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Mirrors > Home > MPE Home > Th. List > nrgtrg | Structured version Visualization version GIF version |
Description: A normed ring is a topological ring. (Contributed by Mario Carneiro, 4-Oct-2015.) (Proof shortened by AV, 31-Oct-2024.) |
Ref | Expression |
---|---|
nrgtrg | ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ TopRing) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nrgtgp 24708 | . 2 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ TopGrp) | |
2 | nrgring 24699 | . 2 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ Ring) | |
3 | eqid 2734 | . . . . 5 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
4 | 3 | ringmgp 20256 | . . . 4 ⊢ (𝑅 ∈ Ring → (mulGrp‘𝑅) ∈ Mnd) |
5 | 2, 4 | syl 17 | . . 3 ⊢ (𝑅 ∈ NrmRing → (mulGrp‘𝑅) ∈ Mnd) |
6 | tgptps 24103 | . . . . . 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 22955 | . . . . 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 20157 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘(mulGrp‘𝑅)) |
13 | 3, 9 | mgptopn 20163 | . . . . 5 ⊢ (TopOpen‘𝑅) = (TopOpen‘(mulGrp‘𝑅)) |
14 | 12, 13 | istps 22955 | . . . 4 ⊢ ((mulGrp‘𝑅) ∈ TopSp ↔ (TopOpen‘𝑅) ∈ (TopOn‘(Base‘𝑅))) |
15 | 11, 14 | sylibr 234 | . . 3 ⊢ (𝑅 ∈ NrmRing → (mulGrp‘𝑅) ∈ TopSp) |
16 | rlmnlm 24724 | . . . 4 ⊢ (𝑅 ∈ NrmRing → (ringLMod‘𝑅) ∈ NrmMod) | |
17 | rlmsca2 21223 | . . . . 5 ⊢ ( I ‘𝑅) = (Scalar‘(ringLMod‘𝑅)) | |
18 | rlmscaf 21231 | . . . . 5 ⊢ (+𝑓‘(mulGrp‘𝑅)) = ( ·sf ‘(ringLMod‘𝑅)) | |
19 | rlmtopn 21225 | . . . . 5 ⊢ (TopOpen‘𝑅) = (TopOpen‘(ringLMod‘𝑅)) | |
20 | baseid 17247 | . . . . . . . . 9 ⊢ Base = Slot (Base‘ndx) | |
21 | 20, 8 | strfvi 17223 | . . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘( I ‘𝑅)) |
22 | 21 | a1i 11 | . . . . . . 7 ⊢ (⊤ → (Base‘𝑅) = (Base‘( I ‘𝑅))) |
23 | tsetid 17398 | . . . . . . . . 9 ⊢ TopSet = Slot (TopSet‘ndx) | |
24 | eqid 2734 | . . . . . . . . 9 ⊢ (TopSet‘𝑅) = (TopSet‘𝑅) | |
25 | 23, 24 | strfvi 17223 | . . . . . . . 8 ⊢ (TopSet‘𝑅) = (TopSet‘( I ‘𝑅)) |
26 | 25 | a1i 11 | . . . . . . 7 ⊢ (⊤ → (TopSet‘𝑅) = (TopSet‘( I ‘𝑅))) |
27 | 22, 26 | topnpropd 17482 | . . . . . 6 ⊢ (⊤ → (TopOpen‘𝑅) = (TopOpen‘( I ‘𝑅))) |
28 | 27 | mptru 1543 | . . . . 5 ⊢ (TopOpen‘𝑅) = (TopOpen‘( I ‘𝑅)) |
29 | 17, 18, 19, 28 | nlmvscn 24723 | . . . 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 24097 | . . 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 1342 | . 2 ⊢ (𝑅 ∈ NrmRing → (mulGrp‘𝑅) ∈ TopMnd) |
34 | 3 | istrg 24187 | . 2 ⊢ (𝑅 ∈ TopRing ↔ (𝑅 ∈ TopGrp ∧ 𝑅 ∈ Ring ∧ (mulGrp‘𝑅) ∈ TopMnd)) |
35 | 1, 2, 33, 34 | syl3anbrc 1342 | 1 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ TopRing) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ⊤wtru 1537 ∈ wcel 2105 I cid 5581 ‘cfv 6562 (class class class)co 7430 ndxcnx 17226 Basecbs 17244 TopSetcts 17303 TopOpenctopn 17467 +𝑓cplusf 18662 Mndcmnd 18759 mulGrpcmgp 20151 Ringcrg 20250 ringLModcrglmod 21188 TopOnctopon 22931 TopSpctps 22953 Cn ccn 23247 ×t ctx 23583 TopMndctmd 24093 TopGrpctgp 24094 TopRingctrg 24179 NrmRingcnrg 24607 NrmModcnlm 24608 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-iin 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-of 7696 df-om 7887 df-1st 8012 df-2nd 8013 df-supp 8184 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-2o 8505 df-er 8743 df-map 8866 df-ixp 8936 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-fsupp 9399 df-fi 9448 df-sup 9479 df-inf 9480 df-oi 9547 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-z 12611 df-dec 12731 df-uz 12876 df-q 12988 df-rp 13032 df-xneg 13151 df-xadd 13152 df-xmul 13153 df-ico 13389 df-icc 13390 df-fz 13544 df-fzo 13691 df-seq 14039 df-exp 14099 df-hash 14366 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-hom 17321 df-cco 17322 df-rest 17468 df-topn 17469 df-0g 17487 df-gsum 17488 df-topgen 17489 df-pt 17490 df-prds 17493 df-xrs 17548 df-qtop 17553 df-imas 17554 df-xps 17556 df-mre 17630 df-mrc 17631 df-acs 17633 df-plusf 18664 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-submnd 18809 df-grp 18966 df-minusg 18967 df-sbg 18968 df-mulg 19098 df-subg 19153 df-cntz 19347 df-cmn 19814 df-abl 19815 df-mgp 20152 df-rng 20170 df-ur 20199 df-ring 20252 df-subrng 20562 df-subrg 20586 df-abv 20826 df-lmod 20876 df-scaf 20877 df-sra 21189 df-rgmod 21190 df-psmet 21373 df-xmet 21374 df-met 21375 df-bl 21376 df-mopn 21377 df-top 22915 df-topon 22932 df-topsp 22954 df-bases 22968 df-cn 23250 df-cnp 23251 df-tx 23585 df-hmeo 23778 df-tmd 24095 df-tgp 24096 df-trg 24183 df-xms 24345 df-ms 24346 df-tms 24347 df-nm 24610 df-ngp 24611 df-nrg 24613 df-nlm 24614 |
This theorem is referenced by: nrgtdrg 24729 nlmtlm 24730 iistmd 33862 qqhcn 33953 |
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