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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 9694 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) |
3 | 2 | recnd 7984 | 1 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2148 ℂcc 7808 ℝ+crp 9651 |
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 7902 |
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 3135 df-ss 3142 df-rp 9652 |
This theorem is referenced by: rpcnne0d 9704 ltaddrp2d 9729 iccf1o 10002 bcp1nk 10737 bcpasc 10741 cvg1nlemcxze 10986 cvg1nlemres 10989 resqrexlemdec 11015 resqrexlemlo 11017 resqrexlemcalc2 11019 resqrexlemcalc3 11020 resqrexlemnm 11022 resqrexlemcvg 11023 resqrexlemoverl 11025 sqrtdiv 11046 absdivap 11074 bdtrilem 11242 isumrpcl 11497 expcnvap0 11505 absgtap 11513 cvgratz 11535 mertenslemi1 11538 effsumlt 11695 pythagtriplem12 12269 pythagtriplem14 12271 pythagtriplem16 12273 limcimolemlt 14064 rpdivcxp 14263 rpcxple2 14269 rpcxplt2 14270 rpcxpsqrt 14273 rpabscxpbnd 14290 logbgcd1irr 14316 iooref1o 14702 trilpolemclim 14704 trilpolemisumle 14706 trilpolemeq1 14708 trilpolemlt1 14709 |
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