ishl2 - Metamath Proof Explorer

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Theorem ishl2 25529

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 25521	. 2 ⊢ (𝑊 ∈ ℂHil ↔ (𝑊 ∈ Ban ∧ 𝑊 ∈ ℂPreHil))
2		df-3an 1105	. . 3 ⊢ ((𝑊 ∈ CMetSp ∧ 𝐾 ∈ {ℝ, ℂ} ∧ 𝑊 ∈ ℂPreHil) ↔ ((𝑊 ∈ CMetSp ∧ 𝐾 ∈ {ℝ, ℂ}) ∧ 𝑊 ∈ ℂPreHil))
3		3ancomb 1116	. . 3 ⊢ ((𝑊 ∈ CMetSp ∧ 𝑊 ∈ ℂPreHil ∧ 𝐾 ∈ {ℝ, ℂ}) ↔ (𝑊 ∈ CMetSp ∧ 𝐾 ∈ {ℝ, ℂ} ∧ 𝑊 ∈ ℂPreHil))
4		cphnvc 25335	. . . . . 6 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmVec)
5		hlress.f	. . . . . . . . 9 ⊢ 𝐹 = (Scalar‘𝑊)
6	5	isbn 25497	. . . . . . . 8 ⊢ (𝑊 ∈ Ban ↔ (𝑊 ∈ NrmVec ∧ 𝑊 ∈ CMetSp ∧ 𝐹 ∈ CMetSp))
7		3anass 1111	. . . . . . . 8 ⊢ ((𝑊 ∈ NrmVec ∧ 𝑊 ∈ CMetSp ∧ 𝐹 ∈ CMetSp) ↔ (𝑊 ∈ NrmVec ∧ (𝑊 ∈ CMetSp ∧ 𝐹 ∈ CMetSp)))
8	6, 7	bitri 278	. . . . . . 7 ⊢ (𝑊 ∈ Ban ↔ (𝑊 ∈ NrmVec ∧ (𝑊 ∈ CMetSp ∧ 𝐹 ∈ CMetSp)))
9	8	baib 544	. . . . . 6 ⊢ (𝑊 ∈ NrmVec → (𝑊 ∈ Ban ↔ (𝑊 ∈ CMetSp ∧ 𝐹 ∈ CMetSp)))
10	4, 9	syl 18	. . . . 5 ⊢ (𝑊 ∈ ℂPreHil → (𝑊 ∈ Ban ↔ (𝑊 ∈ CMetSp ∧ 𝐹 ∈ CMetSp)))
11		hlress.k	. . . . . . . . 9 ⊢ 𝐾 = (Base‘𝐹)
12	5, 11	cphsca 25338	. . . . . . . 8 ⊢ (𝑊 ∈ ℂPreHil → 𝐹 = (ℂ_fld ↾_s 𝐾))
13	12	eleq1d 2848	. . . . . . 7 ⊢ (𝑊 ∈ ℂPreHil → (𝐹 ∈ CMetSp ↔ (ℂ_fld ↾_s 𝐾) ∈ CMetSp))
14	5, 11	cphsubrg 25339	. . . . . . . . 9 ⊢ (𝑊 ∈ ℂPreHil → 𝐾 ∈ (SubRing‘ℂ_fld))
15		cphlvec 25334	. . . . . . . . . . 11 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ LVec)
16	5	lvecdrng 21226	. . . . . . . . . . 11 ⊢ (𝑊 ∈ LVec → 𝐹 ∈ DivRing)
17	15, 16	syl 18	. . . . . . . . . 10 ⊢ (𝑊 ∈ ℂPreHil → 𝐹 ∈ DivRing)
18	12, 17	eqeltrrd 2864	. . . . . . . . 9 ⊢ (𝑊 ∈ ℂPreHil → (ℂ_fld ↾_s 𝐾) ∈ DivRing)
19		eqid 2763	. . . . . . . . . . 11 ⊢ (ℂ_fld ↾_s 𝐾) = (ℂ_fld ↾_s 𝐾)
20	19	cncdrg 25518	. . . . . . . . . 10 ⊢ ((𝐾 ∈ (SubRing‘ℂ_fld) ∧ (ℂ_fld ↾_s 𝐾) ∈ DivRing ∧ (ℂ_fld ↾_s 𝐾) ∈ CMetSp) → 𝐾 ∈ {ℝ, ℂ})
21	20	3expia 1139	. . . . . . . . 9 ⊢ ((𝐾 ∈ (SubRing‘ℂ_fld) ∧ (ℂ_fld ↾_s 𝐾) ∈ DivRing) → ((ℂ_fld ↾_s 𝐾) ∈ CMetSp → 𝐾 ∈ {ℝ, ℂ}))
22	14, 18, 21	syl2anc 595	. . . . . . . 8 ⊢ (𝑊 ∈ ℂPreHil → ((ℂ_fld ↾_s 𝐾) ∈ CMetSp → 𝐾 ∈ {ℝ, ℂ}))
23		elpri 4613	. . . . . . . . 9 ⊢ (𝐾 ∈ {ℝ, ℂ} → (𝐾 = ℝ ∨ 𝐾 = ℂ))
24		oveq2 7418	. . . . . . . . . . 11 ⊢ (𝐾 = ℝ → (ℂ_fld ↾_s 𝐾) = (ℂ_fld ↾_s ℝ))
25		eqid 2763	. . . . . . . . . . . . 13 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
26	25	recld2 24972	. . . . . . . . . . . 12 ⊢ ℝ ∈ (Clsd‘(TopOpen‘ℂ_fld))
27		cncms 25514	. . . . . . . . . . . . 13 ⊢ ℂ_fld ∈ CMetSp
28		ax-resscn 11152	. . . . . . . . . . . . 13 ⊢ ℝ ⊆ ℂ
29		eqid 2763	. . . . . . . . . . . . . 14 ⊢ (ℂ_fld ↾_s ℝ) = (ℂ_fld ↾_s ℝ)
30		cnfldbas 21526	. . . . . . . . . . . . . 14 ⊢ ℂ = (Base‘ℂ_fld)
31	29, 30, 25	cmsss 25510	. . . . . . . . . . . . 13 ⊢ ((ℂ_fld ∈ CMetSp ∧ ℝ ⊆ ℂ) → ((ℂ_fld ↾_s ℝ) ∈ CMetSp ↔ ℝ ∈ (Clsd‘(TopOpen‘ℂ_fld))))
32	27, 28, 31	mp2an 704	. . . . . . . . . . . 12 ⊢ ((ℂ_fld ↾_s ℝ) ∈ CMetSp ↔ ℝ ∈ (Clsd‘(TopOpen‘ℂ_fld)))
33	26, 32	mpbir 234	. . . . . . . . . . 11 ⊢ (ℂ_fld ↾_s ℝ) ∈ CMetSp
34	24, 33	eqeltrdi 2871	. . . . . . . . . 10 ⊢ (𝐾 = ℝ → (ℂ_fld ↾_s 𝐾) ∈ CMetSp)
35		oveq2 7418	. . . . . . . . . . 11 ⊢ (𝐾 = ℂ → (ℂ_fld ↾_s 𝐾) = (ℂ_fld ↾_s ℂ))
36	30	ressid 17299	. . . . . . . . . . . . 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 2859	. . . . . . . . . . 11 ⊢ (ℂ_fld ↾_s ℂ) ∈ CMetSp
39	35, 38	eqeltrdi 2871	. . . . . . . . . 10 ⊢ (𝐾 = ℂ → (ℂ_fld ↾_s 𝐾) ∈ CMetSp)
40	34, 39	jaoi 870	. . . . . . . . 9 ⊢ ((𝐾 = ℝ ∨ 𝐾 = ℂ) → (ℂ_fld ↾_s 𝐾) ∈ CMetSp)
41	23, 40	syl 18	. . . . . . . 8 ⊢ (𝐾 ∈ {ℝ, ℂ} → (ℂ_fld ↾_s 𝐾) ∈ CMetSp)
42	22, 41	impbid1 228	. . . . . . 7 ⊢ (𝑊 ∈ ℂPreHil → ((ℂ_fld ↾_s 𝐾) ∈ CMetSp ↔ 𝐾 ∈ {ℝ, ℂ}))
43	13, 42	bitrd 282	. . . . . 6 ⊢ (𝑊 ∈ ℂPreHil → (𝐹 ∈ CMetSp ↔ 𝐾 ∈ {ℝ, ℂ}))
44	43	anbi2d 641	. . . . 5 ⊢ (𝑊 ∈ ℂPreHil → ((𝑊 ∈ CMetSp ∧ 𝐹 ∈ CMetSp) ↔ (𝑊 ∈ CMetSp ∧ 𝐾 ∈ {ℝ, ℂ})))
45	10, 44	bitrd 282	. . . 4 ⊢ (𝑊 ∈ ℂPreHil → (𝑊 ∈ Ban ↔ (𝑊 ∈ CMetSp ∧ 𝐾 ∈ {ℝ, ℂ})))
46	45	pm5.32ri 585	. . 3 ⊢ ((𝑊 ∈ Ban ∧ 𝑊 ∈ ℂPreHil) ↔ ((𝑊 ∈ CMetSp ∧ 𝐾 ∈ {ℝ, ℂ}) ∧ 𝑊 ∈ ℂPreHil))
47	2, 3, 46	3bitr4ri 307	. 2 ⊢ ((𝑊 ∈ Ban ∧ 𝑊 ∈ ℂPreHil) ↔ (𝑊 ∈ CMetSp ∧ 𝑊 ∈ ℂPreHil ∧ 𝐾 ∈ {ℝ, ℂ}))
48	1, 47	bitri 278	1 ⊢ (𝑊 ∈ ℂHil ↔ (𝑊 ∈ CMetSp ∧ 𝑊 ∈ ℂPreHil ∧ 𝐾 ∈ {ℝ, ℂ}))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 ∨ wo 860 ∧ w3a 1103 = wceq 1570 ∈ wcel 2143 ⊆ wss 3905 {cpr 4591 ‘cfv 6536 (class class class)co 7410 ℂcc 11093 ℝcr 11094 Basecbs 17264 ↾_s cress 17285 Scalarcsca 17308 TopOpenctopn 17469 SubRingcsubrg 20668 DivRingcdr 20827 LVecclvec 21223 ℂ_fldccnfld 21522 Clsdccld 23173 NrmVeccnvc 24738 ℂPreHilccph 25325 CMetSpccms 25491 Bancbn 25492 ℂHilchl 25493

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-cnex 11151 ax-resscn 11152 ax-1cn 11153 ax-icn 11154 ax-addcl 11155 ax-addrcl 11156 ax-mulcl 11157 ax-mulrcl 11158 ax-mulcom 11159 ax-addass 11160 ax-mulass 11161 ax-distr 11162 ax-i2m1 11163 ax-1ne0 11164 ax-1rid 11165 ax-rnegex 11166 ax-rrecex 11167 ax-cnre 11168 ax-pre-lttri 11169 ax-pre-lttrn 11170 ax-pre-ltadd 11171 ax-pre-mulgt0 11172 ax-pre-sup 11173 ax-addf 11174 ax-mulf 11175

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4873 df-int 4913 df-iun 4958 df-iin 4959 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-se 5615 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7674 df-om 7859 df-1st 7982 df-2nd 7983 df-supp 8153 df-tpos 8218 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-2o 8450 df-er 8690 df-map 8822 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9318 df-fi 9367 df-sup 9398 df-inf 9399 df-oi 9468 df-card 9921 df-pnf 11240 df-mnf 11241 df-xr 11242 df-ltxr 11243 df-le 11244 df-sub 11438 df-neg 11439 df-div 11867 df-nn 12229 df-2 12298 df-3 12299 df-4 12300 df-5 12301 df-6 12302 df-7 12303 df-8 12304 df-9 12305 df-n0 12500 df-z 12587 df-dec 12707 df-uz 12858 df-q 12968 df-rp 13012 df-xneg 13132 df-xadd 13133 df-xmul 13134 df-ioo 13371 df-ico 13373 df-icc 13374 df-fz 13531 df-fzo 13679 df-seq 14034 df-exp 14094 df-hash 14363 df-cj 15146 df-re 15147 df-im 15148 df-sqrt 15282 df-abs 15283 df-struct 17202 df-sets 17219 df-slot 17237 df-ndx 17249 df-base 17265 df-ress 17286 df-plusg 17318 df-mulr 17319 df-starv 17320 df-sca 17321 df-vsca 17322 df-ip 17323 df-tset 17324 df-ple 17325 df-ds 17327 df-unif 17328 df-hom 17329 df-cco 17330 df-rest 17470 df-topn 17471 df-0g 17489 df-gsum 17490 df-topgen 17491 df-pt 17492 df-prds 17495 df-xrs 17551 df-qtop 17556 df-imas 17557 df-xps 17559 df-mre 17633 df-mrc 17634 df-acs 17636 df-mgm 18693 df-sgrp 18772 df-mnd 18788 df-submnd 18837 df-grp 18998 df-minusg 18999 df-mulg 19129 df-subg 19184 df-cntz 19382 df-cmn 19847 df-abl 19848 df-mgp 20212 df-rng 20226 df-ur 20259 df-ring 20312 df-cring 20313 df-oppr 20415 df-dvdsr 20435 df-unit 20436 df-invr 20466 df-dvr 20479 df-subrng 20645 df-subrg 20669 df-drng 20829 df-lvec 21224 df-psmet 21514 df-xmet 21515 df-met 21516 df-bl 21517 df-mopn 21518 df-fbas 21519 df-fg 21520 df-cnfld 21523 df-phl 21776 df-top 23051 df-topon 23068 df-topsp 23090 df-bases 23103 df-cld 23176 df-ntr 23177 df-cls 23178 df-nei 23255 df-cn 23384 df-cnp 23385 df-haus 23472 df-cmp 23544 df-tx 23719 df-hmeo 23912 df-fil 24003 df-flim 24096 df-fcls 24098 df-xms 24477 df-ms 24478 df-tms 24479 df-nvc 24744 df-cncf 25037 df-cph 25327 df-cfil 25414 df-cmet 25416 df-cms 25494 df-bn 25495 df-hl 25496

This theorem is referenced by: (None)

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