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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 9690 | . 2 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℝ) | |
2 | 1 | rexrd 8037 | 1 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℝ*) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2160 ℝ*cxr 8021 ℝ+crp 9683 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-rab 2477 df-v 2754 df-un 3148 df-in 3150 df-ss 3157 df-xr 8026 df-rp 9684 |
This theorem is referenced by: xrminrpcl 11314 blcntrps 14372 blcntr 14373 unirnblps 14379 unirnbl 14380 blssexps 14386 blssex 14387 blin2 14389 neibl 14448 blnei 14449 metss 14451 metss2lem 14454 bdmet 14459 bdmopn 14461 mopnex 14462 metrest 14463 xmettx 14467 metcnp3 14468 metcnp 14469 metcnpi3 14474 txmetcnp 14475 limcimolemlt 14590 |
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