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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 9400 | . 2 ⊢ -𝑒+∞ = if(+∞ = +∞, -∞, if(+∞ = -∞, +∞, -+∞)) | |
2 | eqid 2100 | . . 3 ⊢ +∞ = +∞ | |
3 | 2 | iftruei 3427 | . 2 ⊢ if(+∞ = +∞, -∞, if(+∞ = -∞, +∞, -+∞)) = -∞ |
4 | 1, 3 | eqtri 2120 | 1 ⊢ -𝑒+∞ = -∞ |
Colors of variables: wff set class |
Syntax hints: = wceq 1299 ifcif 3421 +∞cpnf 7669 -∞cmnf 7670 -cneg 7805 -𝑒cxne 9397 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-11 1452 ax-4 1455 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 |
This theorem depends on definitions: df-bi 116 df-nf 1405 df-sb 1704 df-clab 2087 df-cleq 2093 df-clel 2096 df-if 3422 df-xneg 9400 |
This theorem is referenced by: xnegcl 9456 xnegneg 9457 xltnegi 9459 xnegid 9483 xnegdi 9492 xaddass2 9494 xsubge0 9505 xposdif 9506 xlesubadd 9507 xblss2ps 12332 xblss2 12333 |
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