![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 10410 | . . 3 ⊢ +∞ ∈ ℝ* | |
2 | xaddval 12342 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (𝐴 +𝑒 +∞) = if(𝐴 = +∞, if(+∞ = -∞, 0, +∞), if(𝐴 = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞)))))) | |
3 | 1, 2 | mpan2 682 | . 2 ⊢ (𝐴 ∈ ℝ* → (𝐴 +𝑒 +∞) = if(𝐴 = +∞, if(+∞ = -∞, 0, +∞), if(𝐴 = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞)))))) |
4 | pnfnemnf 10412 | . . . . 5 ⊢ +∞ ≠ -∞ | |
5 | ifnefalse 4318 | . . . . 5 ⊢ (+∞ ≠ -∞ → if(+∞ = -∞, 0, +∞) = +∞) | |
6 | 4, 5 | mp1i 13 | . . . 4 ⊢ (𝐴 ≠ -∞ → if(+∞ = -∞, 0, +∞) = +∞) |
7 | ifnefalse 4318 | . . . . 5 ⊢ (𝐴 ≠ -∞ → if(𝐴 = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞)))) = if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞)))) | |
8 | eqid 2825 | . . . . . 6 ⊢ +∞ = +∞ | |
9 | 8 | iftruei 4313 | . . . . 5 ⊢ if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞))) = +∞ |
10 | 7, 9 | syl6eq 2877 | . . . 4 ⊢ (𝐴 ≠ -∞ → if(𝐴 = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞)))) = +∞) |
11 | 6, 10 | ifeq12d 4326 | . . 3 ⊢ (𝐴 ≠ -∞ → if(𝐴 = +∞, if(+∞ = -∞, 0, +∞), if(𝐴 = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞))))) = if(𝐴 = +∞, +∞, +∞)) |
12 | ifid 4345 | . . 3 ⊢ if(𝐴 = +∞, +∞, +∞) = +∞ | |
13 | 11, 12 | syl6eq 2877 | . 2 ⊢ (𝐴 ≠ -∞ → if(𝐴 = +∞, if(+∞ = -∞, 0, +∞), if(𝐴 = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞))))) = +∞) |
14 | 3, 13 | sylan9eq 2881 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) → (𝐴 +𝑒 +∞) = +∞) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1656 ∈ wcel 2164 ≠ wne 2999 ifcif 4306 (class class class)co 6905 0cc0 10252 + caddc 10255 +∞cpnf 10388 -∞cmnf 10389 ℝ*cxr 10390 +𝑒 cxad 12230 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-mulcl 10314 ax-i2m1 10320 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-br 4874 df-opab 4936 df-id 5250 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-iota 6086 df-fun 6125 df-fv 6131 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-pnf 10393 df-mnf 10394 df-xr 10395 df-xadd 12233 |
This theorem is referenced by: xnn0xaddcl 12354 xaddnemnf 12355 xaddcom 12359 xnn0xadd0 12365 xnegdi 12366 xaddass 12367 xleadd1a 12371 xlt2add 12378 xsubge0 12379 xlesubadd 12381 xadddilem 12412 xrsdsreclblem 20152 isxmet2d 22502 xrge0iifhom 30517 esumpr2 30663 hasheuni 30681 carsgclctunlem2 30915 ovolsplit 40992 sge0pr 41395 sge0split 41410 sge0xadd 41436 |
Copyright terms: Public domain | W3C validator |