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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 13096 | . 2 ⊢ -𝑒-∞ = if(-∞ = +∞, -∞, if(-∞ = -∞, +∞, --∞)) | |
2 | mnfnepnf 11274 | . . 3 ⊢ -∞ ≠ +∞ | |
3 | ifnefalse 4539 | . . 3 ⊢ (-∞ ≠ +∞ → if(-∞ = +∞, -∞, if(-∞ = -∞, +∞, --∞)) = if(-∞ = -∞, +∞, --∞)) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ if(-∞ = +∞, -∞, if(-∞ = -∞, +∞, --∞)) = if(-∞ = -∞, +∞, --∞) |
5 | eqid 2730 | . . 3 ⊢ -∞ = -∞ | |
6 | 5 | iftruei 4534 | . 2 ⊢ if(-∞ = -∞, +∞, --∞) = +∞ |
7 | 1, 4, 6 | 3eqtri 2762 | 1 ⊢ -𝑒-∞ = +∞ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1539 ≠ wne 2938 ifcif 4527 +∞cpnf 11249 -∞cmnf 11250 -cneg 11449 -𝑒cxne 13093 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-ext 2701 ax-sep 5298 ax-pow 5362 ax-un 7727 ax-cnex 11168 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-tru 1542 df-ex 1780 df-sb 2066 df-clab 2708 df-cleq 2722 df-clel 2808 df-ne 2939 df-rab 3431 df-v 3474 df-un 3952 df-in 3954 df-ss 3964 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-uni 4908 df-pnf 11254 df-mnf 11255 df-xr 11256 df-xneg 13096 |
This theorem is referenced by: xnegcl 13196 xnegneg 13197 xltnegi 13199 xnegid 13221 xnegdi 13231 xsubge0 13244 xmulneg1 13252 xmulpnf1n 13261 xadddi2 13280 xrsdsreclblem 21191 xaddeq0 32233 xrge0npcan 32462 carsgclctunlem2 33616 supminfxr 44472 supminfxr2 44477 liminf0 44807 liminflbuz2 44829 liminfpnfuz 44830 |
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