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| Mirrors > Home > ILE Home > Th. List > pnfaddmnf | Unicode version | ||
| Description: Addition of positive and negative 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 |
|---|---|
| pnfaddmnf |
    
    |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | pnfxr 8079 |
. . 3
  | |
| 2 | mnfxr 8083 |
. . 3
| |
| 3 | xaddval 9920 |
. . 3
![/\](wedge.gif "/\"){kind=small}   
| |
| 4 | 1, 2, 3 | mp2an 426 |
. 2
|
| 5 | eqid 2196 |
. . 3
| |
| 6 | 5 | iftruei 3567 |
. 2
|
| 7 | eqid 2196 |
. . 3
| |
| 8 | 7 | iftruei 3567 |
. 2
|
| 9 | 4, 6, 8 | 3eqtri 2221 |
1
|
| Colors of variables: wff set class |
| Syntax hints: wceq 1364 wcel 2167 cif 3561 (class class class)co 5922
cc0 7879
caddc 7882 cpnf 8058 cmnf 8059 cxr 8060 cxad 9845 |
| 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 4151 ax-pow 4207 ax-pr 4242 ax-un 4468 ax-setind 4573 ax-cnex 7970 ax-resscn 7971 ax-1re 7973 ax-addrcl 7976 ax-rnegex 7988 |
| 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 3562 df-pw 3607 df-sn 3628 df-pr 3629 df-op 3631 df-uni 3840 df-br 4034 df-opab 4095 df-id 4328 df-xp 4669 df-rel 4670 df-cnv 4671 df-co 4672 df-dm 4673 df-iota 5219 df-fun 5260 df-fv 5266 df-ov 5925 df-oprab 5926 df-mpo 5927 df-pnf 8063 df-mnf 8064 df-xr 8065 df-xadd 9848 |
| This theorem is referenced by: xnegid 9934 xaddcom 9936 xnegdi 9943 xsubge0 9956 xposdif 9957 xlesubadd 9958 xrmaxadd 11426 xblss2 14641 |
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