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Theorem rexneg 9185
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  e.  RR  ->  -e
A  =  -u A
)

Proof of Theorem rexneg
StepHypRef Expression
1 df-xneg 9136 . 2  |-  -e
A  =  if ( A  = +oo , -oo ,  if ( A  = -oo , +oo ,  -u A ) )
2 renepnf 7436 . . . 4  |-  ( A  e.  RR  ->  A  =/= +oo )
3 ifnefalse 3384 . . . 4  |-  ( A  =/= +oo  ->  if ( A  = +oo , -oo ,  if ( A  = -oo , +oo ,  -u A ) )  =  if ( A  = -oo , +oo ,  -u A ) )
42, 3syl 14 . . 3  |-  ( A  e.  RR  ->  if ( A  = +oo , -oo ,  if ( A  = -oo , +oo ,  -u A ) )  =  if ( A  = -oo , +oo ,  -u A ) )
5 renemnf 7437 . . . 4  |-  ( A  e.  RR  ->  A  =/= -oo )
6 ifnefalse 3384 . . . 4  |-  ( A  =/= -oo  ->  if ( A  = -oo , +oo ,  -u A )  = 
-u A )
75, 6syl 14 . . 3  |-  ( A  e.  RR  ->  if ( A  = -oo , +oo ,  -u A
)  =  -u A
)
84, 7eqtrd 2115 . 2  |-  ( A  e.  RR  ->  if ( A  = +oo , -oo ,  if ( A  = -oo , +oo ,  -u A ) )  =  -u A )
91, 8syl5eq 2127 1  |-  ( A  e.  RR  ->  -e
A  =  -u A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285    e. wcel 1434    =/= wne 2249   ifcif 3373   RRcr 7250   +oocpnf 7420   -oocmnf 7421   -ucneg 7555    -ecxne 9133
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-un 4223  ax-setind 4315  ax-cnex 7337  ax-resscn 7338
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-rab 2362  df-v 2614  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-if 3374  df-pw 3408  df-sn 3428  df-pr 3429  df-uni 3628  df-pnf 7425  df-mnf 7426  df-xneg 9136
This theorem is referenced by:  xneg0  9186  xnegcl  9187  xnegneg  9188  xltnegi  9190
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