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Mirrors > Home > MPE Home > Th. List > sgn0 | Structured version Visualization version GIF version |
Description: The 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 11285 | . . 3 ⊢ 0 ∈ ℝ* | |
2 | sgnval 15061 | . . 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 2728 | . . 3 ⊢ 0 = 0 | |
5 | 4 | iftruei 4531 | . 2 ⊢ if(0 = 0, 0, if(0 < 0, -1, 1)) = 0 |
6 | 3, 5 | eqtri 2756 | 1 ⊢ (sgn‘0) = 0 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ∈ wcel 2099 ifcif 4524 class class class wbr 5142 ‘cfv 6542 0cc0 11132 1c1 11133 ℝ*cxr 11271 < clt 11272 -cneg 11469 sgncsgn 15059 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pr 5423 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-i2m1 11200 ax-rnegex 11203 ax-cnre 11205 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3472 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-iota 6494 df-fun 6544 df-fv 6550 df-ov 7417 df-xr 11276 df-neg 11471 df-sgn 15060 |
This theorem is referenced by: sgncl 34152 sgnmul 34156 sgnsgn 34162 signstfveq0 34203 |
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