![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > xaddcld | Structured version Visualization version GIF version |
Description: The extended real addition operation is closed in extended reals. (Contributed by Mario Carneiro, 28-May-2016.) |
Ref | Expression |
---|---|
xnegcld.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
xaddcld.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
Ref | Expression |
---|---|
xaddcld | ⊢ (𝜑 → (𝐴 +𝑒 𝐵) ∈ ℝ*) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xnegcld.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ*) | |
2 | xaddcld.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℝ*) | |
3 | xaddcl 13258 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 +𝑒 𝐵) ∈ ℝ*) | |
4 | 1, 2, 3 | syl2anc 582 | 1 ⊢ (𝜑 → (𝐴 +𝑒 𝐵) ∈ ℝ*) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 (class class class)co 7419 ℝ*cxr 11284 +𝑒 cxad 13130 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11201 ax-1cn 11203 ax-addrcl 11206 ax-rnegex 11216 ax-cnre 11218 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-fv 6557 df-ov 7422 df-oprab 7423 df-mpo 7424 df-1st 7994 df-2nd 7995 df-pnf 11287 df-mnf 11288 df-xr 11289 df-xadd 13133 |
This theorem is referenced by: xadd4d 13322 imasdsf1olem 24328 bldisj 24353 xblss2ps 24356 xblss2 24357 blcld 24463 comet 24471 stdbdxmet 24473 metdstri 24816 metdscnlem 24820 iscau3 25255 xlt2addrd 32615 xrge0addcld 32619 xrge0subcld 32620 xrofsup 32624 xrsmulgzz 32830 xrge0adddir 32842 xrge0adddi 32843 esumle 33810 esumlef 33814 omssubadd 34053 inelcarsg 34064 carsgclctunlem2 34072 carsgclctunlem3 34073 carsgclctun 34074 xle2addd 44858 infrpge 44873 xrlexaddrp 44874 infleinflem1 44892 infleinflem2 44893 limsupgtlem 45305 ismbl3 45514 ismbl4 45521 sge0prle 45929 sge0split 45937 sge0iunmptlemre 45943 sge0xaddlem1 45961 omeunle 46044 carageniuncl 46051 ovnsubaddlem1 46098 hspmbl 46157 ovolval5lem1 46180 |
Copyright terms: Public domain | W3C validator |