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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 8372 |
. . 3
  | |
| 2 | pnfxr 8368 |
. . 3
| |
| 3 | xaddval 10226 |
. . 3
![/\](wedge.gif "/\"){kind=small}   
| |
| 4 | 1, 2, 3 | mp2an 430 |
. 2
|
| 5 | mnfnepnf 8371 |
. . . 4
 | |
| 6 | ifnefalse 3648 |
. . . 4
| |
| 7 | 5, 6 | ax-mp 5 |
. . 3
|
| 8 | eqid 2238 |
. . . . 5
| |
| 9 | 8 | iftruei 3643 |
. . . 4
|
| 10 | eqid 2238 |
. . . . 5
| |
| 11 | 10 | iftruei 3643 |
. . . 4
|
| 12 | 9, 11 | eqtri 2259 |
. . 3
|
| 13 | 7, 12 | eqtri 2259 |
. 2
|
| 14 | 4, 13 | eqtri 2259 |
1
|
| Colors of variables: wff set class |
| Syntax hints: wceq 1402 wcel 2209 wne 2420 cif 3635 (class class class)co 6075
cc0 8169
caddc 8172 cpnf 8347 cmnf 8348 cxr 8349 cxad 10151 |
| 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 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-sep 4244 ax-pow 4306 ax-pr 4341 ax-un 4573 ax-setind 4679 ax-cnex 8260 ax-resscn 8261 ax-1re 8263 ax-addrcl 8266 ax-rnegex 8278 |
| This theorem depends on definitions: df-bi 117 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-rab 2537 df-v 2823 df-sbc 3052 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-if 3636 df-pw 3687 df-sn 3711 df-pr 3712 df-op 3714 df-uni 3931 df-br 4126 df-opab 4188 df-id 4433 df-xp 4775 df-rel 4776 df-cnv 4777 df-co 4778 df-dm 4779 df-iota 5332 df-fun 5374 df-fv 5380 df-ov 6078 df-oprab 6079 df-mpo 6080 df-pnf 8352 df-mnf 8353 df-xr 8354 df-xadd 10154 |
| This theorem is referenced by: xnegid 10240 xaddcom 10242 xnegdi 10249 xsubge0 10262 xposdif 10263 xrmaxadd 12005 |
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