nrgtrg - Metamath Proof Explorer

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Theorem nrgtrg 22541

Description: A normed ring is a topological ring. (Contributed by Mario Carneiro, 4-Oct-2015.)

**Assertion**
Ref	Expression
nrgtrg	⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ TopRing)

**Proof of Theorem nrgtrg**
Step	Hyp	Ref	Expression
1		nrgtgp 22523	. 2 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ TopGrp)
2		nrgring 22514	. 2 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ Ring)
3		eqid 2651	. . . . 5 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
4	3	ringmgp 18599	. . . 4 ⊢ (𝑅 ∈ Ring → (mulGrp‘𝑅) ∈ Mnd)
5	2, 4	syl 17	. . 3 ⊢ (𝑅 ∈ NrmRing → (mulGrp‘𝑅) ∈ Mnd)
6		tgptps 21931	. . . . . 6 ⊢ (𝑅 ∈ TopGrp → 𝑅 ∈ TopSp)
7	1, 6	syl 17	. . . . 5 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ TopSp)
8		eqid 2651	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
9		eqid 2651	. . . . . 6 ⊢ (TopOpen‘𝑅) = (TopOpen‘𝑅)
10	8, 9	istps 20786	. . . . 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 18541	. . . . 5 ⊢ (Base‘𝑅) = (Base‘(mulGrp‘𝑅))
13	3, 9	mgptopn 18544	. . . . 5 ⊢ (TopOpen‘𝑅) = (TopOpen‘(mulGrp‘𝑅))
14	12, 13	istps 20786	. . . 4 ⊢ ((mulGrp‘𝑅) ∈ TopSp ↔ (TopOpen‘𝑅) ∈ (TopOn‘(Base‘𝑅)))
15	11, 14	sylibr 224	. . 3 ⊢ (𝑅 ∈ NrmRing → (mulGrp‘𝑅) ∈ TopSp)
16		rlmnlm 22539	. . . 4 ⊢ (𝑅 ∈ NrmRing → (ringLMod‘𝑅) ∈ NrmMod)
17		rlmsca2 19249	. . . . 5 ⊢ ( I ‘𝑅) = (Scalar‘(ringLMod‘𝑅))
18		rlmscaf 19256	. . . . 5 ⊢ (+_𝑓‘(mulGrp‘𝑅)) = ( ·_sf ‘(ringLMod‘𝑅))
19		rlmtopn 19251	. . . . 5 ⊢ (TopOpen‘𝑅) = (TopOpen‘(ringLMod‘𝑅))
20		df-base 15910	. . . . . . . . 9 ⊢ Base = Slot 1
21	20, 8	strfvi 15960	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘( I ‘𝑅))
22	21	a1i 11	. . . . . . 7 ⊢ (⊤ → (Base‘𝑅) = (Base‘( I ‘𝑅)))
23		df-tset 16007	. . . . . . . . 9 ⊢ TopSet = Slot 9
24		eqid 2651	. . . . . . . . 9 ⊢ (TopSet‘𝑅) = (TopSet‘𝑅)
25	23, 24	strfvi 15960	. . . . . . . 8 ⊢ (TopSet‘𝑅) = (TopSet‘( I ‘𝑅))
26	25	a1i 11	. . . . . . 7 ⊢ (⊤ → (TopSet‘𝑅) = (TopSet‘( I ‘𝑅)))
27	22, 26	topnpropd 16144	. . . . . 6 ⊢ (⊤ → (TopOpen‘𝑅) = (TopOpen‘( I ‘𝑅)))
28	27	trud 1533	. . . . 5 ⊢ (TopOpen‘𝑅) = (TopOpen‘( I ‘𝑅))
29	17, 18, 19, 28	nlmvscn 22538	. . . 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 2651	. . . 4 ⊢ (+_𝑓‘(mulGrp‘𝑅)) = (+_𝑓‘(mulGrp‘𝑅))
32	31, 13	istmd 21925	. . 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 1265	. 2 ⊢ (𝑅 ∈ NrmRing → (mulGrp‘𝑅) ∈ TopMnd)
34	3	istrg 22014	. 2 ⊢ (𝑅 ∈ TopRing ↔ (𝑅 ∈ TopGrp ∧ 𝑅 ∈ Ring ∧ (mulGrp‘𝑅) ∈ TopMnd))
35	1, 2, 33, 34	syl3anbrc 1265	1 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ TopRing)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1523 ⊤wtru 1524 ∈ wcel 2030 I cid 5052 ‘cfv 5926 (class class class)co 6690 1c1 9975 9c9 11115 Basecbs 15904 TopSetcts 15994 TopOpenctopn 16129 +_𝑓cplusf 17286 Mndcmnd 17341 mulGrpcmgp 18535 Ringcrg 18593 ringLModcrglmod 19217 TopOnctopon 20763 TopSpctps 20784 Cn ccn 21076 ×_t ctx 21411 TopMndctmd 21921 TopGrpctgp 21922 TopRingctrg 22006 NrmRingcnrg 22431 NrmModcnlm 22432

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-inf2 8576 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 ax-pre-sup 10052

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-iin 4555 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-se 5103 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-isom 5935 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-of 6939 df-om 7108 df-1st 7210 df-2nd 7211 df-supp 7341 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-2o 7606 df-oadd 7609 df-er 7787 df-map 7901 df-ixp 7951 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-fsupp 8317 df-fi 8358 df-sup 8389 df-inf 8390 df-oi 8456 df-card 8803 df-cda 9028 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-div 10723 df-nn 11059 df-2 11117 df-3 11118 df-4 11119 df-5 11120 df-6 11121 df-7 11122 df-8 11123 df-9 11124 df-n0 11331 df-z 11416 df-dec 11532 df-uz 11726 df-q 11827 df-rp 11871 df-xneg 11984 df-xadd 11985 df-xmul 11986 df-ico 12219 df-icc 12220 df-fz 12365 df-fzo 12505 df-seq 12842 df-exp 12901 df-hash 13158 df-cj 13883 df-re 13884 df-im 13885 df-sqrt 14019 df-abs 14020 df-struct 15906 df-ndx 15907 df-slot 15908 df-base 15910 df-sets 15911 df-ress 15912 df-plusg 16001 df-mulr 16002 df-sca 16004 df-vsca 16005 df-ip 16006 df-tset 16007 df-ple 16008 df-ds 16011 df-hom 16013 df-cco 16014 df-rest 16130 df-topn 16131 df-0g 16149 df-gsum 16150 df-topgen 16151 df-pt 16152 df-prds 16155 df-xrs 16209 df-qtop 16214 df-imas 16215 df-xps 16217 df-mre 16293 df-mrc 16294 df-acs 16296 df-plusf 17288 df-mgm 17289 df-sgrp 17331 df-mnd 17342 df-submnd 17383 df-grp 17472 df-minusg 17473 df-sbg 17474 df-mulg 17588 df-subg 17638 df-cntz 17796 df-cmn 18241 df-abl 18242 df-mgp 18536 df-ur 18548 df-ring 18595 df-subrg 18826 df-abv 18865 df-lmod 18913 df-scaf 18914 df-sra 19220 df-rgmod 19221 df-psmet 19786 df-xmet 19787 df-met 19788 df-bl 19789 df-mopn 19790 df-top 20747 df-topon 20764 df-topsp 20785 df-bases 20798 df-cn 21079 df-cnp 21080 df-tx 21413 df-hmeo 21606 df-tmd 21923 df-tgp 21924 df-trg 22010 df-xms 22172 df-ms 22173 df-tms 22174 df-nm 22434 df-ngp 22435 df-nrg 22437 df-nlm 22438

This theorem is referenced by: nrgtdrg 22544 nlmtlm 22545 iistmd 30076 qqhcn 30163

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