Mathbox for Thierry Arnoux |
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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 23318 | . . 3 ⊢ ℂfld ∈ NrmRing | |
2 | cndrng 20504 | . . 3 ⊢ ℂfld ∈ DivRing | |
3 | 1, 2 | pm3.2i 471 | . 2 ⊢ (ℂfld ∈ NrmRing ∧ ℂfld ∈ DivRing) |
4 | cnzh 31111 | . . 3 ⊢ (ℤMod‘ℂfld) ∈ NrmMod | |
5 | df-refld 20679 | . . . . 5 ⊢ ℝfld = (ℂfld ↾s ℝ) | |
6 | 5 | fveq2i 6667 | . . . 4 ⊢ (chr‘ℝfld) = (chr‘(ℂfld ↾s ℝ)) |
7 | reofld 30841 | . . . . 5 ⊢ ℝfld ∈ oField | |
8 | ofldchr 30815 | . . . . 5 ⊢ (ℝfld ∈ oField → (chr‘ℝfld) = 0) | |
9 | 7, 8 | ax-mp 5 | . . . 4 ⊢ (chr‘ℝfld) = 0 |
10 | resubdrg 20682 | . . . . . 6 ⊢ (ℝ ∈ (SubRing‘ℂfld) ∧ ℝfld ∈ DivRing) | |
11 | 10 | simpli 484 | . . . . 5 ⊢ ℝ ∈ (SubRing‘ℂfld) |
12 | subrgchr 30793 | . . . . 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 2853 | . . 3 ⊢ (chr‘ℂfld) = 0 |
15 | 4, 14 | pm3.2i 471 | . 2 ⊢ ((ℤMod‘ℂfld) ∈ NrmMod ∧ (chr‘ℂfld) = 0) |
16 | cnfldcusp 23889 | . . 3 ⊢ ℂfld ∈ CUnifSp | |
17 | eqid 2821 | . . . 4 ⊢ (UnifSt‘ℂfld) = (UnifSt‘ℂfld) | |
18 | 17 | cnflduss 23888 | . . 3 ⊢ (UnifSt‘ℂfld) = (metUnif‘(abs ∘ − )) |
19 | 16, 18 | pm3.2i 471 | . 2 ⊢ (ℂfld ∈ CUnifSp ∧ (UnifSt‘ℂfld) = (metUnif‘(abs ∘ − ))) |
20 | cnfldbas 20479 | . . 3 ⊢ ℂ = (Base‘ℂfld) | |
21 | cnmet 23309 | . . . . . 6 ⊢ (abs ∘ − ) ∈ (Met‘ℂ) | |
22 | metf 22869 | . . . . . 6 ⊢ ((abs ∘ − ) ∈ (Met‘ℂ) → (abs ∘ − ):(ℂ × ℂ)⟶ℝ) | |
23 | ffn 6508 | . . . . . 6 ⊢ ((abs ∘ − ):(ℂ × ℂ)⟶ℝ → (abs ∘ − ) Fn (ℂ × ℂ)) | |
24 | 21, 22, 23 | mp2b 10 | . . . . 5 ⊢ (abs ∘ − ) Fn (ℂ × ℂ) |
25 | fnresdm 6460 | . . . . 5 ⊢ ((abs ∘ − ) Fn (ℂ × ℂ) → ((abs ∘ − ) ↾ (ℂ × ℂ)) = (abs ∘ − )) | |
26 | 24, 25 | ax-mp 5 | . . . 4 ⊢ ((abs ∘ − ) ↾ (ℂ × ℂ)) = (abs ∘ − ) |
27 | cnfldds 20485 | . . . . 5 ⊢ (abs ∘ − ) = (dist‘ℂfld) | |
28 | 27 | reseq1i 5843 | . . . 4 ⊢ ((abs ∘ − ) ↾ (ℂ × ℂ)) = ((dist‘ℂfld) ↾ (ℂ × ℂ)) |
29 | 26, 28 | eqtr3i 2846 | . . 3 ⊢ (abs ∘ − ) = ((dist‘ℂfld) ↾ (ℂ × ℂ)) |
30 | eqid 2821 | . . 3 ⊢ (ℤMod‘ℂfld) = (ℤMod‘ℂfld) | |
31 | 20, 29, 30 | isrrext 31141 | . 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 1333 | 1 ⊢ ℂfld ∈ ℝExt |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1528 ∈ wcel 2105 × cxp 5547 ↾ cres 5551 ∘ ccom 5553 Fn wfn 6344 ⟶wf 6345 ‘cfv 6349 (class class class)co 7145 ℂcc 10524 ℝcr 10525 0cc0 10526 − cmin 10859 abscabs 14583 ↾s cress 16474 distcds 16564 DivRingcdr 19433 SubRingcsubrg 19462 Metcmet 20461 metUnifcmetu 20466 ℂfldccnfld 20475 ℤModczlm 20578 chrcchr 20579 ℝfldcrefld 20678 UnifStcuss 22791 CUnifSpccusp 22835 NrmRingcnrg 23118 NrmModcnlm 23119 oFieldcofld 30797 ℝExt crrext 31135 |
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 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606 |
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 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4833 df-int 4870 df-iun 4914 df-iin 4915 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-om 7569 df-1st 7680 df-2nd 7681 df-supp 7822 df-tpos 7883 df-wrecs 7938 df-recs 7999 df-rdg 8037 df-1o 8093 df-2o 8094 df-oadd 8097 df-er 8279 df-map 8398 df-ixp 8451 df-en 8499 df-dom 8500 df-sdom 8501 df-fin 8502 df-fsupp 8823 df-fi 8864 df-sup 8895 df-inf 8896 df-oi 8963 df-card 9357 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11628 df-2 11689 df-3 11690 df-4 11691 df-5 11692 df-6 11693 df-7 11694 df-8 11695 df-9 11696 df-n0 11887 df-z 11971 df-dec 12088 df-uz 12233 df-q 12338 df-rp 12380 df-xneg 12497 df-xadd 12498 df-xmul 12499 df-ioo 12732 df-ico 12734 df-icc 12735 df-fz 12883 df-fzo 13024 df-seq 13360 df-exp 13420 df-hash 13681 df-cj 14448 df-re 14449 df-im 14450 df-sqrt 14584 df-abs 14585 df-struct 16475 df-ndx 16476 df-slot 16477 df-base 16479 df-sets 16480 df-ress 16481 df-plusg 16568 df-mulr 16569 df-starv 16570 df-sca 16571 df-vsca 16572 df-ip 16573 df-tset 16574 df-ple 16575 df-ds 16577 df-unif 16578 df-hom 16579 df-cco 16580 df-rest 16686 df-topn 16687 df-0g 16705 df-gsum 16706 df-topgen 16707 df-pt 16708 df-prds 16711 df-xrs 16765 df-qtop 16770 df-imas 16771 df-xps 16773 df-mre 16847 df-mrc 16848 df-acs 16850 df-proset 17528 df-poset 17546 df-plt 17558 df-toset 17634 df-ps 17800 df-tsr 17801 df-mgm 17842 df-sgrp 17891 df-mnd 17902 df-submnd 17947 df-grp 18046 df-minusg 18047 df-sbg 18048 df-mulg 18165 df-subg 18216 df-cntz 18387 df-od 18587 df-cmn 18839 df-abl 18840 df-mgp 19171 df-ur 19183 df-ring 19230 df-cring 19231 df-oppr 19304 df-dvdsr 19322 df-unit 19323 df-invr 19353 df-dvr 19364 df-drng 19435 df-field 19436 df-subrg 19464 df-abv 19519 df-lmod 19567 df-psmet 20467 df-xmet 20468 df-met 20469 df-bl 20470 df-mopn 20471 df-fbas 20472 df-fg 20473 df-metu 20474 df-cnfld 20476 df-zring 20548 df-zlm 20582 df-chr 20583 df-refld 20679 df-top 21432 df-topon 21449 df-topsp 21471 df-bases 21484 df-cld 21557 df-ntr 21558 df-cls 21559 df-nei 21636 df-cn 21765 df-cnp 21766 df-haus 21853 df-cmp 21925 df-tx 22100 df-hmeo 22293 df-fil 22384 df-flim 22477 df-fcls 22479 df-ust 22738 df-utop 22769 df-uss 22794 df-usp 22795 df-cfilu 22825 df-cusp 22836 df-xms 22859 df-ms 22860 df-tms 22861 df-nm 23121 df-ngp 23122 df-nrg 23124 df-nlm 23125 df-cncf 23415 df-cfil 23787 df-cmet 23789 df-cms 23867 df-omnd 30628 df-ogrp 30629 df-orng 30798 df-ofld 30799 df-rrext 31140 |
This theorem is referenced by: sitgclcn 31502 |
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