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Theorem blenval 42163
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 42162 . . 3 #b = (𝑛 ∈ V ↦ if(𝑛 = 0, 1, ((⌊‘(2 logb (abs‘𝑛))) + 1)))
21a1i 11 . 2 (𝑁𝑉 → #b = (𝑛 ∈ V ↦ if(𝑛 = 0, 1, ((⌊‘(2 logb (abs‘𝑛))) + 1))))
3 eqeq1 2614 . . . 4 (𝑛 = 𝑁 → (𝑛 = 0 ↔ 𝑁 = 0))
4 fveq2 6103 . . . . . . 7 (𝑛 = 𝑁 → (abs‘𝑛) = (abs‘𝑁))
54oveq2d 6565 . . . . . 6 (𝑛 = 𝑁 → (2 logb (abs‘𝑛)) = (2 logb (abs‘𝑁)))
65fveq2d 6107 . . . . 5 (𝑛 = 𝑁 → (⌊‘(2 logb (abs‘𝑛))) = (⌊‘(2 logb (abs‘𝑁))))
76oveq1d 6564 . . . 4 (𝑛 = 𝑁 → ((⌊‘(2 logb (abs‘𝑛))) + 1) = ((⌊‘(2 logb (abs‘𝑁))) + 1))
83, 7ifbieq2d 4061 . . 3 (𝑛 = 𝑁 → if(𝑛 = 0, 1, ((⌊‘(2 logb (abs‘𝑛))) + 1)) = if(𝑁 = 0, 1, ((⌊‘(2 logb (abs‘𝑁))) + 1)))
98adantl 481 . 2 ((𝑁𝑉𝑛 = 𝑁) → if(𝑛 = 0, 1, ((⌊‘(2 logb (abs‘𝑛))) + 1)) = if(𝑁 = 0, 1, ((⌊‘(2 logb (abs‘𝑁))) + 1)))
10 elex 3185 . 2 (𝑁𝑉𝑁 ∈ V)
11 1ex 9914 . . . 4 1 ∈ V
12 ovex 6577 . . . 4 ((⌊‘(2 logb (abs‘𝑁))) + 1) ∈ V
1311, 12ifex 4106 . . 3 if(𝑁 = 0, 1, ((⌊‘(2 logb (abs‘𝑁))) + 1)) ∈ V
1413a1i 11 . 2 (𝑁𝑉 → if(𝑁 = 0, 1, ((⌊‘(2 logb (abs‘𝑁))) + 1)) ∈ V)
152, 9, 10, 14fvmptd 6197 1 (𝑁𝑉 → (#b𝑁) = if(𝑁 = 0, 1, ((⌊‘(2 logb (abs‘𝑁))) + 1)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1475  wcel 1977  Vcvv 3173  ifcif 4036  cmpt 4643  cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816   + caddc 9818  2c2 10947  cfl 12453  abscabs 13822   logb clogb 24302  #bcblen 42161
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-1cn 9873
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-blen 42162
This theorem is referenced by:  blen0  42164  blenn0  42165
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