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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 26480 | . . . 4 ⊢ (𝐴 ∈ ℝ → (π‘𝐴) ∈ ℕ0) | |
2 | 1 | adantr 481 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 2 ≤ 𝐴) → (π‘𝐴) ∈ ℕ0) |
3 | 2 | nn0zd 12525 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 2 ≤ 𝐴) → (π‘𝐴) ∈ ℤ) |
4 | ppi2 26519 | . . 3 ⊢ (π‘2) = 1 | |
5 | 2re 12227 | . . . 4 ⊢ 2 ∈ ℝ | |
6 | ppiwordi 26511 | . . . 4 ⊢ ((2 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ 2 ≤ 𝐴) → (π‘2) ≤ (π‘𝐴)) | |
7 | 5, 6 | mp3an1 1448 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 2 ≤ 𝐴) → (π‘2) ≤ (π‘𝐴)) |
8 | 4, 7 | eqbrtrrid 5141 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 2 ≤ 𝐴) → 1 ≤ (π‘𝐴)) |
9 | elnnz1 12529 | . 2 ⊢ ((π‘𝐴) ∈ ℕ ↔ ((π‘𝐴) ∈ ℤ ∧ 1 ≤ (π‘𝐴))) | |
10 | 3, 8, 9 | sylanbrc 583 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 2 ≤ 𝐴) → (π‘𝐴) ∈ ℕ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2106 class class class wbr 5105 ‘cfv 6496 ℝcr 11050 1c1 11052 ≤ cle 11190 ℕcn 12153 2c2 12208 ℕ0cn0 12413 ℤcz 12499 πcppi 26443 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-2o 8413 df-oadd 8416 df-er 8648 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-sup 9378 df-inf 9379 df-dju 9837 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-n0 12414 df-xnn0 12486 df-z 12500 df-uz 12764 df-rp 12916 df-icc 13271 df-fz 13425 df-fl 13697 df-seq 13907 df-exp 13968 df-hash 14231 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-dvds 16137 df-prm 16548 df-ppi 26449 |
This theorem is referenced by: ppieq0 26525 chebbnd1lem3 26819 chebbnd1 26820 chtppilimlem1 26821 chtppilimlem2 26822 chtppilim 26823 chebbnd2 26825 chto1lb 26826 pnt 26962 |
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