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Mirrors > Home > ILE Home > Th. List > oddpwdclemndvds | GIF version |
Description: Lemma for oddpwdc 11779. A natural number is not divisible by one more than the highest power of two which divides it. (Contributed by Jim Kingdon, 17-Nov-2021.) |
Ref | Expression |
---|---|
oddpwdclemndvds | ⊢ (𝐴 ∈ ℕ → ¬ (2↑((℩𝑧 ∈ ℕ0 ((2↑𝑧) ∥ 𝐴 ∧ ¬ (2↑(𝑧 + 1)) ∥ 𝐴)) + 1)) ∥ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pw2dvds 11771 | . 2 ⊢ (𝐴 ∈ ℕ → ∃𝑧 ∈ ℕ0 ((2↑𝑧) ∥ 𝐴 ∧ ¬ (2↑(𝑧 + 1)) ∥ 𝐴)) | |
2 | nfv 1493 | . . 3 ⊢ Ⅎ𝑧 𝐴 ∈ ℕ | |
3 | nfcv 2258 | . . . . . 6 ⊢ Ⅎ𝑧2 | |
4 | nfcv 2258 | . . . . . 6 ⊢ Ⅎ𝑧↑ | |
5 | nfriota1 5705 | . . . . . . 7 ⊢ Ⅎ𝑧(℩𝑧 ∈ ℕ0 ((2↑𝑧) ∥ 𝐴 ∧ ¬ (2↑(𝑧 + 1)) ∥ 𝐴)) | |
6 | nfcv 2258 | . . . . . . 7 ⊢ Ⅎ𝑧 + | |
7 | nfcv 2258 | . . . . . . 7 ⊢ Ⅎ𝑧1 | |
8 | 5, 6, 7 | nfov 5769 | . . . . . 6 ⊢ Ⅎ𝑧((℩𝑧 ∈ ℕ0 ((2↑𝑧) ∥ 𝐴 ∧ ¬ (2↑(𝑧 + 1)) ∥ 𝐴)) + 1) |
9 | 3, 4, 8 | nfov 5769 | . . . . 5 ⊢ Ⅎ𝑧(2↑((℩𝑧 ∈ ℕ0 ((2↑𝑧) ∥ 𝐴 ∧ ¬ (2↑(𝑧 + 1)) ∥ 𝐴)) + 1)) |
10 | nfcv 2258 | . . . . 5 ⊢ Ⅎ𝑧 ∥ | |
11 | nfcv 2258 | . . . . 5 ⊢ Ⅎ𝑧𝐴 | |
12 | 9, 10, 11 | nfbr 3944 | . . . 4 ⊢ Ⅎ𝑧(2↑((℩𝑧 ∈ ℕ0 ((2↑𝑧) ∥ 𝐴 ∧ ¬ (2↑(𝑧 + 1)) ∥ 𝐴)) + 1)) ∥ 𝐴 |
13 | 12 | nfn 1621 | . . 3 ⊢ Ⅎ𝑧 ¬ (2↑((℩𝑧 ∈ ℕ0 ((2↑𝑧) ∥ 𝐴 ∧ ¬ (2↑(𝑧 + 1)) ∥ 𝐴)) + 1)) ∥ 𝐴 |
14 | simprrr 514 | . . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ (𝑧 ∈ ℕ0 ∧ ((2↑𝑧) ∥ 𝐴 ∧ ¬ (2↑(𝑧 + 1)) ∥ 𝐴))) → ¬ (2↑(𝑧 + 1)) ∥ 𝐴) | |
15 | pw2dvdseu 11773 | . . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → ∃!𝑧 ∈ ℕ0 ((2↑𝑧) ∥ 𝐴 ∧ ¬ (2↑(𝑧 + 1)) ∥ 𝐴)) | |
16 | riota1 5716 | . . . . . . . . . 10 ⊢ (∃!𝑧 ∈ ℕ0 ((2↑𝑧) ∥ 𝐴 ∧ ¬ (2↑(𝑧 + 1)) ∥ 𝐴) → ((𝑧 ∈ ℕ0 ∧ ((2↑𝑧) ∥ 𝐴 ∧ ¬ (2↑(𝑧 + 1)) ∥ 𝐴)) ↔ (℩𝑧 ∈ ℕ0 ((2↑𝑧) ∥ 𝐴 ∧ ¬ (2↑(𝑧 + 1)) ∥ 𝐴)) = 𝑧)) | |
17 | 15, 16 | syl 14 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → ((𝑧 ∈ ℕ0 ∧ ((2↑𝑧) ∥ 𝐴 ∧ ¬ (2↑(𝑧 + 1)) ∥ 𝐴)) ↔ (℩𝑧 ∈ ℕ0 ((2↑𝑧) ∥ 𝐴 ∧ ¬ (2↑(𝑧 + 1)) ∥ 𝐴)) = 𝑧)) |
18 | 17 | biimpa 294 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ (𝑧 ∈ ℕ0 ∧ ((2↑𝑧) ∥ 𝐴 ∧ ¬ (2↑(𝑧 + 1)) ∥ 𝐴))) → (℩𝑧 ∈ ℕ0 ((2↑𝑧) ∥ 𝐴 ∧ ¬ (2↑(𝑧 + 1)) ∥ 𝐴)) = 𝑧) |
19 | 18 | oveq1d 5757 | . . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ (𝑧 ∈ ℕ0 ∧ ((2↑𝑧) ∥ 𝐴 ∧ ¬ (2↑(𝑧 + 1)) ∥ 𝐴))) → ((℩𝑧 ∈ ℕ0 ((2↑𝑧) ∥ 𝐴 ∧ ¬ (2↑(𝑧 + 1)) ∥ 𝐴)) + 1) = (𝑧 + 1)) |
20 | 19 | oveq2d 5758 | . . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ (𝑧 ∈ ℕ0 ∧ ((2↑𝑧) ∥ 𝐴 ∧ ¬ (2↑(𝑧 + 1)) ∥ 𝐴))) → (2↑((℩𝑧 ∈ ℕ0 ((2↑𝑧) ∥ 𝐴 ∧ ¬ (2↑(𝑧 + 1)) ∥ 𝐴)) + 1)) = (2↑(𝑧 + 1))) |
21 | 20 | breq1d 3909 | . . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ (𝑧 ∈ ℕ0 ∧ ((2↑𝑧) ∥ 𝐴 ∧ ¬ (2↑(𝑧 + 1)) ∥ 𝐴))) → ((2↑((℩𝑧 ∈ ℕ0 ((2↑𝑧) ∥ 𝐴 ∧ ¬ (2↑(𝑧 + 1)) ∥ 𝐴)) + 1)) ∥ 𝐴 ↔ (2↑(𝑧 + 1)) ∥ 𝐴)) |
22 | 14, 21 | mtbird 647 | . . . 4 ⊢ ((𝐴 ∈ ℕ ∧ (𝑧 ∈ ℕ0 ∧ ((2↑𝑧) ∥ 𝐴 ∧ ¬ (2↑(𝑧 + 1)) ∥ 𝐴))) → ¬ (2↑((℩𝑧 ∈ ℕ0 ((2↑𝑧) ∥ 𝐴 ∧ ¬ (2↑(𝑧 + 1)) ∥ 𝐴)) + 1)) ∥ 𝐴) |
23 | 22 | exp32 362 | . . 3 ⊢ (𝐴 ∈ ℕ → (𝑧 ∈ ℕ0 → (((2↑𝑧) ∥ 𝐴 ∧ ¬ (2↑(𝑧 + 1)) ∥ 𝐴) → ¬ (2↑((℩𝑧 ∈ ℕ0 ((2↑𝑧) ∥ 𝐴 ∧ ¬ (2↑(𝑧 + 1)) ∥ 𝐴)) + 1)) ∥ 𝐴))) |
24 | 2, 13, 23 | rexlimd 2523 | . 2 ⊢ (𝐴 ∈ ℕ → (∃𝑧 ∈ ℕ0 ((2↑𝑧) ∥ 𝐴 ∧ ¬ (2↑(𝑧 + 1)) ∥ 𝐴) → ¬ (2↑((℩𝑧 ∈ ℕ0 ((2↑𝑧) ∥ 𝐴 ∧ ¬ (2↑(𝑧 + 1)) ∥ 𝐴)) + 1)) ∥ 𝐴)) |
25 | 1, 24 | mpd 13 | 1 ⊢ (𝐴 ∈ ℕ → ¬ (2↑((℩𝑧 ∈ ℕ0 ((2↑𝑧) ∥ 𝐴 ∧ ¬ (2↑(𝑧 + 1)) ∥ 𝐴)) + 1)) ∥ 𝐴) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1316 ∈ wcel 1465 ∃wrex 2394 ∃!wreu 2395 class class class wbr 3899 ℩crio 5697 (class class class)co 5742 1c1 7589 + caddc 7591 ℕcn 8688 2c2 8739 ℕ0cn0 8945 ↑cexp 10260 ∥ cdvds 11420 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-iinf 4472 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-mulrcl 7687 ax-addcom 7688 ax-mulcom 7689 ax-addass 7690 ax-mulass 7691 ax-distr 7692 ax-i2m1 7693 ax-0lt1 7694 ax-1rid 7695 ax-0id 7696 ax-rnegex 7697 ax-precex 7698 ax-cnre 7699 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 ax-pre-apti 7703 ax-pre-ltadd 7704 ax-pre-mulgt0 7705 ax-pre-mulext 7706 ax-arch 7707 |
This theorem depends on definitions: df-bi 116 df-dc 805 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-reu 2400 df-rmo 2401 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-if 3445 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-tr 3997 df-id 4185 df-po 4188 df-iso 4189 df-iord 4258 df-on 4260 df-ilim 4261 df-suc 4263 df-iom 4475 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-1st 6006 df-2nd 6007 df-recs 6170 df-frec 6256 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-sub 7903 df-neg 7904 df-reap 8305 df-ap 8312 df-div 8401 df-inn 8689 df-2 8747 df-n0 8946 df-z 9023 df-uz 9295 df-q 9380 df-rp 9410 df-fz 9759 df-fl 10011 df-mod 10064 df-seqfrec 10187 df-exp 10261 df-dvds 11421 |
This theorem is referenced by: oddpwdclemodd 11777 |
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