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Theorem nrgtrg 24632

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 24614	. 2 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ TopGrp)
2		nrgring 24605	. 2 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ Ring)
3		eqid 2734	. . . . 5 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
4	3	ringmgp 20172	. . . 4 ⊢ (𝑅 ∈ Ring → (mulGrp‘𝑅) ∈ Mnd)
5	2, 4	syl 17	. . 3 ⊢ (𝑅 ∈ NrmRing → (mulGrp‘𝑅) ∈ Mnd)
6		tgptps 24022	. . . . . 6 ⊢ (𝑅 ∈ TopGrp → 𝑅 ∈ TopSp)
7	1, 6	syl 17	. . . . 5 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ TopSp)
8		eqid 2734	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
9		eqid 2734	. . . . . 6 ⊢ (TopOpen‘𝑅) = (TopOpen‘𝑅)
10	8, 9	istps 22876	. . . . 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 20078	. . . . 5 ⊢ (Base‘𝑅) = (Base‘(mulGrp‘𝑅))
13	3, 9	mgptopn 20081	. . . . 5 ⊢ (TopOpen‘𝑅) = (TopOpen‘(mulGrp‘𝑅))
14	12, 13	istps 22876	. . . 4 ⊢ ((mulGrp‘𝑅) ∈ TopSp ↔ (TopOpen‘𝑅) ∈ (TopOn‘(Base‘𝑅)))
15	11, 14	sylibr 234	. . 3 ⊢ (𝑅 ∈ NrmRing → (mulGrp‘𝑅) ∈ TopSp)
16		rlmnlm 24630	. . . 4 ⊢ (𝑅 ∈ NrmRing → (ringLMod‘𝑅) ∈ NrmMod)
17		rlmsca2 21149	. . . . 5 ⊢ ( I ‘𝑅) = (Scalar‘(ringLMod‘𝑅))
18		rlmscaf 21157	. . . . 5 ⊢ (+_𝑓‘(mulGrp‘𝑅)) = ( ·_sf ‘(ringLMod‘𝑅))
19		rlmtopn 21151	. . . . 5 ⊢ (TopOpen‘𝑅) = (TopOpen‘(ringLMod‘𝑅))
20		baseid 17137	. . . . . . . . 9 ⊢ Base = Slot (Base‘ndx)
21	20, 8	strfvi 17115	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘( I ‘𝑅))
22	21	a1i 11	. . . . . . 7 ⊢ (⊤ → (Base‘𝑅) = (Base‘( I ‘𝑅)))
23		tsetid 17271	. . . . . . . . 9 ⊢ TopSet = Slot (TopSet‘ndx)
24		eqid 2734	. . . . . . . . 9 ⊢ (TopSet‘𝑅) = (TopSet‘𝑅)
25	23, 24	strfvi 17115	. . . . . . . 8 ⊢ (TopSet‘𝑅) = (TopSet‘( I ‘𝑅))
26	25	a1i 11	. . . . . . 7 ⊢ (⊤ → (TopSet‘𝑅) = (TopSet‘( I ‘𝑅)))
27	22, 26	topnpropd 17354	. . . . . 6 ⊢ (⊤ → (TopOpen‘𝑅) = (TopOpen‘( I ‘𝑅)))
28	27	mptru 1548	. . . . 5 ⊢ (TopOpen‘𝑅) = (TopOpen‘( I ‘𝑅))
29	17, 18, 19, 28	nlmvscn 24629	. . . 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 2734	. . . 4 ⊢ (+_𝑓‘(mulGrp‘𝑅)) = (+_𝑓‘(mulGrp‘𝑅))
32	31, 13	istmd 24016	. . 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 1344	. 2 ⊢ (𝑅 ∈ NrmRing → (mulGrp‘𝑅) ∈ TopMnd)
34	3	istrg 24106	. 2 ⊢ (𝑅 ∈ TopRing ↔ (𝑅 ∈ TopGrp ∧ 𝑅 ∈ Ring ∧ (mulGrp‘𝑅) ∈ TopMnd))
35	1, 2, 33, 34	syl3anbrc 1344	1 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ TopRing)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1541 ⊤wtru 1542 ∈ wcel 2113 I cid 5516 ‘cfv 6490 (class class class)co 7356 ndxcnx 17118 Basecbs 17134 TopSetcts 17181 TopOpenctopn 17339 +_𝑓cplusf 18560 Mndcmnd 18657 mulGrpcmgp 20073 Ringcrg 20166 ringLModcrglmod 21122 TopOnctopon 22852 TopSpctps 22874 Cn ccn 23166 ×_t ctx 23502 TopMndctmd 24012 TopGrpctgp 24013 TopRingctrg 24098 NrmRingcnrg 24521 NrmModcnlm 24522

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-rep 5222 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678 ax-cnex 11080 ax-resscn 11081 ax-1cn 11082 ax-icn 11083 ax-addcl 11084 ax-addrcl 11085 ax-mulcl 11086 ax-mulrcl 11087 ax-mulcom 11088 ax-addass 11089 ax-mulass 11090 ax-distr 11091 ax-i2m1 11092 ax-1ne0 11093 ax-1rid 11094 ax-rnegex 11095 ax-rrecex 11096 ax-cnre 11097 ax-pre-lttri 11098 ax-pre-lttrn 11099 ax-pre-ltadd 11100 ax-pre-mulgt0 11101 ax-pre-sup 11102

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-tp 4583 df-op 4585 df-uni 4862 df-int 4901 df-iun 4946 df-iin 4947 df-br 5097 df-opab 5159 df-mpt 5178 df-tr 5204 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-se 5576 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-isom 6499 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7620 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8101 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-er 8633 df-map 8763 df-ixp 8834 df-en 8882 df-dom 8883 df-sdom 8884 df-fin 8885 df-fsupp 9263 df-fi 9312 df-sup 9343 df-inf 9344 df-oi 9413 df-card 9849 df-pnf 11166 df-mnf 11167 df-xr 11168 df-ltxr 11169 df-le 11170 df-sub 11364 df-neg 11365 df-div 11793 df-nn 12144 df-2 12206 df-3 12207 df-4 12208 df-5 12209 df-6 12210 df-7 12211 df-8 12212 df-9 12213 df-n0 12400 df-z 12487 df-dec 12606 df-uz 12750 df-q 12860 df-rp 12904 df-xneg 13024 df-xadd 13025 df-xmul 13026 df-ico 13265 df-icc 13266 df-fz 13422 df-fzo 13569 df-seq 13923 df-exp 13983 df-hash 14252 df-cj 15020 df-re 15021 df-im 15022 df-sqrt 15156 df-abs 15157 df-struct 17072 df-sets 17089 df-slot 17107 df-ndx 17119 df-base 17135 df-ress 17156 df-plusg 17188 df-mulr 17189 df-sca 17191 df-vsca 17192 df-ip 17193 df-tset 17194 df-ple 17195 df-ds 17197 df-hom 17199 df-cco 17200 df-rest 17340 df-topn 17341 df-0g 17359 df-gsum 17360 df-topgen 17361 df-pt 17362 df-prds 17365 df-xrs 17421 df-qtop 17426 df-imas 17427 df-xps 17429 df-mre 17503 df-mrc 17504 df-acs 17506 df-plusf 18562 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-submnd 18707 df-grp 18864 df-minusg 18865 df-sbg 18866 df-mulg 18996 df-subg 19051 df-cntz 19244 df-cmn 19709 df-abl 19710 df-mgp 20074 df-rng 20086 df-ur 20115 df-ring 20168 df-subrng 20477 df-subrg 20501 df-abv 20740 df-lmod 20811 df-scaf 20812 df-sra 21123 df-rgmod 21124 df-psmet 21299 df-xmet 21300 df-met 21301 df-bl 21302 df-mopn 21303 df-top 22836 df-topon 22853 df-topsp 22875 df-bases 22888 df-cn 23169 df-cnp 23170 df-tx 23504 df-hmeo 23697 df-tmd 24014 df-tgp 24015 df-trg 24102 df-xms 24262 df-ms 24263 df-tms 24264 df-nm 24524 df-ngp 24525 df-nrg 24527 df-nlm 24528

This theorem is referenced by: nrgtdrg 24635 nlmtlm 24636 iistmd 34008 qqhcn 34097

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