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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 16129 | . . 3 ⊢ (𝑚 ∈ (bits‘𝑁) ↔ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ0 ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) | |
2 | 1 | simp2bi 1145 | . 2 ⊢ (𝑚 ∈ (bits‘𝑁) → 𝑚 ∈ ℕ0) |
3 | 2 | ssriv 3930 | 1 ⊢ (bits‘𝑁) ⊆ ℕ0 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∈ wcel 2110 ⊆ wss 3892 class class class wbr 5079 ‘cfv 6432 (class class class)co 7271 / cdiv 11632 2c2 12028 ℕ0cn0 12233 ℤcz 12319 ⌊cfl 13508 ↑cexp 13780 ∥ cdvds 15961 bitscbits 16124 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pr 5356 ax-un 7582 ax-cnex 10928 ax-1cn 10930 ax-addcl 10932 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-ral 3071 df-rex 3072 df-reu 3073 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-ov 7274 df-om 7707 df-2nd 7825 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-nn 11974 df-n0 12234 df-bits 16127 |
This theorem is referenced by: bitsinv2 16148 bitsf1ocnv 16149 sadaddlem 16171 sadadd 16172 bitsres 16178 bitsshft 16180 smumullem 16197 smumul 16198 eulerpartlemgc 32325 eulerpartlemgvv 32339 eulerpartlemgh 32341 eulerpartlemgs2 32343 |
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