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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 12118 | . 2 ⊢ · ∈ V | |
2 | cnfldstr 20115 | . . 3 ⊢ ℂfld Struct 〈1, ;13〉 | |
3 | mulrid 16363 | . . 3 ⊢ .r = Slot (.r‘ndx) | |
4 | snsstp3 4569 | . . . 4 ⊢ {〈(.r‘ndx), · 〉} ⊆ {〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} | |
5 | ssun1 4005 | . . . . 5 ⊢ {〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ⊆ ({〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ∪ {〈(*𝑟‘ndx), ∗〉}) | |
6 | ssun1 4005 | . . . . . 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 20114 | . . . . . 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 3863 | . . . . 5 ⊢ ({〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ∪ {〈(*𝑟‘ndx), ∗〉}) ⊆ ℂfld |
9 | 5, 8 | sstri 3836 | . . . 4 ⊢ {〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ⊆ ℂfld |
10 | 4, 9 | sstri 3836 | . . 3 ⊢ {〈(.r‘ndx), · 〉} ⊆ ℂfld |
11 | 2, 3, 10 | strfv 16277 | . 2 ⊢ ( · ∈ V → · = (.r‘ℂfld)) |
12 | 1, 11 | ax-mp 5 | 1 ⊢ · = (.r‘ℂfld) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1656 ∈ wcel 2164 Vcvv 3414 ∪ cun 3796 {csn 4399 {ctp 4403 〈cop 4405 ∘ ccom 5350 ‘cfv 6127 ℂcc 10257 1c1 10260 + caddc 10262 · cmul 10264 ≤ cle 10399 − cmin 10592 3c3 11414 ;cdc 11828 ∗ccj 14220 abscabs 14358 ndxcnx 16226 Basecbs 16229 +gcplusg 16312 .rcmulr 16313 *𝑟cstv 16314 TopSetcts 16318 lecple 16319 distcds 16321 UnifSetcunif 16322 MetOpencmopn 20103 metUnifcmetu 20104 ℂfldccnfld 20113 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 ax-mulf 10339 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-int 4700 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-om 7332 df-1st 7433 df-2nd 7434 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-1o 7831 df-oadd 7835 df-er 8014 df-en 8229 df-dom 8230 df-sdom 8231 df-fin 8232 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-nn 11358 df-2 11421 df-3 11422 df-4 11423 df-5 11424 df-6 11425 df-7 11426 df-8 11427 df-9 11428 df-n0 11626 df-z 11712 df-dec 11829 df-uz 11976 df-fz 12627 df-struct 16231 df-ndx 16232 df-slot 16233 df-base 16235 df-plusg 16325 df-mulr 16326 df-starv 16327 df-tset 16331 df-ple 16332 df-ds 16334 df-unif 16335 df-cnfld 20114 |
This theorem is referenced by: cncrng 20134 cnfld1 20138 cndrng 20142 cnflddiv 20143 cnfldexp 20146 cnsrng 20147 cnsubrglem 20163 absabv 20170 cnsubrg 20173 cnmsubglem 20176 expmhm 20182 nn0srg 20183 rge0srg 20184 zringmulr 20194 expghm 20211 psgnghm 20292 psgnco 20295 evpmodpmf1o 20309 remulr 20325 mdetralt 20789 clmmul 23251 clmmcl 23261 isclmp 23273 cnlmod 23316 cnncvsmulassdemo 23340 cphsubrglem 23353 cphdivcl 23358 cphabscl 23361 cphsqrtcl2 23362 cphsqrtcl3 23363 ipcau2 23409 plypf1 24374 dvply2g 24446 taylply2 24528 reefgim 24610 efabl 24703 efsubm 24704 amgmlem 25136 amgm 25137 wilthlem2 25215 wilthlem3 25216 dchrelbas3 25383 dchrzrhmul 25391 dchrmulcl 25394 dchrn0 25395 dchrinvcl 25398 dchrsum2 25413 sum2dchr 25419 qabvexp 25735 ostthlem2 25737 padicabv 25739 ostth2lem2 25743 ostth3 25747 xrge0slmod 30385 iistmd 30489 xrge0iifmhm 30526 xrge0pluscn 30527 qqhrhm 30574 cnsrexpcl 38573 cnsrplycl 38575 rngunsnply 38581 amgm2d 39336 amgm3d 39337 amgm4d 39338 cnfldsrngmul 42632 aacllem 43453 amgmlemALT 43455 amgmw2d 43456 |
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