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Theorem rexneg 10753
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 10666 . 2  |-  - e A  =  if ( A  =  +oo ,  -oo ,  if ( A  = 
-oo ,  +oo ,  -u A ) )
2 renepnf 9088 . . . 4  |-  ( A  e.  RR  ->  A  =/=  +oo )
3 ifnefalse 3707 . . . 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 9089 . . . 4  |-  ( A  e.  RR  ->  A  =/=  -oo )
6 ifnefalse 3707 . . . 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 2436 . 2  |-  ( A  e.  RR  ->  if ( A  =  +oo , 
-oo ,  if ( A  =  -oo ,  +oo , 
-u A ) )  =  -u A )
91, 8syl5eq 2448 1  |-  ( A  e.  RR  ->  - e A  =  -u A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1721    =/= wne 2567   ifcif 3699   RRcr 8945    +oocpnf 9073    -oocmnf 9074   -ucneg 9248    - ecxne 10663
This theorem is referenced by:  xneg0  10754  xnegcl  10755  xnegneg  10756  xltnegi  10758  rexsub  10775  xnegid  10778  xnegdi  10783  xpncan  10786  xnpcan  10787  xmulneg1  10804  xmulm1  10816  xadddi  10830  xlt2addrd  24077  xrsmulgzz  24153
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-resscn 9003
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xneg 10666
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