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Mirrors > Home > MPE Home > Th. List > xnegmnf | Structured version Visualization version GIF version |
Description: Minus -∞. Remark of [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.) (Revised by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xnegmnf | ⊢ -𝑒-∞ = +∞ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-xneg 12495 | . 2 ⊢ -𝑒-∞ = if(-∞ = +∞, -∞, if(-∞ = -∞, +∞, --∞)) | |
2 | mnfnepnf 10685 | . . 3 ⊢ -∞ ≠ +∞ | |
3 | ifnefalse 4475 | . . 3 ⊢ (-∞ ≠ +∞ → if(-∞ = +∞, -∞, if(-∞ = -∞, +∞, --∞)) = if(-∞ = -∞, +∞, --∞)) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ if(-∞ = +∞, -∞, if(-∞ = -∞, +∞, --∞)) = if(-∞ = -∞, +∞, --∞) |
5 | eqid 2818 | . . 3 ⊢ -∞ = -∞ | |
6 | 5 | iftruei 4470 | . 2 ⊢ if(-∞ = -∞, +∞, --∞) = +∞ |
7 | 1, 4, 6 | 3eqtri 2845 | 1 ⊢ -𝑒-∞ = +∞ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ≠ wne 3013 ifcif 4463 +∞cpnf 10660 -∞cmnf 10661 -cneg 10859 -𝑒cxne 12492 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-pow 5257 ax-un 7450 ax-cnex 10581 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-rex 3141 df-rab 3144 df-v 3494 df-un 3938 df-in 3940 df-ss 3949 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-uni 4831 df-pnf 10665 df-mnf 10666 df-xr 10667 df-xneg 12495 |
This theorem is referenced by: xnegcl 12594 xnegneg 12595 xltnegi 12597 xnegid 12619 xnegdi 12629 xsubge0 12642 xmulneg1 12650 xmulpnf1n 12659 xadddi2 12678 xrsdsreclblem 20519 xaddeq0 30403 xrge0npcan 30608 carsgclctunlem2 31476 supminfxr 41616 supminfxr2 41621 liminf0 41950 liminflbuz2 41972 liminfpnfuz 41973 |
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