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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 11822 | . 2 ⊢ -𝑒+∞ = if(+∞ = +∞, -∞, if(+∞ = -∞, +∞, -+∞)) | |
2 | eqid 2610 | . . 3 ⊢ +∞ = +∞ | |
3 | 2 | iftruei 4043 | . 2 ⊢ if(+∞ = +∞, -∞, if(+∞ = -∞, +∞, -+∞)) = -∞ |
4 | 1, 3 | eqtri 2632 | 1 ⊢ -𝑒+∞ = -∞ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ifcif 4036 +∞cpnf 9950 -∞cmnf 9951 -cneg 10146 -𝑒cxne 11819 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-if 4037 df-xneg 11822 |
This theorem is referenced by: xnegcl 11918 xnegneg 11919 xltnegi 11921 xnegid 11943 xnegdi 11950 xaddass2 11952 xsubge0 11963 xlesubadd 11965 xmulneg1 11971 xmulmnf1 11978 xadddi2 11999 xrsdsreclblem 19611 xblss2ps 22016 xblss2 22017 xaddeq0 28907 |
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