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Mirrors > Home > MPE Home > Th. List > xaddcl | Structured version Visualization version GIF version |
Description: The extended real addition operation is closed in extended reals. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xaddcl | ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 +𝑒 𝐵) ∈ ℝ*) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xaddf 12367 | . 2 ⊢ +𝑒 :(ℝ* × ℝ*)⟶ℝ* | |
2 | 1 | fovcl 7042 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 +𝑒 𝐵) ∈ ℝ*) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∈ wcel 2106 (class class class)co 6922 ℝ*cxr 10410 +𝑒 cxad 12255 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-1cn 10330 ax-addrcl 10333 ax-rnegex 10343 ax-cnre 10345 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-fv 6143 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-1st 7445 df-2nd 7446 df-pnf 10413 df-mnf 10414 df-xr 10415 df-xadd 12258 |
This theorem is referenced by: xaddass 12391 xaddass2 12392 xleadd1a 12395 xleadd1 12397 xltadd1 12398 xaddge0 12400 xle2add 12401 xlt2add 12402 xsubge0 12403 xposdif 12404 xlesubadd 12405 xadddi 12437 xadddir 12438 xadddi2 12439 xadddi2r 12440 xaddcld 12443 ge0xaddcl 12600 xrsmgm 20177 xrs1mnd 20180 xrsds 20185 xrsxmet 23020 xrofsup 30098 supxrgelem 40454 caragenel2d 41666 |
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