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Mirrors > Home > MPE Home > Th. List > sgn0 | Structured version Visualization version GIF version |
Description: Proof that signum of 0 is 0. (Contributed by David A. Wheeler, 15-May-2015.) |
Ref | Expression |
---|---|
sgn0 | ⊢ (sgn‘0) = 0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0xr 9965 | . . 3 ⊢ 0 ∈ ℝ* | |
2 | sgnval 13676 | . . 3 ⊢ (0 ∈ ℝ* → (sgn‘0) = if(0 = 0, 0, if(0 < 0, -1, 1))) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ (sgn‘0) = if(0 = 0, 0, if(0 < 0, -1, 1)) |
4 | eqid 2610 | . . 3 ⊢ 0 = 0 | |
5 | 4 | iftruei 4043 | . 2 ⊢ if(0 = 0, 0, if(0 < 0, -1, 1)) = 0 |
6 | 3, 5 | eqtri 2632 | 1 ⊢ (sgn‘0) = 0 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∈ wcel 1977 ifcif 4036 class class class wbr 4583 ‘cfv 5804 0cc0 9815 1c1 9816 ℝ*cxr 9952 < clt 9953 -cneg 10146 sgncsgn 13674 |
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-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-i2m1 9883 ax-1ne0 9884 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 |
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-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 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-xr 9957 df-neg 10148 df-sgn 13675 |
This theorem is referenced by: sgncl 29927 sgnmul 29931 sgnsgn 29937 signstfveq0 29980 |
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