Theorem xnegpnf 10161

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 10105	. 2 ⊢ -𝑒+∞ = if(+∞ = +∞, -∞, if(+∞ = -∞, +∞, -+∞))
2		eqid 2232	. . 3 ⊢ +∞ = +∞
3	2	iftruei 3628	. 2 ⊢ if(+∞ = +∞, -∞, if(+∞ = -∞, +∞, -+∞)) = -∞
4	1, 3	eqtri 2253	1 ⊢ -𝑒+∞ = -∞

Syntax hints:   = wceq 1398  ifcif 3620  +∞cpnf 8305  -∞cmnf 8306  -cneg 8445  -_𝑒cxne 10102

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 2214

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-if 3621  df-xneg 10105

This theorem is referenced by:  xnegcl  10165  xnegneg  10166  xltnegi  10168  xnegid  10192  xnegdi  10201  xaddass2  10203  xsubge0  10214  xposdif  10215  xlesubadd  10216  xblss2ps  15269  xblss2  15270