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Theorem rexneg 9833
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 9775 . 2 -𝑒𝐴 = if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴))
2 renepnf 8008 . . . 4 (𝐴 ∈ ℝ → 𝐴 ≠ +∞)
3 ifnefalse 3547 . . . 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 8009 . . . 4 (𝐴 ∈ ℝ → 𝐴 ≠ -∞)
6 ifnefalse 3547 . . . 4 (𝐴 ≠ -∞ → if(𝐴 = -∞, +∞, -𝐴) = -𝐴)
75, 6syl 14 . . 3 (𝐴 ∈ ℝ → if(𝐴 = -∞, +∞, -𝐴) = -𝐴)
84, 7eqtrd 2210 . 2 (𝐴 ∈ ℝ → if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴)) = -𝐴)
91, 8eqtrid 2222 1 (𝐴 ∈ ℝ → -𝑒𝐴 = -𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1353  wcel 2148  wne 2347  ifcif 3536  cr 7813  +∞cpnf 7992  -∞cmnf 7993  -cneg 8132  -𝑒cxne 9772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-un 4435  ax-setind 4538  ax-cnex 7905  ax-resscn 7906
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-uni 3812  df-pnf 7997  df-mnf 7998  df-xneg 9775
This theorem is referenced by:  xneg0  9834  xnegcl  9835  xnegneg  9836  xltnegi  9838  rexsub  9856  xnegid  9862  xnegdi  9871  xpncan  9874  xnpcan  9875  xposdif  9885  xrmaxaddlem  11271  xrminrecl  11284  xrminrpcl  11285
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