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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 | β’ bits:β€βΆπ« β0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-bits 16359 | . 2 β’ bits = (π β β€ β¦ {π β β0 β£ Β¬ 2 β₯ (ββ(π / (2βπ)))}) | |
2 | nn0ex 12474 | . . . 4 β’ β0 β V | |
3 | ssrab2 4076 | . . . 4 β’ {π β β0 β£ Β¬ 2 β₯ (ββ(π / (2βπ)))} β β0 | |
4 | 2, 3 | elpwi2 5345 | . . 3 β’ {π β β0 β£ Β¬ 2 β₯ (ββ(π / (2βπ)))} β π« β0 |
5 | 4 | a1i 11 | . 2 β’ (π β β€ β {π β β0 β£ Β¬ 2 β₯ (ββ(π / (2βπ)))} β π« β0) |
6 | 1, 5 | fmpti 7108 | 1 β’ bits:β€βΆπ« β0 |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wcel 2106 {crab 3432 Vcvv 3474 π« cpw 4601 class class class wbr 5147 βΆwf 6536 βcfv 6540 (class class class)co 7405 / cdiv 11867 2c2 12263 β0cn0 12468 β€cz 12554 βcfl 13751 βcexp 14023 β₯ cdvds 16193 bitscbits 16356 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-1cn 11164 ax-addcl 11166 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-ov 7408 df-om 7852 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-nn 12209 df-n0 12469 df-bits 16359 |
This theorem is referenced by: bitsf1ocnv 16381 bitsf1 16383 eulerpartgbij 33359 eulerpartlemmf 33362 |
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