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Mirrors > Home > MPE Home > Th. List > cnfldadd | Structured version Visualization version GIF version |
Description: The addition 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 |
---|---|
cnfldadd | ⊢ + = (+g‘ℂfld) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | addex 12954 | . 2 ⊢ + ∈ V | |
2 | cnfldstr 20880 | . . 3 ⊢ ℂfld Struct 〈1, ;13〉 | |
3 | plusgid 17206 | . . 3 ⊢ +g = Slot (+g‘ndx) | |
4 | snsstp2 4813 | . . . 4 ⊢ {〈(+g‘ndx), + 〉} ⊆ {〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} | |
5 | ssun1 4168 | . . . . 5 ⊢ {〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ⊆ ({〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ∪ {〈(*𝑟‘ndx), ∗〉}) | |
6 | ssun1 4168 | . . . . . 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 20879 | . . . . . 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 4015 | . . . . 5 ⊢ ({〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ∪ {〈(*𝑟‘ndx), ∗〉}) ⊆ ℂfld |
9 | 5, 8 | sstri 3987 | . . . 4 ⊢ {〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ⊆ ℂfld |
10 | 4, 9 | sstri 3987 | . . 3 ⊢ {〈(+g‘ndx), + 〉} ⊆ ℂfld |
11 | 2, 3, 10 | strfv 17119 | . 2 ⊢ ( + ∈ V → + = (+g‘ℂfld)) |
12 | 1, 11 | ax-mp 5 | 1 ⊢ + = (+g‘ℂfld) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ∈ wcel 2106 Vcvv 3473 ∪ cun 3942 {csn 4622 {ctp 4626 〈cop 4628 ∘ ccom 5673 ‘cfv 6532 ℂcc 11090 1c1 11093 + caddc 11095 · cmul 11097 ≤ cle 11231 − cmin 11426 3c3 12250 ;cdc 12659 ∗ccj 15025 abscabs 15163 ndxcnx 17108 Basecbs 17126 +gcplusg 17179 .rcmulr 17180 *𝑟cstv 17181 TopSetcts 17185 lecple 17186 distcds 17188 UnifSetcunif 17189 MetOpencmopn 20868 metUnifcmetu 20869 ℂfldccnfld 20878 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-cnex 11148 ax-resscn 11149 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-mulrcl 11155 ax-mulcom 11156 ax-addass 11157 ax-mulass 11158 ax-distr 11159 ax-i2m1 11160 ax-1ne0 11161 ax-1rid 11162 ax-rnegex 11163 ax-rrecex 11164 ax-cnre 11165 ax-pre-lttri 11166 ax-pre-lttrn 11167 ax-pre-ltadd 11168 ax-pre-mulgt0 11169 ax-addf 11171 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 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 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-tp 4627 df-op 4629 df-uni 4902 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6289 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-om 7839 df-1st 7957 df-2nd 7958 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-rdg 8392 df-1o 8448 df-er 8686 df-en 8923 df-dom 8924 df-sdom 8925 df-fin 8926 df-pnf 11232 df-mnf 11233 df-xr 11234 df-ltxr 11235 df-le 11236 df-sub 11428 df-neg 11429 df-nn 12195 df-2 12257 df-3 12258 df-4 12259 df-5 12260 df-6 12261 df-7 12262 df-8 12263 df-9 12264 df-n0 12455 df-z 12541 df-dec 12660 df-uz 12805 df-fz 13467 df-struct 17062 df-slot 17097 df-ndx 17109 df-base 17127 df-plusg 17192 df-mulr 17193 df-starv 17194 df-tset 17198 df-ple 17199 df-ds 17201 df-unif 17202 df-cnfld 20879 |
This theorem is referenced by: cncrng 20900 cnfld0 20903 cnfldneg 20905 cnfldplusf 20906 cnfldsub 20907 cnfldmulg 20911 cnsrng 20913 cnsubmlem 20927 cnsubglem 20928 absabv 20936 cnsubrg 20939 gsumfsum 20946 regsumfsum 20947 expmhm 20948 nn0srg 20949 rge0srg 20950 zringplusg 20958 replusg 21096 regsumsupp 21108 mhpmulcl 21621 clmadd 24519 clmacl 24529 isclmp 24542 cnlmod 24585 cnncvsaddassdemo 24609 cphsqrtcl2 24632 ipcau2 24680 tdeglem3 25504 tdeglem3OLD 25505 tdeglem4 25506 tdeglem4OLD 25507 taylply2 25809 efgh 25979 efabl 25988 jensenlem1 26418 jensenlem2 26419 amgmlem 26421 qabvle 27055 padicabv 27060 ostth2lem2 27064 ostth3 27068 xrge0slmod 32325 ccfldsrarelvec 32583 ccfldextdgrr 32584 qqhghm 32799 qqhrhm 32800 esumpfinvallem 32903 mhphflem 40956 fsumcnsrcl 41679 rngunsnply 41686 deg1mhm 41720 amgm2d 42721 amgm3d 42722 amgm4d 42723 sge0tsms 44869 cnfldsrngadd 46312 aacllem 47496 amgmw2d 47499 |
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