Theorem xnegpnf 10180

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 10124	. 2 ⊢ -𝑒+∞ = if(+∞ = +∞, -∞, if(+∞ = -∞, +∞, -+∞))
2		eqid 2234	. . 3 ⊢ +∞ = +∞
3	2	iftruei 3632	. 2 ⊢ if(+∞ = +∞, -∞, if(+∞ = -∞, +∞, -+∞)) = -∞
4	1, 3	eqtri 2255	1 ⊢ -𝑒+∞ = -∞

Syntax hints:   = wceq 1398  ifcif 3624  +∞cpnf 8321  -∞cmnf 8322  -cneg 8461  -_𝑒cxne 10121

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 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-11 1555  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-if 3625  df-xneg 10124

This theorem is referenced by:  xnegcl  10184  xnegneg  10185  xltnegi  10187  xnegid  10211  xnegdi  10220  xaddass2  10222  xsubge0  10233  xposdif  10234  xlesubadd  10235  xblss2ps  15395  xblss2  15396