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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 12549 | . 2 ⊢ + ∈ V | |
2 | cnfldstr 20319 | . . 3 ⊢ ℂfld Struct 〈1, ;13〉 | |
3 | plusgid 16780 | . . 3 ⊢ +g = Slot (+g‘ndx) | |
4 | snsstp2 4716 | . . . 4 ⊢ {〈(+g‘ndx), + 〉} ⊆ {〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} | |
5 | ssun1 4072 | . . . . 5 ⊢ {〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ⊆ ({〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ∪ {〈(*𝑟‘ndx), ∗〉}) | |
6 | ssun1 4072 | . . . . . 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 20318 | . . . . . 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 3924 | . . . . 5 ⊢ ({〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ∪ {〈(*𝑟‘ndx), ∗〉}) ⊆ ℂfld |
9 | 5, 8 | sstri 3896 | . . . 4 ⊢ {〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ⊆ ℂfld |
10 | 4, 9 | sstri 3896 | . . 3 ⊢ {〈(+g‘ndx), + 〉} ⊆ ℂfld |
11 | 2, 3, 10 | strfv 16713 | . 2 ⊢ ( + ∈ V → + = (+g‘ℂfld)) |
12 | 1, 11 | ax-mp 5 | 1 ⊢ + = (+g‘ℂfld) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 ∈ wcel 2112 Vcvv 3398 ∪ cun 3851 {csn 4527 {ctp 4531 〈cop 4533 ∘ ccom 5540 ‘cfv 6358 ℂcc 10692 1c1 10695 + caddc 10697 · cmul 10699 ≤ cle 10833 − cmin 11027 3c3 11851 ;cdc 12258 ∗ccj 14624 abscabs 14762 ndxcnx 16663 Basecbs 16666 +gcplusg 16749 .rcmulr 16750 *𝑟cstv 16751 TopSetcts 16755 lecple 16756 distcds 16758 UnifSetcunif 16759 MetOpencmopn 20307 metUnifcmetu 20308 ℂfldccnfld 20317 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 ax-addf 10773 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-2 11858 df-3 11859 df-4 11860 df-5 11861 df-6 11862 df-7 11863 df-8 11864 df-9 11865 df-n0 12056 df-z 12142 df-dec 12259 df-uz 12404 df-fz 13061 df-struct 16668 df-ndx 16669 df-slot 16670 df-base 16672 df-plusg 16762 df-mulr 16763 df-starv 16764 df-tset 16768 df-ple 16769 df-ds 16771 df-unif 16772 df-cnfld 20318 |
This theorem is referenced by: cncrng 20338 cnfld0 20341 cnfldneg 20343 cnfldplusf 20344 cnfldsub 20345 cnfldmulg 20349 cnsrng 20351 cnsubmlem 20365 cnsubglem 20366 absabv 20374 cnsubrg 20377 gsumfsum 20384 regsumfsum 20385 expmhm 20386 nn0srg 20387 rge0srg 20388 zringplusg 20396 replusg 20526 regsumsupp 20538 mhpmulcl 21043 clmadd 23925 clmacl 23935 isclmp 23948 cnlmod 23991 cnncvsaddassdemo 24014 cphsqrtcl2 24037 ipcau2 24085 tdeglem3 24909 tdeglem3OLD 24910 tdeglem4 24911 tdeglem4OLD 24912 taylply2 25214 efgh 25384 efabl 25393 jensenlem1 25823 jensenlem2 25824 amgmlem 25826 qabvle 26460 padicabv 26465 ostth2lem2 26469 ostth3 26473 xrge0slmod 31216 ccfldsrarelvec 31409 ccfldextdgrr 31410 qqhghm 31604 qqhrhm 31605 esumpfinvallem 31708 mhphflem 39935 fsumcnsrcl 40635 rngunsnply 40642 deg1mhm 40676 amgm2d 41428 amgm3d 41429 amgm4d 41430 sge0tsms 43536 cnfldsrngadd 44940 aacllem 46119 amgmw2d 46122 |
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