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Theorem blen0 47831
Description: The binary length of 0. (Contributed by AV, 20-May-2020.)
Assertion
Ref Expression
blen0 (#b‘0) = 1

Proof of Theorem blen0
StepHypRef Expression
1 c0ex 11240 . . 3 0 ∈ V
2 blenval 47830 . . 3 (0 ∈ V → (#b‘0) = if(0 = 0, 1, ((⌊‘(2 logb (abs‘0))) + 1)))
31, 2ax-mp 5 . 2 (#b‘0) = if(0 = 0, 1, ((⌊‘(2 logb (abs‘0))) + 1))
4 eqid 2725 . . 3 0 = 0
54iftruei 4537 . 2 if(0 = 0, 1, ((⌊‘(2 logb (abs‘0))) + 1)) = 1
63, 5eqtri 2753 1 (#b‘0) = 1
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  wcel 2098  Vcvv 3461  ifcif 4530  cfv 6549  (class class class)co 7419  0cc0 11140  1c1 11141   + caddc 11143  2c2 12300  cfl 13791  abscabs 15217   logb clogb 26741  #bcblen 47828
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 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429  ax-1cn 11198  ax-icn 11199  ax-addcl 11200  ax-mulcl 11202  ax-i2m1 11208
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6501  df-fun 6551  df-fv 6557  df-ov 7422  df-blen 47829
This theorem is referenced by:  blennn0elnn  47836  blen1b  47847  nn0sumshdiglem1  47880
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