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Theorem blenval 45350
 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 45349 . 2 #b = (𝑛 ∈ V ↦ if(𝑛 = 0, 1, ((⌊‘(2 logb (abs‘𝑛))) + 1)))
2 eqeq1 2762 . . 3 (𝑛 = 𝑁 → (𝑛 = 0 ↔ 𝑁 = 0))
3 fveq2 6658 . . . . . 6 (𝑛 = 𝑁 → (abs‘𝑛) = (abs‘𝑁))
43oveq2d 7166 . . . . 5 (𝑛 = 𝑁 → (2 logb (abs‘𝑛)) = (2 logb (abs‘𝑁)))
54fveq2d 6662 . . . 4 (𝑛 = 𝑁 → (⌊‘(2 logb (abs‘𝑛))) = (⌊‘(2 logb (abs‘𝑁))))
65oveq1d 7165 . . 3 (𝑛 = 𝑁 → ((⌊‘(2 logb (abs‘𝑛))) + 1) = ((⌊‘(2 logb (abs‘𝑁))) + 1))
72, 6ifbieq2d 4446 . 2 (𝑛 = 𝑁 → if(𝑛 = 0, 1, ((⌊‘(2 logb (abs‘𝑛))) + 1)) = if(𝑁 = 0, 1, ((⌊‘(2 logb (abs‘𝑁))) + 1)))
8 elex 3428 . 2 (𝑁𝑉𝑁 ∈ V)
9 1ex 10675 . . . 4 1 ∈ V
10 ovex 7183 . . . 4 ((⌊‘(2 logb (abs‘𝑁))) + 1) ∈ V
119, 10ifex 4470 . . 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 6782 1 (𝑁𝑉 → (#b𝑁) = if(𝑁 = 0, 1, ((⌊‘(2 logb (abs‘𝑁))) + 1)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  Vcvv 3409  ifcif 4420  ‘cfv 6335  (class class class)co 7150  0cc0 10575  1c1 10576   + caddc 10578  2c2 11729  ⌊cfl 13209  abscabs 14641   logb clogb 25449  #bcblen 45348 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pr 5298  ax-1cn 10633 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3697  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-iota 6294  df-fun 6337  df-fv 6343  df-ov 7153  df-blen 45349 This theorem is referenced by:  blen0  45351  blenn0  45352
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