Nearby theorems
|
|
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
| Mirrors > Home > ILE Home > Th. List > pnfaddmnf | Unicode version | ||
| Description: Addition of positive and negative 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 |
|---|---|
| pnfaddmnf |
    
    |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | pnfxr 7984 |
. . 3
  | |
| 2 | mnfxr 7988 |
. . 3
| |
| 3 | xaddval 9814 |
. . 3
![/\](wedge.gif "/\"){kind=small}   
| |
| 4 | 1, 2, 3 | mp2an 426 |
. 2
|
| 5 | eqid 2175 |
. . 3
| |
| 6 | 5 | iftruei 3538 |
. 2
|
| 7 | eqid 2175 |
. . 3
| |
| 8 | 7 | iftruei 3538 |
. 2
|
| 9 | 4, 6, 8 | 3eqtri 2200 |
1
|
| Colors of variables: wff set class |
| Syntax hints: wceq 1353 wcel 2146 cif 3532 (class class class)co 5865
cc0 7786
caddc 7789 cpnf 7963 cmnf 7964 cxr 7965 cxad 9739 |
| 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 614 ax-in2 615 ax-io 709 ax-5 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-13 2148 ax-14 2149 ax-ext 2157 ax-sep 4116 ax-pow 4169 ax-pr 4203 ax-un 4427 ax-setind 4530 ax-cnex 7877 ax-resscn 7878 ax-1re 7880 ax-addrcl 7883 ax-rnegex 7895 |
| This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1459 df-sb 1761 df-eu 2027 df-mo 2028 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ne 2346 df-nel 2441 df-ral 2458 df-rex 2459 df-rab 2462 df-v 2737 df-sbc 2961 df-dif 3129 df-un 3131 df-in 3133 df-ss 3140 df-if 3533 df-pw 3574 df-sn 3595 df-pr 3596 df-op 3598 df-uni 3806 df-br 3999 df-opab 4060 df-id 4287 df-xp 4626 df-rel 4627 df-cnv 4628 df-co 4629 df-dm 4630 df-iota 5170 df-fun 5210 df-fv 5216 df-ov 5868 df-oprab 5869 df-mpo 5870 df-pnf 7968 df-mnf 7969 df-xr 7970 df-xadd 9742 |
| This theorem is referenced by: xnegid 9828 xaddcom 9830 xnegdi 9837 xsubge0 9850 xposdif 9851 xlesubadd 9852 xrmaxadd 11235 xblss2 13456 |
| Copyright terms: Public domain | W3C validator |