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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 22476 | . 2 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ TopGrp) | |
2 | nrgring 22467 | . 2 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ Ring) | |
3 | eqid 2622 | . . . . 5 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
4 | 3 | ringmgp 18553 | . . . 4 ⊢ (𝑅 ∈ Ring → (mulGrp‘𝑅) ∈ Mnd) |
5 | 2, 4 | syl 17 | . . 3 ⊢ (𝑅 ∈ NrmRing → (mulGrp‘𝑅) ∈ Mnd) |
6 | tgptps 21884 | . . . . . 6 ⊢ (𝑅 ∈ TopGrp → 𝑅 ∈ TopSp) | |
7 | 1, 6 | syl 17 | . . . . 5 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ TopSp) |
8 | eqid 2622 | . . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
9 | eqid 2622 | . . . . . 6 ⊢ (TopOpen‘𝑅) = (TopOpen‘𝑅) | |
10 | 8, 9 | istps 20738 | . . . . 5 ⊢ (𝑅 ∈ TopSp ↔ (TopOpen‘𝑅) ∈ (TopOn‘(Base‘𝑅))) |
11 | 7, 10 | sylib 208 | . . . 4 ⊢ (𝑅 ∈ NrmRing → (TopOpen‘𝑅) ∈ (TopOn‘(Base‘𝑅))) |
12 | 3, 8 | mgpbas 18495 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘(mulGrp‘𝑅)) |
13 | 3, 9 | mgptopn 18498 | . . . . 5 ⊢ (TopOpen‘𝑅) = (TopOpen‘(mulGrp‘𝑅)) |
14 | 12, 13 | istps 20738 | . . . 4 ⊢ ((mulGrp‘𝑅) ∈ TopSp ↔ (TopOpen‘𝑅) ∈ (TopOn‘(Base‘𝑅))) |
15 | 11, 14 | sylibr 224 | . . 3 ⊢ (𝑅 ∈ NrmRing → (mulGrp‘𝑅) ∈ TopSp) |
16 | rlmnlm 22492 | . . . 4 ⊢ (𝑅 ∈ NrmRing → (ringLMod‘𝑅) ∈ NrmMod) | |
17 | rlmsca2 19201 | . . . . 5 ⊢ ( I ‘𝑅) = (Scalar‘(ringLMod‘𝑅)) | |
18 | rlmscaf 19208 | . . . . 5 ⊢ (+𝑓‘(mulGrp‘𝑅)) = ( ·sf ‘(ringLMod‘𝑅)) | |
19 | rlmtopn 19203 | . . . . 5 ⊢ (TopOpen‘𝑅) = (TopOpen‘(ringLMod‘𝑅)) | |
20 | df-base 15863 | . . . . . . . . 9 ⊢ Base = Slot 1 | |
21 | 20, 8 | strfvi 15913 | . . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘( I ‘𝑅)) |
22 | 21 | a1i 11 | . . . . . . 7 ⊢ (⊤ → (Base‘𝑅) = (Base‘( I ‘𝑅))) |
23 | df-tset 15960 | . . . . . . . . 9 ⊢ TopSet = Slot 9 | |
24 | eqid 2622 | . . . . . . . . 9 ⊢ (TopSet‘𝑅) = (TopSet‘𝑅) | |
25 | 23, 24 | strfvi 15913 | . . . . . . . 8 ⊢ (TopSet‘𝑅) = (TopSet‘( I ‘𝑅)) |
26 | 25 | a1i 11 | . . . . . . 7 ⊢ (⊤ → (TopSet‘𝑅) = (TopSet‘( I ‘𝑅))) |
27 | 22, 26 | topnpropd 16097 | . . . . . 6 ⊢ (⊤ → (TopOpen‘𝑅) = (TopOpen‘( I ‘𝑅))) |
28 | 27 | trud 1493 | . . . . 5 ⊢ (TopOpen‘𝑅) = (TopOpen‘( I ‘𝑅)) |
29 | 17, 18, 19, 28 | nlmvscn 22491 | . . . 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 2622 | . . . 4 ⊢ (+𝑓‘(mulGrp‘𝑅)) = (+𝑓‘(mulGrp‘𝑅)) | |
32 | 31, 13 | istmd 21878 | . . 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 1246 | . 2 ⊢ (𝑅 ∈ NrmRing → (mulGrp‘𝑅) ∈ TopMnd) |
34 | 3 | istrg 21967 | . 2 ⊢ (𝑅 ∈ TopRing ↔ (𝑅 ∈ TopGrp ∧ 𝑅 ∈ Ring ∧ (mulGrp‘𝑅) ∈ TopMnd)) |
35 | 1, 2, 33, 34 | syl3anbrc 1246 | 1 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ TopRing) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ⊤wtru 1484 ∈ wcel 1990 I cid 5023 ‘cfv 5888 (class class class)co 6650 1c1 9937 9c9 11077 Basecbs 15857 TopSetcts 15947 TopOpenctopn 16082 +𝑓cplusf 17239 Mndcmnd 17294 mulGrpcmgp 18489 Ringcrg 18547 ringLModcrglmod 19169 TopOnctopon 20715 TopSpctps 20736 Cn ccn 21028 ×t ctx 21363 TopMndctmd 21874 TopGrpctgp 21875 TopRingctrg 21959 NrmRingcnrg 22384 NrmModcnlm 22385 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-pt 16105 df-prds 16108 df-xrs 16162 df-qtop 16167 df-imas 16168 df-xps 16170 df-mre 16246 df-mrc 16247 df-acs 16249 df-plusf 17241 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-grp 17425 df-minusg 17426 df-sbg 17427 df-mulg 17541 df-subg 17591 df-cntz 17750 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-subrg 18778 df-abv 18817 df-lmod 18865 df-scaf 18866 df-sra 19172 df-rgmod 19173 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cn 21031 df-cnp 21032 df-tx 21365 df-hmeo 21558 df-tmd 21876 df-tgp 21877 df-trg 21963 df-xms 22125 df-ms 22126 df-tms 22127 df-nm 22387 df-ngp 22388 df-nrg 22390 df-nlm 22391 |
This theorem is referenced by: nrgtdrg 22497 nlmtlm 22498 iistmd 29948 qqhcn 30035 |
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