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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 2741 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥 = 0 ↔ 𝐴 = 0)) | |
2 | breq1 5113 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 < 0 ↔ 𝐴 < 0)) | |
3 | 2 | ifbid 4514 | . . 3 ⊢ (𝑥 = 𝐴 → if(𝑥 < 0, -1, 1) = if(𝐴 < 0, -1, 1)) |
4 | 1, 3 | ifbieq2d 4517 | . 2 ⊢ (𝑥 = 𝐴 → if(𝑥 = 0, 0, if(𝑥 < 0, -1, 1)) = if(𝐴 = 0, 0, if(𝐴 < 0, -1, 1))) |
5 | df-sgn 14979 | . 2 ⊢ sgn = (𝑥 ∈ ℝ* ↦ if(𝑥 = 0, 0, if(𝑥 < 0, -1, 1))) | |
6 | c0ex 11156 | . . 3 ⊢ 0 ∈ V | |
7 | negex 11406 | . . . 4 ⊢ -1 ∈ V | |
8 | 1ex 11158 | . . . 4 ⊢ 1 ∈ V | |
9 | 7, 8 | ifex 4541 | . . 3 ⊢ if(𝐴 < 0, -1, 1) ∈ V |
10 | 6, 9 | ifex 4541 | . 2 ⊢ if(𝐴 = 0, 0, if(𝐴 < 0, -1, 1)) ∈ V |
11 | 4, 5, 10 | fvmpt 6953 | 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 1542 ∈ wcel 2107 ifcif 4491 class class class wbr 5110 ‘cfv 6501 0cc0 11058 1c1 11059 ℝ*cxr 11195 < clt 11196 -cneg 11393 sgncsgn 14978 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-mulcl 11120 ax-i2m1 11126 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-iota 6453 df-fun 6503 df-fv 6509 df-ov 7365 df-neg 11395 df-sgn 14979 |
This theorem is referenced by: sgn0 14981 sgnp 14982 sgnn 14986 sgnneg 33180 sgn3da 33181 reabssgn 41982 |
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