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Mirrors > Home > MPE Home > Th. List > xaddpnf1 | Structured version Visualization version GIF version |
Description: Addition of positive infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xaddpnf1 | ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) → (𝐴 +𝑒 +∞) = +∞) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pnfxr 11306 | . . 3 ⊢ +∞ ∈ ℝ* | |
2 | xaddval 13242 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (𝐴 +𝑒 +∞) = if(𝐴 = +∞, if(+∞ = -∞, 0, +∞), if(𝐴 = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞)))))) | |
3 | 1, 2 | mpan2 689 | . 2 ⊢ (𝐴 ∈ ℝ* → (𝐴 +𝑒 +∞) = if(𝐴 = +∞, if(+∞ = -∞, 0, +∞), if(𝐴 = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞)))))) |
4 | pnfnemnf 11307 | . . . . 5 ⊢ +∞ ≠ -∞ | |
5 | ifnefalse 4544 | . . . . 5 ⊢ (+∞ ≠ -∞ → if(+∞ = -∞, 0, +∞) = +∞) | |
6 | 4, 5 | mp1i 13 | . . . 4 ⊢ (𝐴 ≠ -∞ → if(+∞ = -∞, 0, +∞) = +∞) |
7 | ifnefalse 4544 | . . . . 5 ⊢ (𝐴 ≠ -∞ → if(𝐴 = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞)))) = if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞)))) | |
8 | eqid 2728 | . . . . . 6 ⊢ +∞ = +∞ | |
9 | 8 | iftruei 4539 | . . . . 5 ⊢ if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞))) = +∞ |
10 | 7, 9 | eqtrdi 2784 | . . . 4 ⊢ (𝐴 ≠ -∞ → if(𝐴 = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞)))) = +∞) |
11 | 6, 10 | ifeq12d 4553 | . . 3 ⊢ (𝐴 ≠ -∞ → if(𝐴 = +∞, if(+∞ = -∞, 0, +∞), if(𝐴 = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞))))) = if(𝐴 = +∞, +∞, +∞)) |
12 | ifid 4572 | . . 3 ⊢ if(𝐴 = +∞, +∞, +∞) = +∞ | |
13 | 11, 12 | eqtrdi 2784 | . 2 ⊢ (𝐴 ≠ -∞ → if(𝐴 = +∞, if(+∞ = -∞, 0, +∞), if(𝐴 = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞))))) = +∞) |
14 | 3, 13 | sylan9eq 2788 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) → (𝐴 +𝑒 +∞) = +∞) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ≠ wne 2937 ifcif 4532 (class class class)co 7426 0cc0 11146 + caddc 11149 +∞cpnf 11283 -∞cmnf 11284 ℝ*cxr 11285 +𝑒 cxad 13130 |
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 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-mulcl 11208 ax-i2m1 11214 |
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 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-iota 6505 df-fun 6555 df-fv 6561 df-ov 7429 df-oprab 7430 df-mpo 7431 df-pnf 11288 df-mnf 11289 df-xr 11290 df-xadd 13133 |
This theorem is referenced by: xnn0xaddcl 13254 xaddnemnf 13255 xaddcom 13259 xnn0xadd0 13266 xnegdi 13267 xaddass 13268 xleadd1a 13272 xlt2add 13279 xsubge0 13280 xlesubadd 13282 xadddilem 13313 xrsdsreclblem 21352 isxmet2d 24253 xrge0iifhom 33571 esumpr2 33719 hasheuni 33737 carsgclctunlem2 33972 ovolsplit 45405 sge0pr 45811 sge0split 45826 sge0xadd 45852 |
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