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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 8100 |
. . 3
  | |
| 2 | pnfxr 8096 |
. . 3
| |
| 3 | xaddval 9937 |
. . 3
![/\](wedge.gif "/\"){kind=small}   
| |
| 4 | 1, 2, 3 | mp2an 426 |
. 2
|
| 5 | mnfnepnf 8099 |
. . . 4
 | |
| 6 | ifnefalse 3573 |
. . . 4
| |
| 7 | 5, 6 | ax-mp 5 |
. . 3
|
| 8 | eqid 2196 |
. . . . 5
| |
| 9 | 8 | iftruei 3568 |
. . . 4
|
| 10 | eqid 2196 |
. . . . 5
| |
| 11 | 10 | iftruei 3568 |
. . . 4
|
| 12 | 9, 11 | eqtri 2217 |
. . 3
|
| 13 | 7, 12 | eqtri 2217 |
. 2
|
| 14 | 4, 13 | eqtri 2217 |
1
|
| Colors of variables: wff set class |
| Syntax hints: wceq 1364 wcel 2167 wne 2367 cif 3562 (class class class)co 5925
cc0 7896
caddc 7899 cpnf 8075 cmnf 8076 cxr 8077 cxad 9862 |
| 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 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-sep 4152 ax-pow 4208 ax-pr 4243 ax-un 4469 ax-setind 4574 ax-cnex 7987 ax-resscn 7988 ax-1re 7990 ax-addrcl 7993 ax-rnegex 8005 |
| This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-rab 2484 df-v 2765 df-sbc 2990 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-if 3563 df-pw 3608 df-sn 3629 df-pr 3630 df-op 3632 df-uni 3841 df-br 4035 df-opab 4096 df-id 4329 df-xp 4670 df-rel 4671 df-cnv 4672 df-co 4673 df-dm 4674 df-iota 5220 df-fun 5261 df-fv 5267 df-ov 5928 df-oprab 5929 df-mpo 5930 df-pnf 8080 df-mnf 8081 df-xr 8082 df-xadd 9865 |
| This theorem is referenced by: xnegid 9951 xaddcom 9953 xnegdi 9960 xsubge0 9973 xposdif 9974 xrmaxadd 11443 |
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