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Mirrors > Home > MPE Home > Th. List > sgnval | Structured version Visualization version GIF version |
Description: Value of the signum function. (Contributed by David A. Wheeler, 15-May-2015.) |
Ref | Expression |
---|---|
sgnval | ⊢ (𝐴 ∈ ℝ* → (sgn‘𝐴) = if(𝐴 = 0, 0, if(𝐴 < 0, -1, 1))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq1 2802 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥 = 0 ↔ 𝐴 = 0)) | |
2 | breq1 5033 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 < 0 ↔ 𝐴 < 0)) | |
3 | 2 | ifbid 4447 | . . 3 ⊢ (𝑥 = 𝐴 → if(𝑥 < 0, -1, 1) = if(𝐴 < 0, -1, 1)) |
4 | 1, 3 | ifbieq2d 4450 | . 2 ⊢ (𝑥 = 𝐴 → if(𝑥 = 0, 0, if(𝑥 < 0, -1, 1)) = if(𝐴 = 0, 0, if(𝐴 < 0, -1, 1))) |
5 | df-sgn 14438 | . 2 ⊢ sgn = (𝑥 ∈ ℝ* ↦ if(𝑥 = 0, 0, if(𝑥 < 0, -1, 1))) | |
6 | c0ex 10624 | . . 3 ⊢ 0 ∈ V | |
7 | negex 10873 | . . . 4 ⊢ -1 ∈ V | |
8 | 1ex 10626 | . . . 4 ⊢ 1 ∈ V | |
9 | 7, 8 | ifex 4473 | . . 3 ⊢ if(𝐴 < 0, -1, 1) ∈ V |
10 | 6, 9 | ifex 4473 | . 2 ⊢ if(𝐴 = 0, 0, if(𝐴 < 0, -1, 1)) ∈ V |
11 | 4, 5, 10 | fvmpt 6745 | 1 ⊢ (𝐴 ∈ ℝ* → (sgn‘𝐴) = if(𝐴 = 0, 0, if(𝐴 < 0, -1, 1))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ifcif 4425 class class class wbr 5030 ‘cfv 6324 0cc0 10526 1c1 10527 ℝ*cxr 10663 < clt 10664 -cneg 10860 sgncsgn 14437 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-mulcl 10588 ax-i2m1 10594 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fv 6332 df-ov 7138 df-neg 10862 df-sgn 14438 |
This theorem is referenced by: sgn0 14440 sgnp 14441 sgnn 14445 sgnneg 31908 sgn3da 31909 reabssgn 40336 |
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