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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 12973 | . 2 ⊢ · ∈ V | |
2 | cnfldstr 20946 | . . 3 ⊢ ℂfld Struct ⟨1, ;13⟩ | |
3 | mulridx 17239 | . . 3 ⊢ .r = Slot (.r‘ndx) | |
4 | snsstp3 4822 | . . . 4 ⊢ {⟨(.r‘ndx), · ⟩} ⊆ {⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩} | |
5 | ssun1 4173 | . . . . 5 ⊢ {⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩} ⊆ ({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩} ∪ {⟨(*𝑟‘ndx), ∗⟩}) | |
6 | ssun1 4173 | . . . . . 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 20945 | . . . . . 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 4020 | . . . . 5 ⊢ ({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩} ∪ {⟨(*𝑟‘ndx), ∗⟩}) ⊆ ℂfld |
9 | 5, 8 | sstri 3992 | . . . 4 ⊢ {⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩} ⊆ ℂfld |
10 | 4, 9 | sstri 3992 | . . 3 ⊢ {⟨(.r‘ndx), · ⟩} ⊆ ℂfld |
11 | 2, 3, 10 | strfv 17137 | . 2 ⊢ ( · ∈ V → · = (.r‘ℂfld)) |
12 | 1, 11 | ax-mp 5 | 1 ⊢ · = (.r‘ℂfld) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2107 Vcvv 3475 ∪ cun 3947 {csn 4629 {ctp 4633 ⟨cop 4635 ∘ ccom 5681 ‘cfv 6544 ℂcc 11108 1c1 11111 + caddc 11113 · cmul 11115 ≤ cle 11249 − cmin 11444 3c3 12268 ;cdc 12677 ∗ccj 15043 abscabs 15181 ndxcnx 17126 Basecbs 17144 +gcplusg 17197 .rcmulr 17198 *𝑟cstv 17199 TopSetcts 17203 lecple 17204 distcds 17206 UnifSetcunif 17207 MetOpencmopn 20934 metUnifcmetu 20935 ℂfldccnfld 20944 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-mulf 11190 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12473 df-z 12559 df-dec 12678 df-uz 12823 df-fz 13485 df-struct 17080 df-slot 17115 df-ndx 17127 df-base 17145 df-plusg 17210 df-mulr 17211 df-starv 17212 df-tset 17216 df-ple 17217 df-ds 17219 df-unif 17220 df-cnfld 20945 |
This theorem is referenced by: cncrng 20966 cnfld1 20970 cndrng 20974 cnflddiv 20975 cnfldexp 20978 cnsrng 20979 cnsubrglem 20995 absabv 21002 cnsubrg 21005 cnmsubglem 21008 expmhm 21014 nn0srg 21015 rge0srg 21016 zringmulr 21027 expghm 21045 psgnghm 21133 psgnco 21136 evpmodpmf1o 21149 remulr 21164 mdetralt 22110 clmmul 24591 clmmcl 24601 isclmp 24613 cnlmod 24656 cnncvsmulassdemo 24681 cphsubrglem 24694 cphdivcl 24699 cphabscl 24702 cphsqrtcl2 24703 cphsqrtcl3 24704 ipcau2 24751 plypf1 25726 dvply2g 25798 taylply2 25880 reefgim 25962 efabl 26059 efsubm 26060 amgmlem 26494 amgm 26495 wilthlem2 26573 wilthlem3 26574 dchrelbas3 26741 dchrzrhmul 26749 dchrmulcl 26752 dchrn0 26753 dchrinvcl 26756 dchrsum2 26771 sum2dchr 26777 qabvexp 27129 ostthlem2 27131 padicabv 27133 ostth2lem2 27137 ostth3 27141 xrge0slmod 32463 ccfldsrarelvec 32745 ccfldextdgrr 32746 iistmd 32882 xrge0iifmhm 32919 xrge0pluscn 32920 qqhrhm 32969 cnsrexpcl 41907 cnsrplycl 41909 rngunsnply 41915 amgm2d 42950 amgm3d 42951 amgm4d 42952 cnfldsrngmul 46541 aacllem 47848 amgmlemALT 47850 amgmw2d 47851 |
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