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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 9774 | . 2 ⊢ -𝑒+∞ = if(+∞ = +∞, -∞, if(+∞ = -∞, +∞, -+∞)) | |
2 | eqid 2177 | . . 3 ⊢ +∞ = +∞ | |
3 | 2 | iftruei 3542 | . 2 ⊢ if(+∞ = +∞, -∞, if(+∞ = -∞, +∞, -+∞)) = -∞ |
4 | 1, 3 | eqtri 2198 | 1 ⊢ -𝑒+∞ = -∞ |
Colors of variables: wff set class |
Syntax hints: = wceq 1353 ifcif 3536 +∞cpnf 7991 -∞cmnf 7992 -cneg 8131 -𝑒cxne 9771 |
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 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-11 1506 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-if 3537 df-xneg 9774 |
This theorem is referenced by: xnegcl 9834 xnegneg 9835 xltnegi 9837 xnegid 9861 xnegdi 9870 xaddass2 9872 xsubge0 9883 xposdif 9884 xlesubadd 9885 xblss2ps 13943 xblss2 13944 |
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