Theorem rexneg 10026

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 9968	. 2 |- -e A = if ( A = +oo , -oo , if ( A = -oo , +oo , -u A ) )
2		renepnf 8194	. . . 4 |- ( A e. RR -> A =/= +oo )
3		ifnefalse 3613	. . . 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 8195	. . . 4 |- ( A e. RR -> A =/= -oo )
6		ifnefalse 3613	. . . 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 2262	. 2 |- ( A e. RR -> if ( A = +oo , -oo , if ( A = -oo , +oo , -u A ) ) = -u A )
9	1, 8	eqtrid 2274	1 |- ( A e. RR -> -e A = -u A )

Syntax hints:   -> wi 4   = wceq 1395   e. wcel 2200   =/= wne 2400   ifcif 3602   RRcr 7998   +oocpnf 8178   -oocmnf 8179   -ucneg 8318   -ecxne 9965

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-un 4524  ax-setind 4629  ax-cnex 8090  ax-resscn 8091

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-uni 3889  df-pnf 8183  df-mnf 8184  df-xneg 9968

This theorem is referenced by:  xneg0  10027  xnegcl  10028  xnegneg  10029  xltnegi  10031  rexsub  10049  xnegid  10055  xnegdi  10064  xpncan  10067  xnpcan  10068  xposdif  10078  xrmaxaddlem  11771  xrminrecl  11784  xrminrpcl  11785