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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 8127 |
. . 3
  | |
| 2 | mnfxr 8131 |
. . 3
| |
| 3 | xaddval 9969 |
. . 3
![/\](wedge.gif "/\"){kind=small}   
| |
| 4 | 1, 2, 3 | mp2an 426 |
. 2
|
| 5 | eqid 2205 |
. . 3
| |
| 6 | 5 | iftruei 3577 |
. 2
|
| 7 | eqid 2205 |
. . 3
| |
| 8 | 7 | iftruei 3577 |
. 2
|
| 9 | 4, 6, 8 | 3eqtri 2230 |
1
|
| Colors of variables: wff set class |
| Syntax hints: wceq 1373 wcel 2176 cif 3571 (class class class)co 5946
cc0 7927
caddc 7930 cpnf 8106 cmnf 8107 cxr 8108 cxad 9894 |
| 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 711 ax-5 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-13 2178 ax-14 2179 ax-ext 2187 ax-sep 4163 ax-pow 4219 ax-pr 4254 ax-un 4481 ax-setind 4586 ax-cnex 8018 ax-resscn 8019 ax-1re 8021 ax-addrcl 8024 ax-rnegex 8036 |
| This theorem depends on definitions: df-bi 117 df-dc 837 df-3or 982 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1484 df-sb 1786 df-eu 2057 df-mo 2058 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-ne 2377 df-nel 2472 df-ral 2489 df-rex 2490 df-rab 2493 df-v 2774 df-sbc 2999 df-dif 3168 df-un 3170 df-in 3172 df-ss 3179 df-if 3572 df-pw 3618 df-sn 3639 df-pr 3640 df-op 3642 df-uni 3851 df-br 4046 df-opab 4107 df-id 4341 df-xp 4682 df-rel 4683 df-cnv 4684 df-co 4685 df-dm 4686 df-iota 5233 df-fun 5274 df-fv 5280 df-ov 5949 df-oprab 5950 df-mpo 5951 df-pnf 8111 df-mnf 8112 df-xr 8113 df-xadd 9897 |
| This theorem is referenced by: xnegid 9983 xaddcom 9985 xnegdi 9992 xsubge0 10005 xposdif 10006 xlesubadd 10007 xrmaxadd 11605 xblss2 14910 |
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