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Mirrors > Home > ILE Home > Th. List > mnfaddpnf | GIF 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 | ⊢ (-∞ +𝑒 +∞) = 0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mnfxr 8010 | . . 3 ⊢ -∞ ∈ ℝ* | |
2 | pnfxr 8006 | . . 3 ⊢ +∞ ∈ ℝ* | |
3 | xaddval 9841 | . . 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 | mnfnepnf 8009 | . . . 4 ⊢ -∞ ≠ +∞ | |
6 | ifnefalse 3545 | . . . 4 ⊢ (-∞ ≠ +∞ → if(-∞ = +∞, if(+∞ = -∞, 0, +∞), if(-∞ = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (-∞ + +∞))))) = if(-∞ = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (-∞ + +∞))))) | |
7 | 5, 6 | ax-mp 5 | . . 3 ⊢ if(-∞ = +∞, if(+∞ = -∞, 0, +∞), if(-∞ = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (-∞ + +∞))))) = if(-∞ = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (-∞ + +∞)))) |
8 | eqid 2177 | . . . . 5 ⊢ -∞ = -∞ | |
9 | 8 | iftruei 3540 | . . . 4 ⊢ if(-∞ = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (-∞ + +∞)))) = if(+∞ = +∞, 0, -∞) |
10 | eqid 2177 | . . . . 5 ⊢ +∞ = +∞ | |
11 | 10 | iftruei 3540 | . . . 4 ⊢ if(+∞ = +∞, 0, -∞) = 0 |
12 | 9, 11 | eqtri 2198 | . . 3 ⊢ if(-∞ = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (-∞ + +∞)))) = 0 |
13 | 7, 12 | eqtri 2198 | . 2 ⊢ if(-∞ = +∞, if(+∞ = -∞, 0, +∞), if(-∞ = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (-∞ + +∞))))) = 0 |
14 | 4, 13 | eqtri 2198 | 1 ⊢ (-∞ +𝑒 +∞) = 0 |
Colors of variables: wff set class |
Syntax hints: = wceq 1353 ∈ wcel 2148 ≠ wne 2347 ifcif 3534 (class class class)co 5872 0cc0 7808 + caddc 7811 +∞cpnf 7985 -∞cmnf 7986 ℝ*cxr 7987 +𝑒 cxad 9766 |
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 4120 ax-pow 4173 ax-pr 4208 ax-un 4432 ax-setind 4535 ax-cnex 7899 ax-resscn 7900 ax-1re 7902 ax-addrcl 7905 ax-rnegex 7917 |
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 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-if 3535 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4003 df-opab 4064 df-id 4292 df-xp 4631 df-rel 4632 df-cnv 4633 df-co 4634 df-dm 4635 df-iota 5177 df-fun 5217 df-fv 5223 df-ov 5875 df-oprab 5876 df-mpo 5877 df-pnf 7990 df-mnf 7991 df-xr 7992 df-xadd 9769 |
This theorem is referenced by: xnegid 9855 xaddcom 9857 xnegdi 9864 xsubge0 9877 xposdif 9878 xrmaxadd 11262 |
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