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Mirrors > Home > MPE Home > Th. List > bitsf | Structured version Visualization version GIF version |
Description: The bits function is a function from integers to subsets of nonnegative integers. (Contributed by Mario Carneiro, 5-Sep-2016.) |
Ref | Expression |
---|---|
bitsf | ⊢ bits:ℤ⟶𝒫 ℕ0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-bits 15765 | . 2 ⊢ bits = (𝑛 ∈ ℤ ↦ {𝑘 ∈ ℕ0 ∣ ¬ 2 ∥ (⌊‘(𝑛 / (2↑𝑘)))}) | |
2 | nn0ex 11897 | . . . 4 ⊢ ℕ0 ∈ V | |
3 | ssrab2 4055 | . . . 4 ⊢ {𝑘 ∈ ℕ0 ∣ ¬ 2 ∥ (⌊‘(𝑛 / (2↑𝑘)))} ⊆ ℕ0 | |
4 | 2, 3 | elpwi2 5241 | . . 3 ⊢ {𝑘 ∈ ℕ0 ∣ ¬ 2 ∥ (⌊‘(𝑛 / (2↑𝑘)))} ∈ 𝒫 ℕ0 |
5 | 4 | a1i 11 | . 2 ⊢ (𝑛 ∈ ℤ → {𝑘 ∈ ℕ0 ∣ ¬ 2 ∥ (⌊‘(𝑛 / (2↑𝑘)))} ∈ 𝒫 ℕ0) |
6 | 1, 5 | fmpti 6870 | 1 ⊢ bits:ℤ⟶𝒫 ℕ0 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∈ wcel 2110 {crab 3142 Vcvv 3494 𝒫 cpw 4538 class class class wbr 5058 ⟶wf 6345 ‘cfv 6349 (class class class)co 7150 / cdiv 11291 2c2 11686 ℕ0cn0 11891 ℤcz 11975 ⌊cfl 13154 ↑cexp 13423 ∥ cdvds 15601 bitscbits 15762 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-1cn 10589 ax-addcl 10591 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 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 3496 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 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 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 7153 df-om 7575 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-nn 11633 df-n0 11892 df-bits 15765 |
This theorem is referenced by: bitsf1ocnv 15787 bitsf1 15789 eulerpartgbij 31625 eulerpartlemmf 31628 |
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