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Mirrors > Home > MPE Home > Th. List > rpcnne0d | Structured version Visualization version GIF version |
Description: A positive real is a nonzero complex number. (Contributed by Mario Carneiro, 28-May-2016.) |
Ref | Expression |
---|---|
rpred.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ+) |
Ref | Expression |
---|---|
rpcnne0d | ⊢ (𝜑 → (𝐴 ∈ ℂ ∧ 𝐴 ≠ 0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rpred.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ+) | |
2 | 1 | rpcnd 11750 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
3 | 1 | rpne0d 11753 | . 2 ⊢ (𝜑 → 𝐴 ≠ 0) |
4 | 2, 3 | jca 553 | 1 ⊢ (𝜑 → (𝐴 ∈ ℂ ∧ 𝐴 ≠ 0)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∈ wcel 1977 ≠ wne 2780 ℂcc 9813 0cc0 9815 ℝ+crp 11708 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-i2m1 9883 ax-1ne0 9884 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-ltxr 9958 df-rp 11709 |
This theorem is referenced by: expcnv 14435 mertenslem1 14455 divgcdcoprm0 15217 ovolscalem1 23088 aalioulem2 23892 aalioulem3 23893 dvsqrt 24283 cxpcn3lem 24288 relogbval 24310 relogbcl 24311 nnlogbexp 24319 divsqrtsumlem 24506 logexprlim 24750 2lgslem3b 24922 2lgslem3c 24923 2lgslem3d 24924 chebbnd1lem3 24960 chebbnd1 24961 chtppilimlem1 24962 chtppilimlem2 24963 chebbnd2 24966 chpchtlim 24968 chpo1ub 24969 rplogsumlem1 24973 rplogsumlem2 24974 rpvmasumlem 24976 dchrvmasumlem1 24984 dchrvmasum2lem 24985 dchrvmasumlem2 24987 dchrisum0fno1 25000 dchrisum0lem1b 25004 dchrisum0lem1 25005 dchrisum0lem2a 25006 dchrisum0lem2 25007 dchrisum0lem3 25008 rplogsum 25016 mulogsum 25021 mulog2sumlem1 25023 selberglem1 25034 pntrmax 25053 pntpbnd1a 25074 pntibndlem2 25080 pntlemc 25084 pntlemb 25086 pntlemn 25089 pntlemr 25091 pntlemj 25092 pntlemf 25094 pntlemk 25095 pntlemo 25096 pnt2 25102 bcm1n 28941 jm2.21 36579 stoweidlem25 38918 stoweidlem42 38935 wallispilem4 38961 stirlinglem10 38976 fourierdlem39 39039 lighneallem3 40062 dignn0flhalflem1 42207 dignn0flhalflem2 42208 |
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