Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > srabn | Structured version Visualization version GIF version |
Description: The subring algebra over a complete normed ring is a Banach space iff the subring is a closed division ring. (Contributed by Mario Carneiro, 15-Oct-2015.) |
Ref | Expression |
---|---|
srabn.a | ⊢ 𝐴 = ((subringAlg ‘𝑊)‘𝑆) |
srabn.j | ⊢ 𝐽 = (TopOpen‘𝑊) |
Ref | Expression |
---|---|
srabn | ⊢ ((𝑊 ∈ NrmRing ∧ 𝑊 ∈ CMetSp ∧ 𝑆 ∈ (SubRing‘𝑊)) → (𝐴 ∈ Ban ↔ (𝑆 ∈ (Clsd‘𝐽) ∧ (𝑊 ↾s 𝑆) ∈ DivRing))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp2 1133 | . . . 4 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑊 ∈ CMetSp ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝑊 ∈ CMetSp) | |
2 | eqidd 2822 | . . . . 5 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑊 ∈ CMetSp ∧ 𝑆 ∈ (SubRing‘𝑊)) → (Base‘𝑊) = (Base‘𝑊)) | |
3 | srabn.a | . . . . . . 7 ⊢ 𝐴 = ((subringAlg ‘𝑊)‘𝑆) | |
4 | 3 | a1i 11 | . . . . . 6 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑊 ∈ CMetSp ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝐴 = ((subringAlg ‘𝑊)‘𝑆)) |
5 | eqid 2821 | . . . . . . . 8 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
6 | 5 | subrgss 19536 | . . . . . . 7 ⊢ (𝑆 ∈ (SubRing‘𝑊) → 𝑆 ⊆ (Base‘𝑊)) |
7 | 6 | 3ad2ant3 1131 | . . . . . 6 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑊 ∈ CMetSp ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝑆 ⊆ (Base‘𝑊)) |
8 | 4, 7 | srabase 19950 | . . . . 5 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑊 ∈ CMetSp ∧ 𝑆 ∈ (SubRing‘𝑊)) → (Base‘𝑊) = (Base‘𝐴)) |
9 | 4, 7 | srads 19958 | . . . . . 6 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑊 ∈ CMetSp ∧ 𝑆 ∈ (SubRing‘𝑊)) → (dist‘𝑊) = (dist‘𝐴)) |
10 | 9 | reseq1d 5852 | . . . . 5 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑊 ∈ CMetSp ∧ 𝑆 ∈ (SubRing‘𝑊)) → ((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊))) = ((dist‘𝐴) ↾ ((Base‘𝑊) × (Base‘𝑊)))) |
11 | 4, 7 | sratopn 19957 | . . . . 5 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑊 ∈ CMetSp ∧ 𝑆 ∈ (SubRing‘𝑊)) → (TopOpen‘𝑊) = (TopOpen‘𝐴)) |
12 | 2, 8, 10, 11 | cmspropd 23952 | . . . 4 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑊 ∈ CMetSp ∧ 𝑆 ∈ (SubRing‘𝑊)) → (𝑊 ∈ CMetSp ↔ 𝐴 ∈ CMetSp)) |
13 | 1, 12 | mpbid 234 | . . 3 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑊 ∈ CMetSp ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝐴 ∈ CMetSp) |
14 | eqid 2821 | . . . . . 6 ⊢ (Scalar‘𝐴) = (Scalar‘𝐴) | |
15 | 14 | isbn 23941 | . . . . 5 ⊢ (𝐴 ∈ Ban ↔ (𝐴 ∈ NrmVec ∧ 𝐴 ∈ CMetSp ∧ (Scalar‘𝐴) ∈ CMetSp)) |
16 | 3anrot 1096 | . . . . 5 ⊢ ((𝐴 ∈ NrmVec ∧ 𝐴 ∈ CMetSp ∧ (Scalar‘𝐴) ∈ CMetSp) ↔ (𝐴 ∈ CMetSp ∧ (Scalar‘𝐴) ∈ CMetSp ∧ 𝐴 ∈ NrmVec)) | |
17 | 3anass 1091 | . . . . 5 ⊢ ((𝐴 ∈ CMetSp ∧ (Scalar‘𝐴) ∈ CMetSp ∧ 𝐴 ∈ NrmVec) ↔ (𝐴 ∈ CMetSp ∧ ((Scalar‘𝐴) ∈ CMetSp ∧ 𝐴 ∈ NrmVec))) | |
18 | 15, 16, 17 | 3bitri 299 | . . . 4 ⊢ (𝐴 ∈ Ban ↔ (𝐴 ∈ CMetSp ∧ ((Scalar‘𝐴) ∈ CMetSp ∧ 𝐴 ∈ NrmVec))) |
19 | 18 | baib 538 | . . 3 ⊢ (𝐴 ∈ CMetSp → (𝐴 ∈ Ban ↔ ((Scalar‘𝐴) ∈ CMetSp ∧ 𝐴 ∈ NrmVec))) |
20 | 13, 19 | syl 17 | . 2 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑊 ∈ CMetSp ∧ 𝑆 ∈ (SubRing‘𝑊)) → (𝐴 ∈ Ban ↔ ((Scalar‘𝐴) ∈ CMetSp ∧ 𝐴 ∈ NrmVec))) |
21 | 4, 7 | srasca 19953 | . . . . 5 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑊 ∈ CMetSp ∧ 𝑆 ∈ (SubRing‘𝑊)) → (𝑊 ↾s 𝑆) = (Scalar‘𝐴)) |
22 | 21 | eleq1d 2897 | . . . 4 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑊 ∈ CMetSp ∧ 𝑆 ∈ (SubRing‘𝑊)) → ((𝑊 ↾s 𝑆) ∈ CMetSp ↔ (Scalar‘𝐴) ∈ CMetSp)) |
23 | eqid 2821 | . . . . . 6 ⊢ (𝑊 ↾s 𝑆) = (𝑊 ↾s 𝑆) | |
24 | srabn.j | . . . . . 6 ⊢ 𝐽 = (TopOpen‘𝑊) | |
25 | 23, 5, 24 | cmsss 23954 | . . . . 5 ⊢ ((𝑊 ∈ CMetSp ∧ 𝑆 ⊆ (Base‘𝑊)) → ((𝑊 ↾s 𝑆) ∈ CMetSp ↔ 𝑆 ∈ (Clsd‘𝐽))) |
26 | 1, 7, 25 | syl2anc 586 | . . . 4 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑊 ∈ CMetSp ∧ 𝑆 ∈ (SubRing‘𝑊)) → ((𝑊 ↾s 𝑆) ∈ CMetSp ↔ 𝑆 ∈ (Clsd‘𝐽))) |
27 | 22, 26 | bitr3d 283 | . . 3 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑊 ∈ CMetSp ∧ 𝑆 ∈ (SubRing‘𝑊)) → ((Scalar‘𝐴) ∈ CMetSp ↔ 𝑆 ∈ (Clsd‘𝐽))) |
28 | 3 | sranlm 23293 | . . . . . 6 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝐴 ∈ NrmMod) |
29 | 28 | 3adant2 1127 | . . . . 5 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑊 ∈ CMetSp ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝐴 ∈ NrmMod) |
30 | 14 | isnvc2 23308 | . . . . . 6 ⊢ (𝐴 ∈ NrmVec ↔ (𝐴 ∈ NrmMod ∧ (Scalar‘𝐴) ∈ DivRing)) |
31 | 30 | baib 538 | . . . . 5 ⊢ (𝐴 ∈ NrmMod → (𝐴 ∈ NrmVec ↔ (Scalar‘𝐴) ∈ DivRing)) |
32 | 29, 31 | syl 17 | . . . 4 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑊 ∈ CMetSp ∧ 𝑆 ∈ (SubRing‘𝑊)) → (𝐴 ∈ NrmVec ↔ (Scalar‘𝐴) ∈ DivRing)) |
33 | 21 | eleq1d 2897 | . . . 4 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑊 ∈ CMetSp ∧ 𝑆 ∈ (SubRing‘𝑊)) → ((𝑊 ↾s 𝑆) ∈ DivRing ↔ (Scalar‘𝐴) ∈ DivRing)) |
34 | 32, 33 | bitr4d 284 | . . 3 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑊 ∈ CMetSp ∧ 𝑆 ∈ (SubRing‘𝑊)) → (𝐴 ∈ NrmVec ↔ (𝑊 ↾s 𝑆) ∈ DivRing)) |
35 | 27, 34 | anbi12d 632 | . 2 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑊 ∈ CMetSp ∧ 𝑆 ∈ (SubRing‘𝑊)) → (((Scalar‘𝐴) ∈ CMetSp ∧ 𝐴 ∈ NrmVec) ↔ (𝑆 ∈ (Clsd‘𝐽) ∧ (𝑊 ↾s 𝑆) ∈ DivRing))) |
36 | 20, 35 | bitrd 281 | 1 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑊 ∈ CMetSp ∧ 𝑆 ∈ (SubRing‘𝑊)) → (𝐴 ∈ Ban ↔ (𝑆 ∈ (Clsd‘𝐽) ∧ (𝑊 ↾s 𝑆) ∈ DivRing))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ⊆ wss 3936 × cxp 5553 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 ↾s cress 16484 Scalarcsca 16568 distcds 16574 TopOpenctopn 16695 DivRingcdr 19502 SubRingcsubrg 19531 subringAlg csra 19940 Clsdccld 21624 NrmRingcnrg 23189 NrmModcnlm 23190 NrmVeccnvc 23191 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-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: rlmbn 23964 |
Copyright terms: Public domain | W3C validator |