ishl2 - Metamath Proof Explorer

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Theorem ishl2 25317

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 25309	. 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 25123	. . . . . 6 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmVec)
5		hlress.f	. . . . . . . . 9 ⊢ 𝐹 = (Scalar‘𝑊)
6	5	isbn 25285	. . . . . . . 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 25126	. . . . . . . 8 ⊢ (𝑊 ∈ ℂPreHil → 𝐹 = (ℂ_fld ↾_s 𝐾))
13	12	eleq1d 2818	. . . . . . 7 ⊢ (𝑊 ∈ ℂPreHil → (𝐹 ∈ CMetSp ↔ (ℂ_fld ↾_s 𝐾) ∈ CMetSp))
14	5, 11	cphsubrg 25127	. . . . . . . . 9 ⊢ (𝑊 ∈ ℂPreHil → 𝐾 ∈ (SubRing‘ℂ_fld))
15		cphlvec 25122	. . . . . . . . . . 11 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ LVec)
16	5	lvecdrng 21048	. . . . . . . . . . 11 ⊢ (𝑊 ∈ LVec → 𝐹 ∈ DivRing)
17	15, 16	syl 17	. . . . . . . . . 10 ⊢ (𝑊 ∈ ℂPreHil → 𝐹 ∈ DivRing)
18	12, 17	eqeltrrd 2834	. . . . . . . . 9 ⊢ (𝑊 ∈ ℂPreHil → (ℂ_fld ↾_s 𝐾) ∈ DivRing)
19		eqid 2733	. . . . . . . . . . 11 ⊢ (ℂ_fld ↾_s 𝐾) = (ℂ_fld ↾_s 𝐾)
20	19	cncdrg 25306	. . . . . . . . . 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 4601	. . . . . . . . 9 ⊢ (𝐾 ∈ {ℝ, ℂ} → (𝐾 = ℝ ∨ 𝐾 = ℂ))
24		oveq2 7363	. . . . . . . . . . 11 ⊢ (𝐾 = ℝ → (ℂ_fld ↾_s 𝐾) = (ℂ_fld ↾_s ℝ))
25		eqid 2733	. . . . . . . . . . . . 13 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
26	25	recld2 24750	. . . . . . . . . . . 12 ⊢ ℝ ∈ (Clsd‘(TopOpen‘ℂ_fld))
27		cncms 25302	. . . . . . . . . . . . 13 ⊢ ℂ_fld ∈ CMetSp
28		ax-resscn 11074	. . . . . . . . . . . . 13 ⊢ ℝ ⊆ ℂ
29		eqid 2733	. . . . . . . . . . . . . 14 ⊢ (ℂ_fld ↾_s ℝ) = (ℂ_fld ↾_s ℝ)
30		cnfldbas 21304	. . . . . . . . . . . . . 14 ⊢ ℂ = (Base‘ℂ_fld)
31	29, 30, 25	cmsss 25298	. . . . . . . . . . . . 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 2841	. . . . . . . . . 10 ⊢ (𝐾 = ℝ → (ℂ_fld ↾_s 𝐾) ∈ CMetSp)
35		oveq2 7363	. . . . . . . . . . 11 ⊢ (𝐾 = ℂ → (ℂ_fld ↾_s 𝐾) = (ℂ_fld ↾_s ℂ))
36	30	ressid 17162	. . . . . . . . . . . . 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 2829	. . . . . . . . . . 11 ⊢ (ℂ_fld ↾_s ℂ) ∈ CMetSp
39	35, 38	eqeltrdi 2841	. . . . . . . . . 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 1541 ∈ wcel 2113 ⊆ wss 3898 {cpr 4579 ‘cfv 6489 (class class class)co 7355 ℂcc 11015 ℝcr 11016 Basecbs 17127 ↾_s cress 17148 Scalarcsca 17171 TopOpenctopn 17332 SubRingcsubrg 20493 DivRingcdr 20653 LVecclvec 21045 ℂ_fldccnfld 21300 Clsdccld 22951 NrmVeccnvc 24516 ℂPreHilccph 25113 CMetSpccms 25279 Bancbn 25280 ℂHilchl 25281

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 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7677 ax-cnex 11073 ax-resscn 11074 ax-1cn 11075 ax-icn 11076 ax-addcl 11077 ax-addrcl 11078 ax-mulcl 11079 ax-mulrcl 11080 ax-mulcom 11081 ax-addass 11082 ax-mulass 11083 ax-distr 11084 ax-i2m1 11085 ax-1ne0 11086 ax-1rid 11087 ax-rnegex 11088 ax-rrecex 11089 ax-cnre 11090 ax-pre-lttri 11091 ax-pre-lttrn 11092 ax-pre-ltadd 11093 ax-pre-mulgt0 11094 ax-pre-sup 11095 ax-addf 11096 ax-mulf 11097

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4861 df-int 4900 df-iun 4945 df-iin 4946 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-se 5575 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-isom 6498 df-riota 7312 df-ov 7358 df-oprab 7359 df-mpo 7360 df-of 7619 df-om 7806 df-1st 7930 df-2nd 7931 df-supp 8100 df-tpos 8165 df-frecs 8220 df-wrecs 8251 df-recs 8300 df-rdg 8338 df-1o 8394 df-2o 8395 df-er 8631 df-map 8761 df-ixp 8832 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-fsupp 9257 df-fi 9306 df-sup 9337 df-inf 9338 df-oi 9407 df-card 9843 df-pnf 11159 df-mnf 11160 df-xr 11161 df-ltxr 11162 df-le 11163 df-sub 11357 df-neg 11358 df-div 11786 df-nn 12137 df-2 12199 df-3 12200 df-4 12201 df-5 12202 df-6 12203 df-7 12204 df-8 12205 df-9 12206 df-n0 12393 df-z 12480 df-dec 12599 df-uz 12743 df-q 12853 df-rp 12897 df-xneg 13017 df-xadd 13018 df-xmul 13019 df-ioo 13256 df-ico 13258 df-icc 13259 df-fz 13415 df-fzo 13562 df-seq 13916 df-exp 13976 df-hash 14245 df-cj 15013 df-re 15014 df-im 15015 df-sqrt 15149 df-abs 15150 df-struct 17065 df-sets 17082 df-slot 17100 df-ndx 17112 df-base 17128 df-ress 17149 df-plusg 17181 df-mulr 17182 df-starv 17183 df-sca 17184 df-vsca 17185 df-ip 17186 df-tset 17187 df-ple 17188 df-ds 17190 df-unif 17191 df-hom 17192 df-cco 17193 df-rest 17333 df-topn 17334 df-0g 17352 df-gsum 17353 df-topgen 17354 df-pt 17355 df-prds 17358 df-xrs 17414 df-qtop 17419 df-imas 17420 df-xps 17422 df-mre 17496 df-mrc 17497 df-acs 17499 df-mgm 18556 df-sgrp 18635 df-mnd 18651 df-submnd 18700 df-grp 18857 df-minusg 18858 df-mulg 18989 df-subg 19044 df-cntz 19237 df-cmn 19702 df-abl 19703 df-mgp 20067 df-rng 20079 df-ur 20108 df-ring 20161 df-cring 20162 df-oppr 20264 df-dvdsr 20284 df-unit 20285 df-invr 20315 df-dvr 20328 df-subrng 20470 df-subrg 20494 df-drng 20655 df-lvec 21046 df-psmet 21292 df-xmet 21293 df-met 21294 df-bl 21295 df-mopn 21296 df-fbas 21297 df-fg 21298 df-cnfld 21301 df-phl 21572 df-top 22829 df-topon 22846 df-topsp 22868 df-bases 22881 df-cld 22954 df-ntr 22955 df-cls 22956 df-nei 23033 df-cn 23162 df-cnp 23163 df-haus 23250 df-cmp 23322 df-tx 23497 df-hmeo 23690 df-fil 23781 df-flim 23874 df-fcls 23876 df-xms 24255 df-ms 24256 df-tms 24257 df-nvc 24522 df-cncf 24818 df-cph 25115 df-cfil 25202 df-cmet 25204 df-cms 25282 df-bn 25283 df-hl 25284

This theorem is referenced by: (None)

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