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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 13152 | . 2 ⊢ -𝑒-∞ = if(-∞ = +∞, -∞, if(-∞ = -∞, +∞, --∞)) | |
2 | mnfnepnf 11315 | . . 3 ⊢ -∞ ≠ +∞ | |
3 | ifnefalse 4543 | . . 3 ⊢ (-∞ ≠ +∞ → if(-∞ = +∞, -∞, if(-∞ = -∞, +∞, --∞)) = if(-∞ = -∞, +∞, --∞)) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ if(-∞ = +∞, -∞, if(-∞ = -∞, +∞, --∞)) = if(-∞ = -∞, +∞, --∞) |
5 | eqid 2735 | . . 3 ⊢ -∞ = -∞ | |
6 | 5 | iftruei 4538 | . 2 ⊢ if(-∞ = -∞, +∞, --∞) = +∞ |
7 | 1, 4, 6 | 3eqtri 2767 | 1 ⊢ -𝑒-∞ = +∞ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ≠ wne 2938 ifcif 4531 +∞cpnf 11290 -∞cmnf 11291 -cneg 11491 -𝑒cxne 13149 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-pow 5371 ax-un 7754 ax-cnex 11209 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-rab 3434 df-v 3480 df-un 3968 df-in 3970 df-ss 3980 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-uni 4913 df-pnf 11295 df-mnf 11296 df-xr 11297 df-xneg 13152 |
This theorem is referenced by: xnegcl 13252 xnegneg 13253 xltnegi 13255 xnegid 13277 xnegdi 13287 xsubge0 13300 xmulneg1 13308 xmulpnf1n 13317 xadddi2 13336 xrsdsreclblem 21448 xaddeq0 32764 xrge0npcan 33008 carsgclctunlem2 34301 supminfxr 45414 supminfxr2 45419 liminf0 45749 liminflbuz2 45771 liminfpnfuz 45772 |
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