Theorem rexneg 9954

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 9896	. 2 |- -e A = if ( A = +oo , -oo , if ( A = -oo , +oo , -u A ) )
2		renepnf 8122	. . . 4 |- ( A e. RR -> A =/= +oo )
3		ifnefalse 3582	. . . 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 8123	. . . 4 |- ( A e. RR -> A =/= -oo )
6		ifnefalse 3582	. . . 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 2238	. 2 |- ( A e. RR -> if ( A = +oo , -oo , if ( A = -oo , +oo , -u A ) ) = -u A )
9	1, 8	eqtrid 2250	1 |- ( A e. RR -> -e A = -u A )

Syntax hints:   -> wi 4   = wceq 1373   e. wcel 2176   =/= wne 2376   ifcif 3571   RRcr 7926   +oocpnf 8106   -oocmnf 8107   -ucneg 8246   -ecxne 9893

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-un 4481  ax-setind 4586  ax-cnex 8018  ax-resscn 8019

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-if 3572  df-pw 3618  df-sn 3639  df-pr 3640  df-uni 3851  df-pnf 8111  df-mnf 8112  df-xneg 9896

This theorem is referenced by:  xneg0  9955  xnegcl  9956  xnegneg  9957  xltnegi  9959  rexsub  9977  xnegid  9983  xnegdi  9992  xpncan  9995  xnpcan  9996  xposdif  10006  xrmaxaddlem  11604  xrminrecl  11617  xrminrpcl  11618