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 8128 |
. . 3
  | |
| 2 | pnfxr 8124 |
. . 3
| |
| 3 | xaddval 9966 |
. . 3
![/\](wedge.gif "/\"){kind=small}   
| |
| 4 | 1, 2, 3 | mp2an 426 |
. 2
|
| 5 | mnfnepnf 8127 |
. . . 4
 | |
| 6 | ifnefalse 3581 |
. . . 4
| |
| 7 | 5, 6 | ax-mp 5 |
. . 3
|
| 8 | eqid 2204 |
. . . . 5
| |
| 9 | 8 | iftruei 3576 |
. . . 4
|
| 10 | eqid 2204 |
. . . . 5
| |
| 11 | 10 | iftruei 3576 |
. . . 4
|
| 12 | 9, 11 | eqtri 2225 |
. . 3
|
| 13 | 7, 12 | eqtri 2225 |
. 2
|
| 14 | 4, 13 | eqtri 2225 |
1
|
| Colors of variables: wff set class |
| Syntax hints: wceq 1372 wcel 2175 wne 2375 cif 3570 (class class class)co 5943
cc0 7924
caddc 7927 cpnf 8103 cmnf 8104 cxr 8105 cxad 9891 |
| 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 1469 ax-7 1470 ax-gen 1471 ax-ie1 1515 ax-ie2 1516 ax-8 1526 ax-10 1527 ax-11 1528 ax-i12 1529 ax-bndl 1531 ax-4 1532 ax-17 1548 ax-i9 1552 ax-ial 1556 ax-i5r 1557 ax-13 2177 ax-14 2178 ax-ext 2186 ax-sep 4161 ax-pow 4217 ax-pr 4252 ax-un 4479 ax-setind 4584 ax-cnex 8015 ax-resscn 8016 ax-1re 8018 ax-addrcl 8021 ax-rnegex 8033 |
| This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1375 df-fal 1378 df-nf 1483 df-sb 1785 df-eu 2056 df-mo 2057 df-clab 2191 df-cleq 2197 df-clel 2200 df-nfc 2336 df-ne 2376 df-nel 2471 df-ral 2488 df-rex 2489 df-rab 2492 df-v 2773 df-sbc 2998 df-dif 3167 df-un 3169 df-in 3171 df-ss 3178 df-if 3571 df-pw 3617 df-sn 3638 df-pr 3639 df-op 3641 df-uni 3850 df-br 4044 df-opab 4105 df-id 4339 df-xp 4680 df-rel 4681 df-cnv 4682 df-co 4683 df-dm 4684 df-iota 5231 df-fun 5272 df-fv 5278 df-ov 5946 df-oprab 5947 df-mpo 5948 df-pnf 8108 df-mnf 8109 df-xr 8110 df-xadd 9894 |
| This theorem is referenced by: xnegid 9980 xaddcom 9982 xnegdi 9989 xsubge0 10002 xposdif 10003 xrmaxadd 11543 |
| Copyright terms: Public domain | W3C validator |