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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 7976 |
. . 3
  | |
| 2 | pnfxr 7972 |
. . 3
| |
| 3 | xaddval 9802 |
. . 3
![/\](wedge.gif "/\"){kind=small}   
| |
| 4 | 1, 2, 3 | mp2an 424 |
. 2
|
| 5 | mnfnepnf 7975 |
. . . 4
 | |
| 6 | ifnefalse 3537 |
. . . 4
| |
| 7 | 5, 6 | ax-mp 5 |
. . 3
|
| 8 | eqid 2170 |
. . . . 5
| |
| 9 | 8 | iftruei 3532 |
. . . 4
|
| 10 | eqid 2170 |
. . . . 5
| |
| 11 | 10 | iftruei 3532 |
. . . 4
|
| 12 | 9, 11 | eqtri 2191 |
. . 3
|
| 13 | 7, 12 | eqtri 2191 |
. 2
|
| 14 | 4, 13 | eqtri 2191 |
1
|
| Colors of variables: wff set class |
| Syntax hints: wceq 1348 wcel 2141 wne 2340 cif 3526 (class class class)co 5853
cc0 7774
caddc 7777 cpnf 7951 cmnf 7952 cxr 7953 cxad 9727 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 ax-1re 7868 ax-addrcl 7871 ax-rnegex 7883 |
| This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-if 3527 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-iota 5160 df-fun 5200 df-fv 5206 df-ov 5856 df-oprab 5857 df-mpo 5858 df-pnf 7956 df-mnf 7957 df-xr 7958 df-xadd 9730 |
| This theorem is referenced by: xnegid 9816 xaddcom 9818 xnegdi 9825 xsubge0 9838 xposdif 9839 xrmaxadd 11224 |
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