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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 12980 | . 2 ⊢ · ∈ V | |
2 | cnfldstr 21236 | . . 3 ⊢ ℂfld Struct 〈1, ;13〉 | |
3 | mulridx 17246 | . . 3 ⊢ .r = Slot (.r‘ndx) | |
4 | snsstp3 4821 | . . . 4 ⊢ {〈(.r‘ndx), · 〉} ⊆ {〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} | |
5 | ssun1 4172 | . . . . 5 ⊢ {〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ⊆ ({〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ∪ {〈(*𝑟‘ndx), ∗〉}) | |
6 | ssun1 4172 | . . . . . 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 21235 | . . . . . 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 4019 | . . . . 5 ⊢ ({〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ∪ {〈(*𝑟‘ndx), ∗〉}) ⊆ ℂfld |
9 | 5, 8 | sstri 3991 | . . . 4 ⊢ {〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ⊆ ℂfld |
10 | 4, 9 | sstri 3991 | . . 3 ⊢ {〈(.r‘ndx), · 〉} ⊆ ℂfld |
11 | 2, 3, 10 | strfv 17144 | . 2 ⊢ ( · ∈ V → · = (.r‘ℂfld)) |
12 | 1, 11 | ax-mp 5 | 1 ⊢ · = (.r‘ℂfld) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1540 ∈ wcel 2105 Vcvv 3473 ∪ cun 3946 {csn 4628 {ctp 4632 〈cop 4634 ∘ ccom 5680 ‘cfv 6543 ℂcc 11114 1c1 11117 + caddc 11119 · cmul 11121 ≤ cle 11256 − cmin 11451 3c3 12275 ;cdc 12684 ∗ccj 15050 abscabs 15188 ndxcnx 17133 Basecbs 17151 +gcplusg 17204 .rcmulr 17205 *𝑟cstv 17206 TopSetcts 17210 lecple 17211 distcds 17213 UnifSetcunif 17214 MetOpencmopn 21224 metUnifcmetu 21225 ℂfldccnfld 21234 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-mulf 11196 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-er 8709 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12480 df-z 12566 df-dec 12685 df-uz 12830 df-fz 13492 df-struct 17087 df-slot 17122 df-ndx 17134 df-base 17152 df-plusg 17217 df-mulr 17218 df-starv 17219 df-tset 17223 df-ple 17224 df-ds 17226 df-unif 17227 df-cnfld 21235 |
This theorem is referenced by: cncrng 21256 cnfld1 21260 cndrng 21264 cnflddiv 21265 cnfldexp 21268 cnsrng 21269 cnsubrglem 21285 absabv 21292 cnsubrg 21295 cnmsubglem 21298 expmhm 21304 nn0srg 21305 rge0srg 21306 zringmulr 21318 expghm 21336 psgnghm 21444 psgnco 21447 evpmodpmf1o 21460 remulr 21475 mdetralt 22431 clmmul 24923 clmmcl 24933 isclmp 24945 cnlmod 24988 cnncvsmulassdemo 25013 cphsubrglem 25026 cphdivcl 25031 cphabscl 25034 cphsqrtcl2 25035 cphsqrtcl3 25036 ipcau2 25083 plypf1 26065 dvply2g 26138 taylply2 26220 reefgim 26303 efabl 26400 efsubm 26401 amgmlem 26837 amgm 26838 wilthlem2 26916 wilthlem3 26917 dchrelbas3 27086 dchrzrhmul 27094 dchrmulcl 27097 dchrn0 27098 dchrinvcl 27101 dchrsum2 27116 sum2dchr 27122 qabvexp 27474 ostthlem2 27476 padicabv 27478 ostth2lem2 27482 ostth3 27486 xrge0slmod 32901 ccfldsrarelvec 33202 ccfldextdgrr 33203 iistmd 33348 xrge0iifmhm 33385 xrge0pluscn 33386 qqhrhm 33435 cnsrexpcl 42373 cnsrplycl 42375 rngunsnply 42381 amgm2d 43416 amgm3d 43417 amgm4d 43418 cnfldsrngmul 47003 aacllem 48013 amgmlemALT 48015 amgmw2d 48016 |
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