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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 13281 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 +𝑒 𝐵) ∈ ℝ*) | |
| 4 | 1, 2, 3 | syl2anc 584 | 1 ⊢ (𝜑 → (𝐴 +𝑒 𝐵) ∈ ℝ*) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 (class class class)co 7431 ℝ*cxr 11294 +𝑒 cxad 13152 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-1cn 11213 ax-addrcl 11216 ax-rnegex 11226 ax-cnre 11228 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-pnf 11297 df-mnf 11298 df-xr 11299 df-xadd 13155 |
| This theorem is referenced by: xadd4d 13345 imasdsf1olem 24383 bldisj 24408 xblss2ps 24411 xblss2 24412 blcld 24518 comet 24526 stdbdxmet 24528 metdstri 24873 metdscnlem 24877 iscau3 25312 xlt2addrd 32762 xrge0addcld 32766 xrge0subcld 32767 xrofsup 32771 xrsmulgzz 33011 xrge0adddir 33023 xrge0adddi 33024 esumle 34059 esumlef 34063 omssubadd 34302 inelcarsg 34313 carsgclctunlem2 34321 carsgclctunlem3 34322 carsgclctun 34323 xle2addd 45347 infrpge 45362 xrlexaddrp 45363 infleinflem1 45381 infleinflem2 45382 limsupgtlem 45792 ismbl3 46001 ismbl4 46008 sge0prle 46416 sge0split 46424 sge0iunmptlemre 46430 sge0xaddlem1 46448 omeunle 46531 carageniuncl 46538 ovnsubaddlem1 46585 hspmbl 46644 ovolval5lem1 46667 |
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