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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 12605 | . 2 ⊢ +𝑒 :(ℝ* × ℝ*)⟶ℝ* | |
2 | 1 | fovcl 7268 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 +𝑒 𝐵) ∈ ℝ*) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2105 (class class class)co 7145 ℝ*cxr 10662 +𝑒 cxad 12493 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-1cn 10583 ax-addrcl 10586 ax-rnegex 10596 ax-cnre 10598 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-1st 7678 df-2nd 7679 df-pnf 10665 df-mnf 10666 df-xr 10667 df-xadd 12496 |
This theorem is referenced by: xaddass 12630 xaddass2 12631 xleadd1a 12634 xleadd1 12636 xltadd1 12637 xaddge0 12639 xle2add 12640 xlt2add 12641 xsubge0 12642 xposdif 12643 xlesubadd 12644 xadddi 12676 xadddir 12677 xadddi2 12678 xadddi2r 12679 xaddcld 12682 ge0xaddcl 12838 xrsmgm 20508 xrs1mnd 20511 xrsds 20516 xrsxmet 23344 xrofsup 30418 supxrgelem 41481 caragenel2d 42691 |
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