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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > blenval | Structured version Visualization version GIF version |
Description: The binary length of an integer. (Contributed by AV, 20-May-2020.) |
Ref | Expression |
---|---|
blenval | ⊢ (𝑁 ∈ 𝑉 → (#b‘𝑁) = if(𝑁 = 0, 1, ((⌊‘(2 logb (abs‘𝑁))) + 1))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-blen 46628 | . 2 ⊢ #b = (𝑛 ∈ V ↦ if(𝑛 = 0, 1, ((⌊‘(2 logb (abs‘𝑛))) + 1))) | |
2 | eqeq1 2740 | . . 3 ⊢ (𝑛 = 𝑁 → (𝑛 = 0 ↔ 𝑁 = 0)) | |
3 | fveq2 6842 | . . . . . 6 ⊢ (𝑛 = 𝑁 → (abs‘𝑛) = (abs‘𝑁)) | |
4 | 3 | oveq2d 7372 | . . . . 5 ⊢ (𝑛 = 𝑁 → (2 logb (abs‘𝑛)) = (2 logb (abs‘𝑁))) |
5 | 4 | fveq2d 6846 | . . . 4 ⊢ (𝑛 = 𝑁 → (⌊‘(2 logb (abs‘𝑛))) = (⌊‘(2 logb (abs‘𝑁)))) |
6 | 5 | oveq1d 7371 | . . 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 3463 | . 2 ⊢ (𝑁 ∈ 𝑉 → 𝑁 ∈ V) | |
9 | 1ex 11150 | . . . 4 ⊢ 1 ∈ V | |
10 | ovex 7389 | . . . 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 6971 | 1 ⊢ (𝑁 ∈ 𝑉 → (#b‘𝑁) = if(𝑁 = 0, 1, ((⌊‘(2 logb (abs‘𝑁))) + 1))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 Vcvv 3445 ifcif 4486 ‘cfv 6496 (class class class)co 7356 0cc0 11050 1c1 11051 + caddc 11053 2c2 12207 ⌊cfl 13694 abscabs 15118 logb clogb 26112 #bcblen 46627 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pr 5384 ax-1cn 11108 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-rab 3408 df-v 3447 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-br 5106 df-opab 5168 df-mpt 5189 df-id 5531 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-iota 6448 df-fun 6498 df-fv 6504 df-ov 7359 df-blen 46628 |
This theorem is referenced by: blen0 46630 blenn0 46631 |
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