Theorem rexneg 10126

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 10068	. 2 |- -e A = if ( A = +oo , -oo , if ( A = -oo , +oo , -u A ) )
2		renepnf 8286	. . . 4 |- ( A e. RR -> A =/= +oo )
3		ifnefalse 3620	. . . 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 8287	. . . 4 |- ( A e. RR -> A =/= -oo )
6		ifnefalse 3620	. . . 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 2264	. 2 |- ( A e. RR -> if ( A = +oo , -oo , if ( A = -oo , +oo , -u A ) ) = -u A )
9	1, 8	eqtrid 2276	1 |- ( A e. RR -> -e A = -u A )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2202   =/= wne 2403   ifcif 3607   RRcr 8091   +oocpnf 8270   -oocmnf 8271   -ucneg 8410   -ecxne 10065

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-un 4536  ax-setind 4641  ax-cnex 8183  ax-resscn 8184

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-uni 3899  df-pnf 8275  df-mnf 8276  df-xneg 10068

This theorem is referenced by:  xneg0  10127  xnegcl  10128  xnegneg  10129  xltnegi  10131  rexsub  10149  xnegid  10155  xnegdi  10164  xpncan  10167  xnpcan  10168  xposdif  10178  xrmaxaddlem  11900  xrminrecl  11913  xrminrpcl  11914