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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 24693 | . 2 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ TopGrp) | |
| 2 | nrgring 24684 | . 2 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ Ring) | |
| 3 | eqid 2737 | . . . . 5 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
| 4 | 3 | ringmgp 20236 | . . . 4 ⊢ (𝑅 ∈ Ring → (mulGrp‘𝑅) ∈ Mnd) |
| 5 | 2, 4 | syl 17 | . . 3 ⊢ (𝑅 ∈ NrmRing → (mulGrp‘𝑅) ∈ Mnd) |
| 6 | tgptps 24088 | . . . . . 6 ⊢ (𝑅 ∈ TopGrp → 𝑅 ∈ TopSp) | |
| 7 | 1, 6 | syl 17 | . . . . 5 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ TopSp) |
| 8 | eqid 2737 | . . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
| 9 | eqid 2737 | . . . . . 6 ⊢ (TopOpen‘𝑅) = (TopOpen‘𝑅) | |
| 10 | 8, 9 | istps 22940 | . . . . 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 20142 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘(mulGrp‘𝑅)) |
| 13 | 3, 9 | mgptopn 20145 | . . . . 5 ⊢ (TopOpen‘𝑅) = (TopOpen‘(mulGrp‘𝑅)) |
| 14 | 12, 13 | istps 22940 | . . . 4 ⊢ ((mulGrp‘𝑅) ∈ TopSp ↔ (TopOpen‘𝑅) ∈ (TopOn‘(Base‘𝑅))) |
| 15 | 11, 14 | sylibr 234 | . . 3 ⊢ (𝑅 ∈ NrmRing → (mulGrp‘𝑅) ∈ TopSp) |
| 16 | rlmnlm 24709 | . . . 4 ⊢ (𝑅 ∈ NrmRing → (ringLMod‘𝑅) ∈ NrmMod) | |
| 17 | rlmsca2 21206 | . . . . 5 ⊢ ( I ‘𝑅) = (Scalar‘(ringLMod‘𝑅)) | |
| 18 | rlmscaf 21214 | . . . . 5 ⊢ (+𝑓‘(mulGrp‘𝑅)) = ( ·sf ‘(ringLMod‘𝑅)) | |
| 19 | rlmtopn 21208 | . . . . 5 ⊢ (TopOpen‘𝑅) = (TopOpen‘(ringLMod‘𝑅)) | |
| 20 | baseid 17250 | . . . . . . . . 9 ⊢ Base = Slot (Base‘ndx) | |
| 21 | 20, 8 | strfvi 17227 | . . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘( I ‘𝑅)) |
| 22 | 21 | a1i 11 | . . . . . . 7 ⊢ (⊤ → (Base‘𝑅) = (Base‘( I ‘𝑅))) |
| 23 | tsetid 17397 | . . . . . . . . 9 ⊢ TopSet = Slot (TopSet‘ndx) | |
| 24 | eqid 2737 | . . . . . . . . 9 ⊢ (TopSet‘𝑅) = (TopSet‘𝑅) | |
| 25 | 23, 24 | strfvi 17227 | . . . . . . . 8 ⊢ (TopSet‘𝑅) = (TopSet‘( I ‘𝑅)) |
| 26 | 25 | a1i 11 | . . . . . . 7 ⊢ (⊤ → (TopSet‘𝑅) = (TopSet‘( I ‘𝑅))) |
| 27 | 22, 26 | topnpropd 17481 | . . . . . 6 ⊢ (⊤ → (TopOpen‘𝑅) = (TopOpen‘( I ‘𝑅))) |
| 28 | 27 | mptru 1547 | . . . . 5 ⊢ (TopOpen‘𝑅) = (TopOpen‘( I ‘𝑅)) |
| 29 | 17, 18, 19, 28 | nlmvscn 24708 | . . . 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 2737 | . . . 4 ⊢ (+𝑓‘(mulGrp‘𝑅)) = (+𝑓‘(mulGrp‘𝑅)) | |
| 32 | 31, 13 | istmd 24082 | . . 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 24172 | . 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 2108 I cid 5577 ‘cfv 6561 (class class class)co 7431 ndxcnx 17230 Basecbs 17247 TopSetcts 17303 TopOpenctopn 17466 +𝑓cplusf 18650 Mndcmnd 18747 mulGrpcmgp 20137 Ringcrg 20230 ringLModcrglmod 21171 TopOnctopon 22916 TopSpctps 22938 Cn ccn 23232 ×t ctx 23568 TopMndctmd 24078 TopGrpctgp 24079 TopRingctrg 24164 NrmRingcnrg 24592 NrmModcnlm 24593 |
| 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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-fi 9451 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 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 17467 df-topn 17468 df-0g 17486 df-gsum 17487 df-topgen 17488 df-pt 17489 df-prds 17492 df-xrs 17547 df-qtop 17552 df-imas 17553 df-xps 17555 df-mre 17629 df-mrc 17630 df-acs 17632 df-plusf 18652 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-grp 18954 df-minusg 18955 df-sbg 18956 df-mulg 19086 df-subg 19141 df-cntz 19335 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-subrng 20546 df-subrg 20570 df-abv 20810 df-lmod 20860 df-scaf 20861 df-sra 21172 df-rgmod 21173 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-cn 23235 df-cnp 23236 df-tx 23570 df-hmeo 23763 df-tmd 24080 df-tgp 24081 df-trg 24168 df-xms 24330 df-ms 24331 df-tms 24332 df-nm 24595 df-ngp 24596 df-nrg 24598 df-nlm 24599 |
| This theorem is referenced by: nrgtdrg 24714 nlmtlm 24715 iistmd 33901 qqhcn 33992 |
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