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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > 0dig2pr01 | Structured version Visualization version GIF version |
Description: The integers 0 and 1 correspond to their last bit. (Contributed by AV, 28-May-2010.) |
Ref | Expression |
---|---|
0dig2pr01 | β’ (π β {0, 1} β (0(digitβ2)π) = π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elpri 4655 | . 2 β’ (π β {0, 1} β (π = 0 β¨ π = 1)) | |
2 | 2nn 12325 | . . . . 5 β’ 2 β β | |
3 | 0z 12609 | . . . . 5 β’ 0 β β€ | |
4 | dig0 47775 | . . . . 5 β’ ((2 β β β§ 0 β β€) β (0(digitβ2)0) = 0) | |
5 | 2, 3, 4 | mp2an 690 | . . . 4 β’ (0(digitβ2)0) = 0 |
6 | oveq2 7434 | . . . 4 β’ (π = 0 β (0(digitβ2)π) = (0(digitβ2)0)) | |
7 | id 22 | . . . 4 β’ (π = 0 β π = 0) | |
8 | 5, 6, 7 | 3eqtr4a 2794 | . . 3 β’ (π = 0 β (0(digitβ2)π) = π) |
9 | 2z 12634 | . . . . 5 β’ 2 β β€ | |
10 | uzid 12877 | . . . . 5 β’ (2 β β€ β 2 β (β€β₯β2)) | |
11 | 0dig1 47778 | . . . . 5 β’ (2 β (β€β₯β2) β (0(digitβ2)1) = 1) | |
12 | 9, 10, 11 | mp2b 10 | . . . 4 β’ (0(digitβ2)1) = 1 |
13 | oveq2 7434 | . . . 4 β’ (π = 1 β (0(digitβ2)π) = (0(digitβ2)1)) | |
14 | id 22 | . . . 4 β’ (π = 1 β π = 1) | |
15 | 12, 13, 14 | 3eqtr4a 2794 | . . 3 β’ (π = 1 β (0(digitβ2)π) = π) |
16 | 8, 15 | jaoi 855 | . 2 β’ ((π = 0 β¨ π = 1) β (0(digitβ2)π) = π) |
17 | 1, 16 | syl 17 | 1 β’ (π β {0, 1} β (0(digitβ2)π) = π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β¨ wo 845 = wceq 1533 β wcel 2098 {cpr 4634 βcfv 6553 (class class class)co 7426 0cc0 11148 1c1 11149 βcn 12252 2c2 12307 β€cz 12598 β€β₯cuz 12862 digitcdig 47764 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 ax-pre-sup 11226 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7879 df-1st 8001 df-2nd 8002 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-er 8733 df-en 8973 df-dom 8974 df-sdom 8975 df-sup 9475 df-inf 9476 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-div 11912 df-nn 12253 df-2 12315 df-n0 12513 df-z 12599 df-uz 12863 df-rp 13017 df-ico 13372 df-fl 13799 df-mod 13877 df-seq 14009 df-exp 14069 df-dig 47765 |
This theorem is referenced by: nn0sumshdiglemB 47789 nn0sumshdiglem2 47791 |
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