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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 23953 | . . . . 5 ⊢ (𝑅 ∈ CMetSp → 𝑅 ∈ MetSp) | |
3 | mstps 23067 | . . . . 5 ⊢ (𝑅 ∈ MetSp → 𝑅 ∈ TopSp) | |
4 | 1, 2, 3 | 3syl 18 | . . . 4 ⊢ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing ∧ 𝑅 ∈ CMetSp) → 𝑅 ∈ TopSp) |
5 | eqid 2823 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
6 | eqid 2823 | . . . . 5 ⊢ (TopOpen‘𝑅) = (TopOpen‘𝑅) | |
7 | 5, 6 | tpsuni 21546 | . . . 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 21547 | . . . 4 ⊢ (𝑅 ∈ TopSp → (TopOpen‘𝑅) ∈ Top) |
10 | eqid 2823 | . . . . 5 ⊢ ∪ (TopOpen‘𝑅) = ∪ (TopOpen‘𝑅) | |
11 | 10 | topcld 21645 | . . . 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 2915 | . 2 ⊢ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing ∧ 𝑅 ∈ CMetSp) → (Base‘𝑅) ∈ (Clsd‘(TopOpen‘𝑅))) |
14 | 5 | ressid 16561 | . . . 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 2915 | . 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 23274 | . . . . 5 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ Ring) | |
20 | 19 | 3ad2ant1 1129 | . . . 4 ⊢ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing ∧ 𝑅 ∈ CMetSp) → 𝑅 ∈ Ring) |
21 | 5 | subrgid 19539 | . . . 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 19965 | . . . 4 ⊢ (ringLMod‘𝑅) = ((subringAlg ‘𝑅)‘(Base‘𝑅)) | |
24 | 23, 6 | srabn 23965 | . . 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 4840 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 ↾s cress 16486 TopOpenctopn 16697 Ringcrg 19299 DivRingcdr 19504 SubRingcsubrg 19533 ringLModcrglmod 19943 Topctop 21503 TopSpctps 21542 Clsdccld 21626 MetSpcms 22930 NrmRingcnrg 23191 CMetSpccms 23937 Bancbn 23938 |
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 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fi 8877 df-sup 8908 df-inf 8909 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-ico 12747 df-icc 12748 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-sca 16583 df-vsca 16584 df-ip 16585 df-tset 16586 df-ds 16589 df-rest 16698 df-topn 16699 df-0g 16717 df-topgen 16719 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-grp 18108 df-minusg 18109 df-sbg 18110 df-subg 18278 df-mgp 19242 df-ur 19254 df-ring 19301 df-subrg 19535 df-abv 19590 df-lmod 19638 df-lvec 19877 df-sra 19946 df-rgmod 19947 df-psmet 20539 df-xmet 20540 df-met 20541 df-bl 20542 df-mopn 20543 df-fbas 20544 df-fg 20545 df-top 21504 df-topon 21521 df-topsp 21543 df-bases 21556 df-cld 21629 df-ntr 21630 df-cls 21631 df-nei 21708 df-haus 21925 df-fil 22456 df-flim 22549 df-xms 22932 df-ms 22933 df-nm 23194 df-ngp 23195 df-nrg 23197 df-nlm 23198 df-nvc 23199 df-cfil 23860 df-cmet 23862 df-cms 23940 df-bn 23941 |
This theorem is referenced by: (None) |
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