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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 10695 | . . 3 ⊢ +∞ ∈ ℝ* | |
2 | xaddval 12617 | . . 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 10696 | . . . . 5 ⊢ +∞ ≠ -∞ | |
5 | ifnefalse 4479 | . . . . 5 ⊢ (+∞ ≠ -∞ → if(+∞ = -∞, 0, +∞) = +∞) | |
6 | 4, 5 | mp1i 13 | . . . 4 ⊢ (𝐴 ≠ -∞ → if(+∞ = -∞, 0, +∞) = +∞) |
7 | ifnefalse 4479 | . . . . 5 ⊢ (𝐴 ≠ -∞ → if(𝐴 = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞)))) = if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞)))) | |
8 | eqid 2821 | . . . . . 6 ⊢ +∞ = +∞ | |
9 | 8 | iftruei 4474 | . . . . 5 ⊢ if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞))) = +∞ |
10 | 7, 9 | syl6eq 2872 | . . . 4 ⊢ (𝐴 ≠ -∞ → if(𝐴 = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞)))) = +∞) |
11 | 6, 10 | ifeq12d 4487 | . . 3 ⊢ (𝐴 ≠ -∞ → if(𝐴 = +∞, if(+∞ = -∞, 0, +∞), if(𝐴 = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞))))) = if(𝐴 = +∞, +∞, +∞)) |
12 | ifid 4506 | . . 3 ⊢ if(𝐴 = +∞, +∞, +∞) = +∞ | |
13 | 11, 12 | syl6eq 2872 | . 2 ⊢ (𝐴 ≠ -∞ → if(𝐴 = +∞, if(+∞ = -∞, 0, +∞), if(𝐴 = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞))))) = +∞) |
14 | 3, 13 | sylan9eq 2876 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) → (𝐴 +𝑒 +∞) = +∞) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ifcif 4467 (class class class)co 7156 0cc0 10537 + caddc 10540 +∞cpnf 10672 -∞cmnf 10673 ℝ*cxr 10674 +𝑒 cxad 12506 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-mulcl 10599 ax-i2m1 10605 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-iota 6314 df-fun 6357 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-pnf 10677 df-mnf 10678 df-xr 10679 df-xadd 12509 |
This theorem is referenced by: xnn0xaddcl 12629 xaddnemnf 12630 xaddcom 12634 xnn0xadd0 12641 xnegdi 12642 xaddass 12643 xleadd1a 12647 xlt2add 12654 xsubge0 12655 xlesubadd 12657 xadddilem 12688 xrsdsreclblem 20591 isxmet2d 22937 xrge0iifhom 31180 esumpr2 31326 hasheuni 31344 carsgclctunlem2 31577 ovolsplit 42322 sge0pr 42725 sge0split 42740 sge0xadd 42766 |
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