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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 23965 | . 2 ⊢ (𝑊 ∈ ℂHil ↔ (𝑊 ∈ Ban ∧ 𝑊 ∈ ℂPreHil)) | |
2 | df-3an 1085 | . . 3 ⊢ ((𝑊 ∈ CMetSp ∧ 𝐾 ∈ {ℝ, ℂ} ∧ 𝑊 ∈ ℂPreHil) ↔ ((𝑊 ∈ CMetSp ∧ 𝐾 ∈ {ℝ, ℂ}) ∧ 𝑊 ∈ ℂPreHil)) | |
3 | 3ancomb 1095 | . . 3 ⊢ ((𝑊 ∈ CMetSp ∧ 𝑊 ∈ ℂPreHil ∧ 𝐾 ∈ {ℝ, ℂ}) ↔ (𝑊 ∈ CMetSp ∧ 𝐾 ∈ {ℝ, ℂ} ∧ 𝑊 ∈ ℂPreHil)) | |
4 | cphnvc 23780 | . . . . . 6 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmVec) | |
5 | hlress.f | . . . . . . . . 9 ⊢ 𝐹 = (Scalar‘𝑊) | |
6 | 5 | isbn 23941 | . . . . . . . 8 ⊢ (𝑊 ∈ Ban ↔ (𝑊 ∈ NrmVec ∧ 𝑊 ∈ CMetSp ∧ 𝐹 ∈ CMetSp)) |
7 | 3anass 1091 | . . . . . . . 8 ⊢ ((𝑊 ∈ NrmVec ∧ 𝑊 ∈ CMetSp ∧ 𝐹 ∈ CMetSp) ↔ (𝑊 ∈ NrmVec ∧ (𝑊 ∈ CMetSp ∧ 𝐹 ∈ CMetSp))) | |
8 | 6, 7 | bitri 277 | . . . . . . 7 ⊢ (𝑊 ∈ Ban ↔ (𝑊 ∈ NrmVec ∧ (𝑊 ∈ CMetSp ∧ 𝐹 ∈ CMetSp))) |
9 | 8 | baib 538 | . . . . . 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 23783 | . . . . . . . 8 ⊢ (𝑊 ∈ ℂPreHil → 𝐹 = (ℂfld ↾s 𝐾)) |
13 | 12 | eleq1d 2897 | . . . . . . 7 ⊢ (𝑊 ∈ ℂPreHil → (𝐹 ∈ CMetSp ↔ (ℂfld ↾s 𝐾) ∈ CMetSp)) |
14 | 5, 11 | cphsubrg 23784 | . . . . . . . . 9 ⊢ (𝑊 ∈ ℂPreHil → 𝐾 ∈ (SubRing‘ℂfld)) |
15 | cphlvec 23779 | . . . . . . . . . . 11 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ LVec) | |
16 | 5 | lvecdrng 19877 | . . . . . . . . . . 11 ⊢ (𝑊 ∈ LVec → 𝐹 ∈ DivRing) |
17 | 15, 16 | syl 17 | . . . . . . . . . 10 ⊢ (𝑊 ∈ ℂPreHil → 𝐹 ∈ DivRing) |
18 | 12, 17 | eqeltrrd 2914 | . . . . . . . . 9 ⊢ (𝑊 ∈ ℂPreHil → (ℂfld ↾s 𝐾) ∈ DivRing) |
19 | eqid 2821 | . . . . . . . . . . 11 ⊢ (ℂfld ↾s 𝐾) = (ℂfld ↾s 𝐾) | |
20 | 19 | cncdrg 23962 | . . . . . . . . . 10 ⊢ ((𝐾 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝐾) ∈ DivRing ∧ (ℂfld ↾s 𝐾) ∈ CMetSp) → 𝐾 ∈ {ℝ, ℂ}) |
21 | 20 | 3expia 1117 | . . . . . . . . 9 ⊢ ((𝐾 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝐾) ∈ DivRing) → ((ℂfld ↾s 𝐾) ∈ CMetSp → 𝐾 ∈ {ℝ, ℂ})) |
22 | 14, 18, 21 | syl2anc 586 | . . . . . . . 8 ⊢ (𝑊 ∈ ℂPreHil → ((ℂfld ↾s 𝐾) ∈ CMetSp → 𝐾 ∈ {ℝ, ℂ})) |
23 | elpri 4589 | . . . . . . . . 9 ⊢ (𝐾 ∈ {ℝ, ℂ} → (𝐾 = ℝ ∨ 𝐾 = ℂ)) | |
24 | oveq2 7164 | . . . . . . . . . . 11 ⊢ (𝐾 = ℝ → (ℂfld ↾s 𝐾) = (ℂfld ↾s ℝ)) | |
25 | eqid 2821 | . . . . . . . . . . . . 13 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
26 | 25 | recld2 23422 | . . . . . . . . . . . 12 ⊢ ℝ ∈ (Clsd‘(TopOpen‘ℂfld)) |
27 | cncms 23958 | . . . . . . . . . . . . 13 ⊢ ℂfld ∈ CMetSp | |
28 | ax-resscn 10594 | . . . . . . . . . . . . 13 ⊢ ℝ ⊆ ℂ | |
29 | eqid 2821 | . . . . . . . . . . . . . 14 ⊢ (ℂfld ↾s ℝ) = (ℂfld ↾s ℝ) | |
30 | cnfldbas 20549 | . . . . . . . . . . . . . 14 ⊢ ℂ = (Base‘ℂfld) | |
31 | 29, 30, 25 | cmsss 23954 | . . . . . . . . . . . . 13 ⊢ ((ℂfld ∈ CMetSp ∧ ℝ ⊆ ℂ) → ((ℂfld ↾s ℝ) ∈ CMetSp ↔ ℝ ∈ (Clsd‘(TopOpen‘ℂfld)))) |
32 | 27, 28, 31 | mp2an 690 | . . . . . . . . . . . 12 ⊢ ((ℂfld ↾s ℝ) ∈ CMetSp ↔ ℝ ∈ (Clsd‘(TopOpen‘ℂfld))) |
33 | 26, 32 | mpbir 233 | . . . . . . . . . . 11 ⊢ (ℂfld ↾s ℝ) ∈ CMetSp |
34 | 24, 33 | eqeltrdi 2921 | . . . . . . . . . 10 ⊢ (𝐾 = ℝ → (ℂfld ↾s 𝐾) ∈ CMetSp) |
35 | oveq2 7164 | . . . . . . . . . . 11 ⊢ (𝐾 = ℂ → (ℂfld ↾s 𝐾) = (ℂfld ↾s ℂ)) | |
36 | 30 | ressid 16559 | . . . . . . . . . . . . 13 ⊢ (ℂfld ∈ CMetSp → (ℂfld ↾s ℂ) = ℂfld) |
37 | 27, 36 | ax-mp 5 | . . . . . . . . . . . 12 ⊢ (ℂfld ↾s ℂ) = ℂfld |
38 | 37, 27 | eqeltri 2909 | . . . . . . . . . . 11 ⊢ (ℂfld ↾s ℂ) ∈ CMetSp |
39 | 35, 38 | eqeltrdi 2921 | . . . . . . . . . 10 ⊢ (𝐾 = ℂ → (ℂfld ↾s 𝐾) ∈ CMetSp) |
40 | 34, 39 | jaoi 853 | . . . . . . . . 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 227 | . . . . . . 7 ⊢ (𝑊 ∈ ℂPreHil → ((ℂfld ↾s 𝐾) ∈ CMetSp ↔ 𝐾 ∈ {ℝ, ℂ})) |
43 | 13, 42 | bitrd 281 | . . . . . 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 281 | . . . 4 ⊢ (𝑊 ∈ ℂPreHil → (𝑊 ∈ Ban ↔ (𝑊 ∈ CMetSp ∧ 𝐾 ∈ {ℝ, ℂ}))) |
46 | 45 | pm5.32ri 578 | . . 3 ⊢ ((𝑊 ∈ Ban ∧ 𝑊 ∈ ℂPreHil) ↔ ((𝑊 ∈ CMetSp ∧ 𝐾 ∈ {ℝ, ℂ}) ∧ 𝑊 ∈ ℂPreHil)) |
47 | 2, 3, 46 | 3bitr4ri 306 | . 2 ⊢ ((𝑊 ∈ Ban ∧ 𝑊 ∈ ℂPreHil) ↔ (𝑊 ∈ CMetSp ∧ 𝑊 ∈ ℂPreHil ∧ 𝐾 ∈ {ℝ, ℂ})) |
48 | 1, 47 | bitri 277 | 1 ⊢ (𝑊 ∈ ℂHil ↔ (𝑊 ∈ CMetSp ∧ 𝑊 ∈ ℂPreHil ∧ 𝐾 ∈ {ℝ, ℂ})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ⊆ wss 3936 {cpr 4569 ‘cfv 6355 (class class class)co 7156 ℂcc 10535 ℝcr 10536 Basecbs 16483 ↾s cress 16484 Scalarcsca 16568 TopOpenctopn 16695 DivRingcdr 19502 SubRingcsubrg 19531 LVecclvec 19874 ℂfldccnfld 20545 Clsdccld 21624 NrmVeccnvc 23191 ℂPreHilccph 23770 CMetSpccms 23935 Bancbn 23936 ℂHilchl 23937 |
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 ax-addf 10616 ax-mulf 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 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-se 5515 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-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-tpos 7892 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-map 8408 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-fi 8875 df-sup 8906 df-inf 8907 df-oi 8974 df-card 9368 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-ioo 12743 df-ico 12745 df-icc 12746 df-fz 12894 df-fzo 13035 df-seq 13371 df-exp 13431 df-hash 13692 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-starv 16580 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-hom 16589 df-cco 16590 df-rest 16696 df-topn 16697 df-0g 16715 df-gsum 16716 df-topgen 16717 df-pt 16718 df-prds 16721 df-xrs 16775 df-qtop 16780 df-imas 16781 df-xps 16783 df-mre 16857 df-mrc 16858 df-acs 16860 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-submnd 17957 df-grp 18106 df-minusg 18107 df-mulg 18225 df-subg 18276 df-cntz 18447 df-cmn 18908 df-mgp 19240 df-ur 19252 df-ring 19299 df-cring 19300 df-oppr 19373 df-dvdsr 19391 df-unit 19392 df-invr 19422 df-dvr 19433 df-drng 19504 df-subrg 19533 df-lvec 19875 df-psmet 20537 df-xmet 20538 df-met 20539 df-bl 20540 df-mopn 20541 df-fbas 20542 df-fg 20543 df-cnfld 20546 df-phl 20770 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-cn 21835 df-cnp 21836 df-haus 21923 df-cmp 21995 df-tx 22170 df-hmeo 22363 df-fil 22454 df-flim 22547 df-fcls 22549 df-xms 22930 df-ms 22931 df-tms 22932 df-nvc 23197 df-cncf 23486 df-cph 23772 df-cfil 23858 df-cmet 23860 df-cms 23938 df-bn 23939 df-hl 23940 |
This theorem is referenced by: (None) |
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