Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 12434 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
3 | 1 | rpne0d 12437 | . 2 ⊢ (𝜑 → 𝐴 ≠ 0) |
4 | 2, 3 | jca 514 | 1 ⊢ (𝜑 → (𝐴 ∈ ℂ ∧ 𝐴 ≠ 0)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2114 ≠ wne 3016 ℂcc 10535 0cc0 10537 ℝ+crp 12390 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-resscn 10594 ax-1cn 10595 ax-addrcl 10598 ax-rnegex 10608 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-po 5474 df-so 5475 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-ltxr 10680 df-rp 12391 |
This theorem is referenced by: expcnv 15219 mertenslem1 15240 divgcdcoprm0 16009 ovolscalem1 24114 aalioulem2 24922 aalioulem3 24923 dvsqrt 25323 cxpcn3lem 25328 relogbval 25350 relogbcl 25351 nnlogbexp 25359 divsqrtsumlem 25557 logexprlim 25801 2lgslem3b 25973 2lgslem3c 25974 2lgslem3d 25975 chebbnd1lem3 26047 chebbnd1 26048 chtppilimlem1 26049 chtppilimlem2 26050 chebbnd2 26053 chpchtlim 26055 chpo1ub 26056 rplogsumlem1 26060 rplogsumlem2 26061 rpvmasumlem 26063 dchrvmasumlem1 26071 dchrvmasum2lem 26072 dchrvmasumlem2 26074 dchrisum0fno1 26087 dchrisum0lem1b 26091 dchrisum0lem1 26092 dchrisum0lem2a 26093 dchrisum0lem2 26094 dchrisum0lem3 26095 rplogsum 26103 mulogsum 26108 mulog2sumlem1 26110 selberglem1 26121 pntrmax 26140 pntpbnd1a 26161 pntibndlem2 26167 pntlemc 26171 pntlemb 26173 pntlemn 26176 pntlemr 26178 pntlemj 26179 pntlemf 26181 pntlemk 26182 pntlemo 26183 pnt2 26189 bcm1n 30518 jm2.21 39640 stoweidlem25 42359 stoweidlem42 42376 wallispilem4 42402 stirlinglem10 42417 fourierdlem39 42480 lighneallem3 43821 dignn0flhalflem1 44724 dignn0flhalflem2 44725 itschlc0xyqsol1 44802 |
Copyright terms: Public domain | W3C validator |