Theorem xnegpnf 9922

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 9866	. 2 ⊢ -𝑒+∞ = if(+∞ = +∞, -∞, if(+∞ = -∞, +∞, -+∞))
2		eqid 2196	. . 3 ⊢ +∞ = +∞
3	2	iftruei 3568	. 2 ⊢ if(+∞ = +∞, -∞, if(+∞ = -∞, +∞, -+∞)) = -∞
4	1, 3	eqtri 2217	1 ⊢ -𝑒+∞ = -∞

Syntax hints:   = wceq 1364  ifcif 3562  +∞cpnf 8077  -∞cmnf 8078  -cneg 8217  -_𝑒cxne 9863

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-11 1520  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-if 3563  df-xneg 9866

This theorem is referenced by:  xnegcl  9926  xnegneg  9927  xltnegi  9929  xnegid  9953  xnegdi  9962  xaddass2  9964  xsubge0  9975  xposdif  9976  xlesubadd  9977  xblss2ps  14748  xblss2  14749