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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 11712 | . . . . 5 ⊢ 2 ∈ ℝ | |
2 | lenlt 10719 | . . . . 5 ⊢ ((2 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (2 ≤ 𝐴 ↔ ¬ 𝐴 < 2)) | |
3 | 1, 2 | mpan 688 | . . . 4 ⊢ (𝐴 ∈ ℝ → (2 ≤ 𝐴 ↔ ¬ 𝐴 < 2)) |
4 | ppinncl 25751 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 2 ≤ 𝐴) → (π‘𝐴) ∈ ℕ) | |
5 | 4 | nnne0d 11688 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 2 ≤ 𝐴) → (π‘𝐴) ≠ 0) |
6 | 5 | ex 415 | . . . 4 ⊢ (𝐴 ∈ ℝ → (2 ≤ 𝐴 → (π‘𝐴) ≠ 0)) |
7 | 3, 6 | sylbird 262 | . . 3 ⊢ (𝐴 ∈ ℝ → (¬ 𝐴 < 2 → (π‘𝐴) ≠ 0)) |
8 | 7 | necon4bd 3036 | . 2 ⊢ (𝐴 ∈ ℝ → ((π‘𝐴) = 0 → 𝐴 < 2)) |
9 | reflcl 13167 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ∈ ℝ) | |
10 | 9 | adantr 483 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 2) → (⌊‘𝐴) ∈ ℝ) |
11 | 1red 10642 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 2) → 1 ∈ ℝ) | |
12 | 2z 12015 | . . . . . . . . . 10 ⊢ 2 ∈ ℤ | |
13 | fllt 13177 | . . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 2 ∈ ℤ) → (𝐴 < 2 ↔ (⌊‘𝐴) < 2)) | |
14 | 12, 13 | mpan2 689 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → (𝐴 < 2 ↔ (⌊‘𝐴) < 2)) |
15 | 14 | biimpa 479 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 2) → (⌊‘𝐴) < 2) |
16 | df-2 11701 | . . . . . . . 8 ⊢ 2 = (1 + 1) | |
17 | 15, 16 | breqtrdi 5107 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 2) → (⌊‘𝐴) < (1 + 1)) |
18 | flcl 13166 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ∈ ℤ) | |
19 | 18 | adantr 483 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 2) → (⌊‘𝐴) ∈ ℤ) |
20 | 1z 12013 | . . . . . . . 8 ⊢ 1 ∈ ℤ | |
21 | zleltp1 12034 | . . . . . . . 8 ⊢ (((⌊‘𝐴) ∈ ℤ ∧ 1 ∈ ℤ) → ((⌊‘𝐴) ≤ 1 ↔ (⌊‘𝐴) < (1 + 1))) | |
22 | 19, 20, 21 | sylancl 588 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 2) → ((⌊‘𝐴) ≤ 1 ↔ (⌊‘𝐴) < (1 + 1))) |
23 | 17, 22 | mpbird 259 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 2) → (⌊‘𝐴) ≤ 1) |
24 | ppiwordi 25739 | . . . . . 6 ⊢ (((⌊‘𝐴) ∈ ℝ ∧ 1 ∈ ℝ ∧ (⌊‘𝐴) ≤ 1) → (π‘(⌊‘𝐴)) ≤ (π‘1)) | |
25 | 10, 11, 23, 24 | syl3anc 1367 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 2) → (π‘(⌊‘𝐴)) ≤ (π‘1)) |
26 | ppifl 25737 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → (π‘(⌊‘𝐴)) = (π‘𝐴)) | |
27 | 26 | adantr 483 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 2) → (π‘(⌊‘𝐴)) = (π‘𝐴)) |
28 | ppi1 25741 | . . . . . 6 ⊢ (π‘1) = 0 | |
29 | 28 | a1i 11 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 2) → (π‘1) = 0) |
30 | 25, 27, 29 | 3brtr3d 5097 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 2) → (π‘𝐴) ≤ 0) |
31 | ppicl 25708 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → (π‘𝐴) ∈ ℕ0) | |
32 | 31 | adantr 483 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 2) → (π‘𝐴) ∈ ℕ0) |
33 | nn0le0eq0 11926 | . . . . 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 234 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 2) → (π‘𝐴) = 0) |
36 | 35 | ex 415 | . 2 ⊢ (𝐴 ∈ ℝ → (𝐴 < 2 → (π‘𝐴) = 0)) |
37 | 8, 36 | impbid 214 | 1 ⊢ (𝐴 ∈ ℝ → ((π‘𝐴) = 0 ↔ 𝐴 < 2)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 class class class wbr 5066 ‘cfv 6355 (class class class)co 7156 ℝcr 10536 0cc0 10537 1c1 10538 + caddc 10540 < clt 10675 ≤ cle 10676 2c2 11693 ℕ0cn0 11898 ℤcz 11982 ⌊cfl 13161 πcppi 25671 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-sup 8906 df-inf 8907 df-dju 9330 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-n0 11899 df-xnn0 11969 df-z 11983 df-uz 12245 df-rp 12391 df-icc 12746 df-fz 12894 df-fl 13163 df-seq 13371 df-exp 13431 df-hash 13692 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-dvds 15608 df-prm 16016 df-ppi 25677 |
This theorem is referenced by: ppiltx 25754 |
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