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Mirrors > Home > MPE Home > Th. List > xaddpnf2 | Structured version Visualization version GIF version |
Description: Addition of positive infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xaddpnf2 | ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) → (+∞ +𝑒 𝐴) = +∞) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pnfxr 11313 | . . 3 ⊢ +∞ ∈ ℝ* | |
2 | xaddval 13262 | . . 3 ⊢ ((+∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ*) → (+∞ +𝑒 𝐴) = if(+∞ = +∞, if(𝐴 = -∞, 0, +∞), if(+∞ = -∞, if(𝐴 = +∞, 0, -∞), if(𝐴 = +∞, +∞, if(𝐴 = -∞, -∞, (+∞ + 𝐴)))))) | |
3 | 1, 2 | mpan 690 | . 2 ⊢ (𝐴 ∈ ℝ* → (+∞ +𝑒 𝐴) = if(+∞ = +∞, if(𝐴 = -∞, 0, +∞), if(+∞ = -∞, if(𝐴 = +∞, 0, -∞), if(𝐴 = +∞, +∞, if(𝐴 = -∞, -∞, (+∞ + 𝐴)))))) |
4 | eqid 2735 | . . . 4 ⊢ +∞ = +∞ | |
5 | 4 | iftruei 4538 | . . 3 ⊢ if(+∞ = +∞, if(𝐴 = -∞, 0, +∞), if(+∞ = -∞, if(𝐴 = +∞, 0, -∞), if(𝐴 = +∞, +∞, if(𝐴 = -∞, -∞, (+∞ + 𝐴))))) = if(𝐴 = -∞, 0, +∞) |
6 | ifnefalse 4543 | . . 3 ⊢ (𝐴 ≠ -∞ → if(𝐴 = -∞, 0, +∞) = +∞) | |
7 | 5, 6 | eqtrid 2787 | . 2 ⊢ (𝐴 ≠ -∞ → if(+∞ = +∞, if(𝐴 = -∞, 0, +∞), if(+∞ = -∞, if(𝐴 = +∞, 0, -∞), if(𝐴 = +∞, +∞, if(𝐴 = -∞, -∞, (+∞ + 𝐴))))) = +∞) |
8 | 3, 7 | sylan9eq 2795 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) → (+∞ +𝑒 𝐴) = +∞) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ifcif 4531 (class class class)co 7431 0cc0 11153 + caddc 11156 +∞cpnf 11290 -∞cmnf 11291 ℝ*cxr 11292 +𝑒 cxad 13150 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-mulcl 11215 ax-i2m1 11221 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-pnf 11295 df-mnf 11296 df-xr 11297 df-xadd 13153 |
This theorem is referenced by: xnn0xaddcl 13274 xaddnemnf 13275 xaddcom 13279 xaddrid 13280 xnn0xadd0 13286 xnegdi 13287 xaddass 13288 xleadd1a 13292 xadddilem 13333 xadddi2 13336 hashinfxadd 14421 xrsdsreclblem 21448 isxmet2d 24353 xaddeq0 32764 xrge0adddir 33006 xrge0iifhom 33898 infrpge 45301 infleinflem1 45320 ovolsplit 45944 sge0pr 46350 sge0split 46365 sge0xaddlem1 46389 sge0xadd 46391 |
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