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Mirrors > Home > MPE Home > Th. List > bitsp1e | Structured version Visualization version GIF version |
Description: The π + 1-th bit of 2π is the π-th bit of π. (Contributed by Mario Carneiro, 5-Sep-2016.) |
Ref | Expression |
---|---|
bitsp1e | β’ ((π β β€ β§ π β β0) β ((π + 1) β (bitsβ(2 Β· π)) β π β (bitsβπ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2z 12540 | . . . . 5 β’ 2 β β€ | |
2 | 1 | a1i 11 | . . . 4 β’ (π β β€ β 2 β β€) |
3 | id 22 | . . . 4 β’ (π β β€ β π β β€) | |
4 | 2, 3 | zmulcld 12618 | . . 3 β’ (π β β€ β (2 Β· π) β β€) |
5 | bitsp1 16316 | . . 3 β’ (((2 Β· π) β β€ β§ π β β0) β ((π + 1) β (bitsβ(2 Β· π)) β π β (bitsβ(ββ((2 Β· π) / 2))))) | |
6 | 4, 5 | sylan 581 | . 2 β’ ((π β β€ β§ π β β0) β ((π + 1) β (bitsβ(2 Β· π)) β π β (bitsβ(ββ((2 Β· π) / 2))))) |
7 | zcn 12509 | . . . . . . . 8 β’ (π β β€ β π β β) | |
8 | 2cnd 12236 | . . . . . . . 8 β’ (π β β€ β 2 β β) | |
9 | 2ne0 12262 | . . . . . . . . 9 β’ 2 β 0 | |
10 | 9 | a1i 11 | . . . . . . . 8 β’ (π β β€ β 2 β 0) |
11 | 7, 8, 10 | divcan3d 11941 | . . . . . . 7 β’ (π β β€ β ((2 Β· π) / 2) = π) |
12 | 11 | fveq2d 6847 | . . . . . 6 β’ (π β β€ β (ββ((2 Β· π) / 2)) = (ββπ)) |
13 | flid 13719 | . . . . . 6 β’ (π β β€ β (ββπ) = π) | |
14 | 12, 13 | eqtrd 2773 | . . . . 5 β’ (π β β€ β (ββ((2 Β· π) / 2)) = π) |
15 | 14 | adantr 482 | . . . 4 β’ ((π β β€ β§ π β β0) β (ββ((2 Β· π) / 2)) = π) |
16 | 15 | fveq2d 6847 | . . 3 β’ ((π β β€ β§ π β β0) β (bitsβ(ββ((2 Β· π) / 2))) = (bitsβπ)) |
17 | 16 | eleq2d 2820 | . 2 β’ ((π β β€ β§ π β β0) β (π β (bitsβ(ββ((2 Β· π) / 2))) β π β (bitsβπ))) |
18 | 6, 17 | bitrd 279 | 1 β’ ((π β β€ β§ π β β0) β ((π + 1) β (bitsβ(2 Β· π)) β π β (bitsβπ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 = wceq 1542 β wcel 2107 β wne 2940 βcfv 6497 (class class class)co 7358 0cc0 11056 1c1 11057 + caddc 11059 Β· cmul 11061 / cdiv 11817 2c2 12213 β0cn0 12418 β€cz 12504 βcfl 13701 bitscbits 16304 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 ax-pre-sup 11134 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-sup 9383 df-inf 9384 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-div 11818 df-nn 12159 df-2 12221 df-n0 12419 df-z 12505 df-uz 12769 df-fl 13703 df-seq 13913 df-exp 13974 df-bits 16307 |
This theorem is referenced by: (None) |
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