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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 13267 | . 2 ⊢ +𝑒 :(ℝ* × ℝ*)⟶ℝ* | |
| 2 | 1 | fovcl 7562 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 +𝑒 𝐵) ∈ ℝ*) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2107 (class class class)co 7432 ℝ*cxr 11295 +𝑒 cxad 13153 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-1cn 11214 ax-addrcl 11217 ax-rnegex 11227 ax-cnre 11229 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-1st 8015 df-2nd 8016 df-pnf 11298 df-mnf 11299 df-xr 11300 df-xadd 13156 | 
| This theorem is referenced by: xaddass 13292 xaddass2 13293 xleadd1a 13296 xleadd1 13298 xltadd1 13299 xaddge0 13301 xle2add 13302 xlt2add 13303 xsubge0 13304 xposdif 13305 xlesubadd 13306 xadddi 13338 xadddir 13339 xadddi2 13340 xadddi2r 13341 xaddcld 13344 ge0xaddcl 13503 xrsmgm 21420 xrs1mnd 21423 xrsds 21428 xrsxmet 24832 xrofsup 32772 supxrgelem 45353 caragenel2d 46552 | 
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