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Theorem rexneg 11284
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 11192 . 2  |-  -e
A  =  if ( A  = +oo , -oo ,  if ( A  = -oo , +oo ,  -u A ) )
2 renepnf 9534 . . . 4  |-  ( A  e.  RR  ->  A  =/= +oo )
3 ifnefalse 3901 . . . 4  |-  ( A  =/= +oo  ->  if ( A  = +oo , -oo ,  if ( A  = -oo , +oo ,  -u A ) )  =  if ( A  = -oo , +oo ,  -u A ) )
42, 3syl 16 . . 3  |-  ( A  e.  RR  ->  if ( A  = +oo , -oo ,  if ( A  = -oo , +oo ,  -u A ) )  =  if ( A  = -oo , +oo ,  -u A ) )
5 renemnf 9535 . . . 4  |-  ( A  e.  RR  ->  A  =/= -oo )
6 ifnefalse 3901 . . . 4  |-  ( A  =/= -oo  ->  if ( A  = -oo , +oo ,  -u A )  = 
-u A )
75, 6syl 16 . . 3  |-  ( A  e.  RR  ->  if ( A  = -oo , +oo ,  -u A
)  =  -u A
)
84, 7eqtrd 2492 . 2  |-  ( A  e.  RR  ->  if ( A  = +oo , -oo ,  if ( A  = -oo , +oo ,  -u A ) )  =  -u A )
91, 8syl5eq 2504 1  |-  ( A  e.  RR  ->  -e
A  =  -u A
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758    =/= wne 2644   ifcif 3891   RRcr 9384   +oocpnf 9518   -oocmnf 9519   -ucneg 9699    -ecxne 11189
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-resscn 9442
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-er 7203  df-en 7413  df-dom 7414  df-sdom 7415  df-pnf 9523  df-mnf 9524  df-xneg 11192
This theorem is referenced by:  xneg0  11285  xnegcl  11286  xnegneg  11287  xltnegi  11289  rexsub  11306  xnegid  11309  xnegdi  11314  xpncan  11317  xnpcan  11318  xmulneg1  11335  xmulm1  11347  xadddi  11361  xlt2addrd  26187  xrsmulgzz  26275
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