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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 9894 | . 2 ⊢ -𝑒+∞ = if(+∞ = +∞, -∞, if(+∞ = -∞, +∞, -+∞)) | |
| 2 | eqid 2205 | . . 3 ⊢ +∞ = +∞ | |
| 3 | 2 | iftruei 3577 | . 2 ⊢ if(+∞ = +∞, -∞, if(+∞ = -∞, +∞, -+∞)) = -∞ |
| 4 | 1, 3 | eqtri 2226 | 1 ⊢ -𝑒+∞ = -∞ |
| Colors of variables: wff set class |
| Syntax hints: = wceq 1373 ifcif 3571 +∞cpnf 8104 -∞cmnf 8105 -cneg 8244 -𝑒cxne 9891 |
| 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 711 ax-5 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-11 1529 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-ext 2187 |
| This theorem depends on definitions: df-bi 117 df-nf 1484 df-sb 1786 df-clab 2192 df-cleq 2198 df-clel 2201 df-if 3572 df-xneg 9894 |
| This theorem is referenced by: xnegcl 9954 xnegneg 9955 xltnegi 9957 xnegid 9981 xnegdi 9990 xaddass2 9992 xsubge0 10003 xposdif 10004 xlesubadd 10005 xblss2ps 14876 xblss2 14877 |
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