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Theorem rexneg 8473
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 8419 . 2 -𝑒A = if(A = +∞, -∞, if(A = -∞, +∞, -A))
2 renepnf 6830 . . . 4 (A ℝ → A ≠ +∞)
3 ifnefalse 3336 . . . 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 6831 . . . 4 (A ℝ → A ≠ -∞)
6 ifnefalse 3336 . . . 4 (A ≠ -∞ → if(A = -∞, +∞, -A) = -A)
75, 6syl 14 . . 3 (A ℝ → if(A = -∞, +∞, -A) = -A)
84, 7eqtrd 2069 . 2 (A ℝ → if(A = +∞, -∞, if(A = -∞, +∞, -A)) = -A)
91, 8syl5eq 2081 1 (A ℝ → -𝑒A = -A)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1242   wcel 1390  wne 2201  ifcif 3325  cr 6670  +∞cpnf 6814  -∞cmnf 6815  -cneg 6940  -𝑒cxne 8416
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-un 4136  ax-setind 4220  ax-cnex 6734  ax-resscn 6735
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-nel 2204  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-if 3326  df-pw 3353  df-sn 3373  df-pr 3374  df-uni 3572  df-pnf 6819  df-mnf 6820  df-xneg 8419
This theorem is referenced by:  xneg0  8474  xnegcl  8475  xnegneg  8476  xltnegi  8478
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