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Theorem blenval 47345
Description: The binary length of an integer. (Contributed by AV, 20-May-2020.)
Assertion
Ref Expression
blenval (𝑁𝑉 → (#b𝑁) = if(𝑁 = 0, 1, ((⌊‘(2 logb (abs‘𝑁))) + 1)))

Proof of Theorem blenval
Dummy variable 𝑛 is distinct from all other variables.
StepHypRef Expression
1 df-blen 47344 . 2 #b = (𝑛 ∈ V ↦ if(𝑛 = 0, 1, ((⌊‘(2 logb (abs‘𝑛))) + 1)))
2 eqeq1 2736 . . 3 (𝑛 = 𝑁 → (𝑛 = 0 ↔ 𝑁 = 0))
3 fveq2 6891 . . . . . 6 (𝑛 = 𝑁 → (abs‘𝑛) = (abs‘𝑁))
43oveq2d 7427 . . . . 5 (𝑛 = 𝑁 → (2 logb (abs‘𝑛)) = (2 logb (abs‘𝑁)))
54fveq2d 6895 . . . 4 (𝑛 = 𝑁 → (⌊‘(2 logb (abs‘𝑛))) = (⌊‘(2 logb (abs‘𝑁))))
65oveq1d 7426 . . 3 (𝑛 = 𝑁 → ((⌊‘(2 logb (abs‘𝑛))) + 1) = ((⌊‘(2 logb (abs‘𝑁))) + 1))
72, 6ifbieq2d 4554 . 2 (𝑛 = 𝑁 → if(𝑛 = 0, 1, ((⌊‘(2 logb (abs‘𝑛))) + 1)) = if(𝑁 = 0, 1, ((⌊‘(2 logb (abs‘𝑁))) + 1)))
8 elex 3492 . 2 (𝑁𝑉𝑁 ∈ V)
9 1ex 11214 . . . 4 1 ∈ V
10 ovex 7444 . . . 4 ((⌊‘(2 logb (abs‘𝑁))) + 1) ∈ V
119, 10ifex 4578 . . 3 if(𝑁 = 0, 1, ((⌊‘(2 logb (abs‘𝑁))) + 1)) ∈ V
1211a1i 11 . 2 (𝑁𝑉 → if(𝑁 = 0, 1, ((⌊‘(2 logb (abs‘𝑁))) + 1)) ∈ V)
131, 7, 8, 12fvmptd3 7021 1 (𝑁𝑉 → (#b𝑁) = if(𝑁 = 0, 1, ((⌊‘(2 logb (abs‘𝑁))) + 1)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2106  Vcvv 3474  ifcif 4528  cfv 6543  (class class class)co 7411  0cc0 11112  1c1 11113   + caddc 11115  2c2 12271  cfl 13759  abscabs 15185   logb clogb 26493  #bcblen 47343
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-1cn 11170
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  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-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7414  df-blen 47344
This theorem is referenced by:  blen0  47346  blenn0  47347
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