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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 11268 | . . 3 ⊢ 0 ∈ ℝ* | |
2 | sgnval 15042 | . . 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 2731 | . . 3 ⊢ 0 = 0 | |
5 | 4 | iftruei 4535 | . 2 ⊢ if(0 = 0, 0, if(0 < 0, -1, 1)) = 0 |
6 | 3, 5 | eqtri 2759 | 1 ⊢ (sgn‘0) = 0 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1540 ∈ wcel 2105 ifcif 4528 class class class wbr 5148 ‘cfv 6543 0cc0 11116 1c1 11117 ℝ*cxr 11254 < clt 11255 -cneg 11452 sgncsgn 15040 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-i2m1 11184 ax-rnegex 11187 ax-cnre 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-iota 6495 df-fun 6545 df-fv 6551 df-ov 7415 df-xr 11259 df-neg 11454 df-sgn 15041 |
This theorem is referenced by: sgncl 33850 sgnmul 33854 sgnsgn 33860 signstfveq0 33901 |
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