Theorem xnegpnf 12040

Description: Minus +oo. Remark of [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.)

Assertion

Ref	Expression
xnegpnf	|- -e +oo = -oo

Proof of Theorem xnegpnf

Step	Hyp	Ref	Expression
1		df-xneg 11946	. 2 |- -e +oo = if ( +oo = +oo , -oo , if ( +oo = -oo , +oo , -u +oo ) )
2		eqid 2622	. . 3 |- +oo = +oo
3	2	iftruei 4093	. 2 |- if ( +oo = +oo , -oo , if ( +oo = -oo , +oo , -u +oo ) ) = -oo
4	1, 3	eqtri 2644	1 |- -e +oo = -oo

Syntax hints:   = wceq 1483   ifcif 4086   +oocpnf 10071   -oocmnf 10072   -ucneg 10267   -ecxne 11943

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-if 4087  df-xneg 11946

This theorem is referenced by:  xnegcl  12044  xnegneg  12045  xltnegi  12047  xnegid  12069  xnegdi  12078  xaddass2  12080  xsubge0  12091  xlesubadd  12093  xmulneg1  12099  xmulmnf1  12106  xadddi2  12127  xrsdsreclblem  19792  xblss2ps  22206  xblss2  22207  xaddeq0  29518  supminfxr  39694