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Mirrors > Home > ILE Home > Th. List > pw2dvds | GIF version |
Description: A natural number has a highest power of two which divides it. (Contributed by Jim Kingdon, 14-Nov-2021.) |
Ref | Expression |
---|---|
pw2dvds | ⊢ (𝑁 ∈ ℕ → ∃𝑚 ∈ ℕ0 ((2↑𝑚) ∥ 𝑁 ∧ ¬ (2↑(𝑚 + 1)) ∥ 𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 19 | . 2 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ) | |
2 | 2nn 8881 | . . . 4 ⊢ 2 ∈ ℕ | |
3 | nnnn0 8984 | . . . 4 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0) | |
4 | nnexpcl 10306 | . . . 4 ⊢ ((2 ∈ ℕ ∧ 𝑁 ∈ ℕ0) → (2↑𝑁) ∈ ℕ) | |
5 | 2, 3, 4 | sylancr 410 | . . 3 ⊢ (𝑁 ∈ ℕ → (2↑𝑁) ∈ ℕ) |
6 | 1zzd 9081 | . . . 4 ⊢ (𝑁 ∈ ℕ → 1 ∈ ℤ) | |
7 | 2z 9082 | . . . . . 6 ⊢ 2 ∈ ℤ | |
8 | zexpcl 10308 | . . . . . 6 ⊢ ((2 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → (2↑𝑁) ∈ ℤ) | |
9 | 7, 3, 8 | sylancr 410 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (2↑𝑁) ∈ ℤ) |
10 | 9, 6 | zsubcld 9178 | . . . 4 ⊢ (𝑁 ∈ ℕ → ((2↑𝑁) − 1) ∈ ℤ) |
11 | nnz 9073 | . . . 4 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ) | |
12 | nnge1 8743 | . . . 4 ⊢ (𝑁 ∈ ℕ → 1 ≤ 𝑁) | |
13 | uzid 9340 | . . . . . . 7 ⊢ (2 ∈ ℤ → 2 ∈ (ℤ≥‘2)) | |
14 | 7, 13 | ax-mp 5 | . . . . . 6 ⊢ 2 ∈ (ℤ≥‘2) |
15 | bernneq3 10414 | . . . . . 6 ⊢ ((2 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) → 𝑁 < (2↑𝑁)) | |
16 | 14, 3, 15 | sylancr 410 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 𝑁 < (2↑𝑁)) |
17 | zltlem1 9111 | . . . . . 6 ⊢ ((𝑁 ∈ ℤ ∧ (2↑𝑁) ∈ ℤ) → (𝑁 < (2↑𝑁) ↔ 𝑁 ≤ ((2↑𝑁) − 1))) | |
18 | 11, 9, 17 | syl2anc 408 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (𝑁 < (2↑𝑁) ↔ 𝑁 ≤ ((2↑𝑁) − 1))) |
19 | 16, 18 | mpbid 146 | . . . 4 ⊢ (𝑁 ∈ ℕ → 𝑁 ≤ ((2↑𝑁) − 1)) |
20 | elfz4 9799 | . . . 4 ⊢ (((1 ∈ ℤ ∧ ((2↑𝑁) − 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (1 ≤ 𝑁 ∧ 𝑁 ≤ ((2↑𝑁) − 1))) → 𝑁 ∈ (1...((2↑𝑁) − 1))) | |
21 | 6, 10, 11, 12, 19, 20 | syl32anc 1224 | . . 3 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ (1...((2↑𝑁) − 1))) |
22 | fzm1ndvds 11554 | . . 3 ⊢ (((2↑𝑁) ∈ ℕ ∧ 𝑁 ∈ (1...((2↑𝑁) − 1))) → ¬ (2↑𝑁) ∥ 𝑁) | |
23 | 5, 21, 22 | syl2anc 408 | . 2 ⊢ (𝑁 ∈ ℕ → ¬ (2↑𝑁) ∥ 𝑁) |
24 | pw2dvdslemn 11843 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ ¬ (2↑𝑁) ∥ 𝑁) → ∃𝑚 ∈ ℕ0 ((2↑𝑚) ∥ 𝑁 ∧ ¬ (2↑(𝑚 + 1)) ∥ 𝑁)) | |
25 | 1, 23, 24 | mpd3an23 1317 | 1 ⊢ (𝑁 ∈ ℕ → ∃𝑚 ∈ ℕ0 ((2↑𝑚) ∥ 𝑁 ∧ ¬ (2↑(𝑚 + 1)) ∥ 𝑁)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 ∈ wcel 1480 ∃wrex 2417 class class class wbr 3929 ‘cfv 5123 (class class class)co 5774 1c1 7621 + caddc 7623 < clt 7800 ≤ cle 7801 − cmin 7933 ℕcn 8720 2c2 8771 ℕ0cn0 8977 ℤcz 9054 ℤ≥cuz 9326 ...cfz 9790 ↑cexp 10292 ∥ cdvds 11493 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 ax-pre-mulext 7738 ax-arch 7739 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-frec 6288 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-reap 8337 df-ap 8344 df-div 8433 df-inn 8721 df-2 8779 df-n0 8978 df-z 9055 df-uz 9327 df-q 9412 df-rp 9442 df-fz 9791 df-fl 10043 df-mod 10096 df-seqfrec 10219 df-exp 10293 df-dvds 11494 |
This theorem is referenced by: pw2dvdseu 11846 oddpwdclemdvds 11848 oddpwdclemndvds 11849 |
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