Theorem blenval 49392

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 49391	. 2 ⊢ #b = (𝑛 ∈ V ↦ if(𝑛 = 0, 1, ((⌊‘(2 logb (abs‘𝑛))) + 1)))
2		eqeq1 2770	. . 3 ⊢ (𝑛 = 𝑁 → (𝑛 = 0 ↔ 𝑁 = 0))
3		fveq2 6888	. . . . . 6 ⊢ (𝑛 = 𝑁 → (abs‘𝑛) = (abs‘𝑁))
4	3	oveq2d 7439	. . . . 5 ⊢ (𝑛 = 𝑁 → (2 logb (abs‘𝑛)) = (2 logb (abs‘𝑁)))
5	4	fveq2d 6892	. . . 4 ⊢ (𝑛 = 𝑁 → (⌊‘(2 logb (abs‘𝑛))) = (⌊‘(2 logb (abs‘𝑁))))
6	5	oveq1d 7438	. . 3 ⊢ (𝑛 = 𝑁 → ((⌊‘(2 logb (abs‘𝑛))) + 1) = ((⌊‘(2 logb (abs‘𝑁))) + 1))
7	2, 6	ifbieq2d 4519	. 2 ⊢ (𝑛 = 𝑁 → if(𝑛 = 0, 1, ((⌊‘(2 logb (abs‘𝑛))) + 1)) = if(𝑁 = 0, 1, ((⌊‘(2 logb (abs‘𝑁))) + 1)))
8		elex 3479	. 2 ⊢ (𝑁 ∈ 𝑉 → 𝑁 ∈ V)
9		1ex 11221	. . . 4 ⊢ 1 ∈ V
10		ovex 7456	. . . 4 ⊢ ((⌊‘(2 logb (abs‘𝑁))) + 1) ∈ V
11	9, 10	ifex 4543	. . 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 7020	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 2146  Vcvv 3458  ifcif 4492  ‘cfv 6543  (class class class)co 7423  0cc0 11118  1c1 11119   + caddc 11121  2c2 12313  ⌊cfl 13843  abscabs 15311   log_b clogb 26966  #_bcblen 49390

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pr 5409  ax-1cn 11176

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 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-iota 6499  df-fun 6545  df-fv 6551  df-ov 7426  df-blen 49391

This theorem is used by:  blen0  49393  blenn0  49394