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Mirrors > Home > MPE Home > Th. List > cnfldbas | Structured version Visualization version Unicode version |
Description: The base set of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) |
Ref | Expression |
---|---|
cnfldbas | ℂfld |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnex 10017 | . 2 | |
2 | cnfldstr 19748 | . . 3 ℂfld Struct ; | |
3 | baseid 15919 | . . 3 Slot | |
4 | snsstp1 4347 | . . . 4 | |
5 | ssun1 3776 | . . . . 5 | |
6 | ssun1 3776 | . . . . . 6 TopSet metUnif | |
7 | df-cnfld 19747 | . . . . . 6 ℂfld TopSet metUnif | |
8 | 6, 7 | sseqtr4i 3638 | . . . . 5 ℂfld |
9 | 5, 8 | sstri 3612 | . . . 4 ℂfld |
10 | 4, 9 | sstri 3612 | . . 3 ℂfld |
11 | 2, 3, 10 | strfv 15907 | . 2 ℂfld |
12 | 1, 11 | ax-mp 5 | 1 ℂfld |
Colors of variables: wff setvar class |
Syntax hints: wceq 1483 wcel 1990 cvv 3200 cun 3572 csn 4177 ctp 4181 cop 4183 ccom 5118 cfv 5888 cc 9934 c1 9937 caddc 9939 cmul 9941 cle 10075 cmin 10266 c3 11071 ;cdc 11493 ccj 13836 cabs 13974 cnx 15854 cbs 15857 cplusg 15941 cmulr 15942 cstv 15943 TopSetcts 15947 cple 15948 cds 15950 cunif 15951 cmopn 19736 metUnifcmetu 19737 ℂfldccnfld 19746 |
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-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
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-reu 2919 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-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 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-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 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-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-fz 12327 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-plusg 15954 df-mulr 15955 df-starv 15956 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-cnfld 19747 |
This theorem is referenced by: cncrng 19767 cnfld0 19770 cnfld1 19771 cnfldneg 19772 cnfldplusf 19773 cnfldsub 19774 cndrng 19775 cnflddiv 19776 cnfldinv 19777 cnfldmulg 19778 cnfldexp 19779 cnsrng 19780 cnsubmlem 19794 cnsubglem 19795 cnsubrglem 19796 cnsubdrglem 19797 absabv 19803 cnsubrg 19806 cnmgpabl 19807 cnmgpid 19808 cnmsubglem 19809 gzrngunit 19812 gsumfsum 19813 regsumfsum 19814 expmhm 19815 nn0srg 19816 rge0srg 19817 zringbas 19824 zring0 19828 zringunit 19836 expghm 19844 cnmsgnbas 19924 psgninv 19928 zrhpsgnmhm 19930 rebase 19952 re0g 19958 regsumsupp 19968 cnfldms 22579 cnfldnm 22582 cnfldtopn 22585 cnfldtopon 22586 clmsscn 22879 cnlmod 22940 cnstrcvs 22941 cnrbas 22942 cncvs 22945 cnncvsaddassdemo 22963 cnncvsmulassdemo 22964 cnncvsabsnegdemo 22965 cphsubrglem 22977 cphreccllem 22978 cphdivcl 22982 cphabscl 22985 cphsqrtcl2 22986 cphsqrtcl3 22987 cphipcl 22991 4cphipval2 23041 cncms 23151 cnflduss 23152 cnfldcusp 23153 resscdrg 23154 ishl2 23166 recms 23168 tdeglem3 23819 tdeglem4 23820 tdeglem2 23821 plypf1 23968 dvply2g 24040 dvply2 24041 dvnply 24043 taylfvallem 24112 taylf 24115 tayl0 24116 taylpfval 24119 taylply2 24122 taylply 24123 efgh 24287 efabl 24296 efsubm 24297 jensenlem1 24713 jensenlem2 24714 jensen 24715 amgmlem 24716 amgm 24717 wilthlem2 24795 wilthlem3 24796 dchrelbas2 24962 dchrelbas3 24963 dchrn0 24975 dchrghm 24981 dchrabs 24985 sum2dchr 24999 lgseisenlem4 25103 qrngbas 25308 cchhllem 25767 cffldtocusgr 26343 xrge0slmod 29844 psgnid 29847 iistmd 29948 xrge0iifmhm 29985 xrge0pluscn 29986 zringnm 30004 cnzh 30014 rezh 30015 cnrrext 30054 esumpfinvallem 30136 cnpwstotbnd 33596 repwsmet 33633 rrnequiv 33634 cnsrexpcl 37735 fsumcnsrcl 37736 cnsrplycl 37737 rngunsnply 37743 proot1ex 37779 deg1mhm 37785 amgm2d 38501 amgm3d 38502 amgm4d 38503 binomcxplemdvbinom 38552 binomcxplemnotnn0 38555 sge0tsms 40597 cnfldsrngbas 41769 2zrng0 41938 aacllem 42547 amgmwlem 42548 amgmlemALT 42549 amgmw2d 42550 |
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