nrgtrg - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > nrgtrg		Structured version Visualization version GIF version

Theorem nrgtrg 24902

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 24884	. 2 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ TopGrp)
2		nrgring 24875	. 2 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ Ring)
3		eqid 2765	. . . . 5 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
4	3	ringmgp 20369	. . . 4 ⊢ (𝑅 ∈ Ring → (mulGrp‘𝑅) ∈ Mnd)
5	2, 4	syl 18	. . 3 ⊢ (𝑅 ∈ NrmRing → (mulGrp‘𝑅) ∈ Mnd)
6		tgptps 24292	. . . . . 6 ⊢ (𝑅 ∈ TopGrp → 𝑅 ∈ TopSp)
7	1, 6	syl 18	. . . . 5 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ TopSp)
8		eqid 2765	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
9		eqid 2765	. . . . . 6 ⊢ (TopOpen‘𝑅) = (TopOpen‘𝑅)
10	8, 9	istps 23145	. . . . 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 20269	. . . . 5 ⊢ (Base‘𝑅) = (Base‘(mulGrp‘𝑅))
13	3, 9	mgptopn 20272	. . . . 5 ⊢ (TopOpen‘𝑅) = (TopOpen‘(mulGrp‘𝑅))
14	12, 13	istps 23145	. . . 4 ⊢ ((mulGrp‘𝑅) ∈ TopSp ↔ (TopOpen‘𝑅) ∈ (TopOn‘(Base‘𝑅)))
15	11, 14	sylibr 237	. . 3 ⊢ (𝑅 ∈ NrmRing → (mulGrp‘𝑅) ∈ TopSp)
16		rlmnlm 24900	. . . 4 ⊢ (𝑅 ∈ NrmRing → (ringLMod‘𝑅) ∈ NrmMod)
17		rlmsca2 21374	. . . . 5 ⊢ ( I ‘𝑅) = (Scalar‘(ringLMod‘𝑅))
18		rlmscaf 21382	. . . . 5 ⊢ (+_𝑓‘(mulGrp‘𝑅)) = ( ·_sf ‘(ringLMod‘𝑅))
19		rlmtopn 21376	. . . . 5 ⊢ (TopOpen‘𝑅) = (TopOpen‘(ringLMod‘𝑅))
20		baseid 17298	. . . . . . . . 9 ⊢ Base = Slot (Base‘ndx)
21	20, 8	strfvi 17276	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘( I ‘𝑅))
22	21	a1i 11	. . . . . . 7 ⊢ (⊤ → (Base‘𝑅) = (Base‘( I ‘𝑅)))
23		tsetid 17432	. . . . . . . . 9 ⊢ TopSet = Slot (TopSet‘ndx)
24		eqid 2765	. . . . . . . . 9 ⊢ (TopSet‘𝑅) = (TopSet‘𝑅)
25	23, 24	strfvi 17276	. . . . . . . 8 ⊢ (TopSet‘𝑅) = (TopSet‘( I ‘𝑅))
26	25	a1i 11	. . . . . . 7 ⊢ (⊤ → (TopSet‘𝑅) = (TopSet‘( I ‘𝑅)))
27	22, 26	topnpropd 17515	. . . . . 6 ⊢ (⊤ → (TopOpen‘𝑅) = (TopOpen‘( I ‘𝑅)))
28	27	mptru 1577	. . . . 5 ⊢ (TopOpen‘𝑅) = (TopOpen‘( I ‘𝑅))
29	17, 18, 19, 28	nlmvscn 24899	. . . 4 ⊢ ((ringLMod‘𝑅) ∈ NrmMod → (+_𝑓‘(mulGrp‘𝑅)) ∈ (((TopOpen‘𝑅) ×_t (TopOpen‘𝑅)) Cn (TopOpen‘𝑅)))
30	16, 29	syl 18	. . 3 ⊢ (𝑅 ∈ NrmRing → (+_𝑓‘(mulGrp‘𝑅)) ∈ (((TopOpen‘𝑅) ×_t (TopOpen‘𝑅)) Cn (TopOpen‘𝑅)))
31		eqid 2765	. . . 4 ⊢ (+_𝑓‘(mulGrp‘𝑅)) = (+_𝑓‘(mulGrp‘𝑅))
32	31, 13	istmd 24286	. . 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 1362	. 2 ⊢ (𝑅 ∈ NrmRing → (mulGrp‘𝑅) ∈ TopMnd)
34	3	istrg 24376	. 2 ⊢ (𝑅 ∈ TopRing ↔ (𝑅 ∈ TopGrp ∧ 𝑅 ∈ Ring ∧ (mulGrp‘𝑅) ∈ TopMnd))
35	1, 2, 33, 34	syl3anbrc 1362	1 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ TopRing)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 = wceq 1570 ⊤wtru 1571 ∈ wcel 2146 I cid 5557 ‘cfv 6540 (class class class)co 7419 ndxcnx 17279 Basecbs 17295 TopSetcts 17342 TopOpenctopn 17500 +_𝑓cplusf 18721 Mndcmnd 18828 mulGrpcmgp 20264 Ringcrg 20363 ringLModcrglmod 21347 TopOnctopon 23121 TopSpctps 23143 Cn ccn 23435 ×_t ctx 23772 TopMndctmd 24282 TopGrpctgp 24283 TopRingctrg 24368 NrmRingcnrg 24791 NrmModcnlm 24792

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-cnex 11175 ax-resscn 11176 ax-1cn 11177 ax-icn 11178 ax-addcl 11179 ax-addrcl 11180 ax-mulcl 11181 ax-mulrcl 11182 ax-mulcom 11183 ax-addass 11184 ax-mulass 11185 ax-distr 11186 ax-i2m1 11187 ax-1ne0 11188 ax-1rid 11189 ax-rnegex 11190 ax-rrecex 11191 ax-cnre 11192 ax-pre-lttri 11193 ax-pre-lttrn 11194 ax-pre-ltadd 11195 ax-pre-mulgt0 11196 ax-pre-sup 11197

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4875 df-int 4915 df-iun 4960 df-iin 4961 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-se 5617 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7684 df-om 7869 df-1st 7992 df-2nd 7993 df-supp 8163 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-2o 8460 df-er 8700 df-map 8832 df-ixp 8902 df-en 8950 df-dom 8951 df-sdom 8952 df-fin 8953 df-fsupp 9329 df-fi 9378 df-sup 9409 df-inf 9410 df-oi 9479 df-card 9941 df-pnf 11264 df-mnf 11265 df-xr 11266 df-ltxr 11267 df-le 11268 df-sub 11462 df-neg 11463 df-div 11891 df-nn 12253 df-2 12322 df-3 12323 df-4 12324 df-5 12325 df-6 12326 df-7 12327 df-8 12328 df-9 12329 df-n0 12524 df-z 12611 df-dec 12732 df-uz 12883 df-q 12993 df-rp 13037 df-xneg 13157 df-xadd 13158 df-xmul 13159 df-ico 13398 df-icc 13399 df-fz 13556 df-fzo 13704 df-seq 14060 df-exp 14120 df-hash 14389 df-cj 15178 df-re 15179 df-im 15180 df-sqrt 15314 df-abs 15315 df-struct 17233 df-sets 17250 df-slot 17268 df-ndx 17280 df-base 17296 df-ress 17317 df-plusg 17349 df-mulr 17350 df-sca 17352 df-vsca 17353 df-ip 17354 df-tset 17355 df-ple 17356 df-ds 17358 df-hom 17360 df-cco 17361 df-rest 17501 df-topn 17502 df-0g 17520 df-gsum 17521 df-topgen 17522 df-pt 17523 df-prds 17526 df-xrs 17582 df-qtop 17587 df-imas 17588 df-xps 17590 df-mre 17664 df-mrc 17665 df-acs 17667 df-plusf 18723 df-mgm 18724 df-sgrp 18813 df-mnd 18829 df-submnd 18883 df-grp 19051 df-minusg 19052 df-sbg 19053 df-mulg 19182 df-subg 19237 df-cntz 19435 df-cmn 19900 df-abl 19901 df-mgp 20265 df-rng 20279 df-ur 20312 df-ring 20365 df-subrng 20699 df-subrg 20723 df-abv 20966 df-lmod 21037 df-scaf 21038 df-sra 21348 df-rgmod 21349 df-psmet 21568 df-xmet 21569 df-met 21570 df-bl 21571 df-mopn 21572 df-top 23105 df-topon 23122 df-topsp 23144 df-bases 23157 df-cn 23438 df-cnp 23439 df-tx 23774 df-hmeo 23967 df-tmd 24284 df-tgp 24285 df-trg 24372 df-xms 24532 df-ms 24533 df-tms 24534 df-nm 24794 df-ngp 24795 df-nrg 24797 df-nlm 24798

This theorem is used by: nrgtdrg 24905 nlmtlm 24906 iistmd 34360 qqhcn 34449

W3C validator