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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 8689 | . 2 ⊢ -𝑒+∞ = if(+∞ = +∞, -∞, if(+∞ = -∞, +∞, -+∞)) | |
2 | eqid 2040 | . . 3 ⊢ +∞ = +∞ | |
3 | 2 | iftruei 3337 | . 2 ⊢ if(+∞ = +∞, -∞, if(+∞ = -∞, +∞, -+∞)) = -∞ |
4 | 1, 3 | eqtri 2060 | 1 ⊢ -𝑒+∞ = -∞ |
Colors of variables: wff set class |
Syntax hints: = wceq 1243 ifcif 3331 +∞cpnf 7057 -∞cmnf 7058 -cneg 7183 -𝑒cxne 8686 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-11 1397 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-if 3332 df-xneg 8689 |
This theorem is referenced by: xnegcl 8745 xnegneg 8746 xltnegi 8748 |
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