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Mirrors > Home > MPE Home > Th. List > ppinncl | Structured version Visualization version GIF version |
Description: Closure of the prime-counting function π in the positive integers. (Contributed by Mario Carneiro, 21-Sep-2014.) |
Ref | Expression |
---|---|
ppinncl | ⊢ ((𝐴 ∈ ℝ ∧ 2 ≤ 𝐴) → (π‘𝐴) ∈ ℕ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ppicl 25425 | . . . 4 ⊢ (𝐴 ∈ ℝ → (π‘𝐴) ∈ ℕ0) | |
2 | 1 | adantr 473 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 2 ≤ 𝐴) → (π‘𝐴) ∈ ℕ0) |
3 | 2 | nn0zd 11904 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 2 ≤ 𝐴) → (π‘𝐴) ∈ ℤ) |
4 | ppi2 25464 | . . 3 ⊢ (π‘2) = 1 | |
5 | 2re 11520 | . . . 4 ⊢ 2 ∈ ℝ | |
6 | ppiwordi 25456 | . . . 4 ⊢ ((2 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ 2 ≤ 𝐴) → (π‘2) ≤ (π‘𝐴)) | |
7 | 5, 6 | mp3an1 1428 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 2 ≤ 𝐴) → (π‘2) ≤ (π‘𝐴)) |
8 | 4, 7 | syl5eqbrr 4970 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 2 ≤ 𝐴) → 1 ≤ (π‘𝐴)) |
9 | elnnz1 11827 | . 2 ⊢ ((π‘𝐴) ∈ ℕ ↔ ((π‘𝐴) ∈ ℤ ∧ 1 ≤ (π‘𝐴))) | |
10 | 3, 8, 9 | sylanbrc 575 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 2 ≤ 𝐴) → (π‘𝐴) ∈ ℕ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 ∈ wcel 2051 class class class wbr 4934 ‘cfv 6193 ℝcr 10340 1c1 10342 ≤ cle 10481 ℕcn 11445 2c2 11501 ℕ0cn0 11713 ℤcz 11799 πcppi 25388 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2752 ax-rep 5053 ax-sep 5064 ax-nul 5071 ax-pow 5123 ax-pr 5190 ax-un 7285 ax-cnex 10397 ax-resscn 10398 ax-1cn 10399 ax-icn 10400 ax-addcl 10401 ax-addrcl 10402 ax-mulcl 10403 ax-mulrcl 10404 ax-mulcom 10405 ax-addass 10406 ax-mulass 10407 ax-distr 10408 ax-i2m1 10409 ax-1ne0 10410 ax-1rid 10411 ax-rnegex 10412 ax-rrecex 10413 ax-cnre 10414 ax-pre-lttri 10415 ax-pre-lttrn 10416 ax-pre-ltadd 10417 ax-pre-mulgt0 10418 ax-pre-sup 10419 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2551 df-eu 2589 df-clab 2761 df-cleq 2773 df-clel 2848 df-nfc 2920 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3419 df-sbc 3684 df-csb 3789 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-pss 3847 df-nul 4182 df-if 4354 df-pw 4427 df-sn 4445 df-pr 4447 df-tp 4449 df-op 4451 df-uni 4718 df-int 4755 df-iun 4799 df-br 4935 df-opab 4997 df-mpt 5014 df-tr 5036 df-id 5316 df-eprel 5321 df-po 5330 df-so 5331 df-fr 5370 df-we 5372 df-xp 5417 df-rel 5418 df-cnv 5419 df-co 5420 df-dm 5421 df-rn 5422 df-res 5423 df-ima 5424 df-pred 5991 df-ord 6037 df-on 6038 df-lim 6039 df-suc 6040 df-iota 6157 df-fun 6195 df-fn 6196 df-f 6197 df-f1 6198 df-fo 6199 df-f1o 6200 df-fv 6201 df-riota 6943 df-ov 6985 df-oprab 6986 df-mpo 6987 df-om 7403 df-1st 7507 df-2nd 7508 df-wrecs 7756 df-recs 7818 df-rdg 7856 df-1o 7911 df-2o 7912 df-oadd 7915 df-er 8095 df-en 8313 df-dom 8314 df-sdom 8315 df-fin 8316 df-sup 8707 df-inf 8708 df-dju 9130 df-card 9168 df-pnf 10482 df-mnf 10483 df-xr 10484 df-ltxr 10485 df-le 10486 df-sub 10678 df-neg 10679 df-div 11105 df-nn 11446 df-2 11509 df-3 11510 df-n0 11714 df-xnn0 11786 df-z 11800 df-uz 12065 df-rp 12211 df-icc 12567 df-fz 12715 df-fl 12983 df-seq 13191 df-exp 13251 df-hash 13512 df-cj 14325 df-re 14326 df-im 14327 df-sqrt 14461 df-abs 14462 df-dvds 15474 df-prm 15878 df-ppi 25394 |
This theorem is referenced by: ppieq0 25470 chebbnd1lem3 25764 chebbnd1 25765 chtppilimlem1 25766 chtppilimlem2 25767 chtppilim 25768 chebbnd2 25770 chto1lb 25771 pnt 25907 |
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