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Mirrors > Home > MPE Home > Th. List > bits0o | Structured version Visualization version GIF version |
Description: The zeroth bit of an odd number is zero. (Contributed by Mario Carneiro, 5-Sep-2016.) |
Ref | Expression |
---|---|
bits0o | ⊢ (𝑁 ∈ ℤ → 0 ∈ (bits‘((2 · 𝑁) + 1))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2z 12646 | . . . 4 ⊢ 2 ∈ ℤ | |
2 | dvdsmul1 16280 | . . . 4 ⊢ ((2 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 2 ∥ (2 · 𝑁)) | |
3 | 1, 2 | mpan 688 | . . 3 ⊢ (𝑁 ∈ ℤ → 2 ∥ (2 · 𝑁)) |
4 | 1 | a1i 11 | . . . . 5 ⊢ (𝑁 ∈ ℤ → 2 ∈ ℤ) |
5 | id 22 | . . . . 5 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℤ) | |
6 | 4, 5 | zmulcld 12724 | . . . 4 ⊢ (𝑁 ∈ ℤ → (2 · 𝑁) ∈ ℤ) |
7 | 2nn 12337 | . . . . 5 ⊢ 2 ∈ ℕ | |
8 | 7 | a1i 11 | . . . 4 ⊢ (𝑁 ∈ ℤ → 2 ∈ ℕ) |
9 | 1lt2 12435 | . . . . 5 ⊢ 1 < 2 | |
10 | 9 | a1i 11 | . . . 4 ⊢ (𝑁 ∈ ℤ → 1 < 2) |
11 | ndvdsp1 16413 | . . . 4 ⊢ (((2 · 𝑁) ∈ ℤ ∧ 2 ∈ ℕ ∧ 1 < 2) → (2 ∥ (2 · 𝑁) → ¬ 2 ∥ ((2 · 𝑁) + 1))) | |
12 | 6, 8, 10, 11 | syl3anc 1368 | . . 3 ⊢ (𝑁 ∈ ℤ → (2 ∥ (2 · 𝑁) → ¬ 2 ∥ ((2 · 𝑁) + 1))) |
13 | 3, 12 | mpd 15 | . 2 ⊢ (𝑁 ∈ ℤ → ¬ 2 ∥ ((2 · 𝑁) + 1)) |
14 | 6 | peano2zd 12721 | . . 3 ⊢ (𝑁 ∈ ℤ → ((2 · 𝑁) + 1) ∈ ℤ) |
15 | bits0 16428 | . . 3 ⊢ (((2 · 𝑁) + 1) ∈ ℤ → (0 ∈ (bits‘((2 · 𝑁) + 1)) ↔ ¬ 2 ∥ ((2 · 𝑁) + 1))) | |
16 | 14, 15 | syl 17 | . 2 ⊢ (𝑁 ∈ ℤ → (0 ∈ (bits‘((2 · 𝑁) + 1)) ↔ ¬ 2 ∥ ((2 · 𝑁) + 1))) |
17 | 13, 16 | mpbird 256 | 1 ⊢ (𝑁 ∈ ℤ → 0 ∈ (bits‘((2 · 𝑁) + 1))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∈ wcel 2099 class class class wbr 5153 ‘cfv 6554 (class class class)co 7424 0cc0 11158 1c1 11159 + caddc 11161 · cmul 11163 < clt 11298 ℕcn 12264 2c2 12319 ℤcz 12610 ∥ cdvds 16256 bitscbits 16419 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 ax-pre-sup 11236 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-1st 8003 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-sup 9485 df-inf 9486 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-div 11922 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12611 df-uz 12875 df-rp 13029 df-fz 13539 df-fl 13812 df-seq 14022 df-exp 14082 df-cj 15104 df-re 15105 df-im 15106 df-sqrt 15240 df-abs 15241 df-dvds 16257 df-bits 16422 |
This theorem is referenced by: (None) |
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