Theorem xnegpnf 9764

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 9708	. 2 ⊢ -𝑒+∞ = if(+∞ = +∞, -∞, if(+∞ = -∞, +∞, -+∞))
2		eqid 2165	. . 3 ⊢ +∞ = +∞
3	2	iftruei 3526	. 2 ⊢ if(+∞ = +∞, -∞, if(+∞ = -∞, +∞, -+∞)) = -∞
4	1, 3	eqtri 2186	1 ⊢ -𝑒+∞ = -∞

Syntax hints:   = wceq 1343  ifcif 3520  +∞cpnf 7930  -∞cmnf 7931  -cneg 8070  -_𝑒cxne 9705

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 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-11 1494  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-if 3521  df-xneg 9708

This theorem is referenced by:  xnegcl  9768  xnegneg  9769  xltnegi  9771  xnegid  9795  xnegdi  9804  xaddass2  9806  xsubge0  9817  xposdif  9818  xlesubadd  9819  xblss2ps  13054  xblss2  13055