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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 10695 | . . 3 ⊢ -∞ ∈ ℝ* | |
2 | pnfxr 10692 | . . 3 ⊢ +∞ ∈ ℝ* | |
3 | xaddval 12614 | . . 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 10694 | . . . 4 ⊢ -∞ ≠ +∞ | |
6 | ifnefalse 4476 | . . . 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 2820 | . . . . 5 ⊢ -∞ = -∞ | |
9 | 8 | iftruei 4471 | . . . 4 ⊢ if(-∞ = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (-∞ + +∞)))) = if(+∞ = +∞, 0, -∞) |
10 | eqid 2820 | . . . . 5 ⊢ +∞ = +∞ | |
11 | 10 | iftruei 4471 | . . . 4 ⊢ if(+∞ = +∞, 0, -∞) = 0 |
12 | 9, 11 | eqtri 2843 | . . 3 ⊢ if(-∞ = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (-∞ + +∞)))) = 0 |
13 | 7, 12 | eqtri 2843 | . 2 ⊢ if(-∞ = +∞, if(+∞ = -∞, 0, +∞), if(-∞ = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (-∞ + +∞))))) = 0 |
14 | 4, 13 | eqtri 2843 | 1 ⊢ (-∞ +𝑒 +∞) = 0 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 ∈ wcel 2113 ≠ wne 3015 ifcif 4464 (class class class)co 7153 0cc0 10534 + caddc 10537 +∞cpnf 10669 -∞cmnf 10670 ℝ*cxr 10671 +𝑒 cxad 12503 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5200 ax-nul 5207 ax-pow 5263 ax-pr 5327 ax-un 7458 ax-cnex 10590 ax-1cn 10592 ax-icn 10593 ax-addcl 10594 ax-mulcl 10596 ax-i2m1 10602 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3495 df-sbc 3771 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4465 df-pw 4538 df-sn 4565 df-pr 4567 df-op 4571 df-uni 4836 df-br 5064 df-opab 5126 df-id 5457 df-xp 5558 df-rel 5559 df-cnv 5560 df-co 5561 df-dm 5562 df-iota 6311 df-fun 6354 df-fv 6360 df-ov 7156 df-oprab 7157 df-mpo 7158 df-pnf 10674 df-mnf 10675 df-xr 10676 df-xadd 12506 |
This theorem is referenced by: xnegid 12629 xaddcom 12631 xnegdi 12639 xsubge0 12652 xadddilem 12685 xrsnsgrp 20577 |
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