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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 12922 | . 2 ⊢ · ∈ V | |
2 | cnfldstr 20821 | . . 3 ⊢ ℂfld Struct ⟨1, ;13⟩ | |
3 | mulrid 17183 | . . 3 ⊢ .r = Slot (.r‘ndx) | |
4 | snsstp3 4782 | . . . 4 ⊢ {⟨(.r‘ndx), · ⟩} ⊆ {⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩} | |
5 | ssun1 4136 | . . . . 5 ⊢ {⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩} ⊆ ({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩} ∪ {⟨(*𝑟‘ndx), ∗⟩}) | |
6 | ssun1 4136 | . . . . . 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 20820 | . . . . . 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 3985 | . . . . 5 ⊢ ({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩} ∪ {⟨(*𝑟‘ndx), ∗⟩}) ⊆ ℂfld |
9 | 5, 8 | sstri 3957 | . . . 4 ⊢ {⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩} ⊆ ℂfld |
10 | 4, 9 | sstri 3957 | . . 3 ⊢ {⟨(.r‘ndx), · ⟩} ⊆ ℂfld |
11 | 2, 3, 10 | strfv 17084 | . 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 3447 ∪ cun 3912 {csn 4590 {ctp 4594 ⟨cop 4596 ∘ ccom 5641 ‘cfv 6500 ℂcc 11057 1c1 11060 + caddc 11062 · cmul 11064 ≤ cle 11198 − cmin 11393 3c3 12217 ;cdc 12626 ∗ccj 14990 abscabs 15128 ndxcnx 17073 Basecbs 17091 +gcplusg 17141 .rcmulr 17142 *𝑟cstv 17143 TopSetcts 17147 lecple 17148 distcds 17150 UnifSetcunif 17151 MetOpencmopn 20809 metUnifcmetu 20810 ℂfldccnfld 20819 |
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 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 ax-mulf 11139 |
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 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-tp 4595 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-er 8654 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-nn 12162 df-2 12224 df-3 12225 df-4 12226 df-5 12227 df-6 12228 df-7 12229 df-8 12230 df-9 12231 df-n0 12422 df-z 12508 df-dec 12627 df-uz 12772 df-fz 13434 df-struct 17027 df-slot 17062 df-ndx 17074 df-base 17092 df-plusg 17154 df-mulr 17155 df-starv 17156 df-tset 17160 df-ple 17161 df-ds 17163 df-unif 17164 df-cnfld 20820 |
This theorem is referenced by: cncrng 20841 cnfld1 20845 cndrng 20849 cnflddiv 20850 cnfldexp 20853 cnsrng 20854 cnsubrglem 20870 absabv 20877 cnsubrg 20880 cnmsubglem 20883 expmhm 20889 nn0srg 20890 rge0srg 20891 zringmulr 20901 expghm 20919 psgnghm 21007 psgnco 21010 evpmodpmf1o 21023 remulr 21038 mdetralt 21980 clmmul 24461 clmmcl 24471 isclmp 24483 cnlmod 24526 cnncvsmulassdemo 24551 cphsubrglem 24564 cphdivcl 24569 cphabscl 24572 cphsqrtcl2 24573 cphsqrtcl3 24574 ipcau2 24621 plypf1 25596 dvply2g 25668 taylply2 25750 reefgim 25832 efabl 25929 efsubm 25930 amgmlem 26362 amgm 26363 wilthlem2 26441 wilthlem3 26442 dchrelbas3 26609 dchrzrhmul 26617 dchrmulcl 26620 dchrn0 26621 dchrinvcl 26624 dchrsum2 26639 sum2dchr 26645 qabvexp 26997 ostthlem2 26999 padicabv 27001 ostth2lem2 27005 ostth3 27009 xrge0slmod 32194 ccfldsrarelvec 32419 ccfldextdgrr 32420 iistmd 32547 xrge0iifmhm 32584 xrge0pluscn 32585 qqhrhm 32634 cnsrexpcl 41539 cnsrplycl 41541 rngunsnply 41547 amgm2d 42563 amgm3d 42564 amgm4d 42565 cnfldsrngmul 46155 aacllem 47338 amgmlemALT 47340 amgmw2d 47341 |
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