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| Mirrors > Home > MPE Home > Th. List > rpcnne0 | Structured version Visualization version GIF version | ||
| Description: A positive real is a nonzero complex number. (Contributed by NM, 11-Nov-2008.) | 
| Ref | Expression | 
|---|---|
| rpcnne0 | ⊢ (𝐴 ∈ ℝ+ → (𝐴 ∈ ℂ ∧ 𝐴 ≠ 0)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | rpcn 13045 | . 2 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℂ) | |
| 2 | rpne0 13051 | . 2 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ≠ 0) | |
| 3 | 1, 2 | jca 511 | 1 ⊢ (𝐴 ∈ ℝ+ → (𝐴 ∈ ℂ ∧ 𝐴 ≠ 0)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 ≠ wne 2940 ℂcc 11153 0cc0 11155 ℝ+crp 13034 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-1cn 11213 ax-addrcl 11216 ax-rnegex 11226 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-ltxr 11300 df-rp 13035 | 
| This theorem is referenced by: rpcndif0 13054 mod0 13916 modlt 13920 modcyc 13946 modmuladdnn0 13956 moddi 13980 modirr 13983 icchmeo 24971 aaliou3lem3 26386 aaliou3lem8 26387 reeff1o 26491 reeflog 26622 relogeftb 26626 rpcxpcl 26718 relogbcxp 26828 rlimcnp 27008 rlimcnp2 27009 divsqrtsumlem 27023 harmonicbnd4 27054 logfacrlim 27268 logexprlim 27269 vmadivsum 27526 dchrmusum2 27538 dchrvmasumlem2 27542 dchrvmasumiflem1 27545 dchrisum0lem2a 27561 mudivsum 27574 mulogsumlem 27575 mulog2sumlem2 27579 selberglem2 27590 selberg2lem 27594 selberg2 27595 pntrsumo1 27609 selbergr 27612 pntibndlem2 27635 pntibndlem3 27636 pntlemb 27641 pntlemr 27646 pntlemf 27649 blocnilem 30823 minvecolem3 30895 itg2addnclem2 37679 fllogbd 48481 | 
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