Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > cnrrext | Structured version Visualization version GIF version |
Description: The field of the complex numbers is an extension of the real numbers. (Contributed by Thierry Arnoux, 2-May-2018.) |
Ref | Expression |
---|---|
cnrrext | ⊢ ℂfld ∈ ℝExt |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnnrg 23632 | . . 3 ⊢ ℂfld ∈ NrmRing | |
2 | cndrng 20346 | . . 3 ⊢ ℂfld ∈ DivRing | |
3 | 1, 2 | pm3.2i 474 | . 2 ⊢ (ℂfld ∈ NrmRing ∧ ℂfld ∈ DivRing) |
4 | cnzh 31586 | . . 3 ⊢ (ℤMod‘ℂfld) ∈ NrmMod | |
5 | df-refld 20521 | . . . . 5 ⊢ ℝfld = (ℂfld ↾s ℝ) | |
6 | 5 | fveq2i 6698 | . . . 4 ⊢ (chr‘ℝfld) = (chr‘(ℂfld ↾s ℝ)) |
7 | reofld 31212 | . . . . 5 ⊢ ℝfld ∈ oField | |
8 | ofldchr 31186 | . . . . 5 ⊢ (ℝfld ∈ oField → (chr‘ℝfld) = 0) | |
9 | 7, 8 | ax-mp 5 | . . . 4 ⊢ (chr‘ℝfld) = 0 |
10 | resubdrg 20524 | . . . . . 6 ⊢ (ℝ ∈ (SubRing‘ℂfld) ∧ ℝfld ∈ DivRing) | |
11 | 10 | simpli 487 | . . . . 5 ⊢ ℝ ∈ (SubRing‘ℂfld) |
12 | subrgchr 31164 | . . . . 5 ⊢ (ℝ ∈ (SubRing‘ℂfld) → (chr‘(ℂfld ↾s ℝ)) = (chr‘ℂfld)) | |
13 | 11, 12 | ax-mp 5 | . . . 4 ⊢ (chr‘(ℂfld ↾s ℝ)) = (chr‘ℂfld) |
14 | 6, 9, 13 | 3eqtr3ri 2768 | . . 3 ⊢ (chr‘ℂfld) = 0 |
15 | 4, 14 | pm3.2i 474 | . 2 ⊢ ((ℤMod‘ℂfld) ∈ NrmMod ∧ (chr‘ℂfld) = 0) |
16 | cnfldcusp 24208 | . . 3 ⊢ ℂfld ∈ CUnifSp | |
17 | eqid 2736 | . . . 4 ⊢ (UnifSt‘ℂfld) = (UnifSt‘ℂfld) | |
18 | 17 | cnflduss 24207 | . . 3 ⊢ (UnifSt‘ℂfld) = (metUnif‘(abs ∘ − )) |
19 | 16, 18 | pm3.2i 474 | . 2 ⊢ (ℂfld ∈ CUnifSp ∧ (UnifSt‘ℂfld) = (metUnif‘(abs ∘ − ))) |
20 | cnfldbas 20321 | . . 3 ⊢ ℂ = (Base‘ℂfld) | |
21 | cnmet 23623 | . . . . . 6 ⊢ (abs ∘ − ) ∈ (Met‘ℂ) | |
22 | metf 23182 | . . . . . 6 ⊢ ((abs ∘ − ) ∈ (Met‘ℂ) → (abs ∘ − ):(ℂ × ℂ)⟶ℝ) | |
23 | ffn 6523 | . . . . . 6 ⊢ ((abs ∘ − ):(ℂ × ℂ)⟶ℝ → (abs ∘ − ) Fn (ℂ × ℂ)) | |
24 | 21, 22, 23 | mp2b 10 | . . . . 5 ⊢ (abs ∘ − ) Fn (ℂ × ℂ) |
25 | fnresdm 6474 | . . . . 5 ⊢ ((abs ∘ − ) Fn (ℂ × ℂ) → ((abs ∘ − ) ↾ (ℂ × ℂ)) = (abs ∘ − )) | |
26 | 24, 25 | ax-mp 5 | . . . 4 ⊢ ((abs ∘ − ) ↾ (ℂ × ℂ)) = (abs ∘ − ) |
27 | cnfldds 20327 | . . . . 5 ⊢ (abs ∘ − ) = (dist‘ℂfld) | |
28 | 27 | reseq1i 5832 | . . . 4 ⊢ ((abs ∘ − ) ↾ (ℂ × ℂ)) = ((dist‘ℂfld) ↾ (ℂ × ℂ)) |
29 | 26, 28 | eqtr3i 2761 | . . 3 ⊢ (abs ∘ − ) = ((dist‘ℂfld) ↾ (ℂ × ℂ)) |
30 | eqid 2736 | . . 3 ⊢ (ℤMod‘ℂfld) = (ℤMod‘ℂfld) | |
31 | 20, 29, 30 | isrrext 31616 | . 2 ⊢ (ℂfld ∈ ℝExt ↔ ((ℂfld ∈ NrmRing ∧ ℂfld ∈ DivRing) ∧ ((ℤMod‘ℂfld) ∈ NrmMod ∧ (chr‘ℂfld) = 0) ∧ (ℂfld ∈ CUnifSp ∧ (UnifSt‘ℂfld) = (metUnif‘(abs ∘ − ))))) |
32 | 3, 15, 19, 31 | mpbir3an 1343 | 1 ⊢ ℂfld ∈ ℝExt |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 = wceq 1543 ∈ wcel 2112 × cxp 5534 ↾ cres 5538 ∘ ccom 5540 Fn wfn 6353 ⟶wf 6354 ‘cfv 6358 (class class class)co 7191 ℂcc 10692 ℝcr 10693 0cc0 10694 − cmin 11027 abscabs 14762 ↾s cress 16667 distcds 16758 DivRingcdr 19721 SubRingcsubrg 19750 Metcmet 20303 metUnifcmetu 20308 ℂfldccnfld 20317 ℤModczlm 20421 chrcchr 20422 ℝfldcrefld 20520 UnifStcuss 23105 CUnifSpccusp 23148 NrmRingcnrg 23431 NrmModcnlm 23432 oFieldcofld 31168 ℝExt crrext 31610 |
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-rep 5164 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-pre-sup 10772 ax-addf 10773 ax-mulf 10774 |
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-rmo 3059 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-int 4846 df-iun 4892 df-iin 4893 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-se 5495 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-isom 6367 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-of 7447 df-om 7623 df-1st 7739 df-2nd 7740 df-supp 7882 df-tpos 7946 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-2o 8181 df-er 8369 df-map 8488 df-ixp 8557 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-fsupp 8964 df-fi 9005 df-sup 9036 df-inf 9037 df-oi 9104 df-card 9520 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-div 11455 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-q 12510 df-rp 12552 df-xneg 12669 df-xadd 12670 df-xmul 12671 df-ioo 12904 df-ico 12906 df-icc 12907 df-fz 13061 df-fzo 13204 df-seq 13540 df-exp 13601 df-hash 13862 df-cj 14627 df-re 14628 df-im 14629 df-sqrt 14763 df-abs 14764 df-struct 16668 df-ndx 16669 df-slot 16670 df-base 16672 df-sets 16673 df-ress 16674 df-plusg 16762 df-mulr 16763 df-starv 16764 df-sca 16765 df-vsca 16766 df-ip 16767 df-tset 16768 df-ple 16769 df-ds 16771 df-unif 16772 df-hom 16773 df-cco 16774 df-rest 16881 df-topn 16882 df-0g 16900 df-gsum 16901 df-topgen 16902 df-pt 16903 df-prds 16906 df-xrs 16961 df-qtop 16966 df-imas 16967 df-xps 16969 df-mre 17043 df-mrc 17044 df-acs 17046 df-proset 17756 df-poset 17774 df-plt 17790 df-toset 17877 df-ps 18026 df-tsr 18027 df-mgm 18068 df-sgrp 18117 df-mnd 18128 df-submnd 18173 df-grp 18322 df-minusg 18323 df-sbg 18324 df-mulg 18443 df-subg 18494 df-cntz 18665 df-od 18874 df-cmn 19126 df-abl 19127 df-mgp 19459 df-ur 19471 df-ring 19518 df-cring 19519 df-oppr 19595 df-dvdsr 19613 df-unit 19614 df-invr 19644 df-dvr 19655 df-drng 19723 df-field 19724 df-subrg 19752 df-abv 19807 df-lmod 19855 df-psmet 20309 df-xmet 20310 df-met 20311 df-bl 20312 df-mopn 20313 df-fbas 20314 df-fg 20315 df-metu 20316 df-cnfld 20318 df-zring 20390 df-zlm 20425 df-chr 20426 df-refld 20521 df-top 21745 df-topon 21762 df-topsp 21784 df-bases 21797 df-cld 21870 df-ntr 21871 df-cls 21872 df-nei 21949 df-cn 22078 df-cnp 22079 df-haus 22166 df-cmp 22238 df-tx 22413 df-hmeo 22606 df-fil 22697 df-flim 22790 df-fcls 22792 df-ust 23052 df-utop 23083 df-uss 23108 df-usp 23109 df-cfilu 23138 df-cusp 23149 df-xms 23172 df-ms 23173 df-tms 23174 df-nm 23434 df-ngp 23435 df-nrg 23437 df-nlm 23438 df-cncf 23729 df-cfil 24106 df-cmet 24108 df-cms 24186 df-omnd 30998 df-ogrp 30999 df-orng 31169 df-ofld 31170 df-rrext 31615 |
This theorem is referenced by: sitgclcn 31977 |
Copyright terms: Public domain | W3C validator |