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| Mirrors > Home > ILE Home > Th. List > mnfaddpnf | Unicode version | ||
| Description: Addition of negative and positive infinity. This is often taken to be a "null" value or out of the domain, but we define it (somewhat arbitrarily) to be zero so that the resulting function is total, which simplifies proofs. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| Ref | Expression |
|---|---|
| mnfaddpnf |
    
    |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | mnfxr 8346 |
. . 3
  | |
| 2 | pnfxr 8342 |
. . 3
| |
| 3 | xaddval 10197 |
. . 3
![/\](wedge.gif "/\"){kind=small}   
| |
| 4 | 1, 2, 3 | mp2an 426 |
. 2
|
| 5 | mnfnepnf 8345 |
. . . 4
 | |
| 6 | ifnefalse 3637 |
. . . 4
| |
| 7 | 5, 6 | ax-mp 5 |
. . 3
|
| 8 | eqid 2234 |
. . . . 5
| |
| 9 | 8 | iftruei 3632 |
. . . 4
|
| 10 | eqid 2234 |
. . . . 5
| |
| 11 | 10 | iftruei 3632 |
. . . 4
|
| 12 | 9, 11 | eqtri 2255 |
. . 3
|
| 13 | 7, 12 | eqtri 2255 |
. 2
|
| 14 | 4, 13 | eqtri 2255 |
1
|
| Colors of variables: wff set class |
| Syntax hints: wceq 1398 wcel 2205 wne 2414 cif 3624 (class class class)co 6058
cc0 8143
caddc 8146 cpnf 8321 cmnf 8322 cxr 8323 cxad 10122 |
| 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-in1 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2207 ax-14 2208 ax-ext 2216 ax-sep 4233 ax-pow 4292 ax-pr 4327 ax-un 4559 ax-setind 4664 ax-cnex 8234 ax-resscn 8235 ax-1re 8237 ax-addrcl 8240 ax-rnegex 8252 |
| This theorem depends on definitions: df-bi 117 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-nel 2510 df-ral 2527 df-rex 2528 df-rab 2531 df-v 2817 df-sbc 3046 df-dif 3216 df-un 3218 df-in 3220 df-ss 3227 df-if 3625 df-pw 3676 df-sn 3700 df-pr 3701 df-op 3703 df-uni 3920 df-br 4115 df-opab 4177 df-id 4419 df-xp 4760 df-rel 4761 df-cnv 4762 df-co 4763 df-dm 4764 df-iota 5317 df-fun 5359 df-fv 5365 df-ov 6061 df-oprab 6062 df-mpo 6063 df-pnf 8326 df-mnf 8327 df-xr 8328 df-xadd 10125 |
| This theorem is referenced by: xnegid 10211 xaddcom 10213 xnegdi 10220 xsubge0 10233 xposdif 10234 xrmaxadd 11971 |
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