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Mirrors > Home > ILE Home > Th. List > rpcnd | GIF version |
Description: A positive real is a complex number. (Contributed by Mario Carneiro, 28-May-2016.) |
Ref | Expression |
---|---|
rpred.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ+) |
Ref | Expression |
---|---|
rpcnd | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rpred.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ+) | |
2 | 1 | rpred 9451 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) |
3 | 2 | recnd 7762 | 1 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1465 ℂcc 7586 ℝ+crp 9409 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-resscn 7680 |
This theorem depends on definitions: df-bi 116 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-rab 2402 df-in 3047 df-ss 3054 df-rp 9410 |
This theorem is referenced by: rpcnne0d 9461 ltaddrp2d 9486 iccf1o 9755 bcp1nk 10476 bcpasc 10480 cvg1nlemcxze 10722 cvg1nlemres 10725 resqrexlemdec 10751 resqrexlemlo 10753 resqrexlemcalc2 10755 resqrexlemcalc3 10756 resqrexlemnm 10758 resqrexlemcvg 10759 resqrexlemoverl 10761 sqrtdiv 10782 absdivap 10810 bdtrilem 10978 isumrpcl 11231 expcnvap0 11239 absgtap 11247 cvgratz 11269 mertenslemi1 11272 effsumlt 11325 limcimolemlt 12729 trilpolemclim 13156 trilpolemisumle 13158 trilpolemeq1 13160 trilpolemlt1 13161 |
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