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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 2825 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥 = 0 ↔ 𝐴 = 0)) | |
2 | breq1 5061 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 < 0 ↔ 𝐴 < 0)) | |
3 | 2 | ifbid 4488 | . . 3 ⊢ (𝑥 = 𝐴 → if(𝑥 < 0, -1, 1) = if(𝐴 < 0, -1, 1)) |
4 | 1, 3 | ifbieq2d 4491 | . 2 ⊢ (𝑥 = 𝐴 → if(𝑥 = 0, 0, if(𝑥 < 0, -1, 1)) = if(𝐴 = 0, 0, if(𝐴 < 0, -1, 1))) |
5 | df-sgn 14440 | . 2 ⊢ sgn = (𝑥 ∈ ℝ* ↦ if(𝑥 = 0, 0, if(𝑥 < 0, -1, 1))) | |
6 | c0ex 10629 | . . 3 ⊢ 0 ∈ V | |
7 | negex 10878 | . . . 4 ⊢ -1 ∈ V | |
8 | 1ex 10631 | . . . 4 ⊢ 1 ∈ V | |
9 | 7, 8 | ifex 4514 | . . 3 ⊢ if(𝐴 < 0, -1, 1) ∈ V |
10 | 6, 9 | ifex 4514 | . 2 ⊢ if(𝐴 = 0, 0, if(𝐴 < 0, -1, 1)) ∈ V |
11 | 4, 5, 10 | fvmpt 6762 | 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 1533 ∈ wcel 2110 ifcif 4466 class class class wbr 5058 ‘cfv 6349 0cc0 10531 1c1 10532 ℝ*cxr 10668 < clt 10669 -cneg 10865 sgncsgn 14439 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-mulcl 10593 ax-i2m1 10599 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-iota 6308 df-fun 6351 df-fv 6357 df-ov 7153 df-neg 10867 df-sgn 14440 |
This theorem is referenced by: sgn0 14442 sgnp 14443 sgnn 14447 sgnneg 31793 sgn3da 31794 |
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