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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 8072 |
. . 3
  | |
| 2 | mnfxr 8076 |
. . 3
| |
| 3 | xaddval 9911 |
. . 3
![/\](wedge.gif "/\"){kind=small}   
| |
| 4 | 1, 2, 3 | mp2an 426 |
. 2
|
| 5 | eqid 2193 |
. . 3
| |
| 6 | 5 | iftruei 3563 |
. 2
|
| 7 | eqid 2193 |
. . 3
| |
| 8 | 7 | iftruei 3563 |
. 2
|
| 9 | 4, 6, 8 | 3eqtri 2218 |
1
|
| Colors of variables: wff set class |
| Syntax hints: wceq 1364 wcel 2164 cif 3557 (class class class)co 5918
cc0 7872
caddc 7875 cpnf 8051 cmnf 8052 cxr 8053 cxad 9836 |
| 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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-sep 4147 ax-pow 4203 ax-pr 4238 ax-un 4464 ax-setind 4569 ax-cnex 7963 ax-resscn 7964 ax-1re 7966 ax-addrcl 7969 ax-rnegex 7981 |
| 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 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-rab 2481 df-v 2762 df-sbc 2986 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-if 3558 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-br 4030 df-opab 4091 df-id 4324 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-iota 5215 df-fun 5256 df-fv 5262 df-ov 5921 df-oprab 5922 df-mpo 5923 df-pnf 8056 df-mnf 8057 df-xr 8058 df-xadd 9839 |
| This theorem is referenced by: xnegid 9925 xaddcom 9927 xnegdi 9934 xsubge0 9947 xposdif 9948 xlesubadd 9949 xrmaxadd 11404 xblss2 14573 |
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