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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 44637 | . 2 ⊢ #b = (𝑛 ∈ V ↦ if(𝑛 = 0, 1, ((⌊‘(2 logb (abs‘𝑛))) + 1))) | |
2 | eqeq1 2827 | . . 3 ⊢ (𝑛 = 𝑁 → (𝑛 = 0 ↔ 𝑁 = 0)) | |
3 | fveq2 6672 | . . . . . 6 ⊢ (𝑛 = 𝑁 → (abs‘𝑛) = (abs‘𝑁)) | |
4 | 3 | oveq2d 7174 | . . . . 5 ⊢ (𝑛 = 𝑁 → (2 logb (abs‘𝑛)) = (2 logb (abs‘𝑁))) |
5 | 4 | fveq2d 6676 | . . . 4 ⊢ (𝑛 = 𝑁 → (⌊‘(2 logb (abs‘𝑛))) = (⌊‘(2 logb (abs‘𝑁)))) |
6 | 5 | oveq1d 7173 | . . 3 ⊢ (𝑛 = 𝑁 → ((⌊‘(2 logb (abs‘𝑛))) + 1) = ((⌊‘(2 logb (abs‘𝑁))) + 1)) |
7 | 2, 6 | ifbieq2d 4494 | . 2 ⊢ (𝑛 = 𝑁 → if(𝑛 = 0, 1, ((⌊‘(2 logb (abs‘𝑛))) + 1)) = if(𝑁 = 0, 1, ((⌊‘(2 logb (abs‘𝑁))) + 1))) |
8 | elex 3514 | . 2 ⊢ (𝑁 ∈ 𝑉 → 𝑁 ∈ V) | |
9 | 1ex 10639 | . . . 4 ⊢ 1 ∈ V | |
10 | ovex 7191 | . . . 4 ⊢ ((⌊‘(2 logb (abs‘𝑁))) + 1) ∈ V | |
11 | 9, 10 | ifex 4517 | . . 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 6793 | 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 1537 ∈ wcel 2114 Vcvv 3496 ifcif 4469 ‘cfv 6357 (class class class)co 7158 0cc0 10539 1c1 10540 + caddc 10542 2c2 11695 ⌊cfl 13163 abscabs 14595 logb clogb 25344 #bcblen 44636 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 ax-1cn 10597 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-iota 6316 df-fun 6359 df-fv 6365 df-ov 7161 df-blen 44637 |
This theorem is referenced by: blen0 44639 blenn0 44640 |
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