Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 15146 | . . 3 ⊢ (𝑚 ∈ (bits‘𝑁) ↔ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ0 ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) | |
2 | 1 | simp2bi 1077 | . 2 ⊢ (𝑚 ∈ (bits‘𝑁) → 𝑚 ∈ ℕ0) |
3 | 2 | ssriv 3607 | 1 ⊢ (bits‘𝑁) ⊆ ℕ0 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∈ wcel 1990 ⊆ wss 3574 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 / cdiv 10684 2c2 11070 ℕ0cn0 11292 ℤcz 11377 ⌊cfl 12591 ↑cexp 12860 ∥ cdvds 14983 bitscbits 15141 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-i2m1 10004 ax-1ne0 10005 ax-rrecex 10008 ax-cnre 10009 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-nn 11021 df-n0 11293 df-bits 15144 |
This theorem is referenced by: bitsinv2 15165 bitsf1ocnv 15166 sadaddlem 15188 sadadd 15189 bitsres 15195 bitsshft 15197 smumullem 15214 smumul 15215 eulerpartlemgc 30424 eulerpartlemgvv 30438 eulerpartlemgh 30440 eulerpartlemgs2 30442 |
Copyright terms: Public domain | W3C validator |