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| Mirrors > Home > MPE Home > Th. List > xnegpnf | Structured version Visualization version 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 13154 | . 2 ⊢ -𝑒+∞ = if(+∞ = +∞, -∞, if(+∞ = -∞, +∞, -+∞)) | |
| 2 | eqid 2737 | . . 3 ⊢ +∞ = +∞ | |
| 3 | 2 | iftruei 4532 | . 2 ⊢ if(+∞ = +∞, -∞, if(+∞ = -∞, +∞, -+∞)) = -∞ |
| 4 | 1, 3 | eqtri 2765 | 1 ⊢ -𝑒+∞ = -∞ |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ifcif 4525 +∞cpnf 11292 -∞cmnf 11293 -cneg 11493 -𝑒cxne 13151 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-if 4526 df-xneg 13154 |
| This theorem is referenced by: xnegcl 13255 xnegneg 13256 xltnegi 13258 xnegid 13280 xnegdi 13290 xaddass2 13292 xsubge0 13303 xlesubadd 13305 xmulneg1 13311 xmulmnf1 13318 xadddi2 13339 xrsdsreclblem 21430 xblss2ps 24411 xblss2 24412 xaddeq0 32757 supminfxr 45475 liminflbuz2 45830 |
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