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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 12620 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 +𝑒 𝐵) ∈ ℝ*) | |
4 | 1, 2, 3 | syl2anc 587 | 1 ⊢ (𝜑 → (𝐴 +𝑒 𝐵) ∈ ℝ*) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2111 (class class class)co 7135 ℝ*cxr 10663 +𝑒 cxad 12493 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-1cn 10584 ax-addrcl 10587 ax-rnegex 10597 ax-cnre 10599 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-pnf 10666 df-mnf 10667 df-xr 10668 df-xadd 12496 |
This theorem is referenced by: xadd4d 12684 imasdsf1olem 22980 bldisj 23005 xblss2ps 23008 xblss2 23009 blcld 23112 comet 23120 stdbdxmet 23122 metdstri 23456 metdscnlem 23460 iscau3 23882 xlt2addrd 30508 xrge0addcld 30512 xrge0subcld 30513 xrofsup 30518 xrsmulgzz 30712 xrge0adddir 30726 xrge0adddi 30727 esumle 31427 esumlef 31431 omssubadd 31668 inelcarsg 31679 carsgclctunlem2 31687 carsgclctunlem3 31688 carsgclctun 31689 xle2addd 41968 infrpge 41983 xrlexaddrp 41984 infleinflem1 42002 infleinflem2 42003 limsupgtlem 42419 ismbl3 42628 ismbl4 42635 sge0prle 43040 sge0split 43048 sge0iunmptlemre 43054 sge0xaddlem1 43072 omeunle 43155 carageniuncl 43162 ovnsubaddlem1 43209 hspmbl 43268 ovolval5lem1 43291 |
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