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Mirrors > Home > ILE Home > Th. List > pnfaddmnf | GIF 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 | ⊢ (+∞ +𝑒 -∞) = 0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pnfxr 7821 | . . 3 ⊢ +∞ ∈ ℝ* | |
2 | mnfxr 7825 | . . 3 ⊢ -∞ ∈ ℝ* | |
3 | xaddval 9631 | . . 3 ⊢ ((+∞ ∈ ℝ* ∧ -∞ ∈ ℝ*) → (+∞ +𝑒 -∞) = if(+∞ = +∞, if(-∞ = -∞, 0, +∞), if(+∞ = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (+∞ + -∞)))))) | |
4 | 1, 2, 3 | mp2an 422 | . 2 ⊢ (+∞ +𝑒 -∞) = if(+∞ = +∞, if(-∞ = -∞, 0, +∞), if(+∞ = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (+∞ + -∞))))) |
5 | eqid 2139 | . . 3 ⊢ +∞ = +∞ | |
6 | 5 | iftruei 3480 | . 2 ⊢ if(+∞ = +∞, if(-∞ = -∞, 0, +∞), if(+∞ = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (+∞ + -∞))))) = if(-∞ = -∞, 0, +∞) |
7 | eqid 2139 | . . 3 ⊢ -∞ = -∞ | |
8 | 7 | iftruei 3480 | . 2 ⊢ if(-∞ = -∞, 0, +∞) = 0 |
9 | 4, 6, 8 | 3eqtri 2164 | 1 ⊢ (+∞ +𝑒 -∞) = 0 |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 ∈ wcel 1480 ifcif 3474 (class class class)co 5774 0cc0 7623 + caddc 7626 +∞cpnf 7800 -∞cmnf 7801 ℝ*cxr 7802 +𝑒 cxad 9560 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7714 ax-resscn 7715 ax-1re 7717 ax-addrcl 7720 ax-rnegex 7732 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pnf 7805 df-mnf 7806 df-xr 7807 df-xadd 9563 |
This theorem is referenced by: xnegid 9645 xaddcom 9647 xnegdi 9654 xsubge0 9667 xposdif 9668 xlesubadd 9669 xrmaxadd 11033 xblss2 12577 |
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