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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 7846 |
. . 3
  | |
| 2 | pnfxr 7842 |
. . 3
| |
| 3 | xaddval 9658 |
. . 3
![/\](wedge.gif "/\"){kind=small}   
| |
| 4 | 1, 2, 3 | mp2an 423 |
. 2
|
| 5 | mnfnepnf 7845 |
. . . 4
 | |
| 6 | ifnefalse 3490 |
. . . 4
| |
| 7 | 5, 6 | ax-mp 5 |
. . 3
|
| 8 | eqid 2140 |
. . . . 5
| |
| 9 | 8 | iftruei 3485 |
. . . 4
|
| 10 | eqid 2140 |
. . . . 5
| |
| 11 | 10 | iftruei 3485 |
. . . 4
|
| 12 | 9, 11 | eqtri 2161 |
. . 3
|
| 13 | 7, 12 | eqtri 2161 |
. 2
|
| 14 | 4, 13 | eqtri 2161 |
1
|
| Colors of variables: wff set class |
| Syntax hints: wceq 1332 wcel 1481 wne 2309 cif 3479 (class class class)co 5782
cc0 7644
caddc 7647 cpnf 7821 cmnf 7822 cxr 7823 cxad 9587 |
| 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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-1re 7738 ax-addrcl 7741 ax-rnegex 7753 |
| This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-rab 2426 df-v 2691 df-sbc 2914 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-if 3480 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-iota 5096 df-fun 5133 df-fv 5139 df-ov 5785 df-oprab 5786 df-mpo 5787 df-pnf 7826 df-mnf 7827 df-xr 7828 df-xadd 9590 |
| This theorem is referenced by: xnegid 9672 xaddcom 9674 xnegdi 9681 xsubge0 9694 xposdif 9695 xrmaxadd 11062 |
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