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Theorem rexneg 9606
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 9552 . 2  |-  -e
A  =  if ( A  = +oo , -oo ,  if ( A  = -oo , +oo ,  -u A ) )
2 renepnf 7806 . . . 4  |-  ( A  e.  RR  ->  A  =/= +oo )
3 ifnefalse 3480 . . . 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 7807 . . . 4  |-  ( A  e.  RR  ->  A  =/= -oo )
6 ifnefalse 3480 . . . 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 2170 . 2  |-  ( A  e.  RR  ->  if ( A  = +oo , -oo ,  if ( A  = -oo , +oo ,  -u A ) )  =  -u A )
91, 8syl5eq 2182 1  |-  ( A  e.  RR  ->  -e
A  =  -u A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1331    e. wcel 1480    =/= wne 2306   ifcif 3469   RRcr 7612   +oocpnf 7790   -oocmnf 7791   -ucneg 7927    -ecxne 9549
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-un 4350  ax-setind 4447  ax-cnex 7704  ax-resscn 7705
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-rab 2423  df-v 2683  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-if 3470  df-pw 3507  df-sn 3528  df-pr 3529  df-uni 3732  df-pnf 7795  df-mnf 7796  df-xneg 9552
This theorem is referenced by:  xneg0  9607  xnegcl  9608  xnegneg  9609  xltnegi  9611  rexsub  9629  xnegid  9635  xnegdi  9644  xpncan  9647  xnpcan  9648  xposdif  9658  xrmaxaddlem  11022  xrminrecl  11035  xrminrpcl  11036
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