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Theorem bpos 16320

Description: Bertrand's postulate: there is a prime between 𝑁 and 2𝑁 for every positive integer 𝑁. This proof follows Erdős's method, for the most part, but with some refinements due to Shigenori Tochiori to save us some calculations of large primes. See http://en.wikipedia.org/wiki/Proof_of_Bertrand%27s_postulate for an overview of the proof strategy. This is Metamath 100 proof #98. (Contributed by Mario Carneiro, 14-Mar-2014.)

**Assertion**
Ref	Expression
bpos	⊢ (𝑁 ∈ ℕ → ∃𝑝 ∈ ℙ (𝑁 < 𝑝 ∧ 𝑝 ≤ (2 · 𝑁)))

Distinct variable group: 𝑁,𝑝

Proof of Theorem bpos Dummy variables 𝑛 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		bpos1 16310	. 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝑁 ≤ ;64) → ∃𝑝 ∈ ℙ (𝑁 < 𝑝 ∧ 𝑝 ≤ (2 · 𝑁)))
2		eqid 2238	. . . . 5 ⊢ (𝑛 ∈ ℕ ↦ ((((√‘2) · ((𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) / 𝑥))‘(√‘𝑛))) + ((9 / 4) · ((𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) / 𝑥))‘(𝑛 / 2)))) + ((log‘2) / (√‘(2 · 𝑛))))) = (𝑛 ∈ ℕ ↦ ((((√‘2) · ((𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) / 𝑥))‘(√‘𝑛))) + ((9 / 4) · ((𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) / 𝑥))‘(𝑛 / 2)))) + ((log‘2) / (√‘(2 · 𝑛)))))
3		eqid 2238	. . . . 5 ⊢ (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) / 𝑥)) = (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) / 𝑥))
4		simpll 531	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ ;64 < 𝑁) ∧ ¬ ∃𝑝 ∈ ℙ (𝑁 < 𝑝 ∧ 𝑝 ≤ (2 · 𝑁))) → 𝑁 ∈ ℕ)
5		simplr 533	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ ;64 < 𝑁) ∧ ¬ ∃𝑝 ∈ ℙ (𝑁 < 𝑝 ∧ 𝑝 ≤ (2 · 𝑁))) → ;64 < 𝑁)
6		simpr 110	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ ;64 < 𝑁) ∧ ¬ ∃𝑝 ∈ ℙ (𝑁 < 𝑝 ∧ 𝑝 ≤ (2 · 𝑁))) → ¬ ∃𝑝 ∈ ℙ (𝑁 < 𝑝 ∧ 𝑝 ≤ (2 · 𝑁)))
7	2, 3, 4, 5, 6	bposlem9 16319	. . . 4 ⊢ (((𝑁 ∈ ℕ ∧ ;64 < 𝑁) ∧ ¬ ∃𝑝 ∈ ℙ (𝑁 < 𝑝 ∧ 𝑝 ≤ (2 · 𝑁))) → ∃𝑝 ∈ ℙ (𝑁 < 𝑝 ∧ 𝑝 ≤ (2 · 𝑁)))
8	7, 6	pm2.65da 671	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ ;64 < 𝑁) → ¬ ¬ ∃𝑝 ∈ ℙ (𝑁 < 𝑝 ∧ 𝑝 ≤ (2 · 𝑁)))
9		nnz 9674	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
10	9	peano2zd 9782	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℤ)
11		2z 9683	. . . . . . . . . 10 ⊢ 2 ∈ ℤ
12	11	a1i 9	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 2 ∈ ℤ)
13	12, 9	zmulcld 9785	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (2 · 𝑁) ∈ ℤ)
14		elfzelz 10445	. . . . . . . . . 10 ⊢ (𝑝 ∈ ((𝑁 + 1)...(2 · 𝑁)) → 𝑝 ∈ ℤ)
15		prmdcz 12934	. . . . . . . . . 10 ⊢ (𝑝 ∈ ℤ → DECID 𝑝 ∈ ℙ)
16	14, 15	syl 14	. . . . . . . . 9 ⊢ (𝑝 ∈ ((𝑁 + 1)...(2 · 𝑁)) → DECID 𝑝 ∈ ℙ)
17	16	adantl 277	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑝 ∈ ((𝑁 + 1)...(2 · 𝑁))) → DECID 𝑝 ∈ ℙ)
18	10, 13, 17	exfzdc 10676	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → DECID ∃𝑝 ∈ ((𝑁 + 1)...(2 · 𝑁))𝑝 ∈ ℙ)
19		ancom 266	. . . . . . . . 9 ⊢ ((𝑝 ∈ ℙ ∧ 𝑝 ∈ ((𝑁 + 1)...(2 · 𝑁))) ↔ (𝑝 ∈ ((𝑁 + 1)...(2 · 𝑁)) ∧ 𝑝 ∈ ℙ))
20	19	rexbii2 2561	. . . . . . . 8 ⊢ (∃𝑝 ∈ ℙ 𝑝 ∈ ((𝑁 + 1)...(2 · 𝑁)) ↔ ∃𝑝 ∈ ((𝑁 + 1)...(2 · 𝑁))𝑝 ∈ ℙ)
21	20	dcbii 852	. . . . . . 7 ⊢ (DECID ∃𝑝 ∈ ℙ 𝑝 ∈ ((𝑁 + 1)...(2 · 𝑁)) ↔ DECID ∃𝑝 ∈ ((𝑁 + 1)...(2 · 𝑁))𝑝 ∈ ℙ)
22	18, 21	sylibr 134	. . . . . 6 ⊢ (𝑁 ∈ ℕ → DECID ∃𝑝 ∈ ℙ 𝑝 ∈ ((𝑁 + 1)...(2 · 𝑁)))
23		elfz2 10435	. . . . . . . . . 10 ⊢ (𝑝 ∈ ((𝑁 + 1)...(2 · 𝑁)) ↔ (((𝑁 + 1) ∈ ℤ ∧ (2 · 𝑁) ∈ ℤ ∧ 𝑝 ∈ ℤ) ∧ ((𝑁 + 1) ≤ 𝑝 ∧ 𝑝 ≤ (2 · 𝑁))))
24		prmz 12914	. . . . . . . . . . 11 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℤ)
25		ibar 301	. . . . . . . . . . 11 ⊢ (((𝑁 + 1) ∈ ℤ ∧ (2 · 𝑁) ∈ ℤ ∧ 𝑝 ∈ ℤ) → (((𝑁 + 1) ≤ 𝑝 ∧ 𝑝 ≤ (2 · 𝑁)) ↔ (((𝑁 + 1) ∈ ℤ ∧ (2 · 𝑁) ∈ ℤ ∧ 𝑝 ∈ ℤ) ∧ ((𝑁 + 1) ≤ 𝑝 ∧ 𝑝 ≤ (2 · 𝑁)))))
26	10, 13, 24, 25	syl2an3an 1339	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑝 ∈ ℙ) → (((𝑁 + 1) ≤ 𝑝 ∧ 𝑝 ≤ (2 · 𝑁)) ↔ (((𝑁 + 1) ∈ ℤ ∧ (2 · 𝑁) ∈ ℤ ∧ 𝑝 ∈ ℤ) ∧ ((𝑁 + 1) ≤ 𝑝 ∧ 𝑝 ≤ (2 · 𝑁)))))
27	23, 26	bitr4id 199	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑝 ∈ ℙ) → (𝑝 ∈ ((𝑁 + 1)...(2 · 𝑁)) ↔ ((𝑁 + 1) ≤ 𝑝 ∧ 𝑝 ≤ (2 · 𝑁))))
28		prmnn 12913	. . . . . . . . . . 11 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℕ)
29		nnltp1le 9716	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑝 ∈ ℕ) → (𝑁 < 𝑝 ↔ (𝑁 + 1) ≤ 𝑝))
30	28, 29	sylan2 286	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑝 ∈ ℙ) → (𝑁 < 𝑝 ↔ (𝑁 + 1) ≤ 𝑝))
31	30	anbi1d 469	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑝 ∈ ℙ) → ((𝑁 < 𝑝 ∧ 𝑝 ≤ (2 · 𝑁)) ↔ ((𝑁 + 1) ≤ 𝑝 ∧ 𝑝 ≤ (2 · 𝑁))))
32	27, 31	bitr4d 191	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑝 ∈ ℙ) → (𝑝 ∈ ((𝑁 + 1)...(2 · 𝑁)) ↔ (𝑁 < 𝑝 ∧ 𝑝 ≤ (2 · 𝑁))))
33	32	rexbidva 2547	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → (∃𝑝 ∈ ℙ 𝑝 ∈ ((𝑁 + 1)...(2 · 𝑁)) ↔ ∃𝑝 ∈ ℙ (𝑁 < 𝑝 ∧ 𝑝 ≤ (2 · 𝑁))))
34	33	dcbid 850	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (DECID ∃𝑝 ∈ ℙ 𝑝 ∈ ((𝑁 + 1)...(2 · 𝑁)) ↔ DECID ∃𝑝 ∈ ℙ (𝑁 < 𝑝 ∧ 𝑝 ≤ (2 · 𝑁))))
35	22, 34	mpbid 147	. . . . 5 ⊢ (𝑁 ∈ ℕ → DECID ∃𝑝 ∈ ℙ (𝑁 < 𝑝 ∧ 𝑝 ≤ (2 · 𝑁)))
36		notnotrdc 855	. . . . 5 ⊢ (DECID ∃𝑝 ∈ ℙ (𝑁 < 𝑝 ∧ 𝑝 ≤ (2 · 𝑁)) → (¬ ¬ ∃𝑝 ∈ ℙ (𝑁 < 𝑝 ∧ 𝑝 ≤ (2 · 𝑁)) → ∃𝑝 ∈ ℙ (𝑁 < 𝑝 ∧ 𝑝 ≤ (2 · 𝑁))))
37	35, 36	syl 14	. . . 4 ⊢ (𝑁 ∈ ℕ → (¬ ¬ ∃𝑝 ∈ ℙ (𝑁 < 𝑝 ∧ 𝑝 ≤ (2 · 𝑁)) → ∃𝑝 ∈ ℙ (𝑁 < 𝑝 ∧ 𝑝 ≤ (2 · 𝑁))))
38	37	adantr 276	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ ;64 < 𝑁) → (¬ ¬ ∃𝑝 ∈ ℙ (𝑁 < 𝑝 ∧ 𝑝 ≤ (2 · 𝑁)) → ∃𝑝 ∈ ℙ (𝑁 < 𝑝 ∧ 𝑝 ≤ (2 · 𝑁))))
39	8, 38	mpd 13	. 2 ⊢ ((𝑁 ∈ ℕ ∧ ;64 < 𝑁) → ∃𝑝 ∈ ℙ (𝑁 < 𝑝 ∧ 𝑝 ≤ (2 · 𝑁)))
40		6nn0 9595	. . . . 5 ⊢ 6 ∈ ℕ₀
41		4nn0 9593	. . . . 5 ⊢ 4 ∈ ℕ₀
42	40, 41	deccl 9802	. . . 4 ⊢ ;64 ∈ ℕ₀
43	42	nn0zi 9677	. . 3 ⊢ ;64 ∈ ℤ
44		zlelttric 9700	. . 3 ⊢ ((𝑁 ∈ ℤ ∧ ;64 ∈ ℤ) → (𝑁 ≤ ;64 ∨ ;64 < 𝑁))
45	9, 43, 44	sylancl 417	. 2 ⊢ (𝑁 ∈ ℕ → (𝑁 ≤ ;64 ∨ ;64 < 𝑁))
46	1, 39, 45	mpjaodan 810	1 ⊢ (𝑁 ∈ ℕ → ∃𝑝 ∈ ℙ (𝑁 < 𝑝 ∧ 𝑝 ≤ (2 · 𝑁)))

Colors of variables: wff set class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 720 DECID wdc 846 ∧ w3a 1009 ∈ wcel 2209 ∃wrex 2529 class class class wbr 4131 ↦ cmpt 4193 ‘cfv 5380 (class class class)co 6088 1c1 8186 + caddc 8188 · cmul 8190 < clt 8366 ≤ cle 8367 / cdiv 9010 ℕcn 9313 2c2 9364 4c4 9366 6c6 9368 9c9 9371 ℤcz 9655 cdc 9788 ℝ⁺crp 10071 ...cfz 10428 √csqrt 11784 ℙcprime 12910 logclog 16085

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4248 ax-sep 4251 ax-nul 4261 ax-pow 4313 ax-pr 4348 ax-un 4580 ax-setind 4686 ax-iinf 4737 ax-cnex 8276 ax-resscn 8277 ax-1cn 8278 ax-1re 8279 ax-icn 8280 ax-addcl 8281 ax-addrcl 8282 ax-mulcl 8283 ax-mulrcl 8284 ax-addcom 8285 ax-mulcom 8286 ax-addass 8287 ax-mulass 8288 ax-distr 8289 ax-i2m1 8290 ax-0lt1 8291 ax-1rid 8292 ax-0id 8293 ax-rnegex 8294 ax-precex 8295 ax-cnre 8296 ax-pre-ltirr 8297 ax-pre-ltwlin 8298 ax-pre-lttrn 8299 ax-pre-apti 8300 ax-pre-ltadd 8301 ax-pre-mulgt0 8302 ax-pre-mulext 8303 ax-arch 8304 ax-caucvg 8305 ax-pre-suploc 8306 ax-addf 8307 ax-mulf 8308

This proof depends on definitions: df-bi 117 df-stab 843 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-xor 1425 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3639 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-disj 4107 df-br 4132 df-opab 4194 df-mpt 4195 df-tr 4232 df-id 4440 df-po 4443 df-iso 4444 df-iord 4513 df-on 4515 df-ilim 4516 df-suc 4518 df-iom 4740 df-xp 4782 df-rel 4783 df-cnv 4784 df-co 4785 df-dm 4786 df-rn 4787 df-res 4788 df-ima 4789 df-iota 5340 df-fun 5382 df-fn 5383 df-f 5384 df-f1 5385 df-fo 5386 df-f1o 5387 df-fv 5388 df-isom 5389 df-riota 6041 df-ov 6091 df-oprab 6092 df-mpo 6093 df-of 6305 df-1st 6378 df-2nd 6379 df-recs 6580 df-irdg 6645 df-frec 6666 df-1o 6691 df-2o 6692 df-oadd 6695 df-er 6811 df-map 6928 df-pm 6929 df-en 7027 df-dom 7028 df-fin 7029 df-sup 7330 df-inf 7331 df-pnf 8368 df-mnf 8369 df-xr 8370 df-ltxr 8371 df-le 8372 df-sub 8506 df-neg 8507 df-reap 8911 df-ap 8918 df-div 9011 df-inn 9314 df-2 9372 df-3 9373 df-4 9374 df-5 9375 df-6 9376 df-7 9377 df-8 9378 df-9 9379 df-n0 9575 df-xnn0 9642 df-z 9656 df-dec 9789 df-uz 9938 df-q 10036 df-rp 10072 df-xneg 10191 df-xadd 10192 df-ioo 10311 df-ico 10313 df-icc 10314 df-fz 10429 df-fzo 10567 df-fl 10722 df-mod 10781 df-seqfrec 10906 df-exp 10997 df-fac 11186 df-bc 11208 df-ihash 11237 df-shft 11602 df-cj 11629 df-re 11630 df-im 11631 df-rsqrt 11786 df-abs 11787 df-clim 12070 df-sumdc 12145 df-ef 12440 df-e 12441 df-dvds 12580 df-gcd 12756 df-prm 12911 df-numer 12988 df-denom 12989 df-pc 13093 df-rest 13654 df-topgen 13673 df-psmet 14989 df-xmet 14990 df-met 14991 df-bl 14992 df-mopn 14993 df-top 15224 df-topon 15237 df-bases 15269 df-ntr 15322 df-cn 15414 df-cnp 15415 df-tx 15479 df-cncf 15797 df-limced 15882 df-dvap 15883 df-relog 16087 df-rpcxp 16088 df-logb 16180 df-cht 16236 df-ppi 16237

This theorem is used by: (None)

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