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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 42162 | . . 3 ⊢ #b = (𝑛 ∈ V ↦ if(𝑛 = 0, 1, ((⌊‘(2 logb (abs‘𝑛))) + 1))) | |
2 | 1 | a1i 11 | . 2 ⊢ (𝑁 ∈ 𝑉 → #b = (𝑛 ∈ V ↦ if(𝑛 = 0, 1, ((⌊‘(2 logb (abs‘𝑛))) + 1)))) |
3 | eqeq1 2614 | . . . 4 ⊢ (𝑛 = 𝑁 → (𝑛 = 0 ↔ 𝑁 = 0)) | |
4 | fveq2 6103 | . . . . . . 7 ⊢ (𝑛 = 𝑁 → (abs‘𝑛) = (abs‘𝑁)) | |
5 | 4 | oveq2d 6565 | . . . . . 6 ⊢ (𝑛 = 𝑁 → (2 logb (abs‘𝑛)) = (2 logb (abs‘𝑁))) |
6 | 5 | fveq2d 6107 | . . . . 5 ⊢ (𝑛 = 𝑁 → (⌊‘(2 logb (abs‘𝑛))) = (⌊‘(2 logb (abs‘𝑁)))) |
7 | 6 | oveq1d 6564 | . . . 4 ⊢ (𝑛 = 𝑁 → ((⌊‘(2 logb (abs‘𝑛))) + 1) = ((⌊‘(2 logb (abs‘𝑁))) + 1)) |
8 | 3, 7 | ifbieq2d 4061 | . . 3 ⊢ (𝑛 = 𝑁 → if(𝑛 = 0, 1, ((⌊‘(2 logb (abs‘𝑛))) + 1)) = if(𝑁 = 0, 1, ((⌊‘(2 logb (abs‘𝑁))) + 1))) |
9 | 8 | adantl 481 | . 2 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑛 = 𝑁) → if(𝑛 = 0, 1, ((⌊‘(2 logb (abs‘𝑛))) + 1)) = if(𝑁 = 0, 1, ((⌊‘(2 logb (abs‘𝑁))) + 1))) |
10 | elex 3185 | . 2 ⊢ (𝑁 ∈ 𝑉 → 𝑁 ∈ V) | |
11 | 1ex 9914 | . . . 4 ⊢ 1 ∈ V | |
12 | ovex 6577 | . . . 4 ⊢ ((⌊‘(2 logb (abs‘𝑁))) + 1) ∈ V | |
13 | 11, 12 | ifex 4106 | . . 3 ⊢ if(𝑁 = 0, 1, ((⌊‘(2 logb (abs‘𝑁))) + 1)) ∈ V |
14 | 13 | a1i 11 | . 2 ⊢ (𝑁 ∈ 𝑉 → if(𝑁 = 0, 1, ((⌊‘(2 logb (abs‘𝑁))) + 1)) ∈ V) |
15 | 2, 9, 10, 14 | fvmptd 6197 | 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 1475 ∈ wcel 1977 Vcvv 3173 ifcif 4036 ↦ cmpt 4643 ‘cfv 5804 (class class class)co 6549 0cc0 9815 1c1 9816 + caddc 9818 2c2 10947 ⌊cfl 12453 abscabs 13822 logb clogb 24302 #bcblen 42161 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-1cn 9873 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-ov 6552 df-blen 42162 |
This theorem is referenced by: blen0 42164 blenn0 42165 |
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