Theorem rexneg 10182

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 10124	. 2 |- -e A = if ( A = +oo , -oo , if ( A = -oo , +oo , -u A ) )
2		renepnf 8337	. . . 4 |- ( A e. RR -> A =/= +oo )
3		ifnefalse 3637	. . . 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 8338	. . . 4 |- ( A e. RR -> A =/= -oo )
6		ifnefalse 3637	. . . 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 2267	. 2 |- ( A e. RR -> if ( A = +oo , -oo , if ( A = -oo , +oo , -u A ) ) = -u A )
9	1, 8	eqtrid 2279	1 |- ( A e. RR -> -e A = -u A )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2205   =/= wne 2414   ifcif 3624   RRcr 8142   +oocpnf 8321   -oocmnf 8322   -ucneg 8461   -ecxne 10121

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 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-uni 3920  df-pnf 8326  df-mnf 8327  df-xneg 10124

This theorem is referenced by:  xneg0  10183  xnegcl  10184  xnegneg  10185  xltnegi  10187  rexsub  10205  xnegid  10211  xnegdi  10220  xpncan  10223  xnpcan  10224  xposdif  10234  xrmaxaddlem  11970  xrminrecl  11983  xrminrpcl  11984