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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 5069 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 < 0 ↔ 𝐴 < 0)) | |
3 | 2 | ifbid 4489 | . . 3 ⊢ (𝑥 = 𝐴 → if(𝑥 < 0, -1, 1) = if(𝐴 < 0, -1, 1)) |
4 | 1, 3 | ifbieq2d 4492 | . 2 ⊢ (𝑥 = 𝐴 → if(𝑥 = 0, 0, if(𝑥 < 0, -1, 1)) = if(𝐴 = 0, 0, if(𝐴 < 0, -1, 1))) |
5 | df-sgn 14446 | . 2 ⊢ sgn = (𝑥 ∈ ℝ* ↦ if(𝑥 = 0, 0, if(𝑥 < 0, -1, 1))) | |
6 | c0ex 10635 | . . 3 ⊢ 0 ∈ V | |
7 | negex 10884 | . . . 4 ⊢ -1 ∈ V | |
8 | 1ex 10637 | . . . 4 ⊢ 1 ∈ V | |
9 | 7, 8 | ifex 4515 | . . 3 ⊢ if(𝐴 < 0, -1, 1) ∈ V |
10 | 6, 9 | ifex 4515 | . 2 ⊢ if(𝐴 = 0, 0, if(𝐴 < 0, -1, 1)) ∈ V |
11 | 4, 5, 10 | fvmpt 6768 | 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 1537 ∈ wcel 2114 ifcif 4467 class class class wbr 5066 ‘cfv 6355 0cc0 10537 1c1 10538 ℝ*cxr 10674 < clt 10675 -cneg 10871 sgncsgn 14445 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-mulcl 10599 ax-i2m1 10605 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-iota 6314 df-fun 6357 df-fv 6363 df-ov 7159 df-neg 10873 df-sgn 14446 |
This theorem is referenced by: sgn0 14448 sgnp 14449 sgnn 14453 sgnneg 31798 sgn3da 31799 |
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