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Theorem rexneg 8743
 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 8689 . 2 -𝑒𝐴 = if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴))
2 renepnf 7073 . . . 4 (𝐴 ∈ ℝ → 𝐴 ≠ +∞)
3 ifnefalse 3342 . . . 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 7074 . . . 4 (𝐴 ∈ ℝ → 𝐴 ≠ -∞)
6 ifnefalse 3342 . . . 4 (𝐴 ≠ -∞ → if(𝐴 = -∞, +∞, -𝐴) = -𝐴)
75, 6syl 14 . . 3 (𝐴 ∈ ℝ → if(𝐴 = -∞, +∞, -𝐴) = -𝐴)
84, 7eqtrd 2072 . 2 (𝐴 ∈ ℝ → if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴)) = -𝐴)
91, 8syl5eq 2084 1 (𝐴 ∈ ℝ → -𝑒𝐴 = -𝐴)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1243   ∈ wcel 1393   ≠ wne 2204  ifcif 3331  ℝcr 6888  +∞cpnf 7057  -∞cmnf 7058  -cneg 7183  -𝑒cxne 8686 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-un 4170  ax-setind 4262  ax-cnex 6975  ax-resscn 6976 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-nel 2207  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-if 3332  df-pw 3361  df-sn 3381  df-pr 3382  df-uni 3581  df-pnf 7062  df-mnf 7063  df-xneg 8689 This theorem is referenced by:  xneg0  8744  xnegcl  8745  xnegneg  8746  xltnegi  8748
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