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Mirrors > Home > MPE Home > Th. List > bitsss | Structured version Visualization version GIF version |
Description: The set of bits of an integer is a subset of ℕ0. (Contributed by Mario Carneiro, 5-Sep-2016.) |
Ref | Expression |
---|---|
bitsss | ⊢ (bits‘𝑁) ⊆ ℕ0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bitsval 15763 | . . 3 ⊢ (𝑚 ∈ (bits‘𝑁) ↔ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ0 ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) | |
2 | 1 | simp2bi 1138 | . 2 ⊢ (𝑚 ∈ (bits‘𝑁) → 𝑚 ∈ ℕ0) |
3 | 2 | ssriv 3970 | 1 ⊢ (bits‘𝑁) ⊆ ℕ0 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∈ wcel 2105 ⊆ wss 3935 class class class wbr 5058 ‘cfv 6349 (class class class)co 7145 / cdiv 11286 2c2 11681 ℕ0cn0 11886 ℤcz 11970 ⌊cfl 13150 ↑cexp 13419 ∥ cdvds 15597 bitscbits 15758 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 ax-cnex 10582 ax-1cn 10584 ax-addcl 10586 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4833 df-iun 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7148 df-om 7569 df-wrecs 7938 df-recs 7999 df-rdg 8037 df-nn 11628 df-n0 11887 df-bits 15761 |
This theorem is referenced by: bitsinv2 15782 bitsf1ocnv 15783 sadaddlem 15805 sadadd 15806 bitsres 15812 bitsshft 15814 smumullem 15831 smumul 15832 eulerpartlemgc 31520 eulerpartlemgvv 31534 eulerpartlemgh 31536 eulerpartlemgs2 31538 |
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