Theorem xnegpnf 9785

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 9729	. 2 ⊢ -𝑒+∞ = if(+∞ = +∞, -∞, if(+∞ = -∞, +∞, -+∞))
2		eqid 2170	. . 3 ⊢ +∞ = +∞
3	2	iftruei 3532	. 2 ⊢ if(+∞ = +∞, -∞, if(+∞ = -∞, +∞, -+∞)) = -∞
4	1, 3	eqtri 2191	1 ⊢ -𝑒+∞ = -∞

Syntax hints:   = wceq 1348  ifcif 3526  +∞cpnf 7951  -∞cmnf 7952  -cneg 8091  -_𝑒cxne 9726

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 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-if 3527  df-xneg 9729

This theorem is referenced by:  xnegcl  9789  xnegneg  9790  xltnegi  9792  xnegid  9816  xnegdi  9825  xaddass2  9827  xsubge0  9838  xposdif  9839  xlesubadd  9840  xblss2ps  13198  xblss2  13199