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Theorem bpos 16249

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 16239	. 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 16248	. . . 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 9668	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
10	9	peano2zd 9776	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℤ)
11		2z 9677	. . . . . . . . . 10 ⊢ 2 ∈ ℤ
12	11	a1i 9	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 2 ∈ ℤ)
13	12, 9	zmulcld 9779	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (2 · 𝑁) ∈ ℤ)
14		elfzelz 10439	. . . . . . . . . 10 ⊢ (𝑝 ∈ ((𝑁 + 1)...(2 · 𝑁)) → 𝑝 ∈ ℤ)
15		prmdcz 12928	. . . . . . . . . 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 10670	. . . . . . 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 10429	. . . . . . . . . 10 ⊢ (𝑝 ∈ ((𝑁 + 1)...(2 · 𝑁)) ↔ (((𝑁 + 1) ∈ ℤ ∧ (2 · 𝑁) ∈ ℤ ∧ 𝑝 ∈ ℤ) ∧ ((𝑁 + 1) ≤ 𝑝 ∧ 𝑝 ≤ (2 · 𝑁))))
24		prmz 12908	. . . . . . . . . . 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 12907	. . . . . . . . . . 11 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℕ)
29		nnltp1le 9710	. . . . . . . . . . 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 9589	. . . . 5 ⊢ 6 ∈ ℕ₀
41		4nn0 9587	. . . . 5 ⊢ 4 ∈ ℕ₀
42	40, 41	deccl 9796	. . . 4 ⊢ ;64 ∈ ℕ₀
43	42	nn0zi 9671	. . 3 ⊢ ;64 ∈ ℤ
44		zlelttric 9694	. . 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 4130 ↦ cmpt 4192 ‘cfv 5377 (class class class)co 6085 1c1 8181 + caddc 8183 · cmul 8185 < clt 8361 ≤ cle 8362 / cdiv 9005 ℕcn 9307 2c2 9358 4c4 9360 6c6 9362 9c9 9365 ℤcz 9649 cdc 9782 ℝ⁺crp 10065 ...cfz 10422 √csqrt 11778 ℙcprime 12904 logclog 16017

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 4246 ax-sep 4249 ax-nul 4259 ax-pow 4311 ax-pr 4346 ax-un 4578 ax-setind 4684 ax-iinf 4735 ax-cnex 8271 ax-resscn 8272 ax-1cn 8273 ax-1re 8274 ax-icn 8275 ax-addcl 8276 ax-addrcl 8277 ax-mulcl 8278 ax-mulrcl 8279 ax-addcom 8280 ax-mulcom 8281 ax-addass 8282 ax-mulass 8283 ax-distr 8284 ax-i2m1 8285 ax-0lt1 8286 ax-1rid 8287 ax-0id 8288 ax-rnegex 8289 ax-precex 8290 ax-cnre 8291 ax-pre-ltirr 8292 ax-pre-ltwlin 8293 ax-pre-lttrn 8294 ax-pre-apti 8295 ax-pre-ltadd 8296 ax-pre-mulgt0 8297 ax-pre-mulext 8298 ax-arch 8299 ax-caucvg 8300 ax-pre-suploc 8301 ax-addf 8302 ax-mulf 8303

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 4131 df-opab 4193 df-mpt 4194 df-tr 4230 df-id 4438 df-po 4441 df-iso 4442 df-iord 4511 df-on 4513 df-ilim 4514 df-suc 4516 df-iom 4738 df-xp 4780 df-rel 4781 df-cnv 4782 df-co 4783 df-dm 4784 df-rn 4785 df-res 4786 df-ima 4787 df-iota 5337 df-fun 5379 df-fn 5380 df-f 5381 df-f1 5382 df-fo 5383 df-f1o 5384 df-fv 5385 df-isom 5386 df-riota 6038 df-ov 6088 df-oprab 6089 df-mpo 6090 df-of 6302 df-1st 6374 df-2nd 6375 df-recs 6576 df-irdg 6641 df-frec 6662 df-1o 6687 df-2o 6688 df-oadd 6691 df-er 6807 df-map 6924 df-pm 6925 df-en 7023 df-dom 7024 df-fin 7025 df-sup 7325 df-inf 7326 df-pnf 8363 df-mnf 8364 df-xr 8365 df-ltxr 8366 df-le 8367 df-sub 8501 df-neg 8502 df-reap 8906 df-ap 8913 df-div 9006 df-inn 9308 df-2 9366 df-3 9367 df-4 9368 df-5 9369 df-6 9370 df-7 9371 df-8 9372 df-9 9373 df-n0 9569 df-xnn0 9636 df-z 9650 df-dec 9783 df-uz 9932 df-q 10030 df-rp 10066 df-xneg 10185 df-xadd 10186 df-ioo 10305 df-ico 10307 df-icc 10308 df-fz 10423 df-fzo 10561 df-fl 10716 df-mod 10775 df-seqfrec 10900 df-exp 10991 df-fac 11180 df-bc 11202 df-ihash 11231 df-shft 11596 df-cj 11623 df-re 11624 df-im 11625 df-rsqrt 11780 df-abs 11781 df-clim 12064 df-sumdc 12139 df-ef 12434 df-e 12435 df-dvds 12574 df-gcd 12750 df-prm 12905 df-numer 12982 df-denom 12983 df-pc 13087 df-rest 13647 df-topgen 13666 df-psmet 14932 df-xmet 14933 df-met 14934 df-bl 14935 df-mopn 14936 df-top 15158 df-topon 15171 df-bases 15203 df-ntr 15256 df-cn 15348 df-cnp 15349 df-tx 15413 df-cncf 15731 df-limced 15816 df-dvap 15817 df-relog 16019 df-rpcxp 16020 df-logb 16109 df-cht 16165 df-ppi 16166

This theorem is used by: (None)

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