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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 8330 |
. . 3
  | |
| 2 | pnfxr 8326 |
. . 3
| |
| 3 | xaddval 10178 |
. . 3
![/\](wedge.gif "/\"){kind=small}   
| |
| 4 | 1, 2, 3 | mp2an 426 |
. 2
|
| 5 | mnfnepnf 8329 |
. . . 4
 | |
| 6 | ifnefalse 3633 |
. . . 4
| |
| 7 | 5, 6 | ax-mp 5 |
. . 3
|
| 8 | eqid 2232 |
. . . . 5
| |
| 9 | 8 | iftruei 3628 |
. . . 4
|
| 10 | eqid 2232 |
. . . . 5
| |
| 11 | 10 | iftruei 3628 |
. . . 4
|
| 12 | 9, 11 | eqtri 2253 |
. . 3
|
| 13 | 7, 12 | eqtri 2253 |
. 2
|
| 14 | 4, 13 | eqtri 2253 |
1
|
| Colors of variables: wff set class |
| Syntax hints: wceq 1398 wcel 2203 wne 2412 cif 3620 (class class class)co 6050
cc0 8127
caddc 8130 cpnf 8305 cmnf 8306 cxr 8307 cxad 10103 |
| 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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2205 ax-14 2206 ax-ext 2214 ax-sep 4228 ax-pow 4287 ax-pr 4322 ax-un 4554 ax-setind 4659 ax-cnex 8218 ax-resscn 8219 ax-1re 8221 ax-addrcl 8224 ax-rnegex 8236 |
| This theorem depends on definitions: df-bi 117 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2083 df-mo 2084 df-clab 2219 df-cleq 2225 df-clel 2228 df-nfc 2373 df-ne 2413 df-nel 2508 df-ral 2525 df-rex 2526 df-rab 2529 df-v 2815 df-sbc 3043 df-dif 3213 df-un 3215 df-in 3217 df-ss 3224 df-if 3621 df-pw 3671 df-sn 3695 df-pr 3696 df-op 3698 df-uni 3915 df-br 4110 df-opab 4172 df-id 4414 df-xp 4755 df-rel 4756 df-cnv 4757 df-co 4758 df-dm 4759 df-iota 5312 df-fun 5354 df-fv 5360 df-ov 6053 df-oprab 6054 df-mpo 6055 df-pnf 8310 df-mnf 8311 df-xr 8312 df-xadd 10106 |
| This theorem is referenced by: xnegid 10192 xaddcom 10194 xnegdi 10201 xsubge0 10214 xposdif 10215 xrmaxadd 11946 |
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