Theorem rexneg 9643

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

Step	Hyp	Ref	Expression
1		df-xneg 9589	. 2 |- -e A = if ( A = +oo , -oo , if ( A = -oo , +oo , -u A ) )
2		renepnf 7837	. . . 4 |- ( A e. RR -> A =/= +oo )
3		ifnefalse 3490	. . . 4 |- ( A =/= +oo -> if ( A = +oo , -oo , if ( A = -oo , +oo , -u A ) ) = if ( A = -oo , +oo , -u A ) )
4	2, 3	syl 14	. . 3 |- ( A e. RR -> if ( A = +oo , -oo , if ( A = -oo , +oo , -u A ) ) = if ( A = -oo , +oo , -u A ) )
5		renemnf 7838	. . . 4 |- ( A e. RR -> A =/= -oo )
6		ifnefalse 3490	. . . 4 |- ( A =/= -oo -> if ( A = -oo , +oo , -u A ) = -u A )
7	5, 6	syl 14	. . 3 |- ( A e. RR -> if ( A = -oo , +oo , -u A ) = -u A )
8	4, 7	eqtrd 2173	. 2 |- ( A e. RR -> if ( A = +oo , -oo , if ( A = -oo , +oo , -u A ) ) = -u A )
9	1, 8	syl5eq 2185	1 |- ( A e. RR -> -e A = -u A )

Syntax hints:   -> wi 4   = wceq 1332   e. wcel 1481   =/= wne 2309   ifcif 3479   RRcr 7643   +oocpnf 7821   -oocmnf 7822   -ucneg 7958   -ecxne 9586

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-un 4363  ax-setind 4460  ax-cnex 7735  ax-resscn 7736

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-if 3480  df-pw 3517  df-sn 3538  df-pr 3539  df-uni 3745  df-pnf 7826  df-mnf 7827  df-xneg 9589

This theorem is referenced by:  xneg0  9644  xnegcl  9645  xnegneg  9646  xltnegi  9648  rexsub  9666  xnegid  9672  xnegdi  9681  xpncan  9684  xnpcan  9685  xposdif  9695  xrmaxaddlem  11061  xrminrecl  11074  xrminrpcl  11075