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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 10370 | . . 3 ⊢ 0 ∈ V | |
2 | blenval 43384 | . . 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 2778 | . . 3 ⊢ 0 = 0 | |
5 | 4 | iftruei 4314 | . 2 ⊢ if(0 = 0, 1, ((⌊‘(2 logb (abs‘0))) + 1)) = 1 |
6 | 3, 5 | eqtri 2802 | 1 ⊢ (#b‘0) = 1 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1601 ∈ wcel 2107 Vcvv 3398 ifcif 4307 ‘cfv 6135 (class class class)co 6922 0cc0 10272 1c1 10273 + caddc 10275 2c2 11430 ⌊cfl 12910 abscabs 14381 logb clogb 24942 #bcblen 43382 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pr 5138 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-mulcl 10334 ax-i2m1 10340 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-iota 6099 df-fun 6137 df-fv 6143 df-ov 6925 df-blen 43383 |
This theorem is referenced by: blennn0elnn 43390 blen1b 43401 nn0sumshdiglem1 43434 |
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