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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > bits0ALTV | Structured version Visualization version GIF version |
Description: Value of the zeroth bit. (Contributed by Mario Carneiro, 5-Sep-2016.) (Revised by AV, 19-Jun-2020.) |
Ref | Expression |
---|---|
bits0ALTV | β’ (π β β€ β (0 β (bitsβπ) β π β Odd )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0nn0 12452 | . . 3 β’ 0 β β0 | |
2 | bitsval2 16331 | . . 3 β’ ((π β β€ β§ 0 β β0) β (0 β (bitsβπ) β Β¬ 2 β₯ (ββ(π / (2β0))))) | |
3 | 1, 2 | mpan2 689 | . 2 β’ (π β β€ β (0 β (bitsβπ) β Β¬ 2 β₯ (ββ(π / (2β0))))) |
4 | 2cn 12252 | . . . . . . . . 9 β’ 2 β β | |
5 | exp0 13996 | . . . . . . . . 9 β’ (2 β β β (2β0) = 1) | |
6 | 4, 5 | ax-mp 5 | . . . . . . . 8 β’ (2β0) = 1 |
7 | 6 | oveq2i 7388 | . . . . . . 7 β’ (π / (2β0)) = (π / 1) |
8 | zcn 12528 | . . . . . . . 8 β’ (π β β€ β π β β) | |
9 | 8 | div1d 11947 | . . . . . . 7 β’ (π β β€ β (π / 1) = π) |
10 | 7, 9 | eqtrid 2783 | . . . . . 6 β’ (π β β€ β (π / (2β0)) = π) |
11 | 10 | fveq2d 6866 | . . . . 5 β’ (π β β€ β (ββ(π / (2β0))) = (ββπ)) |
12 | flid 13738 | . . . . 5 β’ (π β β€ β (ββπ) = π) | |
13 | 11, 12 | eqtrd 2771 | . . . 4 β’ (π β β€ β (ββ(π / (2β0))) = π) |
14 | 13 | breq2d 5137 | . . 3 β’ (π β β€ β (2 β₯ (ββ(π / (2β0))) β 2 β₯ π)) |
15 | 14 | notbid 317 | . 2 β’ (π β β€ β (Β¬ 2 β₯ (ββ(π / (2β0))) β Β¬ 2 β₯ π)) |
16 | isodd3 45997 | . . 3 β’ (π β Odd β (π β β€ β§ Β¬ 2 β₯ π)) | |
17 | 16 | baibr 537 | . 2 β’ (π β β€ β (Β¬ 2 β₯ π β π β Odd )) |
18 | 3, 15, 17 | 3bitrd 304 | 1 β’ (π β β€ β (0 β (bitsβπ) β π β Odd )) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β wb 205 = wceq 1541 β wcel 2106 class class class wbr 5125 βcfv 6516 (class class class)co 7377 βcc 11073 0cc0 11075 1c1 11076 / cdiv 11836 2c2 12232 β0cn0 12437 β€cz 12523 βcfl 13720 βcexp 13992 β₯ cdvds 16162 bitscbits 16325 Odd codd 45970 |
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 2702 ax-sep 5276 ax-nul 5283 ax-pow 5340 ax-pr 5404 ax-un 7692 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153 |
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 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3364 df-reu 3365 df-rab 3419 df-v 3461 df-sbc 3758 df-csb 3874 df-dif 3931 df-un 3933 df-in 3935 df-ss 3945 df-pss 3947 df-nul 4303 df-if 4507 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4886 df-iun 4976 df-br 5126 df-opab 5188 df-mpt 5209 df-tr 5243 df-id 5551 df-eprel 5557 df-po 5565 df-so 5566 df-fr 5608 df-we 5610 df-xp 5659 df-rel 5660 df-cnv 5661 df-co 5662 df-dm 5663 df-rn 5664 df-res 5665 df-ima 5666 df-pred 6273 df-ord 6340 df-on 6341 df-lim 6342 df-suc 6343 df-iota 6468 df-fun 6518 df-fn 6519 df-f 6520 df-f1 6521 df-fo 6522 df-f1o 6523 df-fv 6524 df-riota 7333 df-ov 7380 df-oprab 7381 df-mpo 7382 df-om 7823 df-2nd 7942 df-frecs 8232 df-wrecs 8263 df-recs 8337 df-rdg 8376 df-er 8670 df-en 8906 df-dom 8907 df-sdom 8908 df-sup 9402 df-inf 9403 df-pnf 11215 df-mnf 11216 df-xr 11217 df-ltxr 11218 df-le 11219 df-sub 11411 df-neg 11412 df-div 11837 df-nn 12178 df-2 12240 df-n0 12438 df-z 12524 df-uz 12788 df-fl 13722 df-seq 13932 df-exp 13993 df-dvds 16163 df-bits 16328 df-odd 45972 |
This theorem is referenced by: bits0eALTV 46025 bits0oALTV 46026 |
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