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Theorem rexneg 10540
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 10454 . 2  |-  - e A  =  if ( A  =  +oo ,  -oo ,  if ( A  = 
-oo ,  +oo ,  -u A ) )
2 renepnf 8881 . . . 4  |-  ( A  e.  RR  ->  A  =/=  +oo )
3 ifnefalse 3575 . . . 4  |-  ( A  =/=  +oo  ->  if ( A  =  +oo ,  -oo ,  if ( A  =  -oo ,  +oo , 
-u A ) )  =  if ( A  =  -oo ,  +oo , 
-u A ) )
42, 3syl 15 . . 3  |-  ( A  e.  RR  ->  if ( A  =  +oo , 
-oo ,  if ( A  =  -oo ,  +oo , 
-u A ) )  =  if ( A  =  -oo ,  +oo , 
-u A ) )
5 renemnf 8882 . . . 4  |-  ( A  e.  RR  ->  A  =/=  -oo )
6 ifnefalse 3575 . . . 4  |-  ( A  =/=  -oo  ->  if ( A  =  -oo ,  +oo ,  -u A )  = 
-u A )
75, 6syl 15 . . 3  |-  ( A  e.  RR  ->  if ( A  =  -oo , 
+oo ,  -u A )  =  -u A )
84, 7eqtrd 2317 . 2  |-  ( A  e.  RR  ->  if ( A  =  +oo , 
-oo ,  if ( A  =  -oo ,  +oo , 
-u A ) )  =  -u A )
91, 8syl5eq 2329 1  |-  ( A  e.  RR  ->  - e A  =  -u A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1625    e. wcel 1686    =/= wne 2448   ifcif 3567   RRcr 8738    +oocpnf 8866    -oocmnf 8867   -ucneg 9040    - ecxne 10451
This theorem is referenced by:  xneg0  10541  xnegcl  10542  xnegneg  10543  xltnegi  10545  rexsub  10562  xnegid  10565  xnegdi  10570  xpncan  10573  xnpcan  10574  xmulneg1  10591  xmulm1  10603  xadddi  10617  xlt2addrd  23255  xrsmulgzz  23309
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-resscn 8796
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-er 6662  df-en 6866  df-dom 6867  df-sdom 6868  df-pnf 8871  df-mnf 8872  df-xneg 10454
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