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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 8675 | . . . 4 ⊢ 2 ∈ ℕ | |
3 | nnnn0 8778 | . . . 4 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0) | |
4 | nnexpcl 10083 | . . . 4 ⊢ ((2 ∈ ℕ ∧ 𝑁 ∈ ℕ0) → (2↑𝑁) ∈ ℕ) | |
5 | 2, 3, 4 | sylancr 406 | . . 3 ⊢ (𝑁 ∈ ℕ → (2↑𝑁) ∈ ℕ) |
6 | 1zzd 8875 | . . . 4 ⊢ (𝑁 ∈ ℕ → 1 ∈ ℤ) | |
7 | 2z 8876 | . . . . . 6 ⊢ 2 ∈ ℤ | |
8 | zexpcl 10085 | . . . . . 6 ⊢ ((2 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → (2↑𝑁) ∈ ℤ) | |
9 | 7, 3, 8 | sylancr 406 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (2↑𝑁) ∈ ℤ) |
10 | 9, 6 | zsubcld 8972 | . . . 4 ⊢ (𝑁 ∈ ℕ → ((2↑𝑁) − 1) ∈ ℤ) |
11 | nnz 8867 | . . . 4 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ) | |
12 | nnge1 8543 | . . . 4 ⊢ (𝑁 ∈ ℕ → 1 ≤ 𝑁) | |
13 | uzid 9132 | . . . . . . 7 ⊢ (2 ∈ ℤ → 2 ∈ (ℤ≥‘2)) | |
14 | 7, 13 | ax-mp 7 | . . . . . 6 ⊢ 2 ∈ (ℤ≥‘2) |
15 | bernneq3 10191 | . . . . . 6 ⊢ ((2 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) → 𝑁 < (2↑𝑁)) | |
16 | 14, 3, 15 | sylancr 406 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 𝑁 < (2↑𝑁)) |
17 | zltlem1 8905 | . . . . . 6 ⊢ ((𝑁 ∈ ℤ ∧ (2↑𝑁) ∈ ℤ) → (𝑁 < (2↑𝑁) ↔ 𝑁 ≤ ((2↑𝑁) − 1))) | |
18 | 11, 9, 17 | syl2anc 404 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (𝑁 < (2↑𝑁) ↔ 𝑁 ≤ ((2↑𝑁) − 1))) |
19 | 16, 18 | mpbid 146 | . . . 4 ⊢ (𝑁 ∈ ℕ → 𝑁 ≤ ((2↑𝑁) − 1)) |
20 | elfz4 9582 | . . . 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 1189 | . . 3 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ (1...((2↑𝑁) − 1))) |
22 | fzm1ndvds 11284 | . . 3 ⊢ (((2↑𝑁) ∈ ℕ ∧ 𝑁 ∈ (1...((2↑𝑁) − 1))) → ¬ (2↑𝑁) ∥ 𝑁) | |
23 | 5, 21, 22 | syl2anc 404 | . 2 ⊢ (𝑁 ∈ ℕ → ¬ (2↑𝑁) ∥ 𝑁) |
24 | pw2dvdslemn 11570 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ ¬ (2↑𝑁) ∥ 𝑁) → ∃𝑚 ∈ ℕ0 ((2↑𝑚) ∥ 𝑁 ∧ ¬ (2↑(𝑚 + 1)) ∥ 𝑁)) | |
25 | 1, 23, 24 | mpd3an23 1282 | 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 1445 ∃wrex 2371 class class class wbr 3867 ‘cfv 5049 (class class class)co 5690 1c1 7448 + caddc 7450 < clt 7619 ≤ cle 7620 − cmin 7750 ℕcn 8520 2c2 8571 ℕ0cn0 8771 ℤcz 8848 ℤ≥cuz 9118 ...cfz 9573 ↑cexp 10069 ∥ cdvds 11223 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-13 1456 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-coll 3975 ax-sep 3978 ax-nul 3986 ax-pow 4030 ax-pr 4060 ax-un 4284 ax-setind 4381 ax-iinf 4431 ax-cnex 7533 ax-resscn 7534 ax-1cn 7535 ax-1re 7536 ax-icn 7537 ax-addcl 7538 ax-addrcl 7539 ax-mulcl 7540 ax-mulrcl 7541 ax-addcom 7542 ax-mulcom 7543 ax-addass 7544 ax-mulass 7545 ax-distr 7546 ax-i2m1 7547 ax-0lt1 7548 ax-1rid 7549 ax-0id 7550 ax-rnegex 7551 ax-precex 7552 ax-cnre 7553 ax-pre-ltirr 7554 ax-pre-ltwlin 7555 ax-pre-lttrn 7556 ax-pre-apti 7557 ax-pre-ltadd 7558 ax-pre-mulgt0 7559 ax-pre-mulext 7560 ax-arch 7561 |
This theorem depends on definitions: df-bi 116 df-dc 784 df-3or 928 df-3an 929 df-tru 1299 df-fal 1302 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ne 2263 df-nel 2358 df-ral 2375 df-rex 2376 df-reu 2377 df-rmo 2378 df-rab 2379 df-v 2635 df-sbc 2855 df-csb 2948 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 df-nul 3303 df-if 3414 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-int 3711 df-iun 3754 df-br 3868 df-opab 3922 df-mpt 3923 df-tr 3959 df-id 4144 df-po 4147 df-iso 4148 df-iord 4217 df-on 4219 df-ilim 4220 df-suc 4222 df-iom 4434 df-xp 4473 df-rel 4474 df-cnv 4475 df-co 4476 df-dm 4477 df-rn 4478 df-res 4479 df-ima 4480 df-iota 5014 df-fun 5051 df-fn 5052 df-f 5053 df-f1 5054 df-fo 5055 df-f1o 5056 df-fv 5057 df-riota 5646 df-ov 5693 df-oprab 5694 df-mpt2 5695 df-1st 5949 df-2nd 5950 df-recs 6108 df-frec 6194 df-pnf 7621 df-mnf 7622 df-xr 7623 df-ltxr 7624 df-le 7625 df-sub 7752 df-neg 7753 df-reap 8149 df-ap 8156 df-div 8237 df-inn 8521 df-2 8579 df-n0 8772 df-z 8849 df-uz 9119 df-q 9204 df-rp 9234 df-fz 9574 df-fl 9826 df-mod 9879 df-seqfrec 10001 df-exp 10070 df-dvds 11224 |
This theorem is referenced by: pw2dvdseu 11573 oddpwdclemdvds 11575 oddpwdclemndvds 11576 |
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