Theorem xnegpnf 9831

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 9775	. 2 ⊢ -𝑒+∞ = if(+∞ = +∞, -∞, if(+∞ = -∞, +∞, -+∞))
2		eqid 2177	. . 3 ⊢ +∞ = +∞
3	2	iftruei 3542	. 2 ⊢ if(+∞ = +∞, -∞, if(+∞ = -∞, +∞, -+∞)) = -∞
4	1, 3	eqtri 2198	1 ⊢ -𝑒+∞ = -∞

Syntax hints:   = wceq 1353  ifcif 3536  +∞cpnf 7992  -∞cmnf 7993  -cneg 8132  -_𝑒cxne 9772

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 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-if 3537  df-xneg 9775

This theorem is referenced by:  xnegcl  9835  xnegneg  9836  xltnegi  9838  xnegid  9862  xnegdi  9871  xaddass2  9873  xsubge0  9884  xposdif  9885  xlesubadd  9886  xblss2ps  14044  xblss2  14045