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Theorem nrgtrg 24585

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 24567	. 2 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ TopGrp)
2		nrgring 24558	. 2 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ Ring)
3		eqid 2730	. . . . 5 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
4	3	ringmgp 20155	. . . 4 ⊢ (𝑅 ∈ Ring → (mulGrp‘𝑅) ∈ Mnd)
5	2, 4	syl 17	. . 3 ⊢ (𝑅 ∈ NrmRing → (mulGrp‘𝑅) ∈ Mnd)
6		tgptps 23974	. . . . . 6 ⊢ (𝑅 ∈ TopGrp → 𝑅 ∈ TopSp)
7	1, 6	syl 17	. . . . 5 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ TopSp)
8		eqid 2730	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
9		eqid 2730	. . . . . 6 ⊢ (TopOpen‘𝑅) = (TopOpen‘𝑅)
10	8, 9	istps 22828	. . . . 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 20061	. . . . 5 ⊢ (Base‘𝑅) = (Base‘(mulGrp‘𝑅))
13	3, 9	mgptopn 20064	. . . . 5 ⊢ (TopOpen‘𝑅) = (TopOpen‘(mulGrp‘𝑅))
14	12, 13	istps 22828	. . . 4 ⊢ ((mulGrp‘𝑅) ∈ TopSp ↔ (TopOpen‘𝑅) ∈ (TopOn‘(Base‘𝑅)))
15	11, 14	sylibr 234	. . 3 ⊢ (𝑅 ∈ NrmRing → (mulGrp‘𝑅) ∈ TopSp)
16		rlmnlm 24583	. . . 4 ⊢ (𝑅 ∈ NrmRing → (ringLMod‘𝑅) ∈ NrmMod)
17		rlmsca2 21113	. . . . 5 ⊢ ( I ‘𝑅) = (Scalar‘(ringLMod‘𝑅))
18		rlmscaf 21121	. . . . 5 ⊢ (+_𝑓‘(mulGrp‘𝑅)) = ( ·_sf ‘(ringLMod‘𝑅))
19		rlmtopn 21115	. . . . 5 ⊢ (TopOpen‘𝑅) = (TopOpen‘(ringLMod‘𝑅))
20		baseid 17189	. . . . . . . . 9 ⊢ Base = Slot (Base‘ndx)
21	20, 8	strfvi 17167	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘( I ‘𝑅))
22	21	a1i 11	. . . . . . 7 ⊢ (⊤ → (Base‘𝑅) = (Base‘( I ‘𝑅)))
23		tsetid 17323	. . . . . . . . 9 ⊢ TopSet = Slot (TopSet‘ndx)
24		eqid 2730	. . . . . . . . 9 ⊢ (TopSet‘𝑅) = (TopSet‘𝑅)
25	23, 24	strfvi 17167	. . . . . . . 8 ⊢ (TopSet‘𝑅) = (TopSet‘( I ‘𝑅))
26	25	a1i 11	. . . . . . 7 ⊢ (⊤ → (TopSet‘𝑅) = (TopSet‘( I ‘𝑅)))
27	22, 26	topnpropd 17406	. . . . . 6 ⊢ (⊤ → (TopOpen‘𝑅) = (TopOpen‘( I ‘𝑅)))
28	27	mptru 1547	. . . . 5 ⊢ (TopOpen‘𝑅) = (TopOpen‘( I ‘𝑅))
29	17, 18, 19, 28	nlmvscn 24582	. . . 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 2730	. . . 4 ⊢ (+_𝑓‘(mulGrp‘𝑅)) = (+_𝑓‘(mulGrp‘𝑅))
32	31, 13	istmd 23968	. . 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 24058	. 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 1540 ⊤wtru 1541 ∈ wcel 2109 I cid 5535 ‘cfv 6514 (class class class)co 7390 ndxcnx 17170 Basecbs 17186 TopSetcts 17233 TopOpenctopn 17391 +_𝑓cplusf 18571 Mndcmnd 18668 mulGrpcmgp 20056 Ringcrg 20149 ringLModcrglmod 21086 TopOnctopon 22804 TopSpctps 22826 Cn ccn 23118 ×_t ctx 23454 TopMndctmd 23964 TopGrpctgp 23965 TopRingctrg 24050 NrmRingcnrg 24474 NrmModcnlm 24475

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-tp 4597 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-iin 4961 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-se 5595 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-isom 6523 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-of 7656 df-om 7846 df-1st 7971 df-2nd 7972 df-supp 8143 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-2o 8438 df-er 8674 df-map 8804 df-ixp 8874 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-fsupp 9320 df-fi 9369 df-sup 9400 df-inf 9401 df-oi 9470 df-card 9899 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-div 11843 df-nn 12194 df-2 12256 df-3 12257 df-4 12258 df-5 12259 df-6 12260 df-7 12261 df-8 12262 df-9 12263 df-n0 12450 df-z 12537 df-dec 12657 df-uz 12801 df-q 12915 df-rp 12959 df-xneg 13079 df-xadd 13080 df-xmul 13081 df-ico 13319 df-icc 13320 df-fz 13476 df-fzo 13623 df-seq 13974 df-exp 14034 df-hash 14303 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-struct 17124 df-sets 17141 df-slot 17159 df-ndx 17171 df-base 17187 df-ress 17208 df-plusg 17240 df-mulr 17241 df-sca 17243 df-vsca 17244 df-ip 17245 df-tset 17246 df-ple 17247 df-ds 17249 df-hom 17251 df-cco 17252 df-rest 17392 df-topn 17393 df-0g 17411 df-gsum 17412 df-topgen 17413 df-pt 17414 df-prds 17417 df-xrs 17472 df-qtop 17477 df-imas 17478 df-xps 17480 df-mre 17554 df-mrc 17555 df-acs 17557 df-plusf 18573 df-mgm 18574 df-sgrp 18653 df-mnd 18669 df-submnd 18718 df-grp 18875 df-minusg 18876 df-sbg 18877 df-mulg 19007 df-subg 19062 df-cntz 19256 df-cmn 19719 df-abl 19720 df-mgp 20057 df-rng 20069 df-ur 20098 df-ring 20151 df-subrng 20462 df-subrg 20486 df-abv 20725 df-lmod 20775 df-scaf 20776 df-sra 21087 df-rgmod 21088 df-psmet 21263 df-xmet 21264 df-met 21265 df-bl 21266 df-mopn 21267 df-top 22788 df-topon 22805 df-topsp 22827 df-bases 22840 df-cn 23121 df-cnp 23122 df-tx 23456 df-hmeo 23649 df-tmd 23966 df-tgp 23967 df-trg 24054 df-xms 24215 df-ms 24216 df-tms 24217 df-nm 24477 df-ngp 24478 df-nrg 24480 df-nlm 24481

This theorem is referenced by: nrgtdrg 24588 nlmtlm 24589 iistmd 33899 qqhcn 33988

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