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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 9129 | . . . 4 ⊢ 2 ∈ ℕ | |
3 | nnnn0 9233 | . . . 4 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0) | |
4 | nnexpcl 10597 | . . . 4 ⊢ ((2 ∈ ℕ ∧ 𝑁 ∈ ℕ0) → (2↑𝑁) ∈ ℕ) | |
5 | 2, 3, 4 | sylancr 414 | . . 3 ⊢ (𝑁 ∈ ℕ → (2↑𝑁) ∈ ℕ) |
6 | 1zzd 9330 | . . . 4 ⊢ (𝑁 ∈ ℕ → 1 ∈ ℤ) | |
7 | 2z 9331 | . . . . . 6 ⊢ 2 ∈ ℤ | |
8 | zexpcl 10599 | . . . . . 6 ⊢ ((2 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → (2↑𝑁) ∈ ℤ) | |
9 | 7, 3, 8 | sylancr 414 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (2↑𝑁) ∈ ℤ) |
10 | 9, 6 | zsubcld 9430 | . . . 4 ⊢ (𝑁 ∈ ℕ → ((2↑𝑁) − 1) ∈ ℤ) |
11 | nnz 9322 | . . . 4 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ) | |
12 | nnge1 8991 | . . . 4 ⊢ (𝑁 ∈ ℕ → 1 ≤ 𝑁) | |
13 | uzid 9592 | . . . . . . 7 ⊢ (2 ∈ ℤ → 2 ∈ (ℤ≥‘2)) | |
14 | 7, 13 | ax-mp 5 | . . . . . 6 ⊢ 2 ∈ (ℤ≥‘2) |
15 | bernneq3 10707 | . . . . . 6 ⊢ ((2 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) → 𝑁 < (2↑𝑁)) | |
16 | 14, 3, 15 | sylancr 414 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 𝑁 < (2↑𝑁)) |
17 | zltlem1 9360 | . . . . . 6 ⊢ ((𝑁 ∈ ℤ ∧ (2↑𝑁) ∈ ℤ) → (𝑁 < (2↑𝑁) ↔ 𝑁 ≤ ((2↑𝑁) − 1))) | |
18 | 11, 9, 17 | syl2anc 411 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (𝑁 < (2↑𝑁) ↔ 𝑁 ≤ ((2↑𝑁) − 1))) |
19 | 16, 18 | mpbid 147 | . . . 4 ⊢ (𝑁 ∈ ℕ → 𝑁 ≤ ((2↑𝑁) − 1)) |
20 | elfz4 10070 | . . . 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 1257 | . . 3 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ (1...((2↑𝑁) − 1))) |
22 | fzm1ndvds 11972 | . . 3 ⊢ (((2↑𝑁) ∈ ℕ ∧ 𝑁 ∈ (1...((2↑𝑁) − 1))) → ¬ (2↑𝑁) ∥ 𝑁) | |
23 | 5, 21, 22 | syl2anc 411 | . 2 ⊢ (𝑁 ∈ ℕ → ¬ (2↑𝑁) ∥ 𝑁) |
24 | pw2dvdslemn 12277 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ ¬ (2↑𝑁) ∥ 𝑁) → ∃𝑚 ∈ ℕ0 ((2↑𝑚) ∥ 𝑁 ∧ ¬ (2↑(𝑚 + 1)) ∥ 𝑁)) | |
25 | 1, 23, 24 | mpd3an23 1350 | 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 104 ↔ wb 105 ∈ wcel 2160 ∃wrex 2469 class class class wbr 4025 ‘cfv 5242 (class class class)co 5906 1c1 7859 + caddc 7861 < clt 8040 ≤ cle 8041 − cmin 8176 ℕcn 8968 2c2 9019 ℕ0cn0 9226 ℤcz 9303 ℤ≥cuz 9578 ...cfz 10060 ↑cexp 10583 ∥ cdvds 11904 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4140 ax-sep 4143 ax-nul 4151 ax-pow 4199 ax-pr 4234 ax-un 4458 ax-setind 4561 ax-iinf 4612 ax-cnex 7949 ax-resscn 7950 ax-1cn 7951 ax-1re 7952 ax-icn 7953 ax-addcl 7954 ax-addrcl 7955 ax-mulcl 7956 ax-mulrcl 7957 ax-addcom 7958 ax-mulcom 7959 ax-addass 7960 ax-mulass 7961 ax-distr 7962 ax-i2m1 7963 ax-0lt1 7964 ax-1rid 7965 ax-0id 7966 ax-rnegex 7967 ax-precex 7968 ax-cnre 7969 ax-pre-ltirr 7970 ax-pre-ltwlin 7971 ax-pre-lttrn 7972 ax-pre-apti 7973 ax-pre-ltadd 7974 ax-pre-mulgt0 7975 ax-pre-mulext 7976 ax-arch 7977 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2758 df-sbc 2982 df-csb 3077 df-dif 3151 df-un 3153 df-in 3155 df-ss 3162 df-nul 3443 df-if 3554 df-pw 3599 df-sn 3620 df-pr 3621 df-op 3623 df-uni 3832 df-int 3867 df-iun 3910 df-br 4026 df-opab 4087 df-mpt 4088 df-tr 4124 df-id 4318 df-po 4321 df-iso 4322 df-iord 4391 df-on 4393 df-ilim 4394 df-suc 4396 df-iom 4615 df-xp 4657 df-rel 4658 df-cnv 4659 df-co 4660 df-dm 4661 df-rn 4662 df-res 4663 df-ima 4664 df-iota 5203 df-fun 5244 df-fn 5245 df-f 5246 df-f1 5247 df-fo 5248 df-f1o 5249 df-fv 5250 df-riota 5861 df-ov 5909 df-oprab 5910 df-mpo 5911 df-1st 6180 df-2nd 6181 df-recs 6345 df-frec 6431 df-pnf 8042 df-mnf 8043 df-xr 8044 df-ltxr 8045 df-le 8046 df-sub 8178 df-neg 8179 df-reap 8580 df-ap 8587 df-div 8678 df-inn 8969 df-2 9027 df-n0 9227 df-z 9304 df-uz 9579 df-q 9671 df-rp 9706 df-fz 10061 df-fl 10325 df-mod 10380 df-seqfrec 10505 df-exp 10584 df-dvds 11905 |
This theorem is referenced by: pw2dvdseu 12280 oddpwdclemdvds 12282 oddpwdclemndvds 12283 |
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