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Theorem rexneg 9500
Description: Minus a real number. Remark [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.) (Proof shortened by Mario Carneiro, 20-Aug-2015.)
Assertion
Ref Expression
rexneg (𝐴 ∈ ℝ → -𝑒𝐴 = -𝐴)

Proof of Theorem rexneg
StepHypRef Expression
1 df-xneg 9446 . 2 -𝑒𝐴 = if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴))
2 renepnf 7731 . . . 4 (𝐴 ∈ ℝ → 𝐴 ≠ +∞)
3 ifnefalse 3449 . . . 4 (𝐴 ≠ +∞ → if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴)) = if(𝐴 = -∞, +∞, -𝐴))
42, 3syl 14 . . 3 (𝐴 ∈ ℝ → if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴)) = if(𝐴 = -∞, +∞, -𝐴))
5 renemnf 7732 . . . 4 (𝐴 ∈ ℝ → 𝐴 ≠ -∞)
6 ifnefalse 3449 . . . 4 (𝐴 ≠ -∞ → if(𝐴 = -∞, +∞, -𝐴) = -𝐴)
75, 6syl 14 . . 3 (𝐴 ∈ ℝ → if(𝐴 = -∞, +∞, -𝐴) = -𝐴)
84, 7eqtrd 2145 . 2 (𝐴 ∈ ℝ → if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴)) = -𝐴)
91, 8syl5eq 2157 1 (𝐴 ∈ ℝ → -𝑒𝐴 = -𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1312  wcel 1461  wne 2280  ifcif 3438  cr 7540  +∞cpnf 7715  -∞cmnf 7716  -cneg 7851  -𝑒cxne 9443
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-un 4313  ax-setind 4410  ax-cnex 7630  ax-resscn 7631
This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-fal 1318  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ne 2281  df-nel 2376  df-ral 2393  df-rex 2394  df-rab 2397  df-v 2657  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-if 3439  df-pw 3476  df-sn 3497  df-pr 3498  df-uni 3701  df-pnf 7720  df-mnf 7721  df-xneg 9446
This theorem is referenced by:  xneg0  9501  xnegcl  9502  xnegneg  9503  xltnegi  9505  rexsub  9523  xnegid  9529  xnegdi  9538  xpncan  9541  xnpcan  9542  xposdif  9552  xrmaxaddlem  10915  xrminrecl  10928  xrminrpcl  10929
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