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Mirrors > Home > MPE Home > Th. List > mnfaddpnf | Structured version Visualization version 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 11303 | . . 3 ⊢ -∞ ∈ ℝ* | |
2 | pnfxr 11300 | . . 3 ⊢ +∞ ∈ ℝ* | |
3 | xaddval 13237 | . . 3 ⊢ ((-∞ ∈ ℝ* ∧ +∞ ∈ ℝ*) → (-∞ +𝑒 +∞) = if(-∞ = +∞, if(+∞ = -∞, 0, +∞), if(-∞ = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (-∞ + +∞)))))) | |
4 | 1, 2, 3 | mp2an 690 | . 2 ⊢ (-∞ +𝑒 +∞) = if(-∞ = +∞, if(+∞ = -∞, 0, +∞), if(-∞ = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (-∞ + +∞))))) |
5 | mnfnepnf 11302 | . . . 4 ⊢ -∞ ≠ +∞ | |
6 | ifnefalse 4542 | . . . 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 2725 | . . . . 5 ⊢ -∞ = -∞ | |
9 | 8 | iftruei 4537 | . . . 4 ⊢ if(-∞ = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (-∞ + +∞)))) = if(+∞ = +∞, 0, -∞) |
10 | eqid 2725 | . . . . 5 ⊢ +∞ = +∞ | |
11 | 10 | iftruei 4537 | . . . 4 ⊢ if(+∞ = +∞, 0, -∞) = 0 |
12 | 9, 11 | eqtri 2753 | . . 3 ⊢ if(-∞ = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (-∞ + +∞)))) = 0 |
13 | 7, 12 | eqtri 2753 | . 2 ⊢ if(-∞ = +∞, if(+∞ = -∞, 0, +∞), if(-∞ = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (-∞ + +∞))))) = 0 |
14 | 4, 13 | eqtri 2753 | 1 ⊢ (-∞ +𝑒 +∞) = 0 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2098 ≠ wne 2929 ifcif 4530 (class class class)co 7419 0cc0 11140 + caddc 11143 +∞cpnf 11277 -∞cmnf 11278 ℝ*cxr 11279 +𝑒 cxad 13125 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-mulcl 11202 ax-i2m1 11208 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-sbc 3774 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-iota 6501 df-fun 6551 df-fv 6557 df-ov 7422 df-oprab 7423 df-mpo 7424 df-pnf 11282 df-mnf 11283 df-xr 11284 df-xadd 13128 |
This theorem is referenced by: xnegid 13252 xaddcom 13254 xnegdi 13262 xsubge0 13275 xadddilem 13308 xrsnsgrp 21352 |
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