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Mirrors > Home > ILE Home > Th. List > rpcn | GIF version |
Description: A positive real is a complex number. (Contributed by NM, 11-Nov-2008.) |
Ref | Expression |
---|---|
rpcn | ⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℂ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rpre 9663 | . 2 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℝ) | |
2 | 1 | recnd 7989 | 1 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℂ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2148 ℂcc 7812 ℝ+crp 9656 |
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 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 ax-resscn 7906 |
This theorem depends on definitions: df-bi 117 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-rab 2464 df-in 3137 df-ss 3144 df-rp 9657 |
This theorem is referenced by: rpcnne0 9676 rpcnap0 9677 divge1 9726 sqrtdiv 11054 efgt1p2 11706 efgt1p 11707 pilem1 14388 rpcxp0 14507 rpcxp1 14508 cxprec 14519 rplogbval 14551 rprelogbdiv 14563 |
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