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Theorem nrgtrg 24655

Description: A normed ring is a topological ring. (Contributed by Mario Carneiro, 4-Oct-2015.) (Proof shortened by AV, 31-Oct-2024.)

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

**Proof of Theorem nrgtrg**
Step	Hyp	Ref	Expression
1		nrgtgp 24637	. 2 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ TopGrp)
2		nrgring 24628	. 2 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ Ring)
3		eqid 2736	. . . . 5 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
4	3	ringmgp 20220	. . . 4 ⊢ (𝑅 ∈ Ring → (mulGrp‘𝑅) ∈ Mnd)
5	2, 4	syl 17	. . 3 ⊢ (𝑅 ∈ NrmRing → (mulGrp‘𝑅) ∈ Mnd)
6		tgptps 24045	. . . . . 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 22899	. . . . 5 ⊢ (𝑅 ∈ TopSp ↔ (TopOpen‘𝑅) ∈ (TopOn‘(Base‘𝑅)))
11	7, 10	sylib 218	. . . 4 ⊢ (𝑅 ∈ NrmRing → (TopOpen‘𝑅) ∈ (TopOn‘(Base‘𝑅)))
12	3, 8	mgpbas 20126	. . . . 5 ⊢ (Base‘𝑅) = (Base‘(mulGrp‘𝑅))
13	3, 9	mgptopn 20129	. . . . 5 ⊢ (TopOpen‘𝑅) = (TopOpen‘(mulGrp‘𝑅))
14	12, 13	istps 22899	. . . 4 ⊢ ((mulGrp‘𝑅) ∈ TopSp ↔ (TopOpen‘𝑅) ∈ (TopOn‘(Base‘𝑅)))
15	11, 14	sylibr 234	. . 3 ⊢ (𝑅 ∈ NrmRing → (mulGrp‘𝑅) ∈ TopSp)
16		rlmnlm 24653	. . . 4 ⊢ (𝑅 ∈ NrmRing → (ringLMod‘𝑅) ∈ NrmMod)
17		rlmsca2 21194	. . . . 5 ⊢ ( I ‘𝑅) = (Scalar‘(ringLMod‘𝑅))
18		rlmscaf 21202	. . . . 5 ⊢ (+_𝑓‘(mulGrp‘𝑅)) = ( ·_sf ‘(ringLMod‘𝑅))
19		rlmtopn 21196	. . . . 5 ⊢ (TopOpen‘𝑅) = (TopOpen‘(ringLMod‘𝑅))
20		baseid 17182	. . . . . . . . 9 ⊢ Base = Slot (Base‘ndx)
21	20, 8	strfvi 17160	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘( I ‘𝑅))
22	21	a1i 11	. . . . . . 7 ⊢ (⊤ → (Base‘𝑅) = (Base‘( I ‘𝑅)))
23		tsetid 17316	. . . . . . . . 9 ⊢ TopSet = Slot (TopSet‘ndx)
24		eqid 2736	. . . . . . . . 9 ⊢ (TopSet‘𝑅) = (TopSet‘𝑅)
25	23, 24	strfvi 17160	. . . . . . . 8 ⊢ (TopSet‘𝑅) = (TopSet‘( I ‘𝑅))
26	25	a1i 11	. . . . . . 7 ⊢ (⊤ → (TopSet‘𝑅) = (TopSet‘( I ‘𝑅)))
27	22, 26	topnpropd 17399	. . . . . 6 ⊢ (⊤ → (TopOpen‘𝑅) = (TopOpen‘( I ‘𝑅)))
28	27	mptru 1549	. . . . 5 ⊢ (TopOpen‘𝑅) = (TopOpen‘( I ‘𝑅))
29	17, 18, 19, 28	nlmvscn 24652	. . . 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 24039	. . 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 24129	. 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 1542 ⊤wtru 1543 ∈ wcel 2114 I cid 5525 ‘cfv 6498 (class class class)co 7367 ndxcnx 17163 Basecbs 17179 TopSetcts 17226 TopOpenctopn 17384 +_𝑓cplusf 18605 Mndcmnd 18702 mulGrpcmgp 20121 Ringcrg 20214 ringLModcrglmod 21167 TopOnctopon 22875 TopSpctps 22897 Cn ccn 23189 ×_t ctx 23525 TopMndctmd 24035 TopGrpctgp 24036 TopRingctrg 24121 NrmRingcnrg 24544 NrmModcnlm 24545

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-iin 4936 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-isom 6507 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-of 7631 df-om 7818 df-1st 7942 df-2nd 7943 df-supp 8111 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-er 8643 df-map 8775 df-ixp 8846 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-fsupp 9275 df-fi 9324 df-sup 9355 df-inf 9356 df-oi 9425 df-card 9863 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-z 12525 df-dec 12645 df-uz 12789 df-q 12899 df-rp 12943 df-xneg 13063 df-xadd 13064 df-xmul 13065 df-ico 13304 df-icc 13305 df-fz 13462 df-fzo 13609 df-seq 13964 df-exp 14024 df-hash 14293 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-hom 17244 df-cco 17245 df-rest 17385 df-topn 17386 df-0g 17404 df-gsum 17405 df-topgen 17406 df-pt 17407 df-prds 17410 df-xrs 17466 df-qtop 17471 df-imas 17472 df-xps 17474 df-mre 17548 df-mrc 17549 df-acs 17551 df-plusf 18607 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-submnd 18752 df-grp 18912 df-minusg 18913 df-sbg 18914 df-mulg 19044 df-subg 19099 df-cntz 19292 df-cmn 19757 df-abl 19758 df-mgp 20122 df-rng 20134 df-ur 20163 df-ring 20216 df-subrng 20523 df-subrg 20547 df-abv 20786 df-lmod 20857 df-scaf 20858 df-sra 21168 df-rgmod 21169 df-psmet 21344 df-xmet 21345 df-met 21346 df-bl 21347 df-mopn 21348 df-top 22859 df-topon 22876 df-topsp 22898 df-bases 22911 df-cn 23192 df-cnp 23193 df-tx 23527 df-hmeo 23720 df-tmd 24037 df-tgp 24038 df-trg 24125 df-xms 24285 df-ms 24286 df-tms 24287 df-nm 24547 df-ngp 24548 df-nrg 24550 df-nlm 24551

This theorem is referenced by: nrgtdrg 24658 nlmtlm 24659 iistmd 34046 qqhcn 34135

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