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Mirrors > Home > MPE Home > Th. List > cnring | Structured version Visualization version GIF version |
Description: The complex numbers form a ring. (Contributed by Stefan O'Rear, 27-Nov-2014.) |
Ref | Expression |
---|---|
cnring | ⊢ ℂfld ∈ Ring |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cncrng 20238 | . 2 ⊢ ℂfld ∈ CRing | |
2 | crngring 19428 | . 2 ⊢ (ℂfld ∈ CRing → ℂfld ∈ Ring) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ ℂfld ∈ Ring |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2114 Ringcrg 19416 CRingccrg 19417 ℂfldccnfld 20217 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 ax-addf 10694 ax-mulf 10695 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-om 7600 df-1st 7714 df-2nd 7715 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-1o 8131 df-er 8320 df-en 8556 df-dom 8557 df-sdom 8558 df-fin 8559 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-nn 11717 df-2 11779 df-3 11780 df-4 11781 df-5 11782 df-6 11783 df-7 11784 df-8 11785 df-9 11786 df-n0 11977 df-z 12063 df-dec 12180 df-uz 12325 df-fz 12982 df-struct 16588 df-ndx 16589 df-slot 16590 df-base 16592 df-sets 16593 df-plusg 16681 df-mulr 16682 df-starv 16683 df-tset 16687 df-ple 16688 df-ds 16690 df-unif 16691 df-0g 16818 df-mgm 17968 df-sgrp 18017 df-mnd 18028 df-grp 18222 df-cmn 19026 df-mgp 19359 df-ring 19418 df-cring 19419 df-cnfld 20218 |
This theorem is referenced by: cnfld0 20241 cnfld1 20242 cnfldneg 20243 cnfldsub 20245 cndrng 20246 cnflddiv 20247 cnfldinv 20248 cnfldmulg 20249 cnfldexp 20250 cnsrng 20251 cnsubmlem 20265 cnsubglem 20266 cnsubrglem 20267 cnsubdrglem 20268 absabv 20274 cnmgpid 20279 gsumfsum 20284 expmhm 20286 nn0srg 20287 rge0srg 20288 expghm 20316 zrhpsgnmhm 20400 regsumsupp 20438 mhpmulcl 20943 cnngp 23532 cnfldtgp 23621 cnlmod 23892 cnrlmod 23895 cnncvsaddassdemo 23915 cphsubrglem 23929 tdeglem1 24808 tdeglem1OLD 24809 tdeglem3 24810 tdeglem3OLD 24811 tdeglem4 24812 tdeglem4OLD 24813 tdeglem2 24814 plypf1 24961 dvply2 25034 dvnply 25036 taylfvallem 25105 taylf 25108 tayl0 25109 taylpfval 25112 taylply 25116 efabl 25294 efsubm 25295 jensenlem1 25724 jensenlem2 25725 jensen 25726 amgmlem 25727 amgm 25728 wilthlem2 25806 wilthlem3 25807 dchrelbas3 25974 dchrghm 25992 dchrabs 25996 lgseisenlem4 26114 psgnid 30941 cnmsgn0g 30990 altgnsg 30993 znfermltl 31134 ccfldsrarelvec 31313 xrge0iifmhm 31461 zringnm 31480 rezh 31491 mhphflem 39863 rngunsnply 40570 proot1ex 40598 amgm2d 41356 amgm3d 41357 amgm4d 41358 amgmwlem 45959 amgmlemALT 45960 amgmw2d 45961 |
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