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Mirrors > Home > MPE Home > Th. List > cnfldmul | Structured version Visualization version GIF version |
Description: The multiplication operation 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 |
---|---|
cnfldmul | ⊢ · = (.r‘ℂfld) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mulex 12483 | . 2 ⊢ · ∈ V | |
2 | cnfldstr 20231 | . . 3 ⊢ ℂfld Struct 〈1, ;13〉 | |
3 | mulrid 16731 | . . 3 ⊢ .r = Slot (.r‘ndx) | |
4 | snsstp3 4716 | . . . 4 ⊢ {〈(.r‘ndx), · 〉} ⊆ {〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} | |
5 | ssun1 4072 | . . . . 5 ⊢ {〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ⊆ ({〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ∪ {〈(*𝑟‘ndx), ∗〉}) | |
6 | ssun1 4072 | . . . . . 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 20230 | . . . . . 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 | sseqtrri 3924 | . . . . 5 ⊢ ({〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ∪ {〈(*𝑟‘ndx), ∗〉}) ⊆ ℂfld |
9 | 5, 8 | sstri 3896 | . . . 4 ⊢ {〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ⊆ ℂfld |
10 | 4, 9 | sstri 3896 | . . 3 ⊢ {〈(.r‘ndx), · 〉} ⊆ ℂfld |
11 | 2, 3, 10 | strfv 16646 | . 2 ⊢ ( · ∈ V → · = (.r‘ℂfld)) |
12 | 1, 11 | ax-mp 5 | 1 ⊢ · = (.r‘ℂfld) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2114 Vcvv 3400 ∪ cun 3851 {csn 4526 {ctp 4530 〈cop 4532 ∘ ccom 5539 ‘cfv 6349 ℂcc 10625 1c1 10628 + caddc 10630 · cmul 10632 ≤ cle 10766 − cmin 10960 3c3 11784 ;cdc 12191 ∗ccj 14557 abscabs 14695 ndxcnx 16595 Basecbs 16598 +gcplusg 16680 .rcmulr 16681 *𝑟cstv 16682 TopSetcts 16686 lecple 16687 distcds 16689 UnifSetcunif 16690 MetOpencmopn 20219 metUnifcmetu 20220 ℂfldccnfld 20229 |
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 2711 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7491 ax-cnex 10683 ax-resscn 10684 ax-1cn 10685 ax-icn 10686 ax-addcl 10687 ax-addrcl 10688 ax-mulcl 10689 ax-mulrcl 10690 ax-mulcom 10691 ax-addass 10692 ax-mulass 10693 ax-distr 10694 ax-i2m1 10695 ax-1ne0 10696 ax-1rid 10697 ax-rnegex 10698 ax-rrecex 10699 ax-cnre 10700 ax-pre-lttri 10701 ax-pre-lttrn 10702 ax-pre-ltadd 10703 ax-pre-mulgt0 10704 ax-mulf 10707 |
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 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-uni 4807 df-iun 4893 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5439 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-we 5495 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-pred 6139 df-ord 6185 df-on 6186 df-lim 6187 df-suc 6188 df-iota 6307 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7139 df-ov 7185 df-oprab 7186 df-mpo 7187 df-om 7612 df-1st 7726 df-2nd 7727 df-wrecs 7988 df-recs 8049 df-rdg 8087 df-1o 8143 df-er 8332 df-en 8568 df-dom 8569 df-sdom 8570 df-fin 8571 df-pnf 10767 df-mnf 10768 df-xr 10769 df-ltxr 10770 df-le 10771 df-sub 10962 df-neg 10963 df-nn 11729 df-2 11791 df-3 11792 df-4 11793 df-5 11794 df-6 11795 df-7 11796 df-8 11797 df-9 11798 df-n0 11989 df-z 12075 df-dec 12192 df-uz 12337 df-fz 12994 df-struct 16600 df-ndx 16601 df-slot 16602 df-base 16604 df-plusg 16693 df-mulr 16694 df-starv 16695 df-tset 16699 df-ple 16700 df-ds 16702 df-unif 16703 df-cnfld 20230 |
This theorem is referenced by: cncrng 20250 cnfld1 20254 cndrng 20258 cnflddiv 20259 cnfldexp 20262 cnsrng 20263 cnsubrglem 20279 absabv 20286 cnsubrg 20289 cnmsubglem 20292 expmhm 20298 nn0srg 20299 rge0srg 20300 zringmulr 20310 expghm 20328 psgnghm 20408 psgnco 20411 evpmodpmf1o 20424 remulr 20439 mdetralt 21371 clmmul 23839 clmmcl 23849 isclmp 23861 cnlmod 23904 cnncvsmulassdemo 23928 cphsubrglem 23941 cphdivcl 23946 cphabscl 23949 cphsqrtcl2 23950 cphsqrtcl3 23951 ipcau2 23998 plypf1 24973 dvply2g 25045 taylply2 25127 reefgim 25209 efabl 25306 efsubm 25307 amgmlem 25739 amgm 25740 wilthlem2 25818 wilthlem3 25819 dchrelbas3 25986 dchrzrhmul 25994 dchrmulcl 25997 dchrn0 25998 dchrinvcl 26001 dchrsum2 26016 sum2dchr 26022 qabvexp 26374 ostthlem2 26376 padicabv 26378 ostth2lem2 26382 ostth3 26386 xrge0slmod 31132 ccfldsrarelvec 31325 ccfldextdgrr 31326 iistmd 31436 xrge0iifmhm 31473 xrge0pluscn 31474 qqhrhm 31521 cnsrexpcl 40602 cnsrplycl 40604 rngunsnply 40610 amgm2d 41396 amgm3d 41397 amgm4d 41398 cnfldsrngmul 44906 aacllem 46005 amgmlemALT 46007 amgmw2d 46008 |
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