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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 9441 | . 2 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℝ) | |
2 | 1 | rexrd 7808 | 1 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℝ*) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1480 ℝ*cxr 7792 ℝ+crp 9434 |
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 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-rab 2423 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-xr 7797 df-rp 9435 |
This theorem is referenced by: xrminrpcl 11036 blcntrps 12573 blcntr 12574 unirnblps 12580 unirnbl 12581 blssexps 12587 blssex 12588 blin2 12590 neibl 12649 blnei 12650 metss 12652 metss2lem 12655 bdmet 12660 bdmopn 12662 mopnex 12663 metrest 12664 xmettx 12668 metcnp3 12669 metcnp 12670 metcnpi3 12675 txmetcnp 12676 limcimolemlt 12791 |
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