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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 13019 | . 2 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℂ) | |
2 | rpne0 13025 | . 2 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ≠ 0) | |
3 | 1, 2 | jca 510 | 1 ⊢ (𝐴 ∈ ℝ+ → (𝐴 ∈ ℂ ∧ 𝐴 ≠ 0)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∈ wcel 2098 ≠ wne 2929 ℂcc 11138 0cc0 11140 ℝ+crp 13009 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-resscn 11197 ax-1cn 11198 ax-addrcl 11201 ax-rnegex 11211 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-po 5590 df-so 5591 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11282 df-mnf 11283 df-ltxr 11285 df-rp 13010 |
This theorem is referenced by: rpcndif0 13028 mod0 13877 modlt 13881 modcyc 13907 modmuladdnn0 13916 moddi 13940 modirr 13943 icchmeo 24909 aaliou3lem3 26324 aaliou3lem8 26325 reeff1o 26429 reeflog 26559 relogeftb 26563 rpcxpcl 26655 relogbcxp 26762 rlimcnp 26942 rlimcnp2 26943 divsqrtsumlem 26957 harmonicbnd4 26988 logfacrlim 27202 logexprlim 27203 vmadivsum 27460 dchrmusum2 27472 dchrvmasumlem2 27476 dchrvmasumiflem1 27479 dchrisum0lem2a 27495 mudivsum 27508 mulogsumlem 27509 mulog2sumlem2 27513 selberglem2 27524 selberg2lem 27528 selberg2 27529 pntrsumo1 27543 selbergr 27546 pntibndlem2 27569 pntibndlem3 27570 pntlemb 27575 pntlemr 27580 pntlemf 27583 blocnilem 30686 minvecolem3 30758 itg2addnclem2 37276 fllogbd 47819 |
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