Theorem xnegpnf 9897

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 9841	. 2 ⊢ -𝑒+∞ = if(+∞ = +∞, -∞, if(+∞ = -∞, +∞, -+∞))
2		eqid 2193	. . 3 ⊢ +∞ = +∞
3	2	iftruei 3564	. 2 ⊢ if(+∞ = +∞, -∞, if(+∞ = -∞, +∞, -+∞)) = -∞
4	1, 3	eqtri 2214	1 ⊢ -𝑒+∞ = -∞

Syntax hints:   = wceq 1364  ifcif 3558  +∞cpnf 8053  -∞cmnf 8054  -cneg 8193  -_𝑒cxne 9838

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 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-if 3559  df-xneg 9841

This theorem is referenced by:  xnegcl  9901  xnegneg  9902  xltnegi  9904  xnegid  9928  xnegdi  9937  xaddass2  9939  xsubge0  9950  xposdif  9951  xlesubadd  9952  xblss2ps  14583  xblss2  14584