Nearby theorems
|
|
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
| 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 8045 |
. . 3
  | |
| 2 | pnfxr 8041 |
. . 3
| |
| 3 | xaddval 9877 |
. . 3
![/\](wedge.gif "/\"){kind=small}   
| |
| 4 | 1, 2, 3 | mp2an 426 |
. 2
|
| 5 | mnfnepnf 8044 |
. . . 4
 | |
| 6 | ifnefalse 3560 |
. . . 4
| |
| 7 | 5, 6 | ax-mp 5 |
. . 3
|
| 8 | eqid 2189 |
. . . . 5
| |
| 9 | 8 | iftruei 3555 |
. . . 4
|
| 10 | eqid 2189 |
. . . . 5
| |
| 11 | 10 | iftruei 3555 |
. . . 4
|
| 12 | 9, 11 | eqtri 2210 |
. . 3
|
| 13 | 7, 12 | eqtri 2210 |
. 2
|
| 14 | 4, 13 | eqtri 2210 |
1
|
| Colors of variables: wff set class |
| Syntax hints: wceq 1364 wcel 2160 wne 2360 cif 3549 (class class class)co 5897
cc0 7842
caddc 7845 cpnf 8020 cmnf 8021 cxr 8022 cxad 9802 |
| 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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-cnex 7933 ax-resscn 7934 ax-1re 7936 ax-addrcl 7939 ax-rnegex 7951 |
| This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-if 3550 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-iota 5196 df-fun 5237 df-fv 5243 df-ov 5900 df-oprab 5901 df-mpo 5902 df-pnf 8025 df-mnf 8026 df-xr 8027 df-xadd 9805 |
| This theorem is referenced by: xnegid 9891 xaddcom 9893 xnegdi 9900 xsubge0 9913 xposdif 9914 xrmaxadd 11304 |
| Copyright terms: Public domain | W3C validator |