![]() |
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 10430 | . . 3 ⊢ +∞ ∈ ℝ* | |
2 | xaddval 12366 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (𝐴 +𝑒 +∞) = if(𝐴 = +∞, if(+∞ = -∞, 0, +∞), if(𝐴 = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞)))))) | |
3 | 1, 2 | mpan2 681 | . 2 ⊢ (𝐴 ∈ ℝ* → (𝐴 +𝑒 +∞) = if(𝐴 = +∞, if(+∞ = -∞, 0, +∞), if(𝐴 = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞)))))) |
4 | pnfnemnf 10432 | . . . . 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 2777 | . . . . . 6 ⊢ +∞ = +∞ | |
9 | 8 | iftruei 4313 | . . . . 5 ⊢ if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞))) = +∞ |
10 | 7, 9 | syl6eq 2829 | . . . 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 2829 | . 2 ⊢ (𝐴 ≠ -∞ → if(𝐴 = +∞, if(+∞ = -∞, 0, +∞), if(𝐴 = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞))))) = +∞) |
14 | 3, 13 | sylan9eq 2833 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) → (𝐴 +𝑒 +∞) = +∞) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2106 ≠ wne 2968 ifcif 4306 (class class class)co 6922 0cc0 10272 + caddc 10275 +∞cpnf 10408 -∞cmnf 10409 ℝ*cxr 10410 +𝑒 cxad 12255 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-mulcl 10334 ax-i2m1 10340 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3399 df-sbc 3652 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4672 df-br 4887 df-opab 4949 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-iota 6099 df-fun 6137 df-fv 6143 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-pnf 10413 df-mnf 10414 df-xr 10415 df-xadd 12258 |
This theorem is referenced by: xnn0xaddcl 12378 xaddnemnf 12379 xaddcom 12383 xnn0xadd0 12389 xnegdi 12390 xaddass 12391 xleadd1a 12395 xlt2add 12402 xsubge0 12403 xlesubadd 12405 xadddilem 12436 xrsdsreclblem 20188 isxmet2d 22540 xrge0iifhom 30581 esumpr2 30727 hasheuni 30745 carsgclctunlem2 30979 ovolsplit 41114 sge0pr 41517 sge0split 41532 sge0xadd 41558 |
Copyright terms: Public domain | W3C validator |