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Mirrors > Home > MPE Home > Th. List > nrgtrg | Structured version Visualization version GIF version |
Description: A normed ring is a topological ring. (Contributed by Mario Carneiro, 4-Oct-2015.) |
Ref | Expression |
---|---|
nrgtrg | ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ TopRing) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nrgtgp 22984 | . 2 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ TopGrp) | |
2 | nrgring 22975 | . 2 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ Ring) | |
3 | eqid 2778 | . . . . 5 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
4 | 3 | ringmgp 19026 | . . . 4 ⊢ (𝑅 ∈ Ring → (mulGrp‘𝑅) ∈ Mnd) |
5 | 2, 4 | syl 17 | . . 3 ⊢ (𝑅 ∈ NrmRing → (mulGrp‘𝑅) ∈ Mnd) |
6 | tgptps 22392 | . . . . . 6 ⊢ (𝑅 ∈ TopGrp → 𝑅 ∈ TopSp) | |
7 | 1, 6 | syl 17 | . . . . 5 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ TopSp) |
8 | eqid 2778 | . . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
9 | eqid 2778 | . . . . . 6 ⊢ (TopOpen‘𝑅) = (TopOpen‘𝑅) | |
10 | 8, 9 | istps 21246 | . . . . 5 ⊢ (𝑅 ∈ TopSp ↔ (TopOpen‘𝑅) ∈ (TopOn‘(Base‘𝑅))) |
11 | 7, 10 | sylib 210 | . . . 4 ⊢ (𝑅 ∈ NrmRing → (TopOpen‘𝑅) ∈ (TopOn‘(Base‘𝑅))) |
12 | 3, 8 | mgpbas 18968 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘(mulGrp‘𝑅)) |
13 | 3, 9 | mgptopn 18971 | . . . . 5 ⊢ (TopOpen‘𝑅) = (TopOpen‘(mulGrp‘𝑅)) |
14 | 12, 13 | istps 21246 | . . . 4 ⊢ ((mulGrp‘𝑅) ∈ TopSp ↔ (TopOpen‘𝑅) ∈ (TopOn‘(Base‘𝑅))) |
15 | 11, 14 | sylibr 226 | . . 3 ⊢ (𝑅 ∈ NrmRing → (mulGrp‘𝑅) ∈ TopSp) |
16 | rlmnlm 23000 | . . . 4 ⊢ (𝑅 ∈ NrmRing → (ringLMod‘𝑅) ∈ NrmMod) | |
17 | rlmsca2 19695 | . . . . 5 ⊢ ( I ‘𝑅) = (Scalar‘(ringLMod‘𝑅)) | |
18 | rlmscaf 19702 | . . . . 5 ⊢ (+𝑓‘(mulGrp‘𝑅)) = ( ·sf ‘(ringLMod‘𝑅)) | |
19 | rlmtopn 19697 | . . . . 5 ⊢ (TopOpen‘𝑅) = (TopOpen‘(ringLMod‘𝑅)) | |
20 | df-base 16345 | . . . . . . . . 9 ⊢ Base = Slot 1 | |
21 | 20, 8 | strfvi 16393 | . . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘( I ‘𝑅)) |
22 | 21 | a1i 11 | . . . . . . 7 ⊢ (⊤ → (Base‘𝑅) = (Base‘( I ‘𝑅))) |
23 | df-tset 16440 | . . . . . . . . 9 ⊢ TopSet = Slot 9 | |
24 | eqid 2778 | . . . . . . . . 9 ⊢ (TopSet‘𝑅) = (TopSet‘𝑅) | |
25 | 23, 24 | strfvi 16393 | . . . . . . . 8 ⊢ (TopSet‘𝑅) = (TopSet‘( I ‘𝑅)) |
26 | 25 | a1i 11 | . . . . . . 7 ⊢ (⊤ → (TopSet‘𝑅) = (TopSet‘( I ‘𝑅))) |
27 | 22, 26 | topnpropd 16566 | . . . . . 6 ⊢ (⊤ → (TopOpen‘𝑅) = (TopOpen‘( I ‘𝑅))) |
28 | 27 | mptru 1514 | . . . . 5 ⊢ (TopOpen‘𝑅) = (TopOpen‘( I ‘𝑅)) |
29 | 17, 18, 19, 28 | nlmvscn 22999 | . . . 4 ⊢ ((ringLMod‘𝑅) ∈ NrmMod → (+𝑓‘(mulGrp‘𝑅)) ∈ (((TopOpen‘𝑅) ×t (TopOpen‘𝑅)) Cn (TopOpen‘𝑅))) |
30 | 16, 29 | syl 17 | . . 3 ⊢ (𝑅 ∈ NrmRing → (+𝑓‘(mulGrp‘𝑅)) ∈ (((TopOpen‘𝑅) ×t (TopOpen‘𝑅)) Cn (TopOpen‘𝑅))) |
31 | eqid 2778 | . . . 4 ⊢ (+𝑓‘(mulGrp‘𝑅)) = (+𝑓‘(mulGrp‘𝑅)) | |
32 | 31, 13 | istmd 22386 | . . 3 ⊢ ((mulGrp‘𝑅) ∈ TopMnd ↔ ((mulGrp‘𝑅) ∈ Mnd ∧ (mulGrp‘𝑅) ∈ TopSp ∧ (+𝑓‘(mulGrp‘𝑅)) ∈ (((TopOpen‘𝑅) ×t (TopOpen‘𝑅)) Cn (TopOpen‘𝑅)))) |
33 | 5, 15, 30, 32 | syl3anbrc 1323 | . 2 ⊢ (𝑅 ∈ NrmRing → (mulGrp‘𝑅) ∈ TopMnd) |
34 | 3 | istrg 22475 | . 2 ⊢ (𝑅 ∈ TopRing ↔ (𝑅 ∈ TopGrp ∧ 𝑅 ∈ Ring ∧ (mulGrp‘𝑅) ∈ TopMnd)) |
35 | 1, 2, 33, 34 | syl3anbrc 1323 | 1 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ TopRing) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1507 ⊤wtru 1508 ∈ wcel 2050 I cid 5311 ‘cfv 6188 (class class class)co 6976 1c1 10336 9c9 11502 Basecbs 16339 TopSetcts 16427 TopOpenctopn 16551 +𝑓cplusf 17707 Mndcmnd 17762 mulGrpcmgp 18962 Ringcrg 19020 ringLModcrglmod 19663 TopOnctopon 21222 TopSpctps 21244 Cn ccn 21536 ×t ctx 21872 TopMndctmd 22382 TopGrpctgp 22383 TopRingctrg 22467 NrmRingcnrg 22892 NrmModcnlm 22893 |
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 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-rep 5049 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-cnex 10391 ax-resscn 10392 ax-1cn 10393 ax-icn 10394 ax-addcl 10395 ax-addrcl 10396 ax-mulcl 10397 ax-mulrcl 10398 ax-mulcom 10399 ax-addass 10400 ax-mulass 10401 ax-distr 10402 ax-i2m1 10403 ax-1ne0 10404 ax-1rid 10405 ax-rnegex 10406 ax-rrecex 10407 ax-cnre 10408 ax-pre-lttri 10409 ax-pre-lttrn 10410 ax-pre-ltadd 10411 ax-pre-mulgt0 10412 ax-pre-sup 10413 |
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 2016 df-mo 2547 df-eu 2584 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3417 df-sbc 3682 df-csb 3787 df-dif 3832 df-un 3834 df-in 3836 df-ss 3843 df-pss 3845 df-nul 4179 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-uni 4713 df-int 4750 df-iun 4794 df-iin 4795 df-br 4930 df-opab 4992 df-mpt 5009 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-se 5367 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-pred 5986 df-ord 6032 df-on 6033 df-lim 6034 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-isom 6197 df-riota 6937 df-ov 6979 df-oprab 6980 df-mpo 6981 df-of 7227 df-om 7397 df-1st 7501 df-2nd 7502 df-supp 7634 df-wrecs 7750 df-recs 7812 df-rdg 7850 df-1o 7905 df-2o 7906 df-oadd 7909 df-er 8089 df-map 8208 df-ixp 8260 df-en 8307 df-dom 8308 df-sdom 8309 df-fin 8310 df-fsupp 8629 df-fi 8670 df-sup 8701 df-inf 8702 df-oi 8769 df-card 9162 df-cda 9388 df-pnf 10476 df-mnf 10477 df-xr 10478 df-ltxr 10479 df-le 10480 df-sub 10672 df-neg 10673 df-div 11099 df-nn 11440 df-2 11503 df-3 11504 df-4 11505 df-5 11506 df-6 11507 df-7 11508 df-8 11509 df-9 11510 df-n0 11708 df-z 11794 df-dec 11912 df-uz 12059 df-q 12163 df-rp 12205 df-xneg 12324 df-xadd 12325 df-xmul 12326 df-ico 12560 df-icc 12561 df-fz 12709 df-fzo 12850 df-seq 13185 df-exp 13245 df-hash 13506 df-cj 14319 df-re 14320 df-im 14321 df-sqrt 14455 df-abs 14456 df-struct 16341 df-ndx 16342 df-slot 16343 df-base 16345 df-sets 16346 df-ress 16347 df-plusg 16434 df-mulr 16435 df-sca 16437 df-vsca 16438 df-ip 16439 df-tset 16440 df-ple 16441 df-ds 16443 df-hom 16445 df-cco 16446 df-rest 16552 df-topn 16553 df-0g 16571 df-gsum 16572 df-topgen 16573 df-pt 16574 df-prds 16577 df-xrs 16631 df-qtop 16636 df-imas 16637 df-xps 16639 df-mre 16715 df-mrc 16716 df-acs 16718 df-plusf 17709 df-mgm 17710 df-sgrp 17752 df-mnd 17763 df-submnd 17804 df-grp 17894 df-minusg 17895 df-sbg 17896 df-mulg 18012 df-subg 18060 df-cntz 18218 df-cmn 18668 df-abl 18669 df-mgp 18963 df-ur 18975 df-ring 19022 df-subrg 19256 df-abv 19310 df-lmod 19358 df-scaf 19359 df-sra 19666 df-rgmod 19667 df-psmet 20239 df-xmet 20240 df-met 20241 df-bl 20242 df-mopn 20243 df-top 21206 df-topon 21223 df-topsp 21245 df-bases 21258 df-cn 21539 df-cnp 21540 df-tx 21874 df-hmeo 22067 df-tmd 22384 df-tgp 22385 df-trg 22471 df-xms 22633 df-ms 22634 df-tms 22635 df-nm 22895 df-ngp 22896 df-nrg 22898 df-nlm 22899 |
This theorem is referenced by: nrgtdrg 23005 nlmtlm 23006 iistmd 30795 qqhcn 30882 |
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