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Theorem rexneg 11527
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 11432 . 2  |-  -e
A  =  if ( A  = +oo , -oo ,  if ( A  = -oo , +oo ,  -u A ) )
2 renepnf 9706 . . . 4  |-  ( A  e.  RR  ->  A  =/= +oo )
3 ifnefalse 3884 . . . 4  |-  ( A  =/= +oo  ->  if ( A  = +oo , -oo ,  if ( A  = -oo , +oo ,  -u A ) )  =  if ( A  = -oo , +oo ,  -u A ) )
42, 3syl 17 . . 3  |-  ( A  e.  RR  ->  if ( A  = +oo , -oo ,  if ( A  = -oo , +oo ,  -u A ) )  =  if ( A  = -oo , +oo ,  -u A ) )
5 renemnf 9707 . . . 4  |-  ( A  e.  RR  ->  A  =/= -oo )
6 ifnefalse 3884 . . . 4  |-  ( A  =/= -oo  ->  if ( A  = -oo , +oo ,  -u A )  = 
-u A )
75, 6syl 17 . . 3  |-  ( A  e.  RR  ->  if ( A  = -oo , +oo ,  -u A
)  =  -u A
)
84, 7eqtrd 2505 . 2  |-  ( A  e.  RR  ->  if ( A  = +oo , -oo ,  if ( A  = -oo , +oo ,  -u A ) )  =  -u A )
91, 8syl5eq 2517 1  |-  ( A  e.  RR  ->  -e
A  =  -u A
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1452    e. wcel 1904    =/= wne 2641   ifcif 3872   RRcr 9556   +oocpnf 9690   -oocmnf 9691   -ucneg 9881    -ecxne 11429
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-resscn 9614
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-pnf 9695  df-mnf 9696  df-xneg 11432
This theorem is referenced by:  xneg0  11528  xnegcl  11529  xnegneg  11530  xltnegi  11532  rexsub  11549  xnegid  11553  xnegdi  11559  xpncan  11562  xnpcan  11563  xmulneg1  11580  xmulm1  11592  xadddi  11606  xlt2addrd  28413  xrsmulgzz  28515
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