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Theorem prmreclem5 16953

Description: Lemma for prmrec 16955. Here we show the inequality 𝑁 / 2 < ♯𝑀 by decomposing the set (1...𝑁) into the disjoint union of the set 𝑀 of those numbers that are not divisible by any "large" primes (above 𝐾) and the indexed union over 𝐾 < 𝑘 of the numbers 𝑊‘𝑘 that divide the prime 𝑘. By prmreclem4 16952 the second of these has size less than 𝑁 times the prime reciprocal series, which is less than 1 / 2 by assumption, we find that the complementary part 𝑀 must be at least 𝑁 / 2 large. (Contributed by Mario Carneiro, 6-Aug-2014.)

**Hypotheses**
Ref	Expression
prmrec.1	⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (1 / 𝑛), 0))
prmrec.2	⊢ (𝜑 → 𝐾 ∈ ℕ)
prmrec.3	⊢ (𝜑 → 𝑁 ∈ ℕ)
prmrec.4	⊢ 𝑀 = {𝑛 ∈ (1...𝑁) ∣ ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ 𝑛}
prmrec.5	⊢ (𝜑 → seq1( + , 𝐹) ∈ dom ⇝ )
prmrec.6	⊢ (𝜑 → Σ𝑘 ∈ (ℤ_≥‘(𝐾 + 1))if(𝑘 ∈ ℙ, (1 / 𝑘), 0) < (1 / 2))
prmrec.7	⊢ 𝑊 = (𝑝 ∈ ℕ ↦ {𝑛 ∈ (1...𝑁) ∣ (𝑝 ∈ ℙ ∧ 𝑝 ∥ 𝑛)})

**Assertion**
Ref	Expression
prmreclem5	⊢ (𝜑 → (𝑁 / 2) < ((2↑𝐾) · (√‘𝑁)))

Distinct variable groups: 𝑘,𝑛,𝑝,𝐹 𝑘,𝐾,𝑛,𝑝 𝑘,𝑀,𝑛,𝑝 𝜑,𝑘,𝑛,𝑝 𝑘,𝑊 𝑘,𝑁,𝑛,𝑝 Allowed substitution hints: 𝑊(𝑛,𝑝)

Proof of Theorem prmreclem5 Dummy variables 𝑟 𝑥 𝑞 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prmrec.3	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ)
2	1	nnred 12278	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℝ)
3	2	rehalfcld 12510	. 2 ⊢ (𝜑 → (𝑁 / 2) ∈ ℝ)
4		fzfi 14009	. . . . . 6 ⊢ (1...𝑁) ∈ Fin
5		prmrec.4	. . . . . . 7 ⊢ 𝑀 = {𝑛 ∈ (1...𝑁) ∣ ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ 𝑛}
6	5	ssrab3 4091	. . . . . 6 ⊢ 𝑀 ⊆ (1...𝑁)
7		ssfi 9211	. . . . . 6 ⊢ (((1...𝑁) ∈ Fin ∧ 𝑀 ⊆ (1...𝑁)) → 𝑀 ∈ Fin)
8	4, 6, 7	mp2an 692	. . . . 5 ⊢ 𝑀 ∈ Fin
9		hashcl 14391	. . . . 5 ⊢ (𝑀 ∈ Fin → (♯‘𝑀) ∈ ℕ₀)
10	8, 9	ax-mp 5	. . . 4 ⊢ (♯‘𝑀) ∈ ℕ₀
11	10	nn0rei 12534	. . 3 ⊢ (♯‘𝑀) ∈ ℝ
12	11	a1i 11	. 2 ⊢ (𝜑 → (♯‘𝑀) ∈ ℝ)
13		2nn 12336	. . . . 5 ⊢ 2 ∈ ℕ
14		prmrec.2	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ ℕ)
15	14	nnnn0d 12584	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ ℕ₀)
16		nnexpcl 14111	. . . . 5 ⊢ ((2 ∈ ℕ ∧ 𝐾 ∈ ℕ₀) → (2↑𝐾) ∈ ℕ)
17	13, 15, 16	sylancr 587	. . . 4 ⊢ (𝜑 → (2↑𝐾) ∈ ℕ)
18	17	nnred 12278	. . 3 ⊢ (𝜑 → (2↑𝐾) ∈ ℝ)
19	1	nnrpd 13072	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℝ⁺)
20	19	rpsqrtcld 15446	. . . 4 ⊢ (𝜑 → (√‘𝑁) ∈ ℝ⁺)
21	20	rpred 13074	. . 3 ⊢ (𝜑 → (√‘𝑁) ∈ ℝ)
22	18, 21	remulcld 11288	. 2 ⊢ (𝜑 → ((2↑𝐾) · (√‘𝑁)) ∈ ℝ)
23	2	recnd 11286	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℂ)
24	23	2halvesd 12509	. . . . 5 ⊢ (𝜑 → ((𝑁 / 2) + (𝑁 / 2)) = 𝑁)
25	6	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ⊆ (1...𝑁))
26	14	peano2nnd 12280	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐾 + 1) ∈ ℕ)
27		elfzuz 13556	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ((𝐾 + 1)...𝑁) → 𝑘 ∈ (ℤ_≥‘(𝐾 + 1)))
28		eluznn 12957	. . . . . . . . . . . . 13 ⊢ (((𝐾 + 1) ∈ ℕ ∧ 𝑘 ∈ (ℤ_≥‘(𝐾 + 1))) → 𝑘 ∈ ℕ)
29	26, 27, 28	syl2an 596	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐾 + 1)...𝑁)) → 𝑘 ∈ ℕ)
30		eleq1w 2821	. . . . . . . . . . . . . . . . 17 ⊢ (𝑝 = 𝑘 → (𝑝 ∈ ℙ ↔ 𝑘 ∈ ℙ))
31		breq1 5150	. . . . . . . . . . . . . . . . 17 ⊢ (𝑝 = 𝑘 → (𝑝 ∥ 𝑛 ↔ 𝑘 ∥ 𝑛))
32	30, 31	anbi12d 632	. . . . . . . . . . . . . . . 16 ⊢ (𝑝 = 𝑘 → ((𝑝 ∈ ℙ ∧ 𝑝 ∥ 𝑛) ↔ (𝑘 ∈ ℙ ∧ 𝑘 ∥ 𝑛)))
33	32	rabbidv 3440	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = 𝑘 → {𝑛 ∈ (1...𝑁) ∣ (𝑝 ∈ ℙ ∧ 𝑝 ∥ 𝑛)} = {𝑛 ∈ (1...𝑁) ∣ (𝑘 ∈ ℙ ∧ 𝑘 ∥ 𝑛)})
34		prmrec.7	. . . . . . . . . . . . . . 15 ⊢ 𝑊 = (𝑝 ∈ ℕ ↦ {𝑛 ∈ (1...𝑁) ∣ (𝑝 ∈ ℙ ∧ 𝑝 ∥ 𝑛)})
35		ovex 7463	. . . . . . . . . . . . . . . 16 ⊢ (1...𝑁) ∈ V
36	35	rabex 5344	. . . . . . . . . . . . . . 15 ⊢ {𝑛 ∈ (1...𝑁) ∣ (𝑘 ∈ ℙ ∧ 𝑘 ∥ 𝑛)} ∈ V
37	33, 34, 36	fvmpt 7015	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ → (𝑊‘𝑘) = {𝑛 ∈ (1...𝑁) ∣ (𝑘 ∈ ℙ ∧ 𝑘 ∥ 𝑛)})
38	37	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑊‘𝑘) = {𝑛 ∈ (1...𝑁) ∣ (𝑘 ∈ ℙ ∧ 𝑘 ∥ 𝑛)})
39		ssrab2 4089	. . . . . . . . . . . . 13 ⊢ {𝑛 ∈ (1...𝑁) ∣ (𝑘 ∈ ℙ ∧ 𝑘 ∥ 𝑛)} ⊆ (1...𝑁)
40	38, 39	eqsstrdi 4049	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑊‘𝑘) ⊆ (1...𝑁))
41	29, 40	syldan 591	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐾 + 1)...𝑁)) → (𝑊‘𝑘) ⊆ (1...𝑁))
42	41	ralrimiva 3143	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑘 ∈ ((𝐾 + 1)...𝑁)(𝑊‘𝑘) ⊆ (1...𝑁))
43		iunss 5049	. . . . . . . . . 10 ⊢ (∪ 𝑘 ∈ ((𝐾 + 1)...𝑁)(𝑊‘𝑘) ⊆ (1...𝑁) ↔ ∀𝑘 ∈ ((𝐾 + 1)...𝑁)(𝑊‘𝑘) ⊆ (1...𝑁))
44	42, 43	sylibr 234	. . . . . . . . 9 ⊢ (𝜑 → ∪ 𝑘 ∈ ((𝐾 + 1)...𝑁)(𝑊‘𝑘) ⊆ (1...𝑁))
45	25, 44	unssd 4201	. . . . . . . 8 ⊢ (𝜑 → (𝑀 ∪ ∪ 𝑘 ∈ ((𝐾 + 1)...𝑁)(𝑊‘𝑘)) ⊆ (1...𝑁))
46		breq1 5150	. . . . . . . . . . . . . . . 16 ⊢ (𝑝 = 𝑞 → (𝑝 ∥ 𝑛 ↔ 𝑞 ∥ 𝑛))
47	46	notbid 318	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = 𝑞 → (¬ 𝑝 ∥ 𝑛 ↔ ¬ 𝑞 ∥ 𝑛))
48	47	cbvralvw 3234	. . . . . . . . . . . . . 14 ⊢ (∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ 𝑛 ↔ ∀𝑞 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑞 ∥ 𝑛)
49		breq2 5151	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑥 → (𝑞 ∥ 𝑛 ↔ 𝑞 ∥ 𝑥))
50	49	notbid 318	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑥 → (¬ 𝑞 ∥ 𝑛 ↔ ¬ 𝑞 ∥ 𝑥))
51	50	ralbidv 3175	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑥 → (∀𝑞 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑞 ∥ 𝑛 ↔ ∀𝑞 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑞 ∥ 𝑥))
52	48, 51	bitrid 283	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑥 → (∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ 𝑛 ↔ ∀𝑞 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑞 ∥ 𝑥))
53	52, 5	elrab2 3697	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝑀 ↔ (𝑥 ∈ (1...𝑁) ∧ ∀𝑞 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑞 ∥ 𝑥))
54		elun1 4191	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝑀 → 𝑥 ∈ (𝑀 ∪ ∪ 𝑘 ∈ ((𝐾 + 1)...𝑁)(𝑊‘𝑘)))
55	53, 54	sylbir 235	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (1...𝑁) ∧ ∀𝑞 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑞 ∥ 𝑥) → 𝑥 ∈ (𝑀 ∪ ∪ 𝑘 ∈ ((𝐾 + 1)...𝑁)(𝑊‘𝑘)))
56	55	ex 412	. . . . . . . . . 10 ⊢ (𝑥 ∈ (1...𝑁) → (∀𝑞 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑞 ∥ 𝑥 → 𝑥 ∈ (𝑀 ∪ ∪ 𝑘 ∈ ((𝐾 + 1)...𝑁)(𝑊‘𝑘))))
57	56	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...𝑁)) → (∀𝑞 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑞 ∥ 𝑥 → 𝑥 ∈ (𝑀 ∪ ∪ 𝑘 ∈ ((𝐾 + 1)...𝑁)(𝑊‘𝑘))))
58		dfrex2 3070	. . . . . . . . . 10 ⊢ (∃𝑞 ∈ (ℙ ∖ (1...𝐾))𝑞 ∥ 𝑥 ↔ ¬ ∀𝑞 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑞 ∥ 𝑥)
59	14	nnzd 12637	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐾 ∈ ℤ)
60	59	peano2zd 12722	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐾 + 1) ∈ ℤ)
61	60	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1...𝑁)) ∧ (𝑞 ∈ (ℙ ∖ (1...𝐾)) ∧ 𝑞 ∥ 𝑥)) → (𝐾 + 1) ∈ ℤ)
62	1	nnzd 12637	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑁 ∈ ℤ)
63	62	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1...𝑁)) ∧ (𝑞 ∈ (ℙ ∖ (1...𝐾)) ∧ 𝑞 ∥ 𝑥)) → 𝑁 ∈ ℤ)
64		eldifi 4140	. . . . . . . . . . . . . . . 16 ⊢ (𝑞 ∈ (ℙ ∖ (1...𝐾)) → 𝑞 ∈ ℙ)
65	64	ad2antrl 728	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1...𝑁)) ∧ (𝑞 ∈ (ℙ ∖ (1...𝐾)) ∧ 𝑞 ∥ 𝑥)) → 𝑞 ∈ ℙ)
66		prmz 16708	. . . . . . . . . . . . . . 15 ⊢ (𝑞 ∈ ℙ → 𝑞 ∈ ℤ)
67	65, 66	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1...𝑁)) ∧ (𝑞 ∈ (ℙ ∖ (1...𝐾)) ∧ 𝑞 ∥ 𝑥)) → 𝑞 ∈ ℤ)
68		eldifn 4141	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑞 ∈ (ℙ ∖ (1...𝐾)) → ¬ 𝑞 ∈ (1...𝐾))
69	68	ad2antrl 728	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (1...𝑁)) ∧ (𝑞 ∈ (ℙ ∖ (1...𝐾)) ∧ 𝑞 ∥ 𝑥)) → ¬ 𝑞 ∈ (1...𝐾))
70		prmnn 16707	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑞 ∈ ℙ → 𝑞 ∈ ℕ)
71	65, 70	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ (1...𝑁)) ∧ (𝑞 ∈ (ℙ ∖ (1...𝐾)) ∧ 𝑞 ∥ 𝑥)) → 𝑞 ∈ ℕ)
72		nnuz 12918	. . . . . . . . . . . . . . . . . . 19 ⊢ ℕ = (ℤ_≥‘1)
73	71, 72	eleqtrdi 2848	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ (1...𝑁)) ∧ (𝑞 ∈ (ℙ ∖ (1...𝐾)) ∧ 𝑞 ∥ 𝑥)) → 𝑞 ∈ (ℤ_≥‘1))
74	59	ad2antrr 726	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ (1...𝑁)) ∧ (𝑞 ∈ (ℙ ∖ (1...𝐾)) ∧ 𝑞 ∥ 𝑥)) → 𝐾 ∈ ℤ)
75		elfz5 13552	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑞 ∈ (ℤ_≥‘1) ∧ 𝐾 ∈ ℤ) → (𝑞 ∈ (1...𝐾) ↔ 𝑞 ≤ 𝐾))
76	73, 74, 75	syl2anc 584	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (1...𝑁)) ∧ (𝑞 ∈ (ℙ ∖ (1...𝐾)) ∧ 𝑞 ∥ 𝑥)) → (𝑞 ∈ (1...𝐾) ↔ 𝑞 ≤ 𝐾))
77	69, 76	mtbid 324	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (1...𝑁)) ∧ (𝑞 ∈ (ℙ ∖ (1...𝐾)) ∧ 𝑞 ∥ 𝑥)) → ¬ 𝑞 ≤ 𝐾)
78	14	nnred 12278	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐾 ∈ ℝ)
79	78	ad2antrr 726	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (1...𝑁)) ∧ (𝑞 ∈ (ℙ ∖ (1...𝐾)) ∧ 𝑞 ∥ 𝑥)) → 𝐾 ∈ ℝ)
80	71	nnred 12278	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (1...𝑁)) ∧ (𝑞 ∈ (ℙ ∖ (1...𝐾)) ∧ 𝑞 ∥ 𝑥)) → 𝑞 ∈ ℝ)
81	79, 80	ltnled 11405	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (1...𝑁)) ∧ (𝑞 ∈ (ℙ ∖ (1...𝐾)) ∧ 𝑞 ∥ 𝑥)) → (𝐾 < 𝑞 ↔ ¬ 𝑞 ≤ 𝐾))
82	77, 81	mpbird 257	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1...𝑁)) ∧ (𝑞 ∈ (ℙ ∖ (1...𝐾)) ∧ 𝑞 ∥ 𝑥)) → 𝐾 < 𝑞)
83		zltp1le 12664	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ ℤ ∧ 𝑞 ∈ ℤ) → (𝐾 < 𝑞 ↔ (𝐾 + 1) ≤ 𝑞))
84	74, 67, 83	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1...𝑁)) ∧ (𝑞 ∈ (ℙ ∖ (1...𝐾)) ∧ 𝑞 ∥ 𝑥)) → (𝐾 < 𝑞 ↔ (𝐾 + 1) ≤ 𝑞))
85	82, 84	mpbid 232	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1...𝑁)) ∧ (𝑞 ∈ (ℙ ∖ (1...𝐾)) ∧ 𝑞 ∥ 𝑥)) → (𝐾 + 1) ≤ 𝑞)
86		elfznn 13589	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (1...𝑁) → 𝑥 ∈ ℕ)
87	86	ad2antlr 727	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (1...𝑁)) ∧ (𝑞 ∈ (ℙ ∖ (1...𝐾)) ∧ 𝑞 ∥ 𝑥)) → 𝑥 ∈ ℕ)
88	87	nnred 12278	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1...𝑁)) ∧ (𝑞 ∈ (ℙ ∖ (1...𝐾)) ∧ 𝑞 ∥ 𝑥)) → 𝑥 ∈ ℝ)
89	2	ad2antrr 726	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1...𝑁)) ∧ (𝑞 ∈ (ℙ ∖ (1...𝐾)) ∧ 𝑞 ∥ 𝑥)) → 𝑁 ∈ ℝ)
90		simprr 773	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (1...𝑁)) ∧ (𝑞 ∈ (ℙ ∖ (1...𝐾)) ∧ 𝑞 ∥ 𝑥)) → 𝑞 ∥ 𝑥)
91		dvdsle 16343	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑞 ∈ ℤ ∧ 𝑥 ∈ ℕ) → (𝑞 ∥ 𝑥 → 𝑞 ≤ 𝑥))
92	67, 87, 91	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (1...𝑁)) ∧ (𝑞 ∈ (ℙ ∖ (1...𝐾)) ∧ 𝑞 ∥ 𝑥)) → (𝑞 ∥ 𝑥 → 𝑞 ≤ 𝑥))
93	90, 92	mpd 15	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1...𝑁)) ∧ (𝑞 ∈ (ℙ ∖ (1...𝐾)) ∧ 𝑞 ∥ 𝑥)) → 𝑞 ≤ 𝑥)
94		elfzle2 13564	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (1...𝑁) → 𝑥 ≤ 𝑁)
95	94	ad2antlr 727	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1...𝑁)) ∧ (𝑞 ∈ (ℙ ∖ (1...𝐾)) ∧ 𝑞 ∥ 𝑥)) → 𝑥 ≤ 𝑁)
96	80, 88, 89, 93, 95	letrd 11415	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1...𝑁)) ∧ (𝑞 ∈ (ℙ ∖ (1...𝐾)) ∧ 𝑞 ∥ 𝑥)) → 𝑞 ≤ 𝑁)
97	61, 63, 67, 85, 96	elfzd 13551	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1...𝑁)) ∧ (𝑞 ∈ (ℙ ∖ (1...𝐾)) ∧ 𝑞 ∥ 𝑥)) → 𝑞 ∈ ((𝐾 + 1)...𝑁))
98	49	anbi2d 630	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑥 → ((𝑞 ∈ ℙ ∧ 𝑞 ∥ 𝑛) ↔ (𝑞 ∈ ℙ ∧ 𝑞 ∥ 𝑥)))
99		simplr 769	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1...𝑁)) ∧ (𝑞 ∈ (ℙ ∖ (1...𝐾)) ∧ 𝑞 ∥ 𝑥)) → 𝑥 ∈ (1...𝑁))
100	65, 90	jca 511	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1...𝑁)) ∧ (𝑞 ∈ (ℙ ∖ (1...𝐾)) ∧ 𝑞 ∥ 𝑥)) → (𝑞 ∈ ℙ ∧ 𝑞 ∥ 𝑥))
101	98, 99, 100	elrabd 3696	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1...𝑁)) ∧ (𝑞 ∈ (ℙ ∖ (1...𝐾)) ∧ 𝑞 ∥ 𝑥)) → 𝑥 ∈ {𝑛 ∈ (1...𝑁) ∣ (𝑞 ∈ ℙ ∧ 𝑞 ∥ 𝑛)})
102		eleq1w 2821	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑝 = 𝑞 → (𝑝 ∈ ℙ ↔ 𝑞 ∈ ℙ))
103	102, 46	anbi12d 632	. . . . . . . . . . . . . . . . 17 ⊢ (𝑝 = 𝑞 → ((𝑝 ∈ ℙ ∧ 𝑝 ∥ 𝑛) ↔ (𝑞 ∈ ℙ ∧ 𝑞 ∥ 𝑛)))
104	103	rabbidv 3440	. . . . . . . . . . . . . . . 16 ⊢ (𝑝 = 𝑞 → {𝑛 ∈ (1...𝑁) ∣ (𝑝 ∈ ℙ ∧ 𝑝 ∥ 𝑛)} = {𝑛 ∈ (1...𝑁) ∣ (𝑞 ∈ ℙ ∧ 𝑞 ∥ 𝑛)})
105	35	rabex 5344	. . . . . . . . . . . . . . . 16 ⊢ {𝑛 ∈ (1...𝑁) ∣ (𝑞 ∈ ℙ ∧ 𝑞 ∥ 𝑛)} ∈ V
106	104, 34, 105	fvmpt 7015	. . . . . . . . . . . . . . 15 ⊢ (𝑞 ∈ ℕ → (𝑊‘𝑞) = {𝑛 ∈ (1...𝑁) ∣ (𝑞 ∈ ℙ ∧ 𝑞 ∥ 𝑛)})
107	71, 106	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1...𝑁)) ∧ (𝑞 ∈ (ℙ ∖ (1...𝐾)) ∧ 𝑞 ∥ 𝑥)) → (𝑊‘𝑞) = {𝑛 ∈ (1...𝑁) ∣ (𝑞 ∈ ℙ ∧ 𝑞 ∥ 𝑛)})
108	101, 107	eleqtrrd 2841	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1...𝑁)) ∧ (𝑞 ∈ (ℙ ∖ (1...𝐾)) ∧ 𝑞 ∥ 𝑥)) → 𝑥 ∈ (𝑊‘𝑞))
109		fveq2 6906	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑞 → (𝑊‘𝑘) = (𝑊‘𝑞))
110	109	eliuni 5001	. . . . . . . . . . . . 13 ⊢ ((𝑞 ∈ ((𝐾 + 1)...𝑁) ∧ 𝑥 ∈ (𝑊‘𝑞)) → 𝑥 ∈ ∪ 𝑘 ∈ ((𝐾 + 1)...𝑁)(𝑊‘𝑘))
111	97, 108, 110	syl2anc 584	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1...𝑁)) ∧ (𝑞 ∈ (ℙ ∖ (1...𝐾)) ∧ 𝑞 ∥ 𝑥)) → 𝑥 ∈ ∪ 𝑘 ∈ ((𝐾 + 1)...𝑁)(𝑊‘𝑘))
112		elun2 4192	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ∪ 𝑘 ∈ ((𝐾 + 1)...𝑁)(𝑊‘𝑘) → 𝑥 ∈ (𝑀 ∪ ∪ 𝑘 ∈ ((𝐾 + 1)...𝑁)(𝑊‘𝑘)))
113	111, 112	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (1...𝑁)) ∧ (𝑞 ∈ (ℙ ∖ (1...𝐾)) ∧ 𝑞 ∥ 𝑥)) → 𝑥 ∈ (𝑀 ∪ ∪ 𝑘 ∈ ((𝐾 + 1)...𝑁)(𝑊‘𝑘)))
114	113	rexlimdvaa 3153	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...𝑁)) → (∃𝑞 ∈ (ℙ ∖ (1...𝐾))𝑞 ∥ 𝑥 → 𝑥 ∈ (𝑀 ∪ ∪ 𝑘 ∈ ((𝐾 + 1)...𝑁)(𝑊‘𝑘))))
115	58, 114	biimtrrid 243	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...𝑁)) → (¬ ∀𝑞 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑞 ∥ 𝑥 → 𝑥 ∈ (𝑀 ∪ ∪ 𝑘 ∈ ((𝐾 + 1)...𝑁)(𝑊‘𝑘))))
116	57, 115	pm2.61d 179	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...𝑁)) → 𝑥 ∈ (𝑀 ∪ ∪ 𝑘 ∈ ((𝐾 + 1)...𝑁)(𝑊‘𝑘)))
117	45, 116	eqelssd 4016	. . . . . . 7 ⊢ (𝜑 → (𝑀 ∪ ∪ 𝑘 ∈ ((𝐾 + 1)...𝑁)(𝑊‘𝑘)) = (1...𝑁))
118	117	fveq2d 6910	. . . . . 6 ⊢ (𝜑 → (♯‘(𝑀 ∪ ∪ 𝑘 ∈ ((𝐾 + 1)...𝑁)(𝑊‘𝑘))) = (♯‘(1...𝑁)))
119	1	nnnn0d 12584	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
120		hashfz1 14381	. . . . . . 7 ⊢ (𝑁 ∈ ℕ₀ → (♯‘(1...𝑁)) = 𝑁)
121	119, 120	syl 17	. . . . . 6 ⊢ (𝜑 → (♯‘(1...𝑁)) = 𝑁)
122	118, 121	eqtr2d 2775	. . . . 5 ⊢ (𝜑 → 𝑁 = (♯‘(𝑀 ∪ ∪ 𝑘 ∈ ((𝐾 + 1)...𝑁)(𝑊‘𝑘))))
123	8	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ Fin)
124		ssfi 9211	. . . . . . 7 ⊢ (((1...𝑁) ∈ Fin ∧ ∪ 𝑘 ∈ ((𝐾 + 1)...𝑁)(𝑊‘𝑘) ⊆ (1...𝑁)) → ∪ 𝑘 ∈ ((𝐾 + 1)...𝑁)(𝑊‘𝑘) ∈ Fin)
125	4, 44, 124	sylancr 587	. . . . . 6 ⊢ (𝜑 → ∪ 𝑘 ∈ ((𝐾 + 1)...𝑁)(𝑊‘𝑘) ∈ Fin)
126		breq1 5150	. . . . . . . . . . . . . . . . 17 ⊢ (𝑝 = 𝑘 → (𝑝 ∥ 𝑥 ↔ 𝑘 ∥ 𝑥))
127	126	notbid 318	. . . . . . . . . . . . . . . 16 ⊢ (𝑝 = 𝑘 → (¬ 𝑝 ∥ 𝑥 ↔ ¬ 𝑘 ∥ 𝑥))
128		breq2 5151	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 = 𝑥 → (𝑝 ∥ 𝑛 ↔ 𝑝 ∥ 𝑥))
129	128	notbid 318	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = 𝑥 → (¬ 𝑝 ∥ 𝑛 ↔ ¬ 𝑝 ∥ 𝑥))
130	129	ralbidv 3175	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑥 → (∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ 𝑛 ↔ ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ 𝑥))
131	130, 5	elrab2 3697	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ 𝑀 ↔ (𝑥 ∈ (1...𝑁) ∧ ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ 𝑥))
132	131	simprbi 496	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ 𝑀 → ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ 𝑥)
133	132	ad2antlr 727	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑀) ∧ (𝑘 ∈ ((𝐾 + 1)...𝑁) ∧ 𝑘 ∈ ℙ)) → ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ 𝑥)
134		simprr 773	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑀) ∧ (𝑘 ∈ ((𝐾 + 1)...𝑁) ∧ 𝑘 ∈ ℙ)) → 𝑘 ∈ ℙ)
135		noel 4343	. . . . . . . . . . . . . . . . . 18 ⊢ ¬ 𝑘 ∈ ∅
136		simprl 771	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑀) ∧ (𝑘 ∈ ((𝐾 + 1)...𝑁) ∧ 𝑘 ∈ ℙ)) → 𝑘 ∈ ((𝐾 + 1)...𝑁))
137	136	biantrud 531	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑀) ∧ (𝑘 ∈ ((𝐾 + 1)...𝑁) ∧ 𝑘 ∈ ℙ)) → (𝑘 ∈ (1...𝐾) ↔ (𝑘 ∈ (1...𝐾) ∧ 𝑘 ∈ ((𝐾 + 1)...𝑁))))
138		elin 3978	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ ((1...𝐾) ∩ ((𝐾 + 1)...𝑁)) ↔ (𝑘 ∈ (1...𝐾) ∧ 𝑘 ∈ ((𝐾 + 1)...𝑁)))
139	137, 138	bitr4di 289	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑀) ∧ (𝑘 ∈ ((𝐾 + 1)...𝑁) ∧ 𝑘 ∈ ℙ)) → (𝑘 ∈ (1...𝐾) ↔ 𝑘 ∈ ((1...𝐾) ∩ ((𝐾 + 1)...𝑁))))
140	78	ltp1d 12195	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝐾 < (𝐾 + 1))
141		fzdisj 13587	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐾 < (𝐾 + 1) → ((1...𝐾) ∩ ((𝐾 + 1)...𝑁)) = ∅)
142	140, 141	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ((1...𝐾) ∩ ((𝐾 + 1)...𝑁)) = ∅)
143	142	ad2antrr 726	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑀) ∧ (𝑘 ∈ ((𝐾 + 1)...𝑁) ∧ 𝑘 ∈ ℙ)) → ((1...𝐾) ∩ ((𝐾 + 1)...𝑁)) = ∅)
144	143	eleq2d 2824	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑀) ∧ (𝑘 ∈ ((𝐾 + 1)...𝑁) ∧ 𝑘 ∈ ℙ)) → (𝑘 ∈ ((1...𝐾) ∩ ((𝐾 + 1)...𝑁)) ↔ 𝑘 ∈ ∅))
145	139, 144	bitrd 279	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑀) ∧ (𝑘 ∈ ((𝐾 + 1)...𝑁) ∧ 𝑘 ∈ ℙ)) → (𝑘 ∈ (1...𝐾) ↔ 𝑘 ∈ ∅))
146	135, 145	mtbiri 327	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑀) ∧ (𝑘 ∈ ((𝐾 + 1)...𝑁) ∧ 𝑘 ∈ ℙ)) → ¬ 𝑘 ∈ (1...𝐾))
147	134, 146	eldifd 3973	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑀) ∧ (𝑘 ∈ ((𝐾 + 1)...𝑁) ∧ 𝑘 ∈ ℙ)) → 𝑘 ∈ (ℙ ∖ (1...𝐾)))
148	127, 133, 147	rspcdva 3622	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑀) ∧ (𝑘 ∈ ((𝐾 + 1)...𝑁) ∧ 𝑘 ∈ ℙ)) → ¬ 𝑘 ∥ 𝑥)
149	148	expr 456	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑀) ∧ 𝑘 ∈ ((𝐾 + 1)...𝑁)) → (𝑘 ∈ ℙ → ¬ 𝑘 ∥ 𝑥))
150		imnan 399	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ ℙ → ¬ 𝑘 ∥ 𝑥) ↔ ¬ (𝑘 ∈ ℙ ∧ 𝑘 ∥ 𝑥))
151	149, 150	sylib 218	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑀) ∧ 𝑘 ∈ ((𝐾 + 1)...𝑁)) → ¬ (𝑘 ∈ ℙ ∧ 𝑘 ∥ 𝑥))
152	29	adantlr 715	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑀) ∧ 𝑘 ∈ ((𝐾 + 1)...𝑁)) → 𝑘 ∈ ℕ)
153	152, 37	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑀) ∧ 𝑘 ∈ ((𝐾 + 1)...𝑁)) → (𝑊‘𝑘) = {𝑛 ∈ (1...𝑁) ∣ (𝑘 ∈ ℙ ∧ 𝑘 ∥ 𝑛)})
154	153	eleq2d 2824	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑀) ∧ 𝑘 ∈ ((𝐾 + 1)...𝑁)) → (𝑥 ∈ (𝑊‘𝑘) ↔ 𝑥 ∈ {𝑛 ∈ (1...𝑁) ∣ (𝑘 ∈ ℙ ∧ 𝑘 ∥ 𝑛)}))
155		breq2 5151	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑥 → (𝑘 ∥ 𝑛 ↔ 𝑘 ∥ 𝑥))
156	155	anbi2d 630	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑥 → ((𝑘 ∈ ℙ ∧ 𝑘 ∥ 𝑛) ↔ (𝑘 ∈ ℙ ∧ 𝑘 ∥ 𝑥)))
157	156	elrab 3694	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ {𝑛 ∈ (1...𝑁) ∣ (𝑘 ∈ ℙ ∧ 𝑘 ∥ 𝑛)} ↔ (𝑥 ∈ (1...𝑁) ∧ (𝑘 ∈ ℙ ∧ 𝑘 ∥ 𝑥)))
158	157	simprbi 496	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ {𝑛 ∈ (1...𝑁) ∣ (𝑘 ∈ ℙ ∧ 𝑘 ∥ 𝑛)} → (𝑘 ∈ ℙ ∧ 𝑘 ∥ 𝑥))
159	154, 158	biimtrdi 253	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑀) ∧ 𝑘 ∈ ((𝐾 + 1)...𝑁)) → (𝑥 ∈ (𝑊‘𝑘) → (𝑘 ∈ ℙ ∧ 𝑘 ∥ 𝑥)))
160	151, 159	mtod 198	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑀) ∧ 𝑘 ∈ ((𝐾 + 1)...𝑁)) → ¬ 𝑥 ∈ (𝑊‘𝑘))
161	160	nrexdv 3146	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → ¬ ∃𝑘 ∈ ((𝐾 + 1)...𝑁)𝑥 ∈ (𝑊‘𝑘))
162		eliun 4999	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ∪ 𝑘 ∈ ((𝐾 + 1)...𝑁)(𝑊‘𝑘) ↔ ∃𝑘 ∈ ((𝐾 + 1)...𝑁)𝑥 ∈ (𝑊‘𝑘))
163	161, 162	sylnibr 329	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → ¬ 𝑥 ∈ ∪ 𝑘 ∈ ((𝐾 + 1)...𝑁)(𝑊‘𝑘))
164	163	ex 412	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝑀 → ¬ 𝑥 ∈ ∪ 𝑘 ∈ ((𝐾 + 1)...𝑁)(𝑊‘𝑘)))
165		imnan 399	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑀 → ¬ 𝑥 ∈ ∪ 𝑘 ∈ ((𝐾 + 1)...𝑁)(𝑊‘𝑘)) ↔ ¬ (𝑥 ∈ 𝑀 ∧ 𝑥 ∈ ∪ 𝑘 ∈ ((𝐾 + 1)...𝑁)(𝑊‘𝑘)))
166	164, 165	sylib 218	. . . . . . . 8 ⊢ (𝜑 → ¬ (𝑥 ∈ 𝑀 ∧ 𝑥 ∈ ∪ 𝑘 ∈ ((𝐾 + 1)...𝑁)(𝑊‘𝑘)))
167		elin 3978	. . . . . . . 8 ⊢ (𝑥 ∈ (𝑀 ∩ ∪ 𝑘 ∈ ((𝐾 + 1)...𝑁)(𝑊‘𝑘)) ↔ (𝑥 ∈ 𝑀 ∧ 𝑥 ∈ ∪ 𝑘 ∈ ((𝐾 + 1)...𝑁)(𝑊‘𝑘)))
168	166, 167	sylnibr 329	. . . . . . 7 ⊢ (𝜑 → ¬ 𝑥 ∈ (𝑀 ∩ ∪ 𝑘 ∈ ((𝐾 + 1)...𝑁)(𝑊‘𝑘)))
169	168	eq0rdv 4412	. . . . . 6 ⊢ (𝜑 → (𝑀 ∩ ∪ 𝑘 ∈ ((𝐾 + 1)...𝑁)(𝑊‘𝑘)) = ∅)
170		hashun 14417	. . . . . 6 ⊢ ((𝑀 ∈ Fin ∧ ∪ 𝑘 ∈ ((𝐾 + 1)...𝑁)(𝑊‘𝑘) ∈ Fin ∧ (𝑀 ∩ ∪ 𝑘 ∈ ((𝐾 + 1)...𝑁)(𝑊‘𝑘)) = ∅) → (♯‘(𝑀 ∪ ∪ 𝑘 ∈ ((𝐾 + 1)...𝑁)(𝑊‘𝑘))) = ((♯‘𝑀) + (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑁)(𝑊‘𝑘))))
171	123, 125, 169, 170	syl3anc 1370	. . . . 5 ⊢ (𝜑 → (♯‘(𝑀 ∪ ∪ 𝑘 ∈ ((𝐾 + 1)...𝑁)(𝑊‘𝑘))) = ((♯‘𝑀) + (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑁)(𝑊‘𝑘))))
172	24, 122, 171	3eqtrd 2778	. . . 4 ⊢ (𝜑 → ((𝑁 / 2) + (𝑁 / 2)) = ((♯‘𝑀) + (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑁)(𝑊‘𝑘))))
173		hashcl 14391	. . . . . . 7 ⊢ (∪ 𝑘 ∈ ((𝐾 + 1)...𝑁)(𝑊‘𝑘) ∈ Fin → (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑁)(𝑊‘𝑘)) ∈ ℕ₀)
174	125, 173	syl 17	. . . . . 6 ⊢ (𝜑 → (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑁)(𝑊‘𝑘)) ∈ ℕ₀)
175	174	nn0red 12585	. . . . 5 ⊢ (𝜑 → (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑁)(𝑊‘𝑘)) ∈ ℝ)
176		fzfid 14010	. . . . . . . 8 ⊢ (𝜑 → ((𝐾 + 1)...𝑁) ∈ Fin)
177	26, 28	sylan 580	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘(𝐾 + 1))) → 𝑘 ∈ ℕ)
178		nnrecre 12305	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → (1 / 𝑘) ∈ ℝ)
179		0re 11260	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ
180		ifcl 4575	. . . . . . . . . . 11 ⊢ (((1 / 𝑘) ∈ ℝ ∧ 0 ∈ ℝ) → if(𝑘 ∈ ℙ, (1 / 𝑘), 0) ∈ ℝ)
181	178, 179, 180	sylancl 586	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → if(𝑘 ∈ ℙ, (1 / 𝑘), 0) ∈ ℝ)
182	177, 181	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘(𝐾 + 1))) → if(𝑘 ∈ ℙ, (1 / 𝑘), 0) ∈ ℝ)
183	27, 182	sylan2 593	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐾 + 1)...𝑁)) → if(𝑘 ∈ ℙ, (1 / 𝑘), 0) ∈ ℝ)
184	176, 183	fsumrecl 15766	. . . . . . 7 ⊢ (𝜑 → Σ𝑘 ∈ ((𝐾 + 1)...𝑁)if(𝑘 ∈ ℙ, (1 / 𝑘), 0) ∈ ℝ)
185	2, 184	remulcld 11288	. . . . . 6 ⊢ (𝜑 → (𝑁 · Σ𝑘 ∈ ((𝐾 + 1)...𝑁)if(𝑘 ∈ ℙ, (1 / 𝑘), 0)) ∈ ℝ)
186		prmrec.1	. . . . . . . 8 ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (1 / 𝑛), 0))
187		prmrec.5	. . . . . . . 8 ⊢ (𝜑 → seq1( + , 𝐹) ∈ dom ⇝ )
188		prmrec.6	. . . . . . . 8 ⊢ (𝜑 → Σ𝑘 ∈ (ℤ_≥‘(𝐾 + 1))if(𝑘 ∈ ℙ, (1 / 𝑘), 0) < (1 / 2))
189	186, 14, 1, 5, 187, 188, 34	prmreclem4 16952	. . . . . . 7 ⊢ (𝜑 → (𝑁 ∈ (ℤ_≥‘𝐾) → (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑁)(𝑊‘𝑘)) ≤ (𝑁 · Σ𝑘 ∈ ((𝐾 + 1)...𝑁)if(𝑘 ∈ ℙ, (1 / 𝑘), 0))))
190		eluz 12889	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝐾 ∈ (ℤ_≥‘𝑁) ↔ 𝑁 ≤ 𝐾))
191	62, 59, 190	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → (𝐾 ∈ (ℤ_≥‘𝑁) ↔ 𝑁 ≤ 𝐾))
192		nnleltp1 12670	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ ℕ) → (𝑁 ≤ 𝐾 ↔ 𝑁 < (𝐾 + 1)))
193	1, 14, 192	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → (𝑁 ≤ 𝐾 ↔ 𝑁 < (𝐾 + 1)))
194		fzn 13576	. . . . . . . . . 10 ⊢ (((𝐾 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 < (𝐾 + 1) ↔ ((𝐾 + 1)...𝑁) = ∅))
195	60, 62, 194	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → (𝑁 < (𝐾 + 1) ↔ ((𝐾 + 1)...𝑁) = ∅))
196	191, 193, 195	3bitrd 305	. . . . . . . 8 ⊢ (𝜑 → (𝐾 ∈ (ℤ_≥‘𝑁) ↔ ((𝐾 + 1)...𝑁) = ∅))
197		0le0 12364	. . . . . . . . . 10 ⊢ 0 ≤ 0
198	23	mul01d 11457	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁 · 0) = 0)
199	197, 198	breqtrrid 5185	. . . . . . . . 9 ⊢ (𝜑 → 0 ≤ (𝑁 · 0))
200		iuneq1 5012	. . . . . . . . . . . . 13 ⊢ (((𝐾 + 1)...𝑁) = ∅ → ∪ 𝑘 ∈ ((𝐾 + 1)...𝑁)(𝑊‘𝑘) = ∪ 𝑘 ∈ ∅ (𝑊‘𝑘))
201		0iun 5067	. . . . . . . . . . . . 13 ⊢ ∪ 𝑘 ∈ ∅ (𝑊‘𝑘) = ∅
202	200, 201	eqtrdi 2790	. . . . . . . . . . . 12 ⊢ (((𝐾 + 1)...𝑁) = ∅ → ∪ 𝑘 ∈ ((𝐾 + 1)...𝑁)(𝑊‘𝑘) = ∅)
203	202	fveq2d 6910	. . . . . . . . . . 11 ⊢ (((𝐾 + 1)...𝑁) = ∅ → (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑁)(𝑊‘𝑘)) = (♯‘∅))
204		hash0 14402	. . . . . . . . . . 11 ⊢ (♯‘∅) = 0
205	203, 204	eqtrdi 2790	. . . . . . . . . 10 ⊢ (((𝐾 + 1)...𝑁) = ∅ → (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑁)(𝑊‘𝑘)) = 0)
206		sumeq1 15721	. . . . . . . . . . . 12 ⊢ (((𝐾 + 1)...𝑁) = ∅ → Σ𝑘 ∈ ((𝐾 + 1)...𝑁)if(𝑘 ∈ ℙ, (1 / 𝑘), 0) = Σ𝑘 ∈ ∅ if(𝑘 ∈ ℙ, (1 / 𝑘), 0))
207		sum0 15753	. . . . . . . . . . . 12 ⊢ Σ𝑘 ∈ ∅ if(𝑘 ∈ ℙ, (1 / 𝑘), 0) = 0
208	206, 207	eqtrdi 2790	. . . . . . . . . . 11 ⊢ (((𝐾 + 1)...𝑁) = ∅ → Σ𝑘 ∈ ((𝐾 + 1)...𝑁)if(𝑘 ∈ ℙ, (1 / 𝑘), 0) = 0)
209	208	oveq2d 7446	. . . . . . . . . 10 ⊢ (((𝐾 + 1)...𝑁) = ∅ → (𝑁 · Σ𝑘 ∈ ((𝐾 + 1)...𝑁)if(𝑘 ∈ ℙ, (1 / 𝑘), 0)) = (𝑁 · 0))
210	205, 209	breq12d 5160	. . . . . . . . 9 ⊢ (((𝐾 + 1)...𝑁) = ∅ → ((♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑁)(𝑊‘𝑘)) ≤ (𝑁 · Σ𝑘 ∈ ((𝐾 + 1)...𝑁)if(𝑘 ∈ ℙ, (1 / 𝑘), 0)) ↔ 0 ≤ (𝑁 · 0)))
211	199, 210	syl5ibrcom 247	. . . . . . . 8 ⊢ (𝜑 → (((𝐾 + 1)...𝑁) = ∅ → (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑁)(𝑊‘𝑘)) ≤ (𝑁 · Σ𝑘 ∈ ((𝐾 + 1)...𝑁)if(𝑘 ∈ ℙ, (1 / 𝑘), 0))))
212	196, 211	sylbid 240	. . . . . . 7 ⊢ (𝜑 → (𝐾 ∈ (ℤ_≥‘𝑁) → (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑁)(𝑊‘𝑘)) ≤ (𝑁 · Σ𝑘 ∈ ((𝐾 + 1)...𝑁)if(𝑘 ∈ ℙ, (1 / 𝑘), 0))))
213		uztric 12899	. . . . . . . 8 ⊢ ((𝐾 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ_≥‘𝐾) ∨ 𝐾 ∈ (ℤ_≥‘𝑁)))
214	59, 62, 213	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → (𝑁 ∈ (ℤ_≥‘𝐾) ∨ 𝐾 ∈ (ℤ_≥‘𝑁)))
215	189, 212, 214	mpjaod 860	. . . . . 6 ⊢ (𝜑 → (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑁)(𝑊‘𝑘)) ≤ (𝑁 · Σ𝑘 ∈ ((𝐾 + 1)...𝑁)if(𝑘 ∈ ℙ, (1 / 𝑘), 0)))
216		eqid 2734	. . . . . . . . . 10 ⊢ (ℤ_≥‘(𝐾 + 1)) = (ℤ_≥‘(𝐾 + 1))
217		eleq1w 2821	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑘 → (𝑛 ∈ ℙ ↔ 𝑘 ∈ ℙ))
218		oveq2 7438	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑘 → (1 / 𝑛) = (1 / 𝑘))
219	217, 218	ifbieq1d 4554	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → if(𝑛 ∈ ℙ, (1 / 𝑛), 0) = if(𝑘 ∈ ℙ, (1 / 𝑘), 0))
220		ovex 7463	. . . . . . . . . . . . 13 ⊢ (1 / 𝑘) ∈ V
221		c0ex 11252	. . . . . . . . . . . . 13 ⊢ 0 ∈ V
222	220, 221	ifex 4580	. . . . . . . . . . . 12 ⊢ if(𝑘 ∈ ℙ, (1 / 𝑘), 0) ∈ V
223	219, 186, 222	fvmpt 7015	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → (𝐹‘𝑘) = if(𝑘 ∈ ℙ, (1 / 𝑘), 0))
224	177, 223	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘(𝐾 + 1))) → (𝐹‘𝑘) = if(𝑘 ∈ ℙ, (1 / 𝑘), 0))
225	181	recnd 11286	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ → if(𝑘 ∈ ℙ, (1 / 𝑘), 0) ∈ ℂ)
226	223, 225	eqeltrd 2838	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ → (𝐹‘𝑘) ∈ ℂ)
227	226	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℂ)
228	72, 26, 227	iserex 15689	. . . . . . . . . . 11 ⊢ (𝜑 → (seq1( + , 𝐹) ∈ dom ⇝ ↔ seq(𝐾 + 1)( + , 𝐹) ∈ dom ⇝ ))
229	187, 228	mpbid 232	. . . . . . . . . 10 ⊢ (𝜑 → seq(𝐾 + 1)( + , 𝐹) ∈ dom ⇝ )
230	216, 60, 224, 182, 229	isumrecl 15797	. . . . . . . . 9 ⊢ (𝜑 → Σ𝑘 ∈ (ℤ_≥‘(𝐾 + 1))if(𝑘 ∈ ℙ, (1 / 𝑘), 0) ∈ ℝ)
231		halfre 12477	. . . . . . . . . 10 ⊢ (1 / 2) ∈ ℝ
232	231	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → (1 / 2) ∈ ℝ)
233		fzssuz 13601	. . . . . . . . . . 11 ⊢ ((𝐾 + 1)...𝑁) ⊆ (ℤ_≥‘(𝐾 + 1))
234	233	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐾 + 1)...𝑁) ⊆ (ℤ_≥‘(𝐾 + 1)))
235		nnrp 13043	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℝ⁺)
236	235	rpreccld 13084	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ → (1 / 𝑘) ∈ ℝ⁺)
237	236	rpge0d 13078	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → 0 ≤ (1 / 𝑘))
238		breq2 5151	. . . . . . . . . . . . 13 ⊢ ((1 / 𝑘) = if(𝑘 ∈ ℙ, (1 / 𝑘), 0) → (0 ≤ (1 / 𝑘) ↔ 0 ≤ if(𝑘 ∈ ℙ, (1 / 𝑘), 0)))
239		breq2 5151	. . . . . . . . . . . . 13 ⊢ (0 = if(𝑘 ∈ ℙ, (1 / 𝑘), 0) → (0 ≤ 0 ↔ 0 ≤ if(𝑘 ∈ ℙ, (1 / 𝑘), 0)))
240	238, 239	ifboth 4569	. . . . . . . . . . . 12 ⊢ ((0 ≤ (1 / 𝑘) ∧ 0 ≤ 0) → 0 ≤ if(𝑘 ∈ ℙ, (1 / 𝑘), 0))
241	237, 197, 240	sylancl 586	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → 0 ≤ if(𝑘 ∈ ℙ, (1 / 𝑘), 0))
242	177, 241	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘(𝐾 + 1))) → 0 ≤ if(𝑘 ∈ ℙ, (1 / 𝑘), 0))
243	216, 60, 176, 234, 224, 182, 242, 229	isumless 15877	. . . . . . . . 9 ⊢ (𝜑 → Σ𝑘 ∈ ((𝐾 + 1)...𝑁)if(𝑘 ∈ ℙ, (1 / 𝑘), 0) ≤ Σ𝑘 ∈ (ℤ_≥‘(𝐾 + 1))if(𝑘 ∈ ℙ, (1 / 𝑘), 0))
244	184, 230, 232, 243, 188	lelttrd 11416	. . . . . . . 8 ⊢ (𝜑 → Σ𝑘 ∈ ((𝐾 + 1)...𝑁)if(𝑘 ∈ ℙ, (1 / 𝑘), 0) < (1 / 2))
245	1	nngt0d 12312	. . . . . . . . 9 ⊢ (𝜑 → 0 < 𝑁)
246		ltmul2 12115	. . . . . . . . 9 ⊢ ((Σ𝑘 ∈ ((𝐾 + 1)...𝑁)if(𝑘 ∈ ℙ, (1 / 𝑘), 0) ∈ ℝ ∧ (1 / 2) ∈ ℝ ∧ (𝑁 ∈ ℝ ∧ 0 < 𝑁)) → (Σ𝑘 ∈ ((𝐾 + 1)...𝑁)if(𝑘 ∈ ℙ, (1 / 𝑘), 0) < (1 / 2) ↔ (𝑁 · Σ𝑘 ∈ ((𝐾 + 1)...𝑁)if(𝑘 ∈ ℙ, (1 / 𝑘), 0)) < (𝑁 · (1 / 2))))
247	184, 232, 2, 245, 246	syl112anc 1373	. . . . . . . 8 ⊢ (𝜑 → (Σ𝑘 ∈ ((𝐾 + 1)...𝑁)if(𝑘 ∈ ℙ, (1 / 𝑘), 0) < (1 / 2) ↔ (𝑁 · Σ𝑘 ∈ ((𝐾 + 1)...𝑁)if(𝑘 ∈ ℙ, (1 / 𝑘), 0)) < (𝑁 · (1 / 2))))
248	244, 247	mpbid 232	. . . . . . 7 ⊢ (𝜑 → (𝑁 · Σ𝑘 ∈ ((𝐾 + 1)...𝑁)if(𝑘 ∈ ℙ, (1 / 𝑘), 0)) < (𝑁 · (1 / 2)))
249		2cn 12338	. . . . . . . . 9 ⊢ 2 ∈ ℂ
250		2ne0 12367	. . . . . . . . 9 ⊢ 2 ≠ 0
251		divrec 11935	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → (𝑁 / 2) = (𝑁 · (1 / 2)))
252	249, 250, 251	mp3an23 1452	. . . . . . . 8 ⊢ (𝑁 ∈ ℂ → (𝑁 / 2) = (𝑁 · (1 / 2)))
253	23, 252	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑁 / 2) = (𝑁 · (1 / 2)))
254	248, 253	breqtrrd 5175	. . . . . 6 ⊢ (𝜑 → (𝑁 · Σ𝑘 ∈ ((𝐾 + 1)...𝑁)if(𝑘 ∈ ℙ, (1 / 𝑘), 0)) < (𝑁 / 2))
255	175, 185, 3, 215, 254	lelttrd 11416	. . . . 5 ⊢ (𝜑 → (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑁)(𝑊‘𝑘)) < (𝑁 / 2))
256	175, 3, 12, 255	ltadd2dd 11417	. . . 4 ⊢ (𝜑 → ((♯‘𝑀) + (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑁)(𝑊‘𝑘))) < ((♯‘𝑀) + (𝑁 / 2)))
257	172, 256	eqbrtrd 5169	. . 3 ⊢ (𝜑 → ((𝑁 / 2) + (𝑁 / 2)) < ((♯‘𝑀) + (𝑁 / 2)))
258	3, 12, 3	ltadd1d 11853	. . 3 ⊢ (𝜑 → ((𝑁 / 2) < (♯‘𝑀) ↔ ((𝑁 / 2) + (𝑁 / 2)) < ((♯‘𝑀) + (𝑁 / 2))))
259	257, 258	mpbird 257	. 2 ⊢ (𝜑 → (𝑁 / 2) < (♯‘𝑀))
260		oveq1 7437	. . . . . . . 8 ⊢ (𝑘 = 𝑟 → (𝑘↑2) = (𝑟↑2))
261	260	breq1d 5157	. . . . . . 7 ⊢ (𝑘 = 𝑟 → ((𝑘↑2) ∥ 𝑥 ↔ (𝑟↑2) ∥ 𝑥))
262	261	cbvrabv 3443	. . . . . 6 ⊢ {𝑘 ∈ ℕ ∣ (𝑘↑2) ∥ 𝑥} = {𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑥}
263		breq2 5151	. . . . . . 7 ⊢ (𝑥 = 𝑛 → ((𝑟↑2) ∥ 𝑥 ↔ (𝑟↑2) ∥ 𝑛))
264	263	rabbidv 3440	. . . . . 6 ⊢ (𝑥 = 𝑛 → {𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑥} = {𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑛})
265	262, 264	eqtrid 2786	. . . . 5 ⊢ (𝑥 = 𝑛 → {𝑘 ∈ ℕ ∣ (𝑘↑2) ∥ 𝑥} = {𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑛})
266	265	supeq1d 9483	. . . 4 ⊢ (𝑥 = 𝑛 → sup({𝑘 ∈ ℕ ∣ (𝑘↑2) ∥ 𝑥}, ℝ, < ) = sup({𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑛}, ℝ, < ))
267	266	cbvmptv 5260	. . 3 ⊢ (𝑥 ∈ ℕ ↦ sup({𝑘 ∈ ℕ ∣ (𝑘↑2) ∥ 𝑥}, ℝ, < )) = (𝑛 ∈ ℕ ↦ sup({𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑛}, ℝ, < ))
268	186, 14, 1, 5, 267	prmreclem3 16951	. 2 ⊢ (𝜑 → (♯‘𝑀) ≤ ((2↑𝐾) · (√‘𝑁)))
269	3, 12, 22, 259, 268	ltletrd 11418	1 ⊢ (𝜑 → (𝑁 / 2) < ((2↑𝐾) · (√‘𝑁)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 ∀wral 3058 ∃wrex 3067 {crab 3432 ∖ cdif 3959 ∪ cun 3960 ∩ cin 3961 ⊆ wss 3962 ∅c0 4338 ifcif 4530 ∪ ciun 4995 class class class wbr 5147 ↦ cmpt 5230 dom cdm 5688 ‘cfv 6562 (class class class)co 7430 Fincfn 8983 supcsup 9477 ℂcc 11150 ℝcr 11151 0cc0 11152 1c1 11153 + caddc 11155 · cmul 11157 < clt 11292 ≤ cle 11293 / cdiv 11917 ℕcn 12263 2c2 12318 ℕ₀cn0 12523 ℤcz 12610 ℤ_≥cuz 12875 ...cfz 13543 seqcseq 14038 ↑cexp 14098 ♯chash 14365 √csqrt 15268 ⇝ cli 15516 Σcsu 15718 ∥ cdvds 16286 ℙcprime 16704

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-inf2 9678 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-2o 8505 df-oadd 8508 df-er 8743 df-map 8866 df-pm 8867 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-sup 9479 df-inf 9480 df-oi 9547 df-dju 9938 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-n0 12524 df-xnn0 12597 df-z 12611 df-uz 12876 df-q 12988 df-rp 13032 df-fz 13544 df-fzo 13691 df-fl 13828 df-mod 13906 df-seq 14039 df-exp 14099 df-hash 14366 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-clim 15520 df-rlim 15521 df-sum 15719 df-dvds 16287 df-gcd 16528 df-prm 16705 df-pc 16870

This theorem is referenced by: prmreclem6 16954

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