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Mirrors > Home > MPE Home > Th. List > cnfldbas | Structured version Visualization version GIF 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 | ⊢ ℂ = (Base‘ℂfld) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnex 9896 | . 2 ⊢ ℂ ∈ V | |
2 | cnfldstr 19569 | . . 3 ⊢ ℂfld Struct 〈1, ;13〉 | |
3 | baseid 15747 | . . 3 ⊢ Base = Slot (Base‘ndx) | |
4 | snsstp1 4287 | . . . 4 ⊢ {〈(Base‘ndx), ℂ〉} ⊆ {〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} | |
5 | ssun1 3738 | . . . . 5 ⊢ {〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ⊆ ({〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ∪ {〈(*𝑟‘ndx), ∗〉}) | |
6 | ssun1 3738 | . . . . . 6 ⊢ ({〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ∪ {〈(*𝑟‘ndx), ∗〉}) ⊆ (({〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ∪ {〈(*𝑟‘ndx), ∗〉}) ∪ ({〈(TopSet‘ndx), (MetOpen‘(abs ∘ − ))〉, 〈(le‘ndx), ≤ 〉, 〈(dist‘ndx), (abs ∘ − )〉} ∪ {〈(UnifSet‘ndx), (metUnif‘(abs ∘ − ))〉})) | |
7 | df-cnfld 19568 | . . . . . 6 ⊢ ℂfld = (({〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ∪ {〈(*𝑟‘ndx), ∗〉}) ∪ ({〈(TopSet‘ndx), (MetOpen‘(abs ∘ − ))〉, 〈(le‘ndx), ≤ 〉, 〈(dist‘ndx), (abs ∘ − )〉} ∪ {〈(UnifSet‘ndx), (metUnif‘(abs ∘ − ))〉})) | |
8 | 6, 7 | sseqtr4i 3601 | . . . . 5 ⊢ ({〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ∪ {〈(*𝑟‘ndx), ∗〉}) ⊆ ℂfld |
9 | 5, 8 | sstri 3577 | . . . 4 ⊢ {〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ⊆ ℂfld |
10 | 4, 9 | sstri 3577 | . . 3 ⊢ {〈(Base‘ndx), ℂ〉} ⊆ ℂfld |
11 | 2, 3, 10 | strfv 15735 | . 2 ⊢ (ℂ ∈ V → ℂ = (Base‘ℂfld)) |
12 | 1, 11 | ax-mp 5 | 1 ⊢ ℂ = (Base‘ℂfld) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∪ cun 3538 {csn 4125 {ctp 4129 〈cop 4131 ∘ ccom 5042 ‘cfv 5804 ℂcc 9813 1c1 9816 + caddc 9818 · cmul 9820 ≤ cle 9954 − cmin 10145 3c3 10948 ;cdc 11369 ∗ccj 13684 abscabs 13822 ndxcnx 15692 Basecbs 15695 +gcplusg 15768 .rcmulr 15769 *𝑟cstv 15770 TopSetcts 15774 lecple 15775 distcds 15777 UnifSetcunif 15778 MetOpencmopn 19557 metUnifcmetu 19558 ℂfldccnfld 19567 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-fz 12198 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-plusg 15781 df-mulr 15782 df-starv 15783 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-cnfld 19568 |
This theorem is referenced by: cncrng 19586 cnfld0 19589 cnfld1 19590 cnfldneg 19591 cnfldplusf 19592 cnfldsub 19593 cndrng 19594 cnflddiv 19595 cnfldinv 19596 cnfldmulg 19597 cnfldexp 19598 cnsrng 19599 cnsubmlem 19613 cnsubglem 19614 cnsubrglem 19615 cnsubdrglem 19616 absabv 19622 cnsubrg 19625 cnmgpabl 19626 cnmgpid 19627 cnmsubglem 19628 gzrngunit 19631 gsumfsum 19632 regsumfsum 19633 expmhm 19634 nn0srg 19635 rge0srg 19636 zringbas 19643 zring0 19647 zringunit 19655 expghm 19663 cnmsgnbas 19743 psgninv 19747 zrhpsgnmhm 19749 rebase 19771 re0g 19777 regsumsupp 19787 cnfldms 22389 cnfldnm 22392 cnfldtopn 22395 cnfldtopon 22396 clmsscn 22687 cnlmod 22748 cnstrcvs 22749 cnrbas 22750 cncvs 22753 cnncvsaddassdemo 22771 cnncvsmulassdemo 22772 cnncvsabsnegdemo 22773 cphsubrglem 22785 cphreccllem 22786 cphdivcl 22790 cphabscl 22793 cphsqrtcl2 22794 cphsqrtcl3 22795 cphipcl 22799 4cphipval2 22849 cncms 22959 cnflduss 22960 cnfldcusp 22961 resscdrg 22962 ishl2 22974 recms 22976 tdeglem3 23623 tdeglem4 23624 tdeglem2 23625 plypf1 23772 dvply2g 23844 dvply2 23845 dvnply 23847 taylfvallem 23916 taylf 23919 tayl0 23920 taylpfval 23923 taylply2 23926 taylply 23927 efgh 24091 efabl 24100 efsubm 24101 jensenlem1 24513 jensenlem2 24514 jensen 24515 amgmlem 24516 amgm 24517 wilthlem2 24595 wilthlem3 24596 dchrelbas2 24762 dchrelbas3 24763 dchrn0 24775 dchrghm 24781 dchrabs 24785 sum2dchr 24799 lgseisenlem4 24903 qrngbas 25108 cchhllem 25567 xrge0slmod 29175 psgnid 29178 iistmd 29276 xrge0iifmhm 29313 xrge0pluscn 29314 zringnm 29332 cnzh 29342 rezh 29343 cnrrext 29382 esumpfinvallem 29463 cnpwstotbnd 32766 repwsmet 32803 rrnequiv 32804 cnsrexpcl 36754 fsumcnsrcl 36755 cnsrplycl 36756 rngunsnply 36762 proot1ex 36798 deg1mhm 36804 amgm2d 37523 amgm3d 37524 amgm4d 37525 binomcxplemdvbinom 37574 binomcxplemnotnn0 37577 sge0tsms 39273 cnfldsrngbas 41559 2zrng0 41728 aacllem 42356 amgmwlem 42357 amgmlemALT 42358 amgmw2d 42359 |
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