ishl2 - Metamath Proof Explorer

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Theorem ishl2 24515

Description: A Hilbert space is a complete subcomplex pre-Hilbert space over ℝ or ℂ. (Contributed by Mario Carneiro, 15-Oct-2015.)

**Hypotheses**
Ref	Expression
hlress.f	⊢ 𝐹 = (Scalar‘𝑊)
hlress.k	⊢ 𝐾 = (Base‘𝐹)

**Assertion**
Ref	Expression
ishl2	⊢ (𝑊 ∈ ℂHil ↔ (𝑊 ∈ CMetSp ∧ 𝑊 ∈ ℂPreHil ∧ 𝐾 ∈ {ℝ, ℂ}))

**Proof of Theorem ishl2**
Step	Hyp	Ref	Expression
1		ishl 24507	. 2 ⊢ (𝑊 ∈ ℂHil ↔ (𝑊 ∈ Ban ∧ 𝑊 ∈ ℂPreHil))
2		df-3an 1087	. . 3 ⊢ ((𝑊 ∈ CMetSp ∧ 𝐾 ∈ {ℝ, ℂ} ∧ 𝑊 ∈ ℂPreHil) ↔ ((𝑊 ∈ CMetSp ∧ 𝐾 ∈ {ℝ, ℂ}) ∧ 𝑊 ∈ ℂPreHil))
3		3ancomb 1097	. . 3 ⊢ ((𝑊 ∈ CMetSp ∧ 𝑊 ∈ ℂPreHil ∧ 𝐾 ∈ {ℝ, ℂ}) ↔ (𝑊 ∈ CMetSp ∧ 𝐾 ∈ {ℝ, ℂ} ∧ 𝑊 ∈ ℂPreHil))
4		cphnvc 24321	. . . . . 6 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmVec)
5		hlress.f	. . . . . . . . 9 ⊢ 𝐹 = (Scalar‘𝑊)
6	5	isbn 24483	. . . . . . . 8 ⊢ (𝑊 ∈ Ban ↔ (𝑊 ∈ NrmVec ∧ 𝑊 ∈ CMetSp ∧ 𝐹 ∈ CMetSp))
7		3anass 1093	. . . . . . . 8 ⊢ ((𝑊 ∈ NrmVec ∧ 𝑊 ∈ CMetSp ∧ 𝐹 ∈ CMetSp) ↔ (𝑊 ∈ NrmVec ∧ (𝑊 ∈ CMetSp ∧ 𝐹 ∈ CMetSp)))
8	6, 7	bitri 274	. . . . . . 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 24324	. . . . . . . 8 ⊢ (𝑊 ∈ ℂPreHil → 𝐹 = (ℂ_fld ↾_s 𝐾))
13	12	eleq1d 2824	. . . . . . 7 ⊢ (𝑊 ∈ ℂPreHil → (𝐹 ∈ CMetSp ↔ (ℂ_fld ↾_s 𝐾) ∈ CMetSp))
14	5, 11	cphsubrg 24325	. . . . . . . . 9 ⊢ (𝑊 ∈ ℂPreHil → 𝐾 ∈ (SubRing‘ℂ_fld))
15		cphlvec 24320	. . . . . . . . . . 11 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ LVec)
16	5	lvecdrng 20348	. . . . . . . . . . 11 ⊢ (𝑊 ∈ LVec → 𝐹 ∈ DivRing)
17	15, 16	syl 17	. . . . . . . . . 10 ⊢ (𝑊 ∈ ℂPreHil → 𝐹 ∈ DivRing)
18	12, 17	eqeltrrd 2841	. . . . . . . . 9 ⊢ (𝑊 ∈ ℂPreHil → (ℂ_fld ↾_s 𝐾) ∈ DivRing)
19		eqid 2739	. . . . . . . . . . 11 ⊢ (ℂ_fld ↾_s 𝐾) = (ℂ_fld ↾_s 𝐾)
20	19	cncdrg 24504	. . . . . . . . . 10 ⊢ ((𝐾 ∈ (SubRing‘ℂ_fld) ∧ (ℂ_fld ↾_s 𝐾) ∈ DivRing ∧ (ℂ_fld ↾_s 𝐾) ∈ CMetSp) → 𝐾 ∈ {ℝ, ℂ})
21	20	3expia 1119	. . . . . . . . 9 ⊢ ((𝐾 ∈ (SubRing‘ℂ_fld) ∧ (ℂ_fld ↾_s 𝐾) ∈ DivRing) → ((ℂ_fld ↾_s 𝐾) ∈ CMetSp → 𝐾 ∈ {ℝ, ℂ}))
22	14, 18, 21	syl2anc 583	. . . . . . . 8 ⊢ (𝑊 ∈ ℂPreHil → ((ℂ_fld ↾_s 𝐾) ∈ CMetSp → 𝐾 ∈ {ℝ, ℂ}))
23		elpri 4588	. . . . . . . . 9 ⊢ (𝐾 ∈ {ℝ, ℂ} → (𝐾 = ℝ ∨ 𝐾 = ℂ))
24		oveq2 7276	. . . . . . . . . . 11 ⊢ (𝐾 = ℝ → (ℂ_fld ↾_s 𝐾) = (ℂ_fld ↾_s ℝ))
25		eqid 2739	. . . . . . . . . . . . 13 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
26	25	recld2 23958	. . . . . . . . . . . 12 ⊢ ℝ ∈ (Clsd‘(TopOpen‘ℂ_fld))
27		cncms 24500	. . . . . . . . . . . . 13 ⊢ ℂ_fld ∈ CMetSp
28		ax-resscn 10912	. . . . . . . . . . . . 13 ⊢ ℝ ⊆ ℂ
29		eqid 2739	. . . . . . . . . . . . . 14 ⊢ (ℂ_fld ↾_s ℝ) = (ℂ_fld ↾_s ℝ)
30		cnfldbas 20582	. . . . . . . . . . . . . 14 ⊢ ℂ = (Base‘ℂ_fld)
31	29, 30, 25	cmsss 24496	. . . . . . . . . . . . 13 ⊢ ((ℂ_fld ∈ CMetSp ∧ ℝ ⊆ ℂ) → ((ℂ_fld ↾_s ℝ) ∈ CMetSp ↔ ℝ ∈ (Clsd‘(TopOpen‘ℂ_fld))))
32	27, 28, 31	mp2an 688	. . . . . . . . . . . 12 ⊢ ((ℂ_fld ↾_s ℝ) ∈ CMetSp ↔ ℝ ∈ (Clsd‘(TopOpen‘ℂ_fld)))
33	26, 32	mpbir 230	. . . . . . . . . . 11 ⊢ (ℂ_fld ↾_s ℝ) ∈ CMetSp
34	24, 33	eqeltrdi 2848	. . . . . . . . . 10 ⊢ (𝐾 = ℝ → (ℂ_fld ↾_s 𝐾) ∈ CMetSp)
35		oveq2 7276	. . . . . . . . . . 11 ⊢ (𝐾 = ℂ → (ℂ_fld ↾_s 𝐾) = (ℂ_fld ↾_s ℂ))
36	30	ressid 16935	. . . . . . . . . . . . 13 ⊢ (ℂ_fld ∈ CMetSp → (ℂ_fld ↾_s ℂ) = ℂ_fld)
37	27, 36	ax-mp 5	. . . . . . . . . . . 12 ⊢ (ℂ_fld ↾_s ℂ) = ℂ_fld
38	37, 27	eqeltri 2836	. . . . . . . . . . 11 ⊢ (ℂ_fld ↾_s ℂ) ∈ CMetSp
39	35, 38	eqeltrdi 2848	. . . . . . . . . 10 ⊢ (𝐾 = ℂ → (ℂ_fld ↾_s 𝐾) ∈ CMetSp)
40	34, 39	jaoi 853	. . . . . . . . 9 ⊢ ((𝐾 = ℝ ∨ 𝐾 = ℂ) → (ℂ_fld ↾_s 𝐾) ∈ CMetSp)
41	23, 40	syl 17	. . . . . . . 8 ⊢ (𝐾 ∈ {ℝ, ℂ} → (ℂ_fld ↾_s 𝐾) ∈ CMetSp)
42	22, 41	impbid1 224	. . . . . . 7 ⊢ (𝑊 ∈ ℂPreHil → ((ℂ_fld ↾_s 𝐾) ∈ CMetSp ↔ 𝐾 ∈ {ℝ, ℂ}))
43	13, 42	bitrd 278	. . . . . 6 ⊢ (𝑊 ∈ ℂPreHil → (𝐹 ∈ CMetSp ↔ 𝐾 ∈ {ℝ, ℂ}))
44	43	anbi2d 628	. . . . 5 ⊢ (𝑊 ∈ ℂPreHil → ((𝑊 ∈ CMetSp ∧ 𝐹 ∈ CMetSp) ↔ (𝑊 ∈ CMetSp ∧ 𝐾 ∈ {ℝ, ℂ})))
45	10, 44	bitrd 278	. . . 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 303	. 2 ⊢ ((𝑊 ∈ Ban ∧ 𝑊 ∈ ℂPreHil) ↔ (𝑊 ∈ CMetSp ∧ 𝑊 ∈ ℂPreHil ∧ 𝐾 ∈ {ℝ, ℂ}))
48	1, 47	bitri 274	1 ⊢ (𝑊 ∈ ℂHil ↔ (𝑊 ∈ CMetSp ∧ 𝑊 ∈ ℂPreHil ∧ 𝐾 ∈ {ℝ, ℂ}))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∨ wo 843 ∧ w3a 1085 = wceq 1541 ∈ wcel 2109 ⊆ wss 3891 {cpr 4568 ‘cfv 6430 (class class class)co 7268 ℂcc 10853 ℝcr 10854 Basecbs 16893 ↾_s cress 16922 Scalarcsca 16946 TopOpenctopn 17113 DivRingcdr 19972 SubRingcsubrg 20001 LVecclvec 20345 ℂ_fldccnfld 20578 Clsdccld 22148 NrmVeccnvc 23718 ℂPreHilccph 24311 CMetSpccms 24477 Bancbn 24478 ℂHilchl 24479

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-rep 5213 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 ax-cnex 10911 ax-resscn 10912 ax-1cn 10913 ax-icn 10914 ax-addcl 10915 ax-addrcl 10916 ax-mulcl 10917 ax-mulrcl 10918 ax-mulcom 10919 ax-addass 10920 ax-mulass 10921 ax-distr 10922 ax-i2m1 10923 ax-1ne0 10924 ax-1rid 10925 ax-rnegex 10926 ax-rrecex 10927 ax-cnre 10928 ax-pre-lttri 10929 ax-pre-lttrn 10930 ax-pre-ltadd 10931 ax-pre-mulgt0 10932 ax-pre-sup 10933 ax-addf 10934 ax-mulf 10935

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-reu 3072 df-rmo 3073 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-pss 3910 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4845 df-int 4885 df-iun 4931 df-iin 4932 df-br 5079 df-opab 5141 df-mpt 5162 df-tr 5196 df-id 5488 df-eprel 5494 df-po 5502 df-so 5503 df-fr 5543 df-se 5544 df-we 5545 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-pred 6199 df-ord 6266 df-on 6267 df-lim 6268 df-suc 6269 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-isom 6439 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-of 7524 df-om 7701 df-1st 7817 df-2nd 7818 df-supp 7962 df-tpos 8026 df-frecs 8081 df-wrecs 8112 df-recs 8186 df-rdg 8225 df-1o 8281 df-2o 8282 df-er 8472 df-map 8591 df-ixp 8660 df-en 8708 df-dom 8709 df-sdom 8710 df-fin 8711 df-fsupp 9090 df-fi 9131 df-sup 9162 df-inf 9163 df-oi 9230 df-card 9681 df-pnf 10995 df-mnf 10996 df-xr 10997 df-ltxr 10998 df-le 10999 df-sub 11190 df-neg 11191 df-div 11616 df-nn 11957 df-2 12019 df-3 12020 df-4 12021 df-5 12022 df-6 12023 df-7 12024 df-8 12025 df-9 12026 df-n0 12217 df-z 12303 df-dec 12420 df-uz 12565 df-q 12671 df-rp 12713 df-xneg 12830 df-xadd 12831 df-xmul 12832 df-ioo 13065 df-ico 13067 df-icc 13068 df-fz 13222 df-fzo 13365 df-seq 13703 df-exp 13764 df-hash 14026 df-cj 14791 df-re 14792 df-im 14793 df-sqrt 14927 df-abs 14928 df-struct 16829 df-sets 16846 df-slot 16864 df-ndx 16876 df-base 16894 df-ress 16923 df-plusg 16956 df-mulr 16957 df-starv 16958 df-sca 16959 df-vsca 16960 df-ip 16961 df-tset 16962 df-ple 16963 df-ds 16965 df-unif 16966 df-hom 16967 df-cco 16968 df-rest 17114 df-topn 17115 df-0g 17133 df-gsum 17134 df-topgen 17135 df-pt 17136 df-prds 17139 df-xrs 17194 df-qtop 17199 df-imas 17200 df-xps 17202 df-mre 17276 df-mrc 17277 df-acs 17279 df-mgm 18307 df-sgrp 18356 df-mnd 18367 df-submnd 18412 df-grp 18561 df-minusg 18562 df-mulg 18682 df-subg 18733 df-cntz 18904 df-cmn 19369 df-mgp 19702 df-ur 19719 df-ring 19766 df-cring 19767 df-oppr 19843 df-dvdsr 19864 df-unit 19865 df-invr 19895 df-dvr 19906 df-drng 19974 df-subrg 20003 df-lvec 20346 df-psmet 20570 df-xmet 20571 df-met 20572 df-bl 20573 df-mopn 20574 df-fbas 20575 df-fg 20576 df-cnfld 20579 df-phl 20812 df-top 22024 df-topon 22041 df-topsp 22063 df-bases 22077 df-cld 22151 df-ntr 22152 df-cls 22153 df-nei 22230 df-cn 22359 df-cnp 22360 df-haus 22447 df-cmp 22519 df-tx 22694 df-hmeo 22887 df-fil 22978 df-flim 23071 df-fcls 23073 df-xms 23454 df-ms 23455 df-tms 23456 df-nvc 23724 df-cncf 24022 df-cph 24313 df-cfil 24400 df-cmet 24402 df-cms 24480 df-bn 24481 df-hl 24482

This theorem is referenced by: (None)

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