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Mirrors > Home > MPE Home > Th. List > ishl2 | Structured version Visualization version GIF version |
Description: A Hilbert space is a complete subcomplex pre-Hilbert space over ℝ or ℂ. (Contributed by Mario Carneiro, 15-Oct-2015.) |
Ref | Expression |
---|---|
hlress.f | ⊢ 𝐹 = (Scalar‘𝑊) |
hlress.k | ⊢ 𝐾 = (Base‘𝐹) |
Ref | Expression |
---|---|
ishl2 | ⊢ (𝑊 ∈ ℂHil ↔ (𝑊 ∈ CMetSp ∧ 𝑊 ∈ ℂPreHil ∧ 𝐾 ∈ {ℝ, ℂ})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ishl 25409 | . 2 ⊢ (𝑊 ∈ ℂHil ↔ (𝑊 ∈ Ban ∧ 𝑊 ∈ ℂPreHil)) | |
2 | df-3an 1088 | . . 3 ⊢ ((𝑊 ∈ CMetSp ∧ 𝐾 ∈ {ℝ, ℂ} ∧ 𝑊 ∈ ℂPreHil) ↔ ((𝑊 ∈ CMetSp ∧ 𝐾 ∈ {ℝ, ℂ}) ∧ 𝑊 ∈ ℂPreHil)) | |
3 | 3ancomb 1098 | . . 3 ⊢ ((𝑊 ∈ CMetSp ∧ 𝑊 ∈ ℂPreHil ∧ 𝐾 ∈ {ℝ, ℂ}) ↔ (𝑊 ∈ CMetSp ∧ 𝐾 ∈ {ℝ, ℂ} ∧ 𝑊 ∈ ℂPreHil)) | |
4 | cphnvc 25223 | . . . . . 6 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmVec) | |
5 | hlress.f | . . . . . . . . 9 ⊢ 𝐹 = (Scalar‘𝑊) | |
6 | 5 | isbn 25385 | . . . . . . . 8 ⊢ (𝑊 ∈ Ban ↔ (𝑊 ∈ NrmVec ∧ 𝑊 ∈ CMetSp ∧ 𝐹 ∈ CMetSp)) |
7 | 3anass 1094 | . . . . . . . 8 ⊢ ((𝑊 ∈ NrmVec ∧ 𝑊 ∈ CMetSp ∧ 𝐹 ∈ CMetSp) ↔ (𝑊 ∈ NrmVec ∧ (𝑊 ∈ CMetSp ∧ 𝐹 ∈ CMetSp))) | |
8 | 6, 7 | bitri 275 | . . . . . . 7 ⊢ (𝑊 ∈ Ban ↔ (𝑊 ∈ NrmVec ∧ (𝑊 ∈ CMetSp ∧ 𝐹 ∈ CMetSp))) |
9 | 8 | baib 535 | . . . . . 6 ⊢ (𝑊 ∈ NrmVec → (𝑊 ∈ Ban ↔ (𝑊 ∈ CMetSp ∧ 𝐹 ∈ CMetSp))) |
10 | 4, 9 | syl 17 | . . . . 5 ⊢ (𝑊 ∈ ℂPreHil → (𝑊 ∈ Ban ↔ (𝑊 ∈ CMetSp ∧ 𝐹 ∈ CMetSp))) |
11 | hlress.k | . . . . . . . . 9 ⊢ 𝐾 = (Base‘𝐹) | |
12 | 5, 11 | cphsca 25226 | . . . . . . . 8 ⊢ (𝑊 ∈ ℂPreHil → 𝐹 = (ℂfld ↾s 𝐾)) |
13 | 12 | eleq1d 2823 | . . . . . . 7 ⊢ (𝑊 ∈ ℂPreHil → (𝐹 ∈ CMetSp ↔ (ℂfld ↾s 𝐾) ∈ CMetSp)) |
14 | 5, 11 | cphsubrg 25227 | . . . . . . . . 9 ⊢ (𝑊 ∈ ℂPreHil → 𝐾 ∈ (SubRing‘ℂfld)) |
15 | cphlvec 25222 | . . . . . . . . . . 11 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ LVec) | |
16 | 5 | lvecdrng 21121 | . . . . . . . . . . 11 ⊢ (𝑊 ∈ LVec → 𝐹 ∈ DivRing) |
17 | 15, 16 | syl 17 | . . . . . . . . . 10 ⊢ (𝑊 ∈ ℂPreHil → 𝐹 ∈ DivRing) |
18 | 12, 17 | eqeltrrd 2839 | . . . . . . . . 9 ⊢ (𝑊 ∈ ℂPreHil → (ℂfld ↾s 𝐾) ∈ DivRing) |
19 | eqid 2734 | . . . . . . . . . . 11 ⊢ (ℂfld ↾s 𝐾) = (ℂfld ↾s 𝐾) | |
20 | 19 | cncdrg 25406 | . . . . . . . . . 10 ⊢ ((𝐾 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝐾) ∈ DivRing ∧ (ℂfld ↾s 𝐾) ∈ CMetSp) → 𝐾 ∈ {ℝ, ℂ}) |
21 | 20 | 3expia 1120 | . . . . . . . . 9 ⊢ ((𝐾 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝐾) ∈ DivRing) → ((ℂfld ↾s 𝐾) ∈ CMetSp → 𝐾 ∈ {ℝ, ℂ})) |
22 | 14, 18, 21 | syl2anc 584 | . . . . . . . 8 ⊢ (𝑊 ∈ ℂPreHil → ((ℂfld ↾s 𝐾) ∈ CMetSp → 𝐾 ∈ {ℝ, ℂ})) |
23 | elpri 4653 | . . . . . . . . 9 ⊢ (𝐾 ∈ {ℝ, ℂ} → (𝐾 = ℝ ∨ 𝐾 = ℂ)) | |
24 | oveq2 7438 | . . . . . . . . . . 11 ⊢ (𝐾 = ℝ → (ℂfld ↾s 𝐾) = (ℂfld ↾s ℝ)) | |
25 | eqid 2734 | . . . . . . . . . . . . 13 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
26 | 25 | recld2 24849 | . . . . . . . . . . . 12 ⊢ ℝ ∈ (Clsd‘(TopOpen‘ℂfld)) |
27 | cncms 25402 | . . . . . . . . . . . . 13 ⊢ ℂfld ∈ CMetSp | |
28 | ax-resscn 11209 | . . . . . . . . . . . . 13 ⊢ ℝ ⊆ ℂ | |
29 | eqid 2734 | . . . . . . . . . . . . . 14 ⊢ (ℂfld ↾s ℝ) = (ℂfld ↾s ℝ) | |
30 | cnfldbas 21385 | . . . . . . . . . . . . . 14 ⊢ ℂ = (Base‘ℂfld) | |
31 | 29, 30, 25 | cmsss 25398 | . . . . . . . . . . . . 13 ⊢ ((ℂfld ∈ CMetSp ∧ ℝ ⊆ ℂ) → ((ℂfld ↾s ℝ) ∈ CMetSp ↔ ℝ ∈ (Clsd‘(TopOpen‘ℂfld)))) |
32 | 27, 28, 31 | mp2an 692 | . . . . . . . . . . . 12 ⊢ ((ℂfld ↾s ℝ) ∈ CMetSp ↔ ℝ ∈ (Clsd‘(TopOpen‘ℂfld))) |
33 | 26, 32 | mpbir 231 | . . . . . . . . . . 11 ⊢ (ℂfld ↾s ℝ) ∈ CMetSp |
34 | 24, 33 | eqeltrdi 2846 | . . . . . . . . . 10 ⊢ (𝐾 = ℝ → (ℂfld ↾s 𝐾) ∈ CMetSp) |
35 | oveq2 7438 | . . . . . . . . . . 11 ⊢ (𝐾 = ℂ → (ℂfld ↾s 𝐾) = (ℂfld ↾s ℂ)) | |
36 | 30 | ressid 17289 | . . . . . . . . . . . . 13 ⊢ (ℂfld ∈ CMetSp → (ℂfld ↾s ℂ) = ℂfld) |
37 | 27, 36 | ax-mp 5 | . . . . . . . . . . . 12 ⊢ (ℂfld ↾s ℂ) = ℂfld |
38 | 37, 27 | eqeltri 2834 | . . . . . . . . . . 11 ⊢ (ℂfld ↾s ℂ) ∈ CMetSp |
39 | 35, 38 | eqeltrdi 2846 | . . . . . . . . . 10 ⊢ (𝐾 = ℂ → (ℂfld ↾s 𝐾) ∈ CMetSp) |
40 | 34, 39 | jaoi 857 | . . . . . . . . 9 ⊢ ((𝐾 = ℝ ∨ 𝐾 = ℂ) → (ℂfld ↾s 𝐾) ∈ CMetSp) |
41 | 23, 40 | syl 17 | . . . . . . . 8 ⊢ (𝐾 ∈ {ℝ, ℂ} → (ℂfld ↾s 𝐾) ∈ CMetSp) |
42 | 22, 41 | impbid1 225 | . . . . . . 7 ⊢ (𝑊 ∈ ℂPreHil → ((ℂfld ↾s 𝐾) ∈ CMetSp ↔ 𝐾 ∈ {ℝ, ℂ})) |
43 | 13, 42 | bitrd 279 | . . . . . 6 ⊢ (𝑊 ∈ ℂPreHil → (𝐹 ∈ CMetSp ↔ 𝐾 ∈ {ℝ, ℂ})) |
44 | 43 | anbi2d 630 | . . . . 5 ⊢ (𝑊 ∈ ℂPreHil → ((𝑊 ∈ CMetSp ∧ 𝐹 ∈ CMetSp) ↔ (𝑊 ∈ CMetSp ∧ 𝐾 ∈ {ℝ, ℂ}))) |
45 | 10, 44 | bitrd 279 | . . . 4 ⊢ (𝑊 ∈ ℂPreHil → (𝑊 ∈ Ban ↔ (𝑊 ∈ CMetSp ∧ 𝐾 ∈ {ℝ, ℂ}))) |
46 | 45 | pm5.32ri 575 | . . 3 ⊢ ((𝑊 ∈ Ban ∧ 𝑊 ∈ ℂPreHil) ↔ ((𝑊 ∈ CMetSp ∧ 𝐾 ∈ {ℝ, ℂ}) ∧ 𝑊 ∈ ℂPreHil)) |
47 | 2, 3, 46 | 3bitr4ri 304 | . 2 ⊢ ((𝑊 ∈ Ban ∧ 𝑊 ∈ ℂPreHil) ↔ (𝑊 ∈ CMetSp ∧ 𝑊 ∈ ℂPreHil ∧ 𝐾 ∈ {ℝ, ℂ})) |
48 | 1, 47 | bitri 275 | 1 ⊢ (𝑊 ∈ ℂHil ↔ (𝑊 ∈ CMetSp ∧ 𝑊 ∈ ℂPreHil ∧ 𝐾 ∈ {ℝ, ℂ})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1536 ∈ wcel 2105 ⊆ wss 3962 {cpr 4632 ‘cfv 6562 (class class class)co 7430 ℂcc 11150 ℝcr 11151 Basecbs 17244 ↾s cress 17273 Scalarcsca 17300 TopOpenctopn 17467 SubRingcsubrg 20585 DivRingcdr 20745 LVecclvec 21118 ℂfldccnfld 21381 Clsdccld 23039 NrmVeccnvc 24609 ℂPreHilccph 25213 CMetSpccms 25379 Bancbn 25380 ℂHilchl 25381 |
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 ax-addf 11231 ax-mulf 11232 |
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-tpos 8249 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-ioo 13387 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-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 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-mgm 18665 df-sgrp 18744 df-mnd 18760 df-submnd 18809 df-grp 18966 df-minusg 18967 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-cring 20253 df-oppr 20350 df-dvdsr 20373 df-unit 20374 df-invr 20404 df-dvr 20417 df-subrng 20562 df-subrg 20586 df-drng 20747 df-lvec 21119 df-psmet 21373 df-xmet 21374 df-met 21375 df-bl 21376 df-mopn 21377 df-fbas 21378 df-fg 21379 df-cnfld 21382 df-phl 21661 df-top 22915 df-topon 22932 df-topsp 22954 df-bases 22968 df-cld 23042 df-ntr 23043 df-cls 23044 df-nei 23121 df-cn 23250 df-cnp 23251 df-haus 23338 df-cmp 23410 df-tx 23585 df-hmeo 23778 df-fil 23869 df-flim 23962 df-fcls 23964 df-xms 24345 df-ms 24346 df-tms 24347 df-nvc 24615 df-cncf 24917 df-cph 25215 df-cfil 25302 df-cmet 25304 df-cms 25382 df-bn 25383 df-hl 25384 |
This theorem is referenced by: (None) |
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