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Mirrors > Home > MPE Home > Th. List > xaddpnf2 | Structured version Visualization version GIF version |
Description: Addition of positive infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xaddpnf2 | ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) → (+∞ +𝑒 𝐴) = +∞) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pnfxr 10852 | . . 3 ⊢ +∞ ∈ ℝ* | |
2 | xaddval 12778 | . . 3 ⊢ ((+∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ*) → (+∞ +𝑒 𝐴) = if(+∞ = +∞, if(𝐴 = -∞, 0, +∞), if(+∞ = -∞, if(𝐴 = +∞, 0, -∞), if(𝐴 = +∞, +∞, if(𝐴 = -∞, -∞, (+∞ + 𝐴)))))) | |
3 | 1, 2 | mpan 690 | . 2 ⊢ (𝐴 ∈ ℝ* → (+∞ +𝑒 𝐴) = if(+∞ = +∞, if(𝐴 = -∞, 0, +∞), if(+∞ = -∞, if(𝐴 = +∞, 0, -∞), if(𝐴 = +∞, +∞, if(𝐴 = -∞, -∞, (+∞ + 𝐴)))))) |
4 | eqid 2736 | . . . 4 ⊢ +∞ = +∞ | |
5 | 4 | iftruei 4432 | . . 3 ⊢ if(+∞ = +∞, if(𝐴 = -∞, 0, +∞), if(+∞ = -∞, if(𝐴 = +∞, 0, -∞), if(𝐴 = +∞, +∞, if(𝐴 = -∞, -∞, (+∞ + 𝐴))))) = if(𝐴 = -∞, 0, +∞) |
6 | ifnefalse 4437 | . . 3 ⊢ (𝐴 ≠ -∞ → if(𝐴 = -∞, 0, +∞) = +∞) | |
7 | 5, 6 | syl5eq 2783 | . 2 ⊢ (𝐴 ≠ -∞ → if(+∞ = +∞, if(𝐴 = -∞, 0, +∞), if(+∞ = -∞, if(𝐴 = +∞, 0, -∞), if(𝐴 = +∞, +∞, if(𝐴 = -∞, -∞, (+∞ + 𝐴))))) = +∞) |
8 | 3, 7 | sylan9eq 2791 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) → (+∞ +𝑒 𝐴) = +∞) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ≠ wne 2932 ifcif 4425 (class class class)co 7191 0cc0 10694 + caddc 10697 +∞cpnf 10829 -∞cmnf 10830 ℝ*cxr 10831 +𝑒 cxad 12667 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-mulcl 10756 ax-i2m1 10762 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-iota 6316 df-fun 6360 df-fv 6366 df-ov 7194 df-oprab 7195 df-mpo 7196 df-pnf 10834 df-mnf 10835 df-xr 10836 df-xadd 12670 |
This theorem is referenced by: xnn0xaddcl 12790 xaddnemnf 12791 xaddcom 12795 xaddid1 12796 xnn0xadd0 12802 xnegdi 12803 xaddass 12804 xleadd1a 12808 xadddilem 12849 xadddi2 12852 hashinfxadd 13917 xrsdsreclblem 20363 isxmet2d 23179 xaddeq0 30750 xrge0adddir 30974 xrge0iifhom 31555 infrpge 42504 infleinflem1 42523 ovolsplit 43147 sge0pr 43550 sge0split 43565 sge0xaddlem1 43589 sge0xadd 43591 |
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