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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 8236 |
. . 3
  | |
| 2 | pnfxr 8232 |
. . 3
| |
| 3 | xaddval 10080 |
. . 3
![/\](wedge.gif "/\"){kind=small}   
| |
| 4 | 1, 2, 3 | mp2an 426 |
. 2
|
| 5 | mnfnepnf 8235 |
. . . 4
 | |
| 6 | ifnefalse 3616 |
. . . 4
| |
| 7 | 5, 6 | ax-mp 5 |
. . 3
|
| 8 | eqid 2231 |
. . . . 5
| |
| 9 | 8 | iftruei 3611 |
. . . 4
|
| 10 | eqid 2231 |
. . . . 5
| |
| 11 | 10 | iftruei 3611 |
. . . 4
|
| 12 | 9, 11 | eqtri 2252 |
. . 3
|
| 13 | 7, 12 | eqtri 2252 |
. 2
|
| 14 | 4, 13 | eqtri 2252 |
1
|
| Colors of variables: wff set class |
| Syntax hints: wceq 1397 wcel 2202 wne 2402 cif 3605 (class class class)co 6018
cc0 8032
caddc 8035 cpnf 8211 cmnf 8212 cxr 8213 cxad 10005 |
| 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 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2204 ax-14 2205 ax-ext 2213 ax-sep 4207 ax-pow 4264 ax-pr 4299 ax-un 4530 ax-setind 4635 ax-cnex 8123 ax-resscn 8124 ax-1re 8126 ax-addrcl 8129 ax-rnegex 8141 |
| This theorem depends on definitions: df-bi 117 df-dc 842 df-3or 1005 df-3an 1006 df-tru 1400 df-fal 1403 df-nf 1509 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ne 2403 df-nel 2498 df-ral 2515 df-rex 2516 df-rab 2519 df-v 2804 df-sbc 3032 df-dif 3202 df-un 3204 df-in 3206 df-ss 3213 df-if 3606 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-br 4089 df-opab 4151 df-id 4390 df-xp 4731 df-rel 4732 df-cnv 4733 df-co 4734 df-dm 4735 df-iota 5286 df-fun 5328 df-fv 5334 df-ov 6021 df-oprab 6022 df-mpo 6023 df-pnf 8216 df-mnf 8217 df-xr 8218 df-xadd 10008 |
| This theorem is referenced by: xnegid 10094 xaddcom 10096 xnegdi 10103 xsubge0 10116 xposdif 10117 xrmaxadd 11839 |
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