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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 8382 |
. . 3
  | |
| 2 | pnfxr 8378 |
. . 3
| |
| 3 | xaddval 10247 |
. . 3
![/\](wedge.gif "/\"){kind=small}   
| |
| 4 | 1, 2, 3 | mp2an 430 |
. 2
|
| 5 | mnfnepnf 8381 |
. . . 4
 | |
| 6 | ifnefalse 3651 |
. . . 4
| |
| 7 | 5, 6 | ax-mp 5 |
. . 3
|
| 8 | eqid 2238 |
. . . . 5
| |
| 9 | 8 | iftruei 3646 |
. . . 4
|
| 10 | eqid 2238 |
. . . . 5
| |
| 11 | 10 | iftruei 3646 |
. . . 4
|
| 12 | 9, 11 | eqtri 2259 |
. . 3
|
| 13 | 7, 12 | eqtri 2259 |
. 2
|
| 14 | 4, 13 | eqtri 2259 |
1
|
| Colors of variables: wff set class |
| This proof depends on syntax axioms:
wceq 1402
wcel 2209
wne 2420
cif 3638
(class class class)co 6085 cc0 8179 caddc 8182 cpnf 8357 cmnf 8358 cxr 8359 cxad 10172 |
| This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-sep 4249 ax-pow 4311 ax-pr 4346 ax-un 4578 ax-setind 4684 ax-cnex 8270 ax-resscn 8271 ax-1re 8273 ax-addrcl 8276 ax-rnegex 8288 |
| This proof depends on definitions: df-bi 117 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-rab 2537 df-v 2823 df-sbc 3052 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-if 3639 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-br 4131 df-opab 4193 df-id 4438 df-xp 4780 df-rel 4781 df-cnv 4782 df-co 4783 df-dm 4784 df-iota 5337 df-fun 5379 df-fv 5385 df-ov 6088 df-oprab 6089 df-mpo 6090 df-pnf 8362 df-mnf 8363 df-xr 8364 df-xadd 10175 |
| This theorem is used by: xnegid 10261 xaddcom 10263 xnegdi 10270 xsubge0 10283 xposdif 10284 xrmaxadd 12027 |
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