ishl2 - Metamath Proof Explorer

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Theorem ishl2 25270

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 25262	. 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 25076	. . . . . 6 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmVec)
5		hlress.f	. . . . . . . . 9 ⊢ 𝐹 = (Scalar‘𝑊)
6	5	isbn 25238	. . . . . . . 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 25079	. . . . . . . 8 ⊢ (𝑊 ∈ ℂPreHil → 𝐹 = (ℂ_fld ↾_s 𝐾))
13	12	eleq1d 2813	. . . . . . 7 ⊢ (𝑊 ∈ ℂPreHil → (𝐹 ∈ CMetSp ↔ (ℂ_fld ↾_s 𝐾) ∈ CMetSp))
14	5, 11	cphsubrg 25080	. . . . . . . . 9 ⊢ (𝑊 ∈ ℂPreHil → 𝐾 ∈ (SubRing‘ℂ_fld))
15		cphlvec 25075	. . . . . . . . . . 11 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ LVec)
16	5	lvecdrng 21012	. . . . . . . . . . 11 ⊢ (𝑊 ∈ LVec → 𝐹 ∈ DivRing)
17	15, 16	syl 17	. . . . . . . . . 10 ⊢ (𝑊 ∈ ℂPreHil → 𝐹 ∈ DivRing)
18	12, 17	eqeltrrd 2829	. . . . . . . . 9 ⊢ (𝑊 ∈ ℂPreHil → (ℂ_fld ↾_s 𝐾) ∈ DivRing)
19		eqid 2729	. . . . . . . . . . 11 ⊢ (ℂ_fld ↾_s 𝐾) = (ℂ_fld ↾_s 𝐾)
20	19	cncdrg 25259	. . . . . . . . . 10 ⊢ ((𝐾 ∈ (SubRing‘ℂ_fld) ∧ (ℂ_fld ↾_s 𝐾) ∈ DivRing ∧ (ℂ_fld ↾_s 𝐾) ∈ CMetSp) → 𝐾 ∈ {ℝ, ℂ})
21	20	3expia 1121	. . . . . . . . 9 ⊢ ((𝐾 ∈ (SubRing‘ℂ_fld) ∧ (ℂ_fld ↾_s 𝐾) ∈ DivRing) → ((ℂ_fld ↾_s 𝐾) ∈ CMetSp → 𝐾 ∈ {ℝ, ℂ}))
22	14, 18, 21	syl2anc 584	. . . . . . . 8 ⊢ (𝑊 ∈ ℂPreHil → ((ℂ_fld ↾_s 𝐾) ∈ CMetSp → 𝐾 ∈ {ℝ, ℂ}))
23		elpri 4613	. . . . . . . . 9 ⊢ (𝐾 ∈ {ℝ, ℂ} → (𝐾 = ℝ ∨ 𝐾 = ℂ))
24		oveq2 7395	. . . . . . . . . . 11 ⊢ (𝐾 = ℝ → (ℂ_fld ↾_s 𝐾) = (ℂ_fld ↾_s ℝ))
25		eqid 2729	. . . . . . . . . . . . 13 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
26	25	recld2 24703	. . . . . . . . . . . 12 ⊢ ℝ ∈ (Clsd‘(TopOpen‘ℂ_fld))
27		cncms 25255	. . . . . . . . . . . . 13 ⊢ ℂ_fld ∈ CMetSp
28		ax-resscn 11125	. . . . . . . . . . . . 13 ⊢ ℝ ⊆ ℂ
29		eqid 2729	. . . . . . . . . . . . . 14 ⊢ (ℂ_fld ↾_s ℝ) = (ℂ_fld ↾_s ℝ)
30		cnfldbas 21268	. . . . . . . . . . . . . 14 ⊢ ℂ = (Base‘ℂ_fld)
31	29, 30, 25	cmsss 25251	. . . . . . . . . . . . 13 ⊢ ((ℂ_fld ∈ CMetSp ∧ ℝ ⊆ ℂ) → ((ℂ_fld ↾_s ℝ) ∈ CMetSp ↔ ℝ ∈ (Clsd‘(TopOpen‘ℂ_fld))))
32	27, 28, 31	mp2an 692	. . . . . . . . . . . 12 ⊢ ((ℂ_fld ↾_s ℝ) ∈ CMetSp ↔ ℝ ∈ (Clsd‘(TopOpen‘ℂ_fld)))
33	26, 32	mpbir 231	. . . . . . . . . . 11 ⊢ (ℂ_fld ↾_s ℝ) ∈ CMetSp
34	24, 33	eqeltrdi 2836	. . . . . . . . . 10 ⊢ (𝐾 = ℝ → (ℂ_fld ↾_s 𝐾) ∈ CMetSp)
35		oveq2 7395	. . . . . . . . . . 11 ⊢ (𝐾 = ℂ → (ℂ_fld ↾_s 𝐾) = (ℂ_fld ↾_s ℂ))
36	30	ressid 17214	. . . . . . . . . . . . 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 2824	. . . . . . . . . . 11 ⊢ (ℂ_fld ↾_s ℂ) ∈ CMetSp
39	35, 38	eqeltrdi 2836	. . . . . . . . . 10 ⊢ (𝐾 = ℂ → (ℂ_fld ↾_s 𝐾) ∈ CMetSp)
40	34, 39	jaoi 857	. . . . . . . . 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 225	. . . . . . 7 ⊢ (𝑊 ∈ ℂPreHil → ((ℂ_fld ↾_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 1540 ∈ wcel 2109 ⊆ wss 3914 {cpr 4591 ‘cfv 6511 (class class class)co 7387 ℂcc 11066 ℝcr 11067 Basecbs 17179 ↾_s cress 17200 Scalarcsca 17223 TopOpenctopn 17384 SubRingcsubrg 20478 DivRingcdr 20638 LVecclvec 21009 ℂ_fldccnfld 21264 Clsdccld 22903 NrmVeccnvc 24469 ℂPreHilccph 25066 CMetSpccms 25232 Bancbn 25233 ℂHilchl 25234

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146 ax-addf 11147 ax-mulf 11148

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-iin 4958 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-se 5592 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-isom 6520 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-of 7653 df-om 7843 df-1st 7968 df-2nd 7969 df-supp 8140 df-tpos 8205 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-2o 8435 df-er 8671 df-map 8801 df-ixp 8871 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-fsupp 9313 df-fi 9362 df-sup 9393 df-inf 9394 df-oi 9463 df-card 9892 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-7 12254 df-8 12255 df-9 12256 df-n0 12443 df-z 12530 df-dec 12650 df-uz 12794 df-q 12908 df-rp 12952 df-xneg 13072 df-xadd 13073 df-xmul 13074 df-ioo 13310 df-ico 13312 df-icc 13313 df-fz 13469 df-fzo 13616 df-seq 13967 df-exp 14027 df-hash 14296 df-cj 15065 df-re 15066 df-im 15067 df-sqrt 15201 df-abs 15202 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-rest 17385 df-topn 17386 df-0g 17404 df-gsum 17405 df-topgen 17406 df-pt 17407 df-prds 17410 df-xrs 17465 df-qtop 17470 df-imas 17471 df-xps 17473 df-mre 17547 df-mrc 17548 df-acs 17550 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-submnd 18711 df-grp 18868 df-minusg 18869 df-mulg 19000 df-subg 19055 df-cntz 19249 df-cmn 19712 df-abl 19713 df-mgp 20050 df-rng 20062 df-ur 20091 df-ring 20144 df-cring 20145 df-oppr 20246 df-dvdsr 20266 df-unit 20267 df-invr 20297 df-dvr 20310 df-subrng 20455 df-subrg 20479 df-drng 20640 df-lvec 21010 df-psmet 21256 df-xmet 21257 df-met 21258 df-bl 21259 df-mopn 21260 df-fbas 21261 df-fg 21262 df-cnfld 21265 df-phl 21535 df-top 22781 df-topon 22798 df-topsp 22820 df-bases 22833 df-cld 22906 df-ntr 22907 df-cls 22908 df-nei 22985 df-cn 23114 df-cnp 23115 df-haus 23202 df-cmp 23274 df-tx 23449 df-hmeo 23642 df-fil 23733 df-flim 23826 df-fcls 23828 df-xms 24208 df-ms 24209 df-tms 24210 df-nvc 24475 df-cncf 24771 df-cph 25068 df-cfil 25155 df-cmet 25157 df-cms 25235 df-bn 25236 df-hl 25237

This theorem is referenced by: (None)

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