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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 24296 | . . 3 ⊢ ℂfld ∈ NrmRing | |
2 | cndrng 20973 | . . 3 ⊢ ℂfld ∈ DivRing | |
3 | 1, 2 | pm3.2i 471 | . 2 ⊢ (ℂfld ∈ NrmRing ∧ ℂfld ∈ DivRing) |
4 | cnzh 32945 | . . 3 ⊢ (ℤMod‘ℂfld) ∈ NrmMod | |
5 | df-refld 21157 | . . . . 5 ⊢ ℝfld = (ℂfld ↾s ℝ) | |
6 | 5 | fveq2i 6894 | . . . 4 ⊢ (chr‘ℝfld) = (chr‘(ℂfld ↾s ℝ)) |
7 | reofld 32454 | . . . . 5 ⊢ ℝfld ∈ oField | |
8 | ofldchr 32427 | . . . . 5 ⊢ (ℝfld ∈ oField → (chr‘ℝfld) = 0) | |
9 | 7, 8 | ax-mp 5 | . . . 4 ⊢ (chr‘ℝfld) = 0 |
10 | resubdrg 21160 | . . . . . 6 ⊢ (ℝ ∈ (SubRing‘ℂfld) ∧ ℝfld ∈ DivRing) | |
11 | 10 | simpli 484 | . . . . 5 ⊢ ℝ ∈ (SubRing‘ℂfld) |
12 | subrgchr 32381 | . . . . 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 2769 | . . 3 ⊢ (chr‘ℂfld) = 0 |
15 | 4, 14 | pm3.2i 471 | . 2 ⊢ ((ℤMod‘ℂfld) ∈ NrmMod ∧ (chr‘ℂfld) = 0) |
16 | cnfldcusp 24873 | . . 3 ⊢ ℂfld ∈ CUnifSp | |
17 | eqid 2732 | . . . 4 ⊢ (UnifSt‘ℂfld) = (UnifSt‘ℂfld) | |
18 | 17 | cnflduss 24872 | . . 3 ⊢ (UnifSt‘ℂfld) = (metUnif‘(abs ∘ − )) |
19 | 16, 18 | pm3.2i 471 | . 2 ⊢ (ℂfld ∈ CUnifSp ∧ (UnifSt‘ℂfld) = (metUnif‘(abs ∘ − ))) |
20 | cnfldbas 20947 | . . 3 ⊢ ℂ = (Base‘ℂfld) | |
21 | cnmet 24287 | . . . . . 6 ⊢ (abs ∘ − ) ∈ (Met‘ℂ) | |
22 | metf 23835 | . . . . . 6 ⊢ ((abs ∘ − ) ∈ (Met‘ℂ) → (abs ∘ − ):(ℂ × ℂ)⟶ℝ) | |
23 | ffn 6717 | . . . . . 6 ⊢ ((abs ∘ − ):(ℂ × ℂ)⟶ℝ → (abs ∘ − ) Fn (ℂ × ℂ)) | |
24 | 21, 22, 23 | mp2b 10 | . . . . 5 ⊢ (abs ∘ − ) Fn (ℂ × ℂ) |
25 | fnresdm 6669 | . . . . 5 ⊢ ((abs ∘ − ) Fn (ℂ × ℂ) → ((abs ∘ − ) ↾ (ℂ × ℂ)) = (abs ∘ − )) | |
26 | 24, 25 | ax-mp 5 | . . . 4 ⊢ ((abs ∘ − ) ↾ (ℂ × ℂ)) = (abs ∘ − ) |
27 | cnfldds 20953 | . . . . 5 ⊢ (abs ∘ − ) = (dist‘ℂfld) | |
28 | 27 | reseq1i 5977 | . . . 4 ⊢ ((abs ∘ − ) ↾ (ℂ × ℂ)) = ((dist‘ℂfld) ↾ (ℂ × ℂ)) |
29 | 26, 28 | eqtr3i 2762 | . . 3 ⊢ (abs ∘ − ) = ((dist‘ℂfld) ↾ (ℂ × ℂ)) |
30 | eqid 2732 | . . 3 ⊢ (ℤMod‘ℂfld) = (ℤMod‘ℂfld) | |
31 | 20, 29, 30 | isrrext 32975 | . 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 1341 | 1 ⊢ ℂfld ∈ ℝExt |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1541 ∈ wcel 2106 × cxp 5674 ↾ cres 5678 ∘ ccom 5680 Fn wfn 6538 ⟶wf 6539 ‘cfv 6543 (class class class)co 7408 ℂcc 11107 ℝcr 11108 0cc0 11109 − cmin 11443 abscabs 15180 ↾s cress 17172 distcds 17205 SubRingcsubrg 20314 DivRingcdr 20356 Metcmet 20929 metUnifcmetu 20934 ℂfldccnfld 20943 ℤModczlm 21049 chrcchr 21050 ℝfldcrefld 21156 UnifStcuss 23757 CUnifSpccusp 23801 NrmRingcnrg 24087 NrmModcnlm 24088 oFieldcofld 32409 ℝExt crrext 32969 |
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 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 ax-addf 11188 ax-mulf 11189 |
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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-of 7669 df-om 7855 df-1st 7974 df-2nd 7975 df-supp 8146 df-tpos 8210 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-2o 8466 df-er 8702 df-map 8821 df-ixp 8891 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-fsupp 9361 df-fi 9405 df-sup 9436 df-inf 9437 df-oi 9504 df-card 9933 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-z 12558 df-dec 12677 df-uz 12822 df-q 12932 df-rp 12974 df-xneg 13091 df-xadd 13092 df-xmul 13093 df-ioo 13327 df-ico 13329 df-icc 13330 df-fz 13484 df-fzo 13627 df-seq 13966 df-exp 14027 df-hash 14290 df-cj 15045 df-re 15046 df-im 15047 df-sqrt 15181 df-abs 15182 df-struct 17079 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-ress 17173 df-plusg 17209 df-mulr 17210 df-starv 17211 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-hom 17220 df-cco 17221 df-rest 17367 df-topn 17368 df-0g 17386 df-gsum 17387 df-topgen 17388 df-pt 17389 df-prds 17392 df-xrs 17447 df-qtop 17452 df-imas 17453 df-xps 17455 df-mre 17529 df-mrc 17530 df-acs 17532 df-proset 18247 df-poset 18265 df-plt 18282 df-toset 18369 df-ps 18518 df-tsr 18519 df-mgm 18560 df-sgrp 18609 df-mnd 18625 df-submnd 18671 df-grp 18821 df-minusg 18822 df-sbg 18823 df-mulg 18950 df-subg 19002 df-cntz 19180 df-od 19395 df-cmn 19649 df-abl 19650 df-mgp 19987 df-ur 20004 df-ring 20057 df-cring 20058 df-oppr 20149 df-dvdsr 20170 df-unit 20171 df-invr 20201 df-dvr 20214 df-subrg 20316 df-drng 20358 df-field 20359 df-abv 20424 df-lmod 20472 df-psmet 20935 df-xmet 20936 df-met 20937 df-bl 20938 df-mopn 20939 df-fbas 20940 df-fg 20941 df-metu 20942 df-cnfld 20944 df-zring 21017 df-zlm 21053 df-chr 21054 df-refld 21157 df-top 22395 df-topon 22412 df-topsp 22434 df-bases 22448 df-cld 22522 df-ntr 22523 df-cls 22524 df-nei 22601 df-cn 22730 df-cnp 22731 df-haus 22818 df-cmp 22890 df-tx 23065 df-hmeo 23258 df-fil 23349 df-flim 23442 df-fcls 23444 df-ust 23704 df-utop 23735 df-uss 23760 df-usp 23761 df-cfilu 23791 df-cusp 23802 df-xms 23825 df-ms 23826 df-tms 23827 df-nm 24090 df-ngp 24091 df-nrg 24093 df-nlm 24094 df-cncf 24393 df-cfil 24771 df-cmet 24773 df-cms 24851 df-omnd 32212 df-ogrp 32213 df-orng 32410 df-ofld 32411 df-rrext 32974 |
This theorem is referenced by: sitgclcn 33338 |
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