Theorem xnegpnf 10053

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 9997	. 2 ⊢ -𝑒+∞ = if(+∞ = +∞, -∞, if(+∞ = -∞, +∞, -+∞))
2		eqid 2229	. . 3 ⊢ +∞ = +∞
3	2	iftruei 3609	. 2 ⊢ if(+∞ = +∞, -∞, if(+∞ = -∞, +∞, -+∞)) = -∞
4	1, 3	eqtri 2250	1 ⊢ -𝑒+∞ = -∞

Syntax hints:   = wceq 1395  ifcif 3603  +∞cpnf 8201  -∞cmnf 8202  -cneg 8341  -_𝑒cxne 9994

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 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-if 3604  df-xneg 9997

This theorem is referenced by:  xnegcl  10057  xnegneg  10058  xltnegi  10060  xnegid  10084  xnegdi  10093  xaddass2  10095  xsubge0  10106  xposdif  10107  xlesubadd  10108  xblss2ps  15118  xblss2  15119