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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 8012 |
. . 3
  | |
| 2 | mnfxr 8016 |
. . 3
| |
| 3 | xaddval 9847 |
. . 3
![/\](wedge.gif "/\"){kind=small}   
| |
| 4 | 1, 2, 3 | mp2an 426 |
. 2
|
| 5 | eqid 2177 |
. . 3
| |
| 6 | 5 | iftruei 3542 |
. 2
|
| 7 | eqid 2177 |
. . 3
| |
| 8 | 7 | iftruei 3542 |
. 2
|
| 9 | 4, 6, 8 | 3eqtri 2202 |
1
|
| Colors of variables: wff set class |
| Syntax hints: wceq 1353 wcel 2148 cif 3536 (class class class)co 5877
cc0 7813
caddc 7816 cpnf 7991 cmnf 7992 cxr 7993 cxad 9772 |
| 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 4123 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-cnex 7904 ax-resscn 7905 ax-1re 7907 ax-addrcl 7910 ax-rnegex 7922 |
| 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 2741 df-sbc 2965 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-if 3537 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-br 4006 df-opab 4067 df-id 4295 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-iota 5180 df-fun 5220 df-fv 5226 df-ov 5880 df-oprab 5881 df-mpo 5882 df-pnf 7996 df-mnf 7997 df-xr 7998 df-xadd 9775 |
| This theorem is referenced by: xnegid 9861 xaddcom 9863 xnegdi 9870 xsubge0 9883 xposdif 9884 xlesubadd 9885 xrmaxadd 11271 xblss2 13944 |
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