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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 13079 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
| 3 | 1 | rpne0d 13082 | . 2 ⊢ (𝜑 → 𝐴 ≠ 0) |
| 4 | 2, 3 | jca 511 | 1 ⊢ (𝜑 → (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: expcnv 15900 mertenslem1 15920 divgcdcoprm0 16702 ovolscalem1 25548 aalioulem2 26375 aalioulem3 26376 dvsqrt 26784 cxpcn3lem 26790 relogbval 26815 relogbcl 26816 nnlogbexp 26824 divsqrtsumlem 27023 logexprlim 27269 2lgslem3b 27441 2lgslem3c 27442 2lgslem3d 27443 chebbnd1lem3 27515 chebbnd1 27516 chtppilimlem1 27517 chtppilimlem2 27518 chebbnd2 27521 chpchtlim 27523 chpo1ub 27524 rplogsumlem1 27528 rplogsumlem2 27529 rpvmasumlem 27531 dchrvmasumlem1 27539 dchrvmasum2lem 27540 dchrvmasumlem2 27542 dchrisum0fno1 27555 dchrisum0lem1b 27559 dchrisum0lem1 27560 dchrisum0lem2a 27561 dchrisum0lem2 27562 dchrisum0lem3 27563 rplogsum 27571 mulogsum 27576 mulog2sumlem1 27578 selberglem1 27589 pntrmax 27608 pntpbnd1a 27629 pntibndlem2 27635 pntlemc 27639 pntlemb 27641 pntlemn 27644 pntlemr 27646 pntlemj 27647 pntlemf 27649 pntlemk 27650 pntlemo 27651 pnt2 27657 bcm1n 32797 jm2.21 43006 stoweidlem25 46040 stoweidlem42 46057 wallispilem4 46083 stirlinglem10 46098 fourierdlem39 46161 lighneallem3 47594 dignn0flhalflem1 48536 dignn0flhalflem2 48537 itschlc0xyqsol1 48687 |
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