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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 7694 |
. . 3
  | |
| 2 | pnfxr 7690 |
. . 3
| |
| 3 | xaddval 9469 |
. . 3
![/\](wedge.gif "/\"){kind=small}   
| |
| 4 | 1, 2, 3 | mp2an 420 |
. 2
|
| 5 | mnfnepnf 7693 |
. . . 4
 | |
| 6 | ifnefalse 3432 |
. . . 4
| |
| 7 | 5, 6 | ax-mp 7 |
. . 3
|
| 8 | eqid 2100 |
. . . . 5
| |
| 9 | 8 | iftruei 3427 |
. . . 4
|
| 10 | eqid 2100 |
. . . . 5
| |
| 11 | 10 | iftruei 3427 |
. . . 4
|
| 12 | 9, 11 | eqtri 2120 |
. . 3
|
| 13 | 7, 12 | eqtri 2120 |
. 2
|
| 14 | 4, 13 | eqtri 2120 |
1
|
| Colors of variables: wff set class |
| Syntax hints: wceq 1299 wcel 1448 wne 2267 cif 3421 (class class class)co 5706
cc0 7500
caddc 7503 cpnf 7669 cmnf 7670 cxr 7671 cxad 9398 |
| This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-13 1459 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-sep 3986 ax-pow 4038 ax-pr 4069 ax-un 4293 ax-setind 4390 ax-cnex 7586 ax-resscn 7587 ax-1re 7589 ax-addrcl 7592 ax-rnegex 7604 |
| This theorem depends on definitions: df-bi 116 df-dc 787 df-3or 931 df-3an 932 df-tru 1302 df-fal 1305 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ne 2268 df-nel 2363 df-ral 2380 df-rex 2381 df-rab 2384 df-v 2643 df-sbc 2863 df-dif 3023 df-un 3025 df-in 3027 df-ss 3034 df-if 3422 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-br 3876 df-opab 3930 df-id 4153 df-xp 4483 df-rel 4484 df-cnv 4485 df-co 4486 df-dm 4487 df-iota 5024 df-fun 5061 df-fv 5067 df-ov 5709 df-oprab 5710 df-mpo 5711 df-pnf 7674 df-mnf 7675 df-xr 7676 df-xadd 9401 |
| This theorem is referenced by: xnegid 9483 xaddcom 9485 xnegdi 9492 xsubge0 9505 xposdif 9506 xrmaxadd 10869 |
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