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Mirrors > Home > MPE Home > Th. List > ppieq0 | Structured version Visualization version GIF version |
Description: The prime-counting function π is zero iff its argument is less than 2. (Contributed by Mario Carneiro, 22-Sep-2014.) |
Ref | Expression |
---|---|
ppieq0 | ⊢ (𝐴 ∈ ℝ → ((π‘𝐴) = 0 ↔ 𝐴 < 2)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2re 12337 | . . . . 5 ⊢ 2 ∈ ℝ | |
2 | lenlt 11336 | . . . . 5 ⊢ ((2 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (2 ≤ 𝐴 ↔ ¬ 𝐴 < 2)) | |
3 | 1, 2 | mpan 690 | . . . 4 ⊢ (𝐴 ∈ ℝ → (2 ≤ 𝐴 ↔ ¬ 𝐴 < 2)) |
4 | ppinncl 27231 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 2 ≤ 𝐴) → (π‘𝐴) ∈ ℕ) | |
5 | 4 | nnne0d 12313 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 2 ≤ 𝐴) → (π‘𝐴) ≠ 0) |
6 | 5 | ex 412 | . . . 4 ⊢ (𝐴 ∈ ℝ → (2 ≤ 𝐴 → (π‘𝐴) ≠ 0)) |
7 | 3, 6 | sylbird 260 | . . 3 ⊢ (𝐴 ∈ ℝ → (¬ 𝐴 < 2 → (π‘𝐴) ≠ 0)) |
8 | 7 | necon4bd 2957 | . 2 ⊢ (𝐴 ∈ ℝ → ((π‘𝐴) = 0 → 𝐴 < 2)) |
9 | reflcl 13832 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ∈ ℝ) | |
10 | 9 | adantr 480 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 2) → (⌊‘𝐴) ∈ ℝ) |
11 | 1red 11259 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 2) → 1 ∈ ℝ) | |
12 | 2z 12646 | . . . . . . . . . 10 ⊢ 2 ∈ ℤ | |
13 | fllt 13842 | . . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 2 ∈ ℤ) → (𝐴 < 2 ↔ (⌊‘𝐴) < 2)) | |
14 | 12, 13 | mpan2 691 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → (𝐴 < 2 ↔ (⌊‘𝐴) < 2)) |
15 | 14 | biimpa 476 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 2) → (⌊‘𝐴) < 2) |
16 | df-2 12326 | . . . . . . . 8 ⊢ 2 = (1 + 1) | |
17 | 15, 16 | breqtrdi 5188 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 2) → (⌊‘𝐴) < (1 + 1)) |
18 | flcl 13831 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ∈ ℤ) | |
19 | 18 | adantr 480 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 2) → (⌊‘𝐴) ∈ ℤ) |
20 | 1z 12644 | . . . . . . . 8 ⊢ 1 ∈ ℤ | |
21 | zleltp1 12665 | . . . . . . . 8 ⊢ (((⌊‘𝐴) ∈ ℤ ∧ 1 ∈ ℤ) → ((⌊‘𝐴) ≤ 1 ↔ (⌊‘𝐴) < (1 + 1))) | |
22 | 19, 20, 21 | sylancl 586 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 2) → ((⌊‘𝐴) ≤ 1 ↔ (⌊‘𝐴) < (1 + 1))) |
23 | 17, 22 | mpbird 257 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 2) → (⌊‘𝐴) ≤ 1) |
24 | ppiwordi 27219 | . . . . . 6 ⊢ (((⌊‘𝐴) ∈ ℝ ∧ 1 ∈ ℝ ∧ (⌊‘𝐴) ≤ 1) → (π‘(⌊‘𝐴)) ≤ (π‘1)) | |
25 | 10, 11, 23, 24 | syl3anc 1370 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 2) → (π‘(⌊‘𝐴)) ≤ (π‘1)) |
26 | ppifl 27217 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → (π‘(⌊‘𝐴)) = (π‘𝐴)) | |
27 | 26 | adantr 480 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 2) → (π‘(⌊‘𝐴)) = (π‘𝐴)) |
28 | ppi1 27221 | . . . . . 6 ⊢ (π‘1) = 0 | |
29 | 28 | a1i 11 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 2) → (π‘1) = 0) |
30 | 25, 27, 29 | 3brtr3d 5178 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 2) → (π‘𝐴) ≤ 0) |
31 | ppicl 27188 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → (π‘𝐴) ∈ ℕ0) | |
32 | 31 | adantr 480 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 2) → (π‘𝐴) ∈ ℕ0) |
33 | nn0le0eq0 12551 | . . . . 5 ⊢ ((π‘𝐴) ∈ ℕ0 → ((π‘𝐴) ≤ 0 ↔ (π‘𝐴) = 0)) | |
34 | 32, 33 | syl 17 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 2) → ((π‘𝐴) ≤ 0 ↔ (π‘𝐴) = 0)) |
35 | 30, 34 | mpbid 232 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 2) → (π‘𝐴) = 0) |
36 | 35 | ex 412 | . 2 ⊢ (𝐴 ∈ ℝ → (𝐴 < 2 → (π‘𝐴) = 0)) |
37 | 8, 36 | impbid 212 | 1 ⊢ (𝐴 ∈ ℝ → ((π‘𝐴) = 0 ↔ 𝐴 < 2)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 class class class wbr 5147 ‘cfv 6562 (class class class)co 7430 ℝcr 11151 0cc0 11152 1c1 11153 + caddc 11155 < clt 11292 ≤ cle 11293 2c2 12318 ℕ0cn0 12523 ℤcz 12610 ⌊cfl 13826 πcppi 27151 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-2o 8505 df-oadd 8508 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-sup 9479 df-inf 9480 df-dju 9938 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-n0 12524 df-xnn0 12597 df-z 12611 df-uz 12876 df-rp 13032 df-icc 13390 df-fz 13544 df-fl 13828 df-seq 14039 df-exp 14099 df-hash 14366 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-dvds 16287 df-prm 16705 df-ppi 27157 |
This theorem is referenced by: ppiltx 27234 |
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