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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 27083 | . . . 4 ⊢ (𝐴 ∈ ℝ → (π‘𝐴) ∈ ℕ0) | |
2 | 1 | adantr 479 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 2 ≤ 𝐴) → (π‘𝐴) ∈ ℕ0) |
3 | 2 | nn0zd 12622 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 2 ≤ 𝐴) → (π‘𝐴) ∈ ℤ) |
4 | ppi2 27122 | . . 3 ⊢ (π‘2) = 1 | |
5 | 2re 12324 | . . . 4 ⊢ 2 ∈ ℝ | |
6 | ppiwordi 27114 | . . . 4 ⊢ ((2 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ 2 ≤ 𝐴) → (π‘2) ≤ (π‘𝐴)) | |
7 | 5, 6 | mp3an1 1444 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 2 ≤ 𝐴) → (π‘2) ≤ (π‘𝐴)) |
8 | 4, 7 | eqbrtrrid 5188 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 2 ≤ 𝐴) → 1 ≤ (π‘𝐴)) |
9 | elnnz1 12626 | . 2 ⊢ ((π‘𝐴) ∈ ℕ ↔ ((π‘𝐴) ∈ ℤ ∧ 1 ≤ (π‘𝐴))) | |
10 | 3, 8, 9 | sylanbrc 581 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 2 ≤ 𝐴) → (π‘𝐴) ∈ ℕ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∈ wcel 2098 class class class wbr 5152 ‘cfv 6553 ℝcr 11145 1c1 11147 ≤ cle 11287 ℕcn 12250 2c2 12305 ℕ0cn0 12510 ℤcz 12596 πcppi 27046 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-2o 8494 df-oadd 8497 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-sup 9473 df-inf 9474 df-dju 9932 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-n0 12511 df-xnn0 12583 df-z 12597 df-uz 12861 df-rp 13015 df-icc 13371 df-fz 13525 df-fl 13797 df-seq 14007 df-exp 14067 df-hash 14330 df-cj 15086 df-re 15087 df-im 15088 df-sqrt 15222 df-abs 15223 df-dvds 16239 df-prm 16650 df-ppi 27052 |
This theorem is referenced by: ppieq0 27128 chebbnd1lem3 27424 chebbnd1 27425 chtppilimlem1 27426 chtppilimlem2 27427 chtppilim 27428 chebbnd2 27430 chto1lb 27431 pnt 27567 |
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