Theorem xnegpnf 10209

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 10153	. 2 ⊢ -𝑒+∞ = if(+∞ = +∞, -∞, if(+∞ = -∞, +∞, -+∞))
2		eqid 2238	. . 3 ⊢ +∞ = +∞
3	2	iftruei 3643	. 2 ⊢ if(+∞ = +∞, -∞, if(+∞ = -∞, +∞, -+∞)) = -∞
4	1, 3	eqtri 2259	1 ⊢ -𝑒+∞ = -∞

Syntax hints:   = wceq 1402  ifcif 3635  +∞cpnf 8347  -∞cmnf 8348  -cneg 8488  -_𝑒cxne 10150

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 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-if 3636  df-xneg 10153

This theorem is referenced by:  xnegcl  10213  xnegneg  10214  xltnegi  10216  xnegid  10240  xnegdi  10249  xaddass2  10251  xsubge0  10262  xposdif  10263  xlesubadd  10264  xblss2ps  15428  xblss2  15429