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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > blen0 | Structured version Visualization version GIF version |
Description: The binary length of 0. (Contributed by AV, 20-May-2020.) |
Ref | Expression |
---|---|
blen0 | ⊢ (#b‘0) = 1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | c0ex 11240 | . . 3 ⊢ 0 ∈ V | |
2 | blenval 47830 | . . 3 ⊢ (0 ∈ V → (#b‘0) = if(0 = 0, 1, ((⌊‘(2 logb (abs‘0))) + 1))) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ (#b‘0) = if(0 = 0, 1, ((⌊‘(2 logb (abs‘0))) + 1)) |
4 | eqid 2725 | . . 3 ⊢ 0 = 0 | |
5 | 4 | iftruei 4537 | . 2 ⊢ if(0 = 0, 1, ((⌊‘(2 logb (abs‘0))) + 1)) = 1 |
6 | 3, 5 | eqtri 2753 | 1 ⊢ (#b‘0) = 1 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2098 Vcvv 3461 ifcif 4530 ‘cfv 6549 (class class class)co 7419 0cc0 11140 1c1 11141 + caddc 11143 2c2 12300 ⌊cfl 13791 abscabs 15217 logb clogb 26741 #bcblen 47828 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-mulcl 11202 ax-i2m1 11208 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-iota 6501 df-fun 6551 df-fv 6557 df-ov 7422 df-blen 47829 |
This theorem is referenced by: blennn0elnn 47836 blen1b 47847 nn0sumshdiglem1 47880 |
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