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Mirrors > Home > MPE Home > Th. List > bitsval2 | Structured version Visualization version GIF version |
Description: Expand the definition of the bits of an integer. (Contributed by Mario Carneiro, 5-Sep-2016.) |
Ref | Expression |
---|---|
bitsval2 | β’ ((π β β€ β§ π β β0) β (π β (bitsβπ) β Β¬ 2 β₯ (ββ(π / (2βπ))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bitsval 16364 | . . 3 β’ (π β (bitsβπ) β (π β β€ β§ π β β0 β§ Β¬ 2 β₯ (ββ(π / (2βπ))))) | |
2 | df-3an 1089 | . . 3 β’ ((π β β€ β§ π β β0 β§ Β¬ 2 β₯ (ββ(π / (2βπ)))) β ((π β β€ β§ π β β0) β§ Β¬ 2 β₯ (ββ(π / (2βπ))))) | |
3 | 1, 2 | bitri 274 | . 2 β’ (π β (bitsβπ) β ((π β β€ β§ π β β0) β§ Β¬ 2 β₯ (ββ(π / (2βπ))))) |
4 | 3 | baib 536 | 1 β’ ((π β β€ β§ π β β0) β (π β (bitsβπ) β Β¬ 2 β₯ (ββ(π / (2βπ))))) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β wb 205 β§ wa 396 β§ w3a 1087 β wcel 2106 class class class wbr 5148 βcfv 6543 (class class class)co 7408 / cdiv 11870 2c2 12266 β0cn0 12471 β€cz 12557 βcfl 13754 βcexp 14026 β₯ cdvds 16196 bitscbits 16359 |
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 5299 ax-nul 5306 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-1cn 11167 ax-addcl 11169 |
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 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7411 df-om 7855 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-nn 12212 df-n0 12472 df-bits 16362 |
This theorem is referenced by: bits0 16368 bitsp1 16371 bitsfzolem 16374 bitsfzo 16375 bitsmod 16376 bitscmp 16378 bitsinv1lem 16381 bitsshft 16415 bits0ALTV 46337 dig2bits 47290 |
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