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Theorem sgnval 14051
Description: Value of Signum function. Pronounced "signum" . See df-sgn 14050. (Contributed by David A. Wheeler, 15-May-2015.)
Assertion
Ref Expression
sgnval (𝐴 ∈ ℝ* → (sgn‘𝐴) = if(𝐴 = 0, 0, if(𝐴 < 0, -1, 1)))

Proof of Theorem sgnval
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eqeq1 2810 . . 3 (𝑥 = 𝐴 → (𝑥 = 0 ↔ 𝐴 = 0))
2 breq1 4847 . . . 4 (𝑥 = 𝐴 → (𝑥 < 0 ↔ 𝐴 < 0))
32ifbid 4301 . . 3 (𝑥 = 𝐴 → if(𝑥 < 0, -1, 1) = if(𝐴 < 0, -1, 1))
41, 3ifbieq2d 4304 . 2 (𝑥 = 𝐴 → if(𝑥 = 0, 0, if(𝑥 < 0, -1, 1)) = if(𝐴 = 0, 0, if(𝐴 < 0, -1, 1)))
5 df-sgn 14050 . 2 sgn = (𝑥 ∈ ℝ* ↦ if(𝑥 = 0, 0, if(𝑥 < 0, -1, 1)))
6 c0ex 10319 . . 3 0 ∈ V
7 negex 10564 . . . 4 -1 ∈ V
8 1ex 10321 . . . 4 1 ∈ V
97, 8ifex 4327 . . 3 if(𝐴 < 0, -1, 1) ∈ V
106, 9ifex 4327 . 2 if(𝐴 = 0, 0, if(𝐴 < 0, -1, 1)) ∈ V
114, 5, 10fvmpt 6503 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 1637  wcel 2156  ifcif 4279   class class class wbr 4844  cfv 6101  0cc0 10221  1c1 10222  *cxr 10358   < clt 10359  -cneg 10552  sgncsgn 14049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pr 5096  ax-1cn 10279  ax-icn 10280  ax-addcl 10281  ax-mulcl 10283  ax-i2m1 10289
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ral 3101  df-rex 3102  df-rab 3105  df-v 3393  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-iota 6064  df-fun 6103  df-fv 6109  df-ov 6877  df-neg 10554  df-sgn 14050
This theorem is referenced by:  sgn0  14052  sgnp  14053  sgnn  14057  sgnneg  30927  sgn3da  30928
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