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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 7822 |
. . 3
  | |
| 2 | pnfxr 7818 |
. . 3
| |
| 3 | xaddval 9628 |
. . 3
![/\](wedge.gif "/\"){kind=small}   
| |
| 4 | 1, 2, 3 | mp2an 422 |
. 2
|
| 5 | mnfnepnf 7821 |
. . . 4
 | |
| 6 | ifnefalse 3485 |
. . . 4
| |
| 7 | 5, 6 | ax-mp 5 |
. . 3
|
| 8 | eqid 2139 |
. . . . 5
| |
| 9 | 8 | iftruei 3480 |
. . . 4
|
| 10 | eqid 2139 |
. . . . 5
| |
| 11 | 10 | iftruei 3480 |
. . . 4
|
| 12 | 9, 11 | eqtri 2160 |
. . 3
|
| 13 | 7, 12 | eqtri 2160 |
. 2
|
| 14 | 4, 13 | eqtri 2160 |
1
|
| Colors of variables: wff set class |
| Syntax hints: wceq 1331 wcel 1480 wne 2308 cif 3474 (class class class)co 5774
cc0 7620
caddc 7623 cpnf 7797 cmnf 7798 cxr 7799 cxad 9557 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1re 7714 ax-addrcl 7717 ax-rnegex 7729 |
| This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pnf 7802 df-mnf 7803 df-xr 7804 df-xadd 9560 |
| This theorem is referenced by: xnegid 9642 xaddcom 9644 xnegdi 9651 xsubge0 9664 xposdif 9665 xrmaxadd 11030 |
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