Theorem rexneg 9801

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 9743	. 2 |- -e A = if ( A = +oo , -oo , if ( A = -oo , +oo , -u A ) )
2		renepnf 7979	. . . 4 |- ( A e. RR -> A =/= +oo )
3		ifnefalse 3543	. . . 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 7980	. . . 4 |- ( A e. RR -> A =/= -oo )
6		ifnefalse 3543	. . . 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 2208	. 2 |- ( A e. RR -> if ( A = +oo , -oo , if ( A = -oo , +oo , -u A ) ) = -u A )
9	1, 8	eqtrid 2220	1 |- ( A e. RR -> -e A = -u A )

Syntax hints:   -> wi 4   = wceq 1353   e. wcel 2146   =/= wne 2345   ifcif 3532   RRcr 7785   +oocpnf 7963   -oocmnf 7964   -ucneg 8103   -ecxne 9740

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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-un 4427  ax-setind 4530  ax-cnex 7877  ax-resscn 7878

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-rab 2462  df-v 2737  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-if 3533  df-pw 3574  df-sn 3595  df-pr 3596  df-uni 3806  df-pnf 7968  df-mnf 7969  df-xneg 9743

This theorem is referenced by:  xneg0  9802  xnegcl  9803  xnegneg  9804  xltnegi  9806  rexsub  9824  xnegid  9830  xnegdi  9839  xpncan  9842  xnpcan  9843  xposdif  9853  xrmaxaddlem  11236  xrminrecl  11249  xrminrpcl  11250