19	14, 18	rexeqbidv 3342	. . . . . . . 8 ⊢ (𝑑 = 2 → (∃𝑓 ∈ (ℙ ↑_m (1...𝑑))(𝑑 ≤ 3 ∧ 𝑁 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘)) ↔ ∃𝑓 ∈ (ℙ ↑_m {1, 2})(2 ≤ 3 ∧ 𝑁 = Σ𝑘 ∈ {1, 2} (𝑓‘𝑘))))
20	19	adantl 481	. . . . . . 7 ⊢ ((((𝑝 ∈ ℙ ∧ 𝑞 ∈ ℙ) ∧ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑁 = (𝑝 + 𝑞))) ∧ 𝑑 = 2) → (∃𝑓 ∈ (ℙ ↑_m (1...𝑑))(𝑑 ≤ 3 ∧ 𝑁 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘)) ↔ ∃𝑓 ∈ (ℙ ↑_m {1, 2})(2 ≤ 3 ∧ 𝑁 = Σ𝑘 ∈ {1, 2} (𝑓‘𝑘))))
21		1ne2 12425	. . . . . . . . . . . . 13 ⊢ 1 ≠ 2
22		1ex 11215	. . . . . . . . . . . . . 14 ⊢ 1 ∈ V
23		2ex 12294	. . . . . . . . . . . . . 14 ⊢ 2 ∈ V
24		vex 3477	. . . . . . . . . . . . . 14 ⊢ 𝑝 ∈ V
25		vex 3477	. . . . . . . . . . . . . 14 ⊢ 𝑞 ∈ V
26	22, 23, 24, 25	fpr 7154	. . . . . . . . . . . . 13 ⊢ (1 ≠ 2 → {⟨1, 𝑝⟩, ⟨2, 𝑞⟩}:{1, 2}⟶{𝑝, 𝑞})
27	21, 26	mp1i 13	. . . . . . . . . . . 12 ⊢ ((𝑝 ∈ ℙ ∧ 𝑞 ∈ ℙ) → {⟨1, 𝑝⟩, ⟨2, 𝑞⟩}:{1, 2}⟶{𝑝, 𝑞})
28		prssi 4824	. . . . . . . . . . . 12 ⊢ ((𝑝 ∈ ℙ ∧ 𝑞 ∈ ℙ) → {𝑝, 𝑞} ⊆ ℙ)
29	27, 28	fssd 6735	. . . . . . . . . . 11 ⊢ ((𝑝 ∈ ℙ ∧ 𝑞 ∈ ℙ) → {⟨1, 𝑝⟩, ⟨2, 𝑞⟩}:{1, 2}⟶ℙ)
30		prmex 16619	. . . . . . . . . . . . 13 ⊢ ℙ ∈ V
31		prex 5432	. . . . . . . . . . . . 13 ⊢ {1, 2} ∈ V
32	30, 31	pm3.2i 470	. . . . . . . . . . . 12 ⊢ (ℙ ∈ V ∧ {1, 2} ∈ V)
33		elmapg 8837	. . . . . . . . . . . 12 ⊢ ((ℙ ∈ V ∧ {1, 2} ∈ V) → ({⟨1, 𝑝⟩, ⟨2, 𝑞⟩} ∈ (ℙ ↑_m {1, 2}) ↔ {⟨1, 𝑝⟩, ⟨2, 𝑞⟩}:{1, 2}⟶ℙ))
34	32, 33	mp1i 13	. . . . . . . . . . 11 ⊢ ((𝑝 ∈ ℙ ∧ 𝑞 ∈ ℙ) → ({⟨1, 𝑝⟩, ⟨2, 𝑞⟩} ∈ (ℙ ↑_m {1, 2}) ↔ {⟨1, 𝑝⟩, ⟨2, 𝑞⟩}:{1, 2}⟶ℙ))
35	29, 34	mpbird 257	. . . . . . . . . 10 ⊢ ((𝑝 ∈ ℙ ∧ 𝑞 ∈ ℙ) → {⟨1, 𝑝⟩, ⟨2, 𝑞⟩} ∈ (ℙ ↑_m {1, 2}))
36		fveq1 6890	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = {⟨1, 𝑝⟩, ⟨2, 𝑞⟩} → (𝑓‘𝑘) = ({⟨1, 𝑝⟩, ⟨2, 𝑞⟩}‘𝑘))
37	36	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝑓 = {⟨1, 𝑝⟩, ⟨2, 𝑞⟩} ∧ 𝑘 ∈ {1, 2}) → (𝑓‘𝑘) = ({⟨1, 𝑝⟩, ⟨2, 𝑞⟩}‘𝑘))
38	37	sumeq2dv 15654	. . . . . . . . . . . . 13 ⊢ (𝑓 = {⟨1, 𝑝⟩, ⟨2, 𝑞⟩} → Σ𝑘 ∈ {1, 2} (𝑓‘𝑘) = Σ𝑘 ∈ {1, 2} ({⟨1, 𝑝⟩, ⟨2, 𝑞⟩}‘𝑘))
39	38	eqeq1d 2733	. . . . . . . . . . . 12 ⊢ (𝑓 = {⟨1, 𝑝⟩, ⟨2, 𝑞⟩} → (Σ𝑘 ∈ {1, 2} (𝑓‘𝑘) = (𝑝 + 𝑞) ↔ Σ𝑘 ∈ {1, 2} ({⟨1, 𝑝⟩, ⟨2, 𝑞⟩}‘𝑘) = (𝑝 + 𝑞)))
40	39	anbi2d 628	. . . . . . . . . . 11 ⊢ (𝑓 = {⟨1, 𝑝⟩, ⟨2, 𝑞⟩} → ((2 ≤ 3 ∧ Σ𝑘 ∈ {1, 2} (𝑓‘𝑘) = (𝑝 + 𝑞)) ↔ (2 ≤ 3 ∧ Σ𝑘 ∈ {1, 2} ({⟨1, 𝑝⟩, ⟨2, 𝑞⟩}‘𝑘) = (𝑝 + 𝑞))))
41	40	adantl 481	. . . . . . . . . 10 ⊢ (((𝑝 ∈ ℙ ∧ 𝑞 ∈ ℙ) ∧ 𝑓 = {⟨1, 𝑝⟩, ⟨2, 𝑞⟩}) → ((2 ≤ 3 ∧ Σ𝑘 ∈ {1, 2} (𝑓‘𝑘) = (𝑝 + 𝑞)) ↔ (2 ≤ 3 ∧ Σ𝑘 ∈ {1, 2} ({⟨1, 𝑝⟩, ⟨2, 𝑞⟩}‘𝑘) = (𝑝 + 𝑞))))
42		prmz 16617	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℤ)
43		prmz 16617	. . . . . . . . . . . 12 ⊢ (𝑞 ∈ ℙ → 𝑞 ∈ ℤ)
44		fveq2 6891	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 1 → ({⟨1, 𝑝⟩, ⟨2, 𝑞⟩}‘𝑘) = ({⟨1, 𝑝⟩, ⟨2, 𝑞⟩}‘1))
45	22, 24	fvpr1 7193	. . . . . . . . . . . . . . 15 ⊢ (1 ≠ 2 → ({⟨1, 𝑝⟩, ⟨2, 𝑞⟩}‘1) = 𝑝)
46	21, 45	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ({⟨1, 𝑝⟩, ⟨2, 𝑞⟩}‘1) = 𝑝
47	44, 46	eqtrdi 2787	. . . . . . . . . . . . 13 ⊢ (𝑘 = 1 → ({⟨1, 𝑝⟩, ⟨2, 𝑞⟩}‘𝑘) = 𝑝)
48		fveq2 6891	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 2 → ({⟨1, 𝑝⟩, ⟨2, 𝑞⟩}‘𝑘) = ({⟨1, 𝑝⟩, ⟨2, 𝑞⟩}‘2))
49	23, 25	fvpr2 7195	. . . . . . . . . . . . . . 15 ⊢ (1 ≠ 2 → ({⟨1, 𝑝⟩, ⟨2, 𝑞⟩}‘2) = 𝑞)
50	21, 49	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ({⟨1, 𝑝⟩, ⟨2, 𝑞⟩}‘2) = 𝑞
51	48, 50	eqtrdi 2787	. . . . . . . . . . . . 13 ⊢ (𝑘 = 2 → ({⟨1, 𝑝⟩, ⟨2, 𝑞⟩}‘𝑘) = 𝑞)
52		zcn 12568	. . . . . . . . . . . . . 14 ⊢ (𝑝 ∈ ℤ → 𝑝 ∈ ℂ)
53		zcn 12568	. . . . . . . . . . . . . 14 ⊢ (𝑞 ∈ ℤ → 𝑞 ∈ ℂ)
54	52, 53	anim12i 612	. . . . . . . . . . . . 13 ⊢ ((𝑝 ∈ ℤ ∧ 𝑞 ∈ ℤ) → (𝑝 ∈ ℂ ∧ 𝑞 ∈ ℂ))
55	7, 2	pm3.2i 470	. . . . . . . . . . . . . 14 ⊢ (1 ∈ ℤ ∧ 2 ∈ ℕ)
56	55	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝑝 ∈ ℤ ∧ 𝑞 ∈ ℤ) → (1 ∈ ℤ ∧ 2 ∈ ℕ))
57	21	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝑝 ∈ ℤ ∧ 𝑞 ∈ ℤ) → 1 ≠ 2)
58	47, 51, 54, 56, 57	sumpr 15699	. . . . . . . . . . . 12 ⊢ ((𝑝 ∈ ℤ ∧ 𝑞 ∈ ℤ) → Σ𝑘 ∈ {1, 2} ({⟨1, 𝑝⟩, ⟨2, 𝑞⟩}‘𝑘) = (𝑝 + 𝑞))
59	42, 43, 58	syl2an 595	. . . . . . . . . . 11 ⊢ ((𝑝 ∈ ℙ ∧ 𝑞 ∈ ℙ) → Σ𝑘 ∈ {1, 2} ({⟨1, 𝑝⟩, ⟨2, 𝑞⟩}‘𝑘) = (𝑝 + 𝑞))
60		2re 12291	. . . . . . . . . . . 12 ⊢ 2 ∈ ℝ
61		3re 12297	. . . . . . . . . . . 12 ⊢ 3 ∈ ℝ
62		2lt3 12389	. . . . . . . . . . . 12 ⊢ 2 < 3
63	60, 61, 62	ltleii 11342	. . . . . . . . . . 11 ⊢ 2 ≤ 3
64	59, 63	jctil 519	. . . . . . . . . 10 ⊢ ((𝑝 ∈ ℙ ∧ 𝑞 ∈ ℙ) → (2 ≤ 3 ∧ Σ𝑘 ∈ {1, 2} ({⟨1, 𝑝⟩, ⟨2, 𝑞⟩}‘𝑘) = (𝑝 + 𝑞)))
65	35, 41, 64	rspcedvd 3614	. . . . . . . . 9 ⊢ ((𝑝 ∈ ℙ ∧ 𝑞 ∈ ℙ) → ∃𝑓 ∈ (ℙ ↑_m {1, 2})(2 ≤ 3 ∧ Σ𝑘 ∈ {1, 2} (𝑓‘𝑘) = (𝑝 + 𝑞)))
66	65	adantr 480	. . . . . . . 8 ⊢ (((𝑝 ∈ ℙ ∧ 𝑞 ∈ ℙ) ∧ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑁 = (𝑝 + 𝑞))) → ∃𝑓 ∈ (ℙ ↑_m {1, 2})(2 ≤ 3 ∧ Σ𝑘 ∈ {1, 2} (𝑓‘𝑘) = (𝑝 + 𝑞)))
67		eqeq1 2735	. . . . . . . . . . . . 13 ⊢ (𝑁 = (𝑝 + 𝑞) → (𝑁 = Σ𝑘 ∈ {1, 2} (𝑓‘𝑘) ↔ (𝑝 + 𝑞) = Σ𝑘 ∈ {1, 2} (𝑓‘𝑘)))
68		eqcom 2738	. . . . . . . . . . . . 13 ⊢ ((𝑝 + 𝑞) = Σ𝑘 ∈ {1, 2} (𝑓‘𝑘) ↔ Σ𝑘 ∈ {1, 2} (𝑓‘𝑘) = (𝑝 + 𝑞))
69	67, 68	bitrdi 287	. . . . . . . . . . . 12 ⊢ (𝑁 = (𝑝 + 𝑞) → (𝑁 = Σ𝑘 ∈ {1, 2} (𝑓‘𝑘) ↔ Σ𝑘 ∈ {1, 2} (𝑓‘𝑘) = (𝑝 + 𝑞)))
70	69	anbi2d 628	. . . . . . . . . . 11 ⊢ (𝑁 = (𝑝 + 𝑞) → ((2 ≤ 3 ∧ 𝑁 = Σ𝑘 ∈ {1, 2} (𝑓‘𝑘)) ↔ (2 ≤ 3 ∧ Σ𝑘 ∈ {1, 2} (𝑓‘𝑘) = (𝑝 + 𝑞))))
71	70	rexbidv 3177	. . . . . . . . . 10 ⊢ (𝑁 = (𝑝 + 𝑞) → (∃𝑓 ∈ (ℙ ↑_m {1, 2})(2 ≤ 3 ∧ 𝑁 = Σ𝑘 ∈ {1, 2} (𝑓‘𝑘)) ↔ ∃𝑓 ∈ (ℙ ↑_m {1, 2})(2 ≤ 3 ∧ Σ𝑘 ∈ {1, 2} (𝑓‘𝑘) = (𝑝 + 𝑞))))
72	71	3ad2ant3 1134	. . . . . . . . 9 ⊢ ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑁 = (𝑝 + 𝑞)) → (∃𝑓 ∈ (ℙ ↑_m {1, 2})(2 ≤ 3 ∧ 𝑁 = Σ𝑘 ∈ {1, 2} (𝑓‘𝑘)) ↔ ∃𝑓 ∈ (ℙ ↑_m {1, 2})(2 ≤ 3 ∧ Σ𝑘 ∈ {1, 2} (𝑓‘𝑘) = (𝑝 + 𝑞))))
73	72	adantl 481	. . . . . . . 8 ⊢ (((𝑝 ∈ ℙ ∧ 𝑞 ∈ ℙ) ∧ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑁 = (𝑝 + 𝑞))) → (∃𝑓 ∈ (ℙ ↑_m {1, 2})(2 ≤ 3 ∧ 𝑁 = Σ𝑘 ∈ {1, 2} (𝑓‘𝑘)) ↔ ∃𝑓 ∈ (ℙ ↑_m {1, 2})(2 ≤ 3 ∧ Σ𝑘 ∈ {1, 2} (𝑓‘𝑘) = (𝑝 + 𝑞))))
74	66, 73	mpbird 257	. . . . . . 7 ⊢ (((𝑝 ∈ ℙ ∧ 𝑞 ∈ ℙ) ∧ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑁 = (𝑝 + 𝑞))) → ∃𝑓 ∈ (ℙ ↑_m {1, 2})(2 ≤ 3 ∧ 𝑁 = Σ𝑘 ∈ {1, 2} (𝑓‘𝑘)))
75	3, 20, 74	rspcedvd 3614	. . . . . 6 ⊢ (((𝑝 ∈ ℙ ∧ 𝑞 ∈ ℙ) ∧ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑁 = (𝑝 + 𝑞))) → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑_m (1...𝑑))(𝑑 ≤ 3 ∧ 𝑁 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘)))
76	75	a1d 25	. . . . 5 ⊢ (((𝑝 ∈ ℙ ∧ 𝑞 ∈ ℙ) ∧ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑁 = (𝑝 + 𝑞))) → (𝑁 ∈ Even → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑_m (1...𝑑))(𝑑 ≤ 3 ∧ 𝑁 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘))))
77	76	ex 412	. . . 4 ⊢ ((𝑝 ∈ ℙ ∧ 𝑞 ∈ ℙ) → ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑁 = (𝑝 + 𝑞)) → (𝑁 ∈ Even → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑_m (1...𝑑))(𝑑 ≤ 3 ∧ 𝑁 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘)))))
78	77	rexlimivv 3198	. . 3 ⊢ (∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑁 = (𝑝 + 𝑞)) → (𝑁 ∈ Even → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑_m (1...𝑑))(𝑑 ≤ 3 ∧ 𝑁 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘))))
79	78	impcom 407	. 2 ⊢ ((𝑁 ∈ Even ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑁 = (𝑝 + 𝑞))) → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑_m (1...𝑑))(𝑑 ≤ 3 ∧ 𝑁 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘)))
80	1, 79	sylbi 216	1 ⊢ (𝑁 ∈ GoldbachEven → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑_m (1...𝑑))(𝑑 ≤ 3 ∧ 𝑁 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 ≠ wne 2939 ∃wrex 3069 Vcvv 3473 {cpr 4630 ⟨cop 4634 class class class wbr 5148 ⟶wf 6539 ‘cfv 6543 (class class class)co 7412 ↑_m cmap 8824 ℂcc 11112 1c1 11115 + caddc 11117 ≤ cle 11254 ℕcn 12217 2c2 12272 3c3 12273 ℤcz 12563 ...cfz 13489 Σcsu 15637 ℙcprime 16613 Even ceven 46591 Odd codd 46592 GoldbachEven cgbe 46712

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-inf2 9640 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 ax-pre-sup 11192

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-er 8707 df-map 8826 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-sup 9441 df-oi 9509 df-card 9938 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-div 11877 df-nn 12218 df-2 12280 df-3 12281 df-n0 12478 df-z 12564 df-uz 12828 df-rp 12980 df-fz 13490 df-fzo 13633 df-seq 13972 df-exp 14033 df-hash 14296 df-cj 15051 df-re 15052 df-im 15053 df-sqrt 15187 df-abs 15188 df-clim 15437 df-sum 15638 df-prm 16614 df-gbe 46715

This theorem is referenced by: nnsum4primesgbe 46760 nnsum3primesle9 46761 bgoldbnnsum3prm 46771

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