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Mirrors > Home > ILE Home > Th. List > rpxr | GIF version |
Description: A positive real is an extended real. (Contributed by Mario Carneiro, 21-Aug-2015.) |
Ref | Expression |
---|---|
rpxr | ⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℝ*) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rpre 9448 | . 2 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℝ) | |
2 | 1 | rexrd 7815 | 1 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℝ*) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1480 ℝ*cxr 7799 ℝ+crp 9441 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-rab 2425 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-xr 7804 df-rp 9442 |
This theorem is referenced by: xrminrpcl 11043 blcntrps 12584 blcntr 12585 unirnblps 12591 unirnbl 12592 blssexps 12598 blssex 12599 blin2 12601 neibl 12660 blnei 12661 metss 12663 metss2lem 12666 bdmet 12671 bdmopn 12673 mopnex 12674 metrest 12675 xmettx 12679 metcnp3 12680 metcnp 12681 metcnpi3 12686 txmetcnp 12687 limcimolemlt 12802 |
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