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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 23524 | . 2 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ TopGrp) | |
2 | nrgring 23515 | . 2 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ Ring) | |
3 | eqid 2736 | . . . . 5 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
4 | 3 | ringmgp 19522 | . . . 4 ⊢ (𝑅 ∈ Ring → (mulGrp‘𝑅) ∈ Mnd) |
5 | 2, 4 | syl 17 | . . 3 ⊢ (𝑅 ∈ NrmRing → (mulGrp‘𝑅) ∈ Mnd) |
6 | tgptps 22931 | . . . . . 6 ⊢ (𝑅 ∈ TopGrp → 𝑅 ∈ TopSp) | |
7 | 1, 6 | syl 17 | . . . . 5 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ TopSp) |
8 | eqid 2736 | . . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
9 | eqid 2736 | . . . . . 6 ⊢ (TopOpen‘𝑅) = (TopOpen‘𝑅) | |
10 | 8, 9 | istps 21785 | . . . . 5 ⊢ (𝑅 ∈ TopSp ↔ (TopOpen‘𝑅) ∈ (TopOn‘(Base‘𝑅))) |
11 | 7, 10 | sylib 221 | . . . 4 ⊢ (𝑅 ∈ NrmRing → (TopOpen‘𝑅) ∈ (TopOn‘(Base‘𝑅))) |
12 | 3, 8 | mgpbas 19464 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘(mulGrp‘𝑅)) |
13 | 3, 9 | mgptopn 19467 | . . . . 5 ⊢ (TopOpen‘𝑅) = (TopOpen‘(mulGrp‘𝑅)) |
14 | 12, 13 | istps 21785 | . . . 4 ⊢ ((mulGrp‘𝑅) ∈ TopSp ↔ (TopOpen‘𝑅) ∈ (TopOn‘(Base‘𝑅))) |
15 | 11, 14 | sylibr 237 | . . 3 ⊢ (𝑅 ∈ NrmRing → (mulGrp‘𝑅) ∈ TopSp) |
16 | rlmnlm 23540 | . . . 4 ⊢ (𝑅 ∈ NrmRing → (ringLMod‘𝑅) ∈ NrmMod) | |
17 | rlmsca2 20192 | . . . . 5 ⊢ ( I ‘𝑅) = (Scalar‘(ringLMod‘𝑅)) | |
18 | rlmscaf 20200 | . . . . 5 ⊢ (+𝑓‘(mulGrp‘𝑅)) = ( ·sf ‘(ringLMod‘𝑅)) | |
19 | rlmtopn 20194 | . . . . 5 ⊢ (TopOpen‘𝑅) = (TopOpen‘(ringLMod‘𝑅)) | |
20 | df-base 16672 | . . . . . . . . 9 ⊢ Base = Slot 1 | |
21 | 20, 8 | strfvi 16718 | . . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘( I ‘𝑅)) |
22 | 21 | a1i 11 | . . . . . . 7 ⊢ (⊤ → (Base‘𝑅) = (Base‘( I ‘𝑅))) |
23 | df-tset 16768 | . . . . . . . . 9 ⊢ TopSet = Slot 9 | |
24 | eqid 2736 | . . . . . . . . 9 ⊢ (TopSet‘𝑅) = (TopSet‘𝑅) | |
25 | 23, 24 | strfvi 16718 | . . . . . . . 8 ⊢ (TopSet‘𝑅) = (TopSet‘( I ‘𝑅)) |
26 | 25 | a1i 11 | . . . . . . 7 ⊢ (⊤ → (TopSet‘𝑅) = (TopSet‘( I ‘𝑅))) |
27 | 22, 26 | topnpropd 16895 | . . . . . 6 ⊢ (⊤ → (TopOpen‘𝑅) = (TopOpen‘( I ‘𝑅))) |
28 | 27 | mptru 1550 | . . . . 5 ⊢ (TopOpen‘𝑅) = (TopOpen‘( I ‘𝑅)) |
29 | 17, 18, 19, 28 | nlmvscn 23539 | . . . 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 2736 | . . . 4 ⊢ (+𝑓‘(mulGrp‘𝑅)) = (+𝑓‘(mulGrp‘𝑅)) | |
32 | 31, 13 | istmd 22925 | . . 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 1345 | . 2 ⊢ (𝑅 ∈ NrmRing → (mulGrp‘𝑅) ∈ TopMnd) |
34 | 3 | istrg 23015 | . 2 ⊢ (𝑅 ∈ TopRing ↔ (𝑅 ∈ TopGrp ∧ 𝑅 ∈ Ring ∧ (mulGrp‘𝑅) ∈ TopMnd)) |
35 | 1, 2, 33, 34 | syl3anbrc 1345 | 1 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ TopRing) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ⊤wtru 1544 ∈ wcel 2112 I cid 5439 ‘cfv 6358 (class class class)co 7191 1c1 10695 9c9 11857 Basecbs 16666 TopSetcts 16755 TopOpenctopn 16880 +𝑓cplusf 18065 Mndcmnd 18127 mulGrpcmgp 19458 Ringcrg 19516 ringLModcrglmod 20160 TopOnctopon 21761 TopSpctps 21783 Cn ccn 22075 ×t ctx 22411 TopMndctmd 22921 TopGrpctgp 22922 TopRingctrg 23007 NrmRingcnrg 23431 NrmModcnlm 23432 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 ax-pre-sup 10772 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-iin 4893 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-se 5495 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-of 7447 df-om 7623 df-1st 7739 df-2nd 7740 df-supp 7882 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-2o 8181 df-er 8369 df-map 8488 df-ixp 8557 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-fsupp 8964 df-fi 9005 df-sup 9036 df-inf 9037 df-oi 9104 df-card 9520 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-div 11455 df-nn 11796 df-2 11858 df-3 11859 df-4 11860 df-5 11861 df-6 11862 df-7 11863 df-8 11864 df-9 11865 df-n0 12056 df-z 12142 df-dec 12259 df-uz 12404 df-q 12510 df-rp 12552 df-xneg 12669 df-xadd 12670 df-xmul 12671 df-ico 12906 df-icc 12907 df-fz 13061 df-fzo 13204 df-seq 13540 df-exp 13601 df-hash 13862 df-cj 14627 df-re 14628 df-im 14629 df-sqrt 14763 df-abs 14764 df-struct 16668 df-ndx 16669 df-slot 16670 df-base 16672 df-sets 16673 df-ress 16674 df-plusg 16762 df-mulr 16763 df-sca 16765 df-vsca 16766 df-ip 16767 df-tset 16768 df-ple 16769 df-ds 16771 df-hom 16773 df-cco 16774 df-rest 16881 df-topn 16882 df-0g 16900 df-gsum 16901 df-topgen 16902 df-pt 16903 df-prds 16906 df-xrs 16961 df-qtop 16966 df-imas 16967 df-xps 16969 df-mre 17043 df-mrc 17044 df-acs 17046 df-plusf 18067 df-mgm 18068 df-sgrp 18117 df-mnd 18128 df-submnd 18173 df-grp 18322 df-minusg 18323 df-sbg 18324 df-mulg 18443 df-subg 18494 df-cntz 18665 df-cmn 19126 df-abl 19127 df-mgp 19459 df-ur 19471 df-ring 19518 df-subrg 19752 df-abv 19807 df-lmod 19855 df-scaf 19856 df-sra 20163 df-rgmod 20164 df-psmet 20309 df-xmet 20310 df-met 20311 df-bl 20312 df-mopn 20313 df-top 21745 df-topon 21762 df-topsp 21784 df-bases 21797 df-cn 22078 df-cnp 22079 df-tx 22413 df-hmeo 22606 df-tmd 22923 df-tgp 22924 df-trg 23011 df-xms 23172 df-ms 23173 df-tms 23174 df-nm 23434 df-ngp 23435 df-nrg 23437 df-nlm 23438 |
This theorem is referenced by: nrgtdrg 23545 nlmtlm 23546 iistmd 31520 qqhcn 31607 |
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