Theorem blenval 49334

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 49333	. 2 ⊢ #b = (𝑛 ∈ V ↦ if(𝑛 = 0, 1, ((⌊‘(2 logb (abs‘𝑛))) + 1)))
2		eqeq1 2767	. . 3 ⊢ (𝑛 = 𝑁 → (𝑛 = 0 ↔ 𝑁 = 0))
3		fveq2 6883	. . . . . 6 ⊢ (𝑛 = 𝑁 → (abs‘𝑛) = (abs‘𝑁))
4	3	oveq2d 7428	. . . . 5 ⊢ (𝑛 = 𝑁 → (2 logb (abs‘𝑛)) = (2 logb (abs‘𝑁)))
5	4	fveq2d 6887	. . . 4 ⊢ (𝑛 = 𝑁 → (⌊‘(2 logb (abs‘𝑛))) = (⌊‘(2 logb (abs‘𝑁))))
6	5	oveq1d 7427	. . 3 ⊢ (𝑛 = 𝑁 → ((⌊‘(2 logb (abs‘𝑛))) + 1) = ((⌊‘(2 logb (abs‘𝑁))) + 1))
7	2, 6	ifbieq2d 4515	. 2 ⊢ (𝑛 = 𝑁 → if(𝑛 = 0, 1, ((⌊‘(2 logb (abs‘𝑛))) + 1)) = if(𝑁 = 0, 1, ((⌊‘(2 logb (abs‘𝑁))) + 1)))
8		elex 3476	. 2 ⊢ (𝑁 ∈ 𝑉 → 𝑁 ∈ V)
9		1ex 11204	. . . 4 ⊢ 1 ∈ V
10		ovex 7445	. . . 4 ⊢ ((⌊‘(2 logb (abs‘𝑁))) + 1) ∈ V
11	9, 10	ifex 4539	. . 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 7015	1 ⊢ (𝑁 ∈ 𝑉 → (#b‘𝑁) = if(𝑁 = 0, 1, ((⌊‘(2 logb (abs‘𝑁))) + 1)))

Syntax hints:   → wi 4   = wceq 1570   ∈ wcel 2143  Vcvv 3455  ifcif 4488  ‘cfv 6538  (class class class)co 7412  0cc0 11101  1c1 11102   + caddc 11104  2c2 12296  ⌊cfl 13825  abscabs 15287   log_b clogb 26910  #_bcblen 49332

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pr 5406  ax-1cn 11159

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-iota 6494  df-fun 6540  df-fv 6546  df-ov 7415  df-blen 49333

This theorem is referenced by:  blen0  49335  blenn0  49336