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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 8014 |
. . 3
  | |
| 2 | pnfxr 8010 |
. . 3
| |
| 3 | xaddval 9845 |
. . 3
![/\](wedge.gif "/\"){kind=small}   
| |
| 4 | 1, 2, 3 | mp2an 426 |
. 2
|
| 5 | mnfnepnf 8013 |
. . . 4
 | |
| 6 | ifnefalse 3546 |
. . . 4
| |
| 7 | 5, 6 | ax-mp 5 |
. . 3
|
| 8 | eqid 2177 |
. . . . 5
| |
| 9 | 8 | iftruei 3541 |
. . . 4
|
| 10 | eqid 2177 |
. . . . 5
| |
| 11 | 10 | iftruei 3541 |
. . . 4
|
| 12 | 9, 11 | eqtri 2198 |
. . 3
|
| 13 | 7, 12 | eqtri 2198 |
. 2
|
| 14 | 4, 13 | eqtri 2198 |
1
|
| Colors of variables: wff set class |
| Syntax hints: wceq 1353 wcel 2148 wne 2347 cif 3535 (class class class)co 5875
cc0 7811
caddc 7814 cpnf 7989 cmnf 7990 cxr 7991 cxad 9770 |
| 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 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4122 ax-pow 4175 ax-pr 4210 ax-un 4434 ax-setind 4537 ax-cnex 7902 ax-resscn 7903 ax-1re 7905 ax-addrcl 7908 ax-rnegex 7920 |
| 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 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2740 df-sbc 2964 df-dif 3132 df-un 3134 df-in 3136 df-ss 3143 df-if 3536 df-pw 3578 df-sn 3599 df-pr 3600 df-op 3602 df-uni 3811 df-br 4005 df-opab 4066 df-id 4294 df-xp 4633 df-rel 4634 df-cnv 4635 df-co 4636 df-dm 4637 df-iota 5179 df-fun 5219 df-fv 5225 df-ov 5878 df-oprab 5879 df-mpo 5880 df-pnf 7994 df-mnf 7995 df-xr 7996 df-xadd 9773 |
| This theorem is referenced by: xnegid 9859 xaddcom 9861 xnegdi 9868 xsubge0 9881 xposdif 9882 xrmaxadd 11269 |
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