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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 11286 | . . 3 ⊢ 0 ∈ V | |
2 | blenval 48307 | . . 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 2740 | . . 3 ⊢ 0 = 0 | |
5 | 4 | iftruei 4555 | . 2 ⊢ if(0 = 0, 1, ((⌊‘(2 logb (abs‘0))) + 1)) = 1 |
6 | 3, 5 | eqtri 2768 | 1 ⊢ (#b‘0) = 1 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2108 Vcvv 3488 ifcif 4548 ‘cfv 6575 (class class class)co 7450 0cc0 11186 1c1 11187 + caddc 11189 2c2 12350 ⌊cfl 13843 abscabs 15285 logb clogb 26827 #bcblen 48305 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pr 5447 ax-1cn 11244 ax-icn 11245 ax-addcl 11246 ax-mulcl 11248 ax-i2m1 11254 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-dif 3979 df-un 3981 df-ss 3993 df-nul 4353 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-iota 6527 df-fun 6577 df-fv 6583 df-ov 7453 df-blen 48306 |
This theorem is referenced by: blennn0elnn 48313 blen1b 48324 nn0sumshdiglem1 48357 |
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