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Mirrors > Home > ILE Home > Th. List > xnegpnf | GIF version |
Description: Minus +∞. Remark of [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.) |
Ref | Expression |
---|---|
xnegpnf | ⊢ -𝑒+∞ = -∞ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-xneg 9699 | . 2 ⊢ -𝑒+∞ = if(+∞ = +∞, -∞, if(+∞ = -∞, +∞, -+∞)) | |
2 | eqid 2164 | . . 3 ⊢ +∞ = +∞ | |
3 | 2 | iftruei 3521 | . 2 ⊢ if(+∞ = +∞, -∞, if(+∞ = -∞, +∞, -+∞)) = -∞ |
4 | 1, 3 | eqtri 2185 | 1 ⊢ -𝑒+∞ = -∞ |
Colors of variables: wff set class |
Syntax hints: = wceq 1342 ifcif 3515 +∞cpnf 7921 -∞cmnf 7922 -cneg 8061 -𝑒cxne 9696 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-11 1493 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-if 3516 df-xneg 9699 |
This theorem is referenced by: xnegcl 9759 xnegneg 9760 xltnegi 9762 xnegid 9786 xnegdi 9795 xaddass2 9797 xsubge0 9808 xposdif 9809 xlesubadd 9810 xblss2ps 12951 xblss2 12952 |
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