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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 12375 | . 2 ⊢ + ∈ V | |
2 | cnfldstr 20475 | . . 3 ⊢ ℂfld Struct 〈1, ;13〉 | |
3 | plusgid 16584 | . . 3 ⊢ +g = Slot (+g‘ndx) | |
4 | snsstp2 4742 | . . . 4 ⊢ {〈(+g‘ndx), + 〉} ⊆ {〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} | |
5 | ssun1 4145 | . . . . 5 ⊢ {〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ⊆ ({〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ∪ {〈(*𝑟‘ndx), ∗〉}) | |
6 | ssun1 4145 | . . . . . 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 20474 | . . . . . 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 4001 | . . . . 5 ⊢ ({〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ∪ {〈(*𝑟‘ndx), ∗〉}) ⊆ ℂfld |
9 | 5, 8 | sstri 3973 | . . . 4 ⊢ {〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ⊆ ℂfld |
10 | 4, 9 | sstri 3973 | . . 3 ⊢ {〈(+g‘ndx), + 〉} ⊆ ℂfld |
11 | 2, 3, 10 | strfv 16519 | . 2 ⊢ ( + ∈ V → + = (+g‘ℂfld)) |
12 | 1, 11 | ax-mp 5 | 1 ⊢ + = (+g‘ℂfld) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ∈ wcel 2105 Vcvv 3492 ∪ cun 3931 {csn 4557 {ctp 4561 〈cop 4563 ∘ ccom 5552 ‘cfv 6348 ℂcc 10523 1c1 10526 + caddc 10528 · cmul 10530 ≤ cle 10664 − cmin 10858 3c3 11681 ;cdc 12086 ∗ccj 14443 abscabs 14581 ndxcnx 16468 Basecbs 16471 +gcplusg 16553 .rcmulr 16554 *𝑟cstv 16555 TopSetcts 16559 lecple 16560 distcds 16562 UnifSetcunif 16563 MetOpencmopn 20463 metUnifcmetu 20464 ℂfldccnfld 20473 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-addf 10604 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12881 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-plusg 16566 df-mulr 16567 df-starv 16568 df-tset 16572 df-ple 16573 df-ds 16575 df-unif 16576 df-cnfld 20474 |
This theorem is referenced by: cncrng 20494 cnfld0 20497 cnfldneg 20499 cnfldplusf 20500 cnfldsub 20501 cnfldmulg 20505 cnsrng 20507 cnsubmlem 20521 cnsubglem 20522 absabv 20530 cnsubrg 20533 gsumfsum 20540 regsumfsum 20541 expmhm 20542 nn0srg 20543 rge0srg 20544 zringplusg 20552 replusg 20682 regsumsupp 20694 clmadd 23605 clmacl 23615 isclmp 23628 cnlmod 23671 cnncvsaddassdemo 23694 cphsqrtcl2 23717 ipcau2 23764 tdeglem3 24580 tdeglem4 24581 taylply2 24883 efgh 25052 efabl 25061 jensenlem1 25491 jensenlem2 25492 amgmlem 25494 qabvle 26128 padicabv 26133 ostth2lem2 26137 ostth3 26141 xrge0slmod 30844 ccfldsrarelvec 30955 ccfldextdgrr 30956 qqhghm 31128 qqhrhm 31129 esumpfinvallem 31232 fsumcnsrcl 39644 rngunsnply 39651 deg1mhm 39685 amgm2d 40429 amgm3d 40430 amgm4d 40431 sge0tsms 42539 cnfldsrngadd 43914 aacllem 44830 amgmw2d 44833 |
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