Theorem blenval 47305

Description: The binary length of an integer. (Contributed by AV, 20-May-2020.)

Assertion

Ref	Expression
blenval	⊢ (𝑁 ∈ 𝑉 → (#b‘𝑁) = if(𝑁 = 0, 1, ((⌊‘(2 logb (abs‘𝑁))) + 1)))

Proof of Theorem blenval

Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-blen 47304	. 2 ⊢ #b = (𝑛 ∈ V ↦ if(𝑛 = 0, 1, ((⌊‘(2 logb (abs‘𝑛))) + 1)))
2		eqeq1 2737	. . 3 ⊢ (𝑛 = 𝑁 → (𝑛 = 0 ↔ 𝑁 = 0))
3		fveq2 6892	. . . . . 6 ⊢ (𝑛 = 𝑁 → (abs‘𝑛) = (abs‘𝑁))
4	3	oveq2d 7425	. . . . 5 ⊢ (𝑛 = 𝑁 → (2 logb (abs‘𝑛)) = (2 logb (abs‘𝑁)))
5	4	fveq2d 6896	. . . 4 ⊢ (𝑛 = 𝑁 → (⌊‘(2 logb (abs‘𝑛))) = (⌊‘(2 logb (abs‘𝑁))))
6	5	oveq1d 7424	. . 3 ⊢ (𝑛 = 𝑁 → ((⌊‘(2 logb (abs‘𝑛))) + 1) = ((⌊‘(2 logb (abs‘𝑁))) + 1))
7	2, 6	ifbieq2d 4555	. 2 ⊢ (𝑛 = 𝑁 → if(𝑛 = 0, 1, ((⌊‘(2 logb (abs‘𝑛))) + 1)) = if(𝑁 = 0, 1, ((⌊‘(2 logb (abs‘𝑁))) + 1)))
8		elex 3493	. 2 ⊢ (𝑁 ∈ 𝑉 → 𝑁 ∈ V)
9		1ex 11210	. . . 4 ⊢ 1 ∈ V
10		ovex 7442	. . . 4 ⊢ ((⌊‘(2 logb (abs‘𝑁))) + 1) ∈ V
11	9, 10	ifex 4579	. . 3 ⊢ if(𝑁 = 0, 1, ((⌊‘(2 logb (abs‘𝑁))) + 1)) ∈ V
12	11	a1i 11	. 2 ⊢ (𝑁 ∈ 𝑉 → if(𝑁 = 0, 1, ((⌊‘(2 logb (abs‘𝑁))) + 1)) ∈ V)
13	1, 7, 8, 12	fvmptd3 7022	1 ⊢ (𝑁 ∈ 𝑉 → (#b‘𝑁) = if(𝑁 = 0, 1, ((⌊‘(2 logb (abs‘𝑁))) + 1)))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2107  Vcvv 3475  ifcif 4529  ‘cfv 6544  (class class class)co 7409  0cc0 11110  1c1 11111   + caddc 11113  2c2 12267  ⌊cfl 13755  abscabs 15181   log_b clogb 26269  #_bcblen 47303

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-1cn 11168

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552  df-ov 7412  df-blen 47304

This theorem is referenced by:  blen0  47306  blenn0  47307