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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 12382 | . 2 ⊢ · ∈ V | |
2 | cnfldstr 20541 | . . 3 ⊢ ℂfld Struct 〈1, ;13〉 | |
3 | mulrid 16610 | . . 3 ⊢ .r = Slot (.r‘ndx) | |
4 | snsstp3 4744 | . . . 4 ⊢ {〈(.r‘ndx), · 〉} ⊆ {〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} | |
5 | ssun1 4147 | . . . . 5 ⊢ {〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ⊆ ({〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ∪ {〈(*𝑟‘ndx), ∗〉}) | |
6 | ssun1 4147 | . . . . . 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 20540 | . . . . . 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 4003 | . . . . 5 ⊢ ({〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ∪ {〈(*𝑟‘ndx), ∗〉}) ⊆ ℂfld |
9 | 5, 8 | sstri 3975 | . . . 4 ⊢ {〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ⊆ ℂfld |
10 | 4, 9 | sstri 3975 | . . 3 ⊢ {〈(.r‘ndx), · 〉} ⊆ ℂfld |
11 | 2, 3, 10 | strfv 16525 | . 2 ⊢ ( · ∈ V → · = (.r‘ℂfld)) |
12 | 1, 11 | ax-mp 5 | 1 ⊢ · = (.r‘ℂfld) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2110 Vcvv 3494 ∪ cun 3933 {csn 4560 {ctp 4564 〈cop 4566 ∘ ccom 5553 ‘cfv 6349 ℂcc 10529 1c1 10532 + caddc 10534 · cmul 10536 ≤ cle 10670 − cmin 10864 3c3 11687 ;cdc 12092 ∗ccj 14449 abscabs 14587 ndxcnx 16474 Basecbs 16477 +gcplusg 16559 .rcmulr 16560 *𝑟cstv 16561 TopSetcts 16565 lecple 16566 distcds 16568 UnifSetcunif 16569 MetOpencmopn 20529 metUnifcmetu 20530 ℂfldccnfld 20539 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-mulf 10611 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-fz 12887 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-plusg 16572 df-mulr 16573 df-starv 16574 df-tset 16578 df-ple 16579 df-ds 16581 df-unif 16582 df-cnfld 20540 |
This theorem is referenced by: cncrng 20560 cnfld1 20564 cndrng 20568 cnflddiv 20569 cnfldexp 20572 cnsrng 20573 cnsubrglem 20589 absabv 20596 cnsubrg 20599 cnmsubglem 20602 expmhm 20608 nn0srg 20609 rge0srg 20610 zringmulr 20620 expghm 20637 psgnghm 20718 psgnco 20721 evpmodpmf1o 20734 remulr 20749 mdetralt 21211 clmmul 23673 clmmcl 23683 isclmp 23695 cnlmod 23738 cnncvsmulassdemo 23762 cphsubrglem 23775 cphdivcl 23780 cphabscl 23783 cphsqrtcl2 23784 cphsqrtcl3 23785 ipcau2 23831 plypf1 24796 dvply2g 24868 taylply2 24950 reefgim 25032 efabl 25128 efsubm 25129 amgmlem 25561 amgm 25562 wilthlem2 25640 wilthlem3 25641 dchrelbas3 25808 dchrzrhmul 25816 dchrmulcl 25819 dchrn0 25820 dchrinvcl 25823 dchrsum2 25838 sum2dchr 25844 qabvexp 26196 ostthlem2 26198 padicabv 26200 ostth2lem2 26204 ostth3 26208 xrge0slmod 30912 ccfldsrarelvec 31051 ccfldextdgrr 31052 iistmd 31140 xrge0iifmhm 31177 xrge0pluscn 31178 qqhrhm 31225 cnsrexpcl 39758 cnsrplycl 39760 rngunsnply 39766 amgm2d 40544 amgm3d 40545 amgm4d 40546 cnfldsrngmul 44032 aacllem 44896 amgmlemALT 44898 amgmw2d 44899 |
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