MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rexneg Structured version   Unicode version

Theorem rexneg 11455
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 11360 . 2  |-  -e
A  =  if ( A  = +oo , -oo ,  if ( A  = -oo , +oo ,  -u A ) )
2 renepnf 9639 . . . 4  |-  ( A  e.  RR  ->  A  =/= +oo )
3 ifnefalse 3866 . . . 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 9640 . . . 4  |-  ( A  e.  RR  ->  A  =/= -oo )
6 ifnefalse 3866 . . . 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 2462 . 2  |-  ( A  e.  RR  ->  if ( A  = +oo , -oo ,  if ( A  = -oo , +oo ,  -u A ) )  =  -u A )
91, 8syl5eq 2474 1  |-  ( A  e.  RR  ->  -e
A  =  -u A
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437    e. wcel 1872    =/= wne 2599   ifcif 3854   RRcr 9489   +oocpnf 9623   -oocmnf 9624   -ucneg 9812    -ecxne 11357
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541  ax-resscn 9547
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-nel 2602  df-ral 2719  df-rex 2720  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-op 3948  df-uni 4163  df-br 4367  df-opab 4426  df-mpt 4427  df-id 4711  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-er 7318  df-en 7525  df-dom 7526  df-sdom 7527  df-pnf 9628  df-mnf 9629  df-xneg 11360
This theorem is referenced by:  xneg0  11456  xnegcl  11457  xnegneg  11458  xltnegi  11460  rexsub  11477  xnegid  11480  xnegdi  11485  xpncan  11488  xnpcan  11489  xmulneg1  11506  xmulm1  11518  xadddi  11532  xlt2addrd  28288  xrsmulgzz  28391
  Copyright terms: Public domain W3C validator