Theorem xnegpnf 9611

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 9559	. 2 ⊢ -𝑒+∞ = if(+∞ = +∞, -∞, if(+∞ = -∞, +∞, -+∞))
2		eqid 2139	. . 3 ⊢ +∞ = +∞
3	2	iftruei 3480	. 2 ⊢ if(+∞ = +∞, -∞, if(+∞ = -∞, +∞, -+∞)) = -∞
4	1, 3	eqtri 2160	1 ⊢ -𝑒+∞ = -∞

Syntax hints:   = wceq 1331  ifcif 3474  +∞cpnf 7797  -∞cmnf 7798  -cneg 7934  -_𝑒cxne 9556

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-if 3475  df-xneg 9559

This theorem is referenced by:  xnegcl  9615  xnegneg  9616  xltnegi  9618  xnegid  9642  xnegdi  9651  xaddass2  9653  xsubge0  9664  xposdif  9665  xlesubadd  9666  xblss2ps  12573  xblss2  12574