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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 11874 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
3 | 1 | rpne0d 11877 | . 2 ⊢ (𝜑 → 𝐴 ≠ 0) |
4 | 2, 3 | jca 554 | 1 ⊢ (𝜑 → (𝐴 ∈ ℂ ∧ 𝐴 ≠ 0)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∈ wcel 1990 ≠ wne 2794 ℂcc 9934 0cc0 9936 ℝ+crp 11832 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-i2m1 10004 ax-1ne0 10005 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-ltxr 10079 df-rp 11833 |
This theorem is referenced by: expcnv 14596 mertenslem1 14616 divgcdcoprm0 15379 ovolscalem1 23281 aalioulem2 24088 aalioulem3 24089 dvsqrt 24483 cxpcn3lem 24488 relogbval 24510 relogbcl 24511 nnlogbexp 24519 divsqrtsumlem 24706 logexprlim 24950 2lgslem3b 25122 2lgslem3c 25123 2lgslem3d 25124 chebbnd1lem3 25160 chebbnd1 25161 chtppilimlem1 25162 chtppilimlem2 25163 chebbnd2 25166 chpchtlim 25168 chpo1ub 25169 rplogsumlem1 25173 rplogsumlem2 25174 rpvmasumlem 25176 dchrvmasumlem1 25184 dchrvmasum2lem 25185 dchrvmasumlem2 25187 dchrisum0fno1 25200 dchrisum0lem1b 25204 dchrisum0lem1 25205 dchrisum0lem2a 25206 dchrisum0lem2 25207 dchrisum0lem3 25208 rplogsum 25216 mulogsum 25221 mulog2sumlem1 25223 selberglem1 25234 pntrmax 25253 pntpbnd1a 25274 pntibndlem2 25280 pntlemc 25284 pntlemb 25286 pntlemn 25289 pntlemr 25291 pntlemj 25292 pntlemf 25294 pntlemk 25295 pntlemo 25296 pnt2 25302 bcm1n 29554 jm2.21 37561 stoweidlem25 40242 stoweidlem42 40259 wallispilem4 40285 stirlinglem10 40300 fourierdlem39 40363 lighneallem3 41524 dignn0flhalflem1 42409 dignn0flhalflem2 42410 |
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