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Mirrors > Home > MPE Home > Th. List > xaddmnf2 | Structured version Visualization version GIF version |
Description: Addition of negative infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xaddmnf2 | ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞) → (-∞ +𝑒 𝐴) = -∞) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mnfxr 10695 | . . 3 ⊢ -∞ ∈ ℝ* | |
2 | xaddval 12614 | . . 3 ⊢ ((-∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ*) → (-∞ +𝑒 𝐴) = if(-∞ = +∞, if(𝐴 = -∞, 0, +∞), if(-∞ = -∞, if(𝐴 = +∞, 0, -∞), if(𝐴 = +∞, +∞, if(𝐴 = -∞, -∞, (-∞ + 𝐴)))))) | |
3 | 1, 2 | mpan 688 | . 2 ⊢ (𝐴 ∈ ℝ* → (-∞ +𝑒 𝐴) = if(-∞ = +∞, if(𝐴 = -∞, 0, +∞), if(-∞ = -∞, if(𝐴 = +∞, 0, -∞), if(𝐴 = +∞, +∞, if(𝐴 = -∞, -∞, (-∞ + 𝐴)))))) |
4 | mnfnepnf 10694 | . . . . 5 ⊢ -∞ ≠ +∞ | |
5 | ifnefalse 4476 | . . . . 5 ⊢ (-∞ ≠ +∞ → if(-∞ = +∞, if(𝐴 = -∞, 0, +∞), if(-∞ = -∞, if(𝐴 = +∞, 0, -∞), if(𝐴 = +∞, +∞, if(𝐴 = -∞, -∞, (-∞ + 𝐴))))) = if(-∞ = -∞, if(𝐴 = +∞, 0, -∞), if(𝐴 = +∞, +∞, if(𝐴 = -∞, -∞, (-∞ + 𝐴))))) | |
6 | 4, 5 | ax-mp 5 | . . . 4 ⊢ if(-∞ = +∞, if(𝐴 = -∞, 0, +∞), if(-∞ = -∞, if(𝐴 = +∞, 0, -∞), if(𝐴 = +∞, +∞, if(𝐴 = -∞, -∞, (-∞ + 𝐴))))) = if(-∞ = -∞, if(𝐴 = +∞, 0, -∞), if(𝐴 = +∞, +∞, if(𝐴 = -∞, -∞, (-∞ + 𝐴)))) |
7 | eqid 2820 | . . . . 5 ⊢ -∞ = -∞ | |
8 | 7 | iftruei 4471 | . . . 4 ⊢ if(-∞ = -∞, if(𝐴 = +∞, 0, -∞), if(𝐴 = +∞, +∞, if(𝐴 = -∞, -∞, (-∞ + 𝐴)))) = if(𝐴 = +∞, 0, -∞) |
9 | 6, 8 | eqtri 2843 | . . 3 ⊢ if(-∞ = +∞, if(𝐴 = -∞, 0, +∞), if(-∞ = -∞, if(𝐴 = +∞, 0, -∞), if(𝐴 = +∞, +∞, if(𝐴 = -∞, -∞, (-∞ + 𝐴))))) = if(𝐴 = +∞, 0, -∞) |
10 | ifnefalse 4476 | . . 3 ⊢ (𝐴 ≠ +∞ → if(𝐴 = +∞, 0, -∞) = -∞) | |
11 | 9, 10 | syl5eq 2867 | . 2 ⊢ (𝐴 ≠ +∞ → if(-∞ = +∞, if(𝐴 = -∞, 0, +∞), if(-∞ = -∞, if(𝐴 = +∞, 0, -∞), if(𝐴 = +∞, +∞, if(𝐴 = -∞, -∞, (-∞ + 𝐴))))) = -∞) |
12 | 3, 11 | sylan9eq 2875 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞) → (-∞ +𝑒 𝐴) = -∞) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = 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: xaddnepnf 12628 xaddcom 12631 xaddid1 12632 xnegdi 12639 xpncan 12642 xleadd1a 12644 xlt2add 12651 xadddilem 12685 xadddi2 12688 xrsnsgrp 20577 xaddeq0 30477 supxrgelem 41679 supxrge 41680 xrlexaddrp 41694 infleinflem2 41713 |
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