Theorem xnegpnf 10107

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 10051	. 2 ⊢ -𝑒+∞ = if(+∞ = +∞, -∞, if(+∞ = -∞, +∞, -+∞))
2		eqid 2231	. . 3 ⊢ +∞ = +∞
3	2	iftruei 3615	. 2 ⊢ if(+∞ = +∞, -∞, if(+∞ = -∞, +∞, -+∞)) = -∞
4	1, 3	eqtri 2252	1 ⊢ -𝑒+∞ = -∞

Syntax hints:   = wceq 1398  ifcif 3607  +∞cpnf 8253  -∞cmnf 8254  -cneg 8393  -_𝑒cxne 10048

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 2213

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-if 3608  df-xneg 10051

This theorem is referenced by:  xnegcl  10111  xnegneg  10112  xltnegi  10114  xnegid  10138  xnegdi  10147  xaddass2  10149  xsubge0  10160  xposdif  10161  xlesubadd  10162  xblss2ps  15198  xblss2  15199