Theorem xnegpnf 9641

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 9589	. 2 |- -e +oo = if ( +oo = +oo , -oo , if ( +oo = -oo , +oo , -u +oo ) )
2		eqid 2140	. . 3 |- +oo = +oo
3	2	iftruei 3485	. 2 |- if ( +oo = +oo , -oo , if ( +oo = -oo , +oo , -u +oo ) ) = -oo
4	1, 3	eqtri 2161	1 |- -e +oo = -oo

Syntax hints:   = wceq 1332   ifcif 3479   +oocpnf 7821   -oocmnf 7822   -ucneg 7958   -ecxne 9586

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-11 1485  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-if 3480  df-xneg 9589

This theorem is referenced by:  xnegcl  9645  xnegneg  9646  xltnegi  9648  xnegid  9672  xnegdi  9681  xaddass2  9683  xsubge0  9694  xposdif  9695  xlesubadd  9696  xblss2ps  12612  xblss2  12613