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 10684 | . . 3 ⊢ +∞ ∈ ℝ* | |
2 | xaddval 12604 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (𝐴 +𝑒 +∞) = if(𝐴 = +∞, if(+∞ = -∞, 0, +∞), if(𝐴 = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞)))))) | |
3 | 1, 2 | mpan2 690 | . 2 ⊢ (𝐴 ∈ ℝ* → (𝐴 +𝑒 +∞) = if(𝐴 = +∞, if(+∞ = -∞, 0, +∞), if(𝐴 = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞)))))) |
4 | pnfnemnf 10685 | . . . . 5 ⊢ +∞ ≠ -∞ | |
5 | ifnefalse 4437 | . . . . 5 ⊢ (+∞ ≠ -∞ → if(+∞ = -∞, 0, +∞) = +∞) | |
6 | 4, 5 | mp1i 13 | . . . 4 ⊢ (𝐴 ≠ -∞ → if(+∞ = -∞, 0, +∞) = +∞) |
7 | ifnefalse 4437 | . . . . 5 ⊢ (𝐴 ≠ -∞ → if(𝐴 = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞)))) = if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞)))) | |
8 | eqid 2798 | . . . . . 6 ⊢ +∞ = +∞ | |
9 | 8 | iftruei 4432 | . . . . 5 ⊢ if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞))) = +∞ |
10 | 7, 9 | eqtrdi 2849 | . . . 4 ⊢ (𝐴 ≠ -∞ → if(𝐴 = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞)))) = +∞) |
11 | 6, 10 | ifeq12d 4445 | . . 3 ⊢ (𝐴 ≠ -∞ → if(𝐴 = +∞, if(+∞ = -∞, 0, +∞), if(𝐴 = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞))))) = if(𝐴 = +∞, +∞, +∞)) |
12 | ifid 4464 | . . 3 ⊢ if(𝐴 = +∞, +∞, +∞) = +∞ | |
13 | 11, 12 | eqtrdi 2849 | . 2 ⊢ (𝐴 ≠ -∞ → if(𝐴 = +∞, if(+∞ = -∞, 0, +∞), if(𝐴 = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞))))) = +∞) |
14 | 3, 13 | sylan9eq 2853 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) → (𝐴 +𝑒 +∞) = +∞) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ifcif 4425 (class class class)co 7135 0cc0 10526 + caddc 10529 +∞cpnf 10661 -∞cmnf 10662 ℝ*cxr 10663 +𝑒 cxad 12493 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-mulcl 10588 ax-i2m1 10594 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-pnf 10666 df-mnf 10667 df-xr 10668 df-xadd 12496 |
This theorem is referenced by: xnn0xaddcl 12616 xaddnemnf 12617 xaddcom 12621 xnn0xadd0 12628 xnegdi 12629 xaddass 12630 xleadd1a 12634 xlt2add 12641 xsubge0 12642 xlesubadd 12644 xadddilem 12675 xrsdsreclblem 20137 isxmet2d 22934 xrge0iifhom 31290 esumpr2 31436 hasheuni 31454 carsgclctunlem2 31687 ovolsplit 42630 sge0pr 43033 sge0split 43048 sge0xadd 43074 |
Copyright terms: Public domain | W3C validator |