Theorem xnegpnf 9985

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 9929	. 2 ⊢ -𝑒+∞ = if(+∞ = +∞, -∞, if(+∞ = -∞, +∞, -+∞))
2		eqid 2207	. . 3 ⊢ +∞ = +∞
3	2	iftruei 3585	. 2 ⊢ if(+∞ = +∞, -∞, if(+∞ = -∞, +∞, -+∞)) = -∞
4	1, 3	eqtri 2228	1 ⊢ -𝑒+∞ = -∞

Syntax hints:   = wceq 1373  ifcif 3579  +∞cpnf 8139  -∞cmnf 8140  -cneg 8279  -_𝑒cxne 9926

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-if 3580  df-xneg 9929

This theorem is referenced by:  xnegcl  9989  xnegneg  9990  xltnegi  9992  xnegid  10016  xnegdi  10025  xaddass2  10027  xsubge0  10038  xposdif  10039  xlesubadd  10040  xblss2ps  14991  xblss2  14992