![]() |
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 12319 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 +𝑒 𝐵) ∈ ℝ*) | |
4 | 1, 2, 3 | syl2anc 580 | 1 ⊢ (𝜑 → (𝐴 +𝑒 𝐵) ∈ ℝ*) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2157 (class class class)co 6878 ℝ*cxr 10362 +𝑒 cxad 12191 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-cnex 10280 ax-1cn 10282 ax-addrcl 10285 ax-rnegex 10295 ax-cnre 10297 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-op 4375 df-uni 4629 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-id 5220 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-fv 6109 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-1st 7401 df-2nd 7402 df-pnf 10365 df-mnf 10366 df-xr 10367 df-xadd 12194 |
This theorem is referenced by: xadd4d 12382 imasdsf1olem 22506 bldisj 22531 xblss2ps 22534 xblss2 22535 blcld 22638 comet 22646 stdbdxmet 22648 metdstri 22982 metdscnlem 22986 iscau3 23404 xlt2addrd 30041 xrge0addcld 30045 xrge0subcld 30046 xrofsup 30051 xrsmulgzz 30194 xrge0adddir 30208 xrge0adddi 30209 esumle 30636 esumlef 30640 omssubadd 30878 inelcarsg 30889 carsgclctunlem2 30897 carsgclctunlem3 30898 carsgclctun 30899 xle2addd 40296 infrpge 40311 xrlexaddrp 40312 infleinflem1 40330 infleinflem2 40331 limsupgtlem 40753 ismbl3 40946 ismbl4 40953 sge0prle 41361 sge0split 41369 sge0iunmptlemre 41375 sge0xaddlem1 41393 omeunle 41476 carageniuncl 41483 ovnsubaddlem1 41530 hspmbl 41589 ovolval5lem1 41612 |
Copyright terms: Public domain | W3C validator |