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Mirrors > Home > MPE Home > Th. List > nrgtdrg | Structured version Visualization version GIF version |
Description: A normed division ring is a topological division ring. (Contributed by Mario Carneiro, 6-Oct-2015.) |
Ref | Expression |
---|---|
nrgtdrg | ⊢ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing) → 𝑅 ∈ TopDRing) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nrgtrg 22992 | . . 3 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ TopRing) | |
2 | 1 | adantr 473 | . 2 ⊢ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing) → 𝑅 ∈ TopRing) |
3 | simpr 477 | . 2 ⊢ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing) → 𝑅 ∈ DivRing) | |
4 | nrgring 22965 | . . . . 5 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ Ring) | |
5 | 4 | adantr 473 | . . . 4 ⊢ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing) → 𝑅 ∈ Ring) |
6 | eqid 2772 | . . . . 5 ⊢ (Unit‘𝑅) = (Unit‘𝑅) | |
7 | eqid 2772 | . . . . 5 ⊢ ((mulGrp‘𝑅) ↾s (Unit‘𝑅)) = ((mulGrp‘𝑅) ↾s (Unit‘𝑅)) | |
8 | 6, 7 | unitgrp 19130 | . . . 4 ⊢ (𝑅 ∈ Ring → ((mulGrp‘𝑅) ↾s (Unit‘𝑅)) ∈ Grp) |
9 | 5, 8 | syl 17 | . . 3 ⊢ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing) → ((mulGrp‘𝑅) ↾s (Unit‘𝑅)) ∈ Grp) |
10 | eqid 2772 | . . . . . 6 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
11 | 10 | trgtmd 22466 | . . . . 5 ⊢ (𝑅 ∈ TopRing → (mulGrp‘𝑅) ∈ TopMnd) |
12 | 2, 11 | syl 17 | . . . 4 ⊢ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing) → (mulGrp‘𝑅) ∈ TopMnd) |
13 | 6, 10 | unitsubm 19133 | . . . . 5 ⊢ (𝑅 ∈ Ring → (Unit‘𝑅) ∈ (SubMnd‘(mulGrp‘𝑅))) |
14 | 5, 13 | syl 17 | . . . 4 ⊢ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing) → (Unit‘𝑅) ∈ (SubMnd‘(mulGrp‘𝑅))) |
15 | 7 | submtmd 22406 | . . . 4 ⊢ (((mulGrp‘𝑅) ∈ TopMnd ∧ (Unit‘𝑅) ∈ (SubMnd‘(mulGrp‘𝑅))) → ((mulGrp‘𝑅) ↾s (Unit‘𝑅)) ∈ TopMnd) |
16 | 12, 14, 15 | syl2anc 576 | . . 3 ⊢ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing) → ((mulGrp‘𝑅) ↾s (Unit‘𝑅)) ∈ TopMnd) |
17 | eqid 2772 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
18 | eqid 2772 | . . . . 5 ⊢ (invr‘𝑅) = (invr‘𝑅) | |
19 | eqid 2772 | . . . . 5 ⊢ (TopOpen‘𝑅) = (TopOpen‘𝑅) | |
20 | 17, 6, 18, 19 | nrginvrcn 22994 | . . . 4 ⊢ (𝑅 ∈ NrmRing → (invr‘𝑅) ∈ (((TopOpen‘𝑅) ↾t (Unit‘𝑅)) Cn ((TopOpen‘𝑅) ↾t (Unit‘𝑅)))) |
21 | 20 | adantr 473 | . . 3 ⊢ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing) → (invr‘𝑅) ∈ (((TopOpen‘𝑅) ↾t (Unit‘𝑅)) Cn ((TopOpen‘𝑅) ↾t (Unit‘𝑅)))) |
22 | 10, 19 | mgptopn 18961 | . . . . 5 ⊢ (TopOpen‘𝑅) = (TopOpen‘(mulGrp‘𝑅)) |
23 | 7, 22 | resstopn 21488 | . . . 4 ⊢ ((TopOpen‘𝑅) ↾t (Unit‘𝑅)) = (TopOpen‘((mulGrp‘𝑅) ↾s (Unit‘𝑅))) |
24 | 6, 7, 18 | invrfval 19136 | . . . 4 ⊢ (invr‘𝑅) = (invg‘((mulGrp‘𝑅) ↾s (Unit‘𝑅))) |
25 | 23, 24 | istgp 22379 | . . 3 ⊢ (((mulGrp‘𝑅) ↾s (Unit‘𝑅)) ∈ TopGrp ↔ (((mulGrp‘𝑅) ↾s (Unit‘𝑅)) ∈ Grp ∧ ((mulGrp‘𝑅) ↾s (Unit‘𝑅)) ∈ TopMnd ∧ (invr‘𝑅) ∈ (((TopOpen‘𝑅) ↾t (Unit‘𝑅)) Cn ((TopOpen‘𝑅) ↾t (Unit‘𝑅))))) |
26 | 9, 16, 21, 25 | syl3anbrc 1323 | . 2 ⊢ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing) → ((mulGrp‘𝑅) ↾s (Unit‘𝑅)) ∈ TopGrp) |
27 | 10, 6 | istdrg 22467 | . 2 ⊢ (𝑅 ∈ TopDRing ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ ((mulGrp‘𝑅) ↾s (Unit‘𝑅)) ∈ TopGrp)) |
28 | 2, 3, 26, 27 | syl3anbrc 1323 | 1 ⊢ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing) → 𝑅 ∈ TopDRing) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 ∈ wcel 2048 ‘cfv 6182 (class class class)co 6970 Basecbs 16329 ↾s cress 16330 ↾t crest 16540 TopOpenctopn 16541 SubMndcsubmnd 17792 Grpcgrp 17881 mulGrpcmgp 18952 Ringcrg 19010 Unitcui 19102 invrcinvr 19134 DivRingcdr 19215 Cn ccn 21526 TopMndctmd 22372 TopGrpctgp 22373 TopRingctrg 22457 TopDRingctdrg 22458 NrmRingcnrg 22882 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10383 ax-resscn 10384 ax-1cn 10385 ax-icn 10386 ax-addcl 10387 ax-addrcl 10388 ax-mulcl 10389 ax-mulrcl 10390 ax-mulcom 10391 ax-addass 10392 ax-mulass 10393 ax-distr 10394 ax-i2m1 10395 ax-1ne0 10396 ax-1rid 10397 ax-rnegex 10398 ax-rrecex 10399 ax-cnre 10400 ax-pre-lttri 10401 ax-pre-lttrn 10402 ax-pre-ltadd 10403 ax-pre-mulgt0 10404 ax-pre-sup 10405 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-pss 3841 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-int 4744 df-iun 4788 df-iin 4789 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5305 df-eprel 5310 df-po 5319 df-so 5320 df-fr 5359 df-se 5360 df-we 5361 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-isom 6191 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-of 7221 df-om 7391 df-1st 7494 df-2nd 7495 df-supp 7627 df-tpos 7688 df-wrecs 7743 df-recs 7805 df-rdg 7843 df-1o 7897 df-2o 7898 df-oadd 7901 df-er 8081 df-map 8200 df-ixp 8252 df-en 8299 df-dom 8300 df-sdom 8301 df-fin 8302 df-fsupp 8621 df-fi 8662 df-sup 8693 df-inf 8694 df-oi 8761 df-card 9154 df-cda 9380 df-pnf 10468 df-mnf 10469 df-xr 10470 df-ltxr 10471 df-le 10472 df-sub 10664 df-neg 10665 df-div 11091 df-nn 11432 df-2 11496 df-3 11497 df-4 11498 df-5 11499 df-6 11500 df-7 11501 df-8 11502 df-9 11503 df-n0 11701 df-z 11787 df-dec 11905 df-uz 12052 df-q 12156 df-rp 12198 df-xneg 12317 df-xadd 12318 df-xmul 12319 df-ico 12553 df-icc 12554 df-fz 12702 df-fzo 12843 df-seq 13178 df-exp 13238 df-hash 13499 df-cj 14309 df-re 14310 df-im 14311 df-sqrt 14445 df-abs 14446 df-struct 16331 df-ndx 16332 df-slot 16333 df-base 16335 df-sets 16336 df-ress 16337 df-plusg 16424 df-mulr 16425 df-sca 16427 df-vsca 16428 df-ip 16429 df-tset 16430 df-ple 16431 df-ds 16433 df-hom 16435 df-cco 16436 df-rest 16542 df-topn 16543 df-0g 16561 df-gsum 16562 df-topgen 16563 df-pt 16564 df-prds 16567 df-xrs 16621 df-qtop 16626 df-imas 16627 df-xps 16629 df-mre 16705 df-mrc 16706 df-acs 16708 df-plusf 17699 df-mgm 17700 df-sgrp 17742 df-mnd 17753 df-submnd 17794 df-grp 17884 df-minusg 17885 df-sbg 17886 df-mulg 18002 df-subg 18050 df-cntz 18208 df-cmn 18658 df-abl 18659 df-mgp 18953 df-ur 18965 df-ring 19012 df-oppr 19086 df-dvdsr 19104 df-unit 19105 df-invr 19135 df-subrg 19246 df-abv 19300 df-lmod 19348 df-scaf 19349 df-sra 19656 df-rgmod 19657 df-nzr 19742 df-psmet 20229 df-xmet 20230 df-met 20231 df-bl 20232 df-mopn 20233 df-top 21196 df-topon 21213 df-topsp 21235 df-bases 21248 df-cn 21529 df-cnp 21530 df-tx 21864 df-hmeo 22057 df-tmd 22374 df-tgp 22375 df-trg 22461 df-tdrg 22462 df-xms 22623 df-ms 22624 df-tms 22625 df-nm 22885 df-ngp 22886 df-nrg 22888 df-nlm 22889 |
This theorem is referenced by: nvctvc 23002 |
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