![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 8013 | . . 3 ⊢ +∞ ∈ ℝ* | |
2 | mnfxr 8017 | . . 3 ⊢ -∞ ∈ ℝ* | |
3 | xaddval 9848 | . . 3 ⊢ ((+∞ ∈ ℝ* ∧ -∞ ∈ ℝ*) → (+∞ +𝑒 -∞) = if(+∞ = +∞, if(-∞ = -∞, 0, +∞), if(+∞ = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (+∞ + -∞)))))) | |
4 | 1, 2, 3 | mp2an 426 | . 2 ⊢ (+∞ +𝑒 -∞) = if(+∞ = +∞, if(-∞ = -∞, 0, +∞), if(+∞ = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (+∞ + -∞))))) |
5 | eqid 2177 | . . 3 ⊢ +∞ = +∞ | |
6 | 5 | iftruei 3542 | . 2 ⊢ if(+∞ = +∞, if(-∞ = -∞, 0, +∞), if(+∞ = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (+∞ + -∞))))) = if(-∞ = -∞, 0, +∞) |
7 | eqid 2177 | . . 3 ⊢ -∞ = -∞ | |
8 | 7 | iftruei 3542 | . 2 ⊢ if(-∞ = -∞, 0, +∞) = 0 |
9 | 4, 6, 8 | 3eqtri 2202 | 1 ⊢ (+∞ +𝑒 -∞) = 0 |
Colors of variables: wff set class |
Syntax hints: = wceq 1353 ∈ wcel 2148 ifcif 3536 (class class class)co 5878 0cc0 7814 + caddc 7817 +∞cpnf 7992 -∞cmnf 7993 ℝ*cxr 7994 +𝑒 cxad 9773 |
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 7905 ax-resscn 7906 ax-1re 7908 ax-addrcl 7911 ax-rnegex 7923 |
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 5881 df-oprab 5882 df-mpo 5883 df-pnf 7997 df-mnf 7998 df-xr 7999 df-xadd 9776 |
This theorem is referenced by: xnegid 9862 xaddcom 9864 xnegdi 9871 xsubge0 9884 xposdif 9885 xlesubadd 9886 xrmaxadd 11272 xblss2 14066 |
Copyright terms: Public domain | W3C validator |