Theorem blenval 49509

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 49508	. 2 ⊢ #b = (𝑛 ∈ V ↦ if(𝑛 = 0, 1, ((⌊‘(2 logb (abs‘𝑛))) + 1)))
2		eqeq1 2766	. . 3 ⊢ (𝑛 = 𝑁 → (𝑛 = 0 ↔ 𝑁 = 0))
3		fveq2 6882	. . . . . 6 ⊢ (𝑛 = 𝑁 → (abs‘𝑛) = (abs‘𝑁))
4	3	oveq2d 7433	. . . . 5 ⊢ (𝑛 = 𝑁 → (2 logb (abs‘𝑛)) = (2 logb (abs‘𝑁)))
5	4	fveq2d 6886	. . . 4 ⊢ (𝑛 = 𝑁 → (⌊‘(2 logb (abs‘𝑛))) = (⌊‘(2 logb (abs‘𝑁))))
6	5	oveq1d 7432	. . 3 ⊢ (𝑛 = 𝑁 → ((⌊‘(2 logb (abs‘𝑛))) + 1) = ((⌊‘(2 logb (abs‘𝑁))) + 1))
7	2, 6	ifbieq2d 4512	. 2 ⊢ (𝑛 = 𝑁 → if(𝑛 = 0, 1, ((⌊‘(2 logb (abs‘𝑛))) + 1)) = if(𝑁 = 0, 1, ((⌊‘(2 logb (abs‘𝑁))) + 1)))
8		elex 3474	. 2 ⊢ (𝑁 ∈ 𝑉 → 𝑁 ∈ V)
9		1ex 11231	. . . 4 ⊢ 1 ∈ V
10		ovex 7450	. . . 4 ⊢ ((⌊‘(2 logb (abs‘𝑁))) + 1) ∈ V
11	9, 10	ifex 4536	. . 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 7014	1 ⊢ (𝑁 ∈ 𝑉 → (#b‘𝑁) = if(𝑁 = 0, 1, ((⌊‘(2 logb (abs‘𝑁))) + 1)))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  Vcvv 3453  ifcif 4485  ‘cfv 6537  (class class class)co 7417  0cc0 11128  1c1 11129   + caddc 11131  2c2 12323  ⌊cfl 13855  abscabs 15325   log_b clogb 27009  #_bcblen 49507

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402  ax-1cn 11186

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-iota 6493  df-fun 6539  df-fv 6545  df-ov 7420  df-blen 49508

This theorem is used by:  blen0  49510  blenn0  49511