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 7955 |
. . 3
  | |
| 2 | pnfxr 7951 |
. . 3
| |
| 3 | xaddval 9781 |
. . 3
![/\](wedge.gif "/\"){kind=small}   
| |
| 4 | 1, 2, 3 | mp2an 423 |
. 2
|
| 5 | mnfnepnf 7954 |
. . . 4
 | |
| 6 | ifnefalse 3531 |
. . . 4
| |
| 7 | 5, 6 | ax-mp 5 |
. . 3
|
| 8 | eqid 2165 |
. . . . 5
| |
| 9 | 8 | iftruei 3526 |
. . . 4
|
| 10 | eqid 2165 |
. . . . 5
| |
| 11 | 10 | iftruei 3526 |
. . . 4
|
| 12 | 9, 11 | eqtri 2186 |
. . 3
|
| 13 | 7, 12 | eqtri 2186 |
. 2
|
| 14 | 4, 13 | eqtri 2186 |
1
|
| Colors of variables: wff set class |
| Syntax hints: wceq 1343 wcel 2136 wne 2336 cif 3520 (class class class)co 5842
cc0 7753
caddc 7756 cpnf 7930 cmnf 7931 cxr 7932 cxad 9706 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-sep 4100 ax-pow 4153 ax-pr 4187 ax-un 4411 ax-setind 4514 ax-cnex 7844 ax-resscn 7845 ax-1re 7847 ax-addrcl 7850 ax-rnegex 7862 |
| This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 969 df-3an 970 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ne 2337 df-nel 2432 df-ral 2449 df-rex 2450 df-rab 2453 df-v 2728 df-sbc 2952 df-dif 3118 df-un 3120 df-in 3122 df-ss 3129 df-if 3521 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-br 3983 df-opab 4044 df-id 4271 df-xp 4610 df-rel 4611 df-cnv 4612 df-co 4613 df-dm 4614 df-iota 5153 df-fun 5190 df-fv 5196 df-ov 5845 df-oprab 5846 df-mpo 5847 df-pnf 7935 df-mnf 7936 df-xr 7937 df-xadd 9709 |
| This theorem is referenced by: xnegid 9795 xaddcom 9797 xnegdi 9804 xsubge0 9817 xposdif 9818 xrmaxadd 11202 |
| Copyright terms: Public domain | W3C validator |