Theorem xnegpnf 9949

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

Assertion

Ref	Expression
xnegpnf	⊢ -𝑒+∞ = -∞

Proof of Theorem xnegpnf

Step	Hyp	Ref	Expression
1		df-xneg 9893	. 2 ⊢ -𝑒+∞ = if(+∞ = +∞, -∞, if(+∞ = -∞, +∞, -+∞))
2		eqid 2204	. . 3 ⊢ +∞ = +∞
3	2	iftruei 3576	. 2 ⊢ if(+∞ = +∞, -∞, if(+∞ = -∞, +∞, -+∞)) = -∞
4	1, 3	eqtri 2225	1 ⊢ -𝑒+∞ = -∞

Syntax hints:   = wceq 1372  ifcif 3570  +∞cpnf 8103  -∞cmnf 8104  -cneg 8243  -_𝑒cxne 9890

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-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-11 1528  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-if 3571  df-xneg 9893

This theorem is referenced by:  xnegcl  9953  xnegneg  9954  xltnegi  9956  xnegid  9980  xnegdi  9989  xaddass2  9991  xsubge0  10002  xposdif  10003  xlesubadd  10004  xblss2ps  14847  xblss2  14848