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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 9913 | . . 3 ⊢ 0 ∈ V | |
2 | blenval 42163 | . . 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 2610 | . . 3 ⊢ 0 = 0 | |
5 | 4 | iftruei 4043 | . 2 ⊢ if(0 = 0, 1, ((⌊‘(2 logb (abs‘0))) + 1)) = 1 |
6 | 3, 5 | eqtri 2632 | 1 ⊢ (#b‘0) = 1 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∈ wcel 1977 Vcvv 3173 ifcif 4036 ‘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 ax-icn 9874 ax-addcl 9875 ax-mulcl 9877 ax-i2m1 9883 |
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: blennn0elnn 42169 blen1b 42180 nn0sumshdiglem1 42213 |
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