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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 8193 | . . . 4 ⊢ 2 ∈ ℕ | |
3 | nnnn0 8295 | . . . 4 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0) | |
4 | nnexpcl 9489 | . . . 4 ⊢ ((2 ∈ ℕ ∧ 𝑁 ∈ ℕ0) → (2↑𝑁) ∈ ℕ) | |
5 | 2, 3, 4 | sylancr 405 | . . 3 ⊢ (𝑁 ∈ ℕ → (2↑𝑁) ∈ ℕ) |
6 | 1zzd 8378 | . . . 4 ⊢ (𝑁 ∈ ℕ → 1 ∈ ℤ) | |
7 | 2z 8379 | . . . . . 6 ⊢ 2 ∈ ℤ | |
8 | zexpcl 9491 | . . . . . 6 ⊢ ((2 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → (2↑𝑁) ∈ ℤ) | |
9 | 7, 3, 8 | sylancr 405 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (2↑𝑁) ∈ ℤ) |
10 | 9, 6 | zsubcld 8474 | . . . 4 ⊢ (𝑁 ∈ ℕ → ((2↑𝑁) − 1) ∈ ℤ) |
11 | nnz 8370 | . . . 4 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ) | |
12 | nnge1 8062 | . . . 4 ⊢ (𝑁 ∈ ℕ → 1 ≤ 𝑁) | |
13 | uzid 8633 | . . . . . . 7 ⊢ (2 ∈ ℤ → 2 ∈ (ℤ≥‘2)) | |
14 | 7, 13 | ax-mp 7 | . . . . . 6 ⊢ 2 ∈ (ℤ≥‘2) |
15 | bernneq3 9595 | . . . . . 6 ⊢ ((2 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) → 𝑁 < (2↑𝑁)) | |
16 | 14, 3, 15 | sylancr 405 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 𝑁 < (2↑𝑁)) |
17 | zltlem1 8408 | . . . . . 6 ⊢ ((𝑁 ∈ ℤ ∧ (2↑𝑁) ∈ ℤ) → (𝑁 < (2↑𝑁) ↔ 𝑁 ≤ ((2↑𝑁) − 1))) | |
18 | 11, 9, 17 | syl2anc 403 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (𝑁 < (2↑𝑁) ↔ 𝑁 ≤ ((2↑𝑁) − 1))) |
19 | 16, 18 | mpbid 145 | . . . 4 ⊢ (𝑁 ∈ ℕ → 𝑁 ≤ ((2↑𝑁) − 1)) |
20 | elfz4 9038 | . . . 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 1177 | . . 3 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ (1...((2↑𝑁) − 1))) |
22 | fzm1ndvds 10256 | . . 3 ⊢ (((2↑𝑁) ∈ ℕ ∧ 𝑁 ∈ (1...((2↑𝑁) − 1))) → ¬ (2↑𝑁) ∥ 𝑁) | |
23 | 5, 21, 22 | syl2anc 403 | . 2 ⊢ (𝑁 ∈ ℕ → ¬ (2↑𝑁) ∥ 𝑁) |
24 | pw2dvdslemn 10543 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ ¬ (2↑𝑁) ∥ 𝑁) → ∃𝑚 ∈ ℕ0 ((2↑𝑚) ∥ 𝑁 ∧ ¬ (2↑(𝑚 + 1)) ∥ 𝑁)) | |
25 | 1, 23, 24 | mpd3an23 1270 | 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 102 ↔ wb 103 ∈ wcel 1433 ∃wrex 2349 class class class wbr 3785 ‘cfv 4922 (class class class)co 5532 1c1 6982 + caddc 6984 < clt 7153 ≤ cle 7154 − cmin 7279 ℕcn 8039 2c2 8089 ℕ0cn0 8288 ℤcz 8351 ℤ≥cuz 8619 ...cfz 9029 ↑cexp 9475 ∥ cdvds 10195 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-mulrcl 7075 ax-addcom 7076 ax-mulcom 7077 ax-addass 7078 ax-mulass 7079 ax-distr 7080 ax-i2m1 7081 ax-0lt1 7082 ax-1rid 7083 ax-0id 7084 ax-rnegex 7085 ax-precex 7086 ax-cnre 7087 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 ax-pre-apti 7091 ax-pre-ltadd 7092 ax-pre-mulgt0 7093 ax-pre-mulext 7094 ax-arch 7095 |
This theorem depends on definitions: df-bi 115 df-dc 776 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rmo 2356 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-if 3352 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-id 4048 df-po 4051 df-iso 4052 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-recs 5943 df-frec 6001 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-sub 7281 df-neg 7282 df-reap 7675 df-ap 7682 df-div 7761 df-inn 8040 df-2 8098 df-n0 8289 df-z 8352 df-uz 8620 df-q 8705 df-rp 8735 df-fz 9030 df-fl 9274 df-mod 9325 df-iseq 9432 df-iexp 9476 df-dvds 10196 |
This theorem is referenced by: pw2dvdseu 10546 oddpwdclemdvds 10548 oddpwdclemndvds 10549 |
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