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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 8214 |
. . 3
  | |
| 2 | pnfxr 8210 |
. . 3
| |
| 3 | xaddval 10053 |
. . 3
![/\](wedge.gif "/\"){kind=small}   
| |
| 4 | 1, 2, 3 | mp2an 426 |
. 2
|
| 5 | mnfnepnf 8213 |
. . . 4
 | |
| 6 | ifnefalse 3613 |
. . . 4
| |
| 7 | 5, 6 | ax-mp 5 |
. . 3
|
| 8 | eqid 2229 |
. . . . 5
| |
| 9 | 8 | iftruei 3608 |
. . . 4
|
| 10 | eqid 2229 |
. . . . 5
| |
| 11 | 10 | iftruei 3608 |
. . . 4
|
| 12 | 9, 11 | eqtri 2250 |
. . 3
|
| 13 | 7, 12 | eqtri 2250 |
. 2
|
| 14 | 4, 13 | eqtri 2250 |
1
|
| Colors of variables: wff set class |
| Syntax hints: wceq 1395 wcel 2200 wne 2400 cif 3602 (class class class)co 6007
cc0 8010
caddc 8013 cpnf 8189 cmnf 8190 cxr 8191 cxad 9978 |
| 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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-sep 4202 ax-pow 4258 ax-pr 4293 ax-un 4524 ax-setind 4629 ax-cnex 8101 ax-resscn 8102 ax-1re 8104 ax-addrcl 8107 ax-rnegex 8119 |
| This theorem depends on definitions: df-bi 117 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-rab 2517 df-v 2801 df-sbc 3029 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-if 3603 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3889 df-br 4084 df-opab 4146 df-id 4384 df-xp 4725 df-rel 4726 df-cnv 4727 df-co 4728 df-dm 4729 df-iota 5278 df-fun 5320 df-fv 5326 df-ov 6010 df-oprab 6011 df-mpo 6012 df-pnf 8194 df-mnf 8195 df-xr 8196 df-xadd 9981 |
| This theorem is referenced by: xnegid 10067 xaddcom 10069 xnegdi 10076 xsubge0 10089 xposdif 10090 xrmaxadd 11787 |
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