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Mirrors > Home > MPE Home > Th. List > rlmbn | Structured version Visualization version GIF version |
Description: The ring module over a complete normed division ring is a Banach space. (Contributed by Mario Carneiro, 15-Oct-2015.) |
Ref | Expression |
---|---|
rlmbn | ⊢ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing ∧ 𝑅 ∈ CMetSp) → (ringLMod‘𝑅) ∈ Ban) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp3 1134 | . . . . 5 ⊢ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing ∧ 𝑅 ∈ CMetSp) → 𝑅 ∈ CMetSp) | |
2 | cmsms 23951 | . . . . 5 ⊢ (𝑅 ∈ CMetSp → 𝑅 ∈ MetSp) | |
3 | mstps 23065 | . . . . 5 ⊢ (𝑅 ∈ MetSp → 𝑅 ∈ TopSp) | |
4 | 1, 2, 3 | 3syl 18 | . . . 4 ⊢ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing ∧ 𝑅 ∈ CMetSp) → 𝑅 ∈ TopSp) |
5 | eqid 2821 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
6 | eqid 2821 | . . . . 5 ⊢ (TopOpen‘𝑅) = (TopOpen‘𝑅) | |
7 | 5, 6 | tpsuni 21544 | . . . 4 ⊢ (𝑅 ∈ TopSp → (Base‘𝑅) = ∪ (TopOpen‘𝑅)) |
8 | 4, 7 | syl 17 | . . 3 ⊢ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing ∧ 𝑅 ∈ CMetSp) → (Base‘𝑅) = ∪ (TopOpen‘𝑅)) |
9 | 6 | tpstop 21545 | . . . 4 ⊢ (𝑅 ∈ TopSp → (TopOpen‘𝑅) ∈ Top) |
10 | eqid 2821 | . . . . 5 ⊢ ∪ (TopOpen‘𝑅) = ∪ (TopOpen‘𝑅) | |
11 | 10 | topcld 21643 | . . . 4 ⊢ ((TopOpen‘𝑅) ∈ Top → ∪ (TopOpen‘𝑅) ∈ (Clsd‘(TopOpen‘𝑅))) |
12 | 4, 9, 11 | 3syl 18 | . . 3 ⊢ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing ∧ 𝑅 ∈ CMetSp) → ∪ (TopOpen‘𝑅) ∈ (Clsd‘(TopOpen‘𝑅))) |
13 | 8, 12 | eqeltrd 2913 | . 2 ⊢ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing ∧ 𝑅 ∈ CMetSp) → (Base‘𝑅) ∈ (Clsd‘(TopOpen‘𝑅))) |
14 | 5 | ressid 16559 | . . . 4 ⊢ (𝑅 ∈ NrmRing → (𝑅 ↾s (Base‘𝑅)) = 𝑅) |
15 | 14 | 3ad2ant1 1129 | . . 3 ⊢ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing ∧ 𝑅 ∈ CMetSp) → (𝑅 ↾s (Base‘𝑅)) = 𝑅) |
16 | simp2 1133 | . . 3 ⊢ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing ∧ 𝑅 ∈ CMetSp) → 𝑅 ∈ DivRing) | |
17 | 15, 16 | eqeltrd 2913 | . 2 ⊢ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing ∧ 𝑅 ∈ CMetSp) → (𝑅 ↾s (Base‘𝑅)) ∈ DivRing) |
18 | simp1 1132 | . . 3 ⊢ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing ∧ 𝑅 ∈ CMetSp) → 𝑅 ∈ NrmRing) | |
19 | nrgring 23272 | . . . . 5 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ Ring) | |
20 | 19 | 3ad2ant1 1129 | . . . 4 ⊢ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing ∧ 𝑅 ∈ CMetSp) → 𝑅 ∈ Ring) |
21 | 5 | subrgid 19537 | . . . 4 ⊢ (𝑅 ∈ Ring → (Base‘𝑅) ∈ (SubRing‘𝑅)) |
22 | 20, 21 | syl 17 | . . 3 ⊢ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing ∧ 𝑅 ∈ CMetSp) → (Base‘𝑅) ∈ (SubRing‘𝑅)) |
23 | rlmval 19963 | . . . 4 ⊢ (ringLMod‘𝑅) = ((subringAlg ‘𝑅)‘(Base‘𝑅)) | |
24 | 23, 6 | srabn 23963 | . . 3 ⊢ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ CMetSp ∧ (Base‘𝑅) ∈ (SubRing‘𝑅)) → ((ringLMod‘𝑅) ∈ Ban ↔ ((Base‘𝑅) ∈ (Clsd‘(TopOpen‘𝑅)) ∧ (𝑅 ↾s (Base‘𝑅)) ∈ DivRing))) |
25 | 18, 1, 22, 24 | syl3anc 1367 | . 2 ⊢ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing ∧ 𝑅 ∈ CMetSp) → ((ringLMod‘𝑅) ∈ Ban ↔ ((Base‘𝑅) ∈ (Clsd‘(TopOpen‘𝑅)) ∧ (𝑅 ↾s (Base‘𝑅)) ∈ DivRing))) |
26 | 13, 17, 25 | mpbir2and 711 | 1 ⊢ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing ∧ 𝑅 ∈ CMetSp) → (ringLMod‘𝑅) ∈ Ban) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∪ cuni 4838 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 ↾s cress 16484 TopOpenctopn 16695 Ringcrg 19297 DivRingcdr 19502 SubRingcsubrg 19531 ringLModcrglmod 19941 Topctop 21501 TopSpctps 21540 Clsdccld 21624 MetSpcms 22928 NrmRingcnrg 23189 CMetSpccms 23935 Bancbn 23936 |
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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fi 8875 df-sup 8906 df-inf 8907 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-q 12350 df-rp 12391 df-xneg 12508 df-xadd 12509 df-xmul 12510 df-ico 12745 df-icc 12746 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ds 16587 df-rest 16696 df-topn 16697 df-0g 16715 df-topgen 16717 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-grp 18106 df-minusg 18107 df-sbg 18108 df-subg 18276 df-mgp 19240 df-ur 19252 df-ring 19299 df-subrg 19533 df-abv 19588 df-lmod 19636 df-lvec 19875 df-sra 19944 df-rgmod 19945 df-psmet 20537 df-xmet 20538 df-met 20539 df-bl 20540 df-mopn 20541 df-fbas 20542 df-fg 20543 df-top 21502 df-topon 21519 df-topsp 21541 df-bases 21554 df-cld 21627 df-ntr 21628 df-cls 21629 df-nei 21706 df-haus 21923 df-fil 22454 df-flim 22547 df-xms 22930 df-ms 22931 df-nm 23192 df-ngp 23193 df-nrg 23195 df-nlm 23196 df-nvc 23197 df-cfil 23858 df-cmet 23860 df-cms 23938 df-bn 23939 |
This theorem is referenced by: (None) |
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