ishl2 - Metamath Proof Explorer

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Theorem ishl2 25404

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 25396	. 2 ⊢ (𝑊 ∈ ℂHil ↔ (𝑊 ∈ Ban ∧ 𝑊 ∈ ℂPreHil))
2		df-3an 1089	. . 3 ⊢ ((𝑊 ∈ CMetSp ∧ 𝐾 ∈ {ℝ, ℂ} ∧ 𝑊 ∈ ℂPreHil) ↔ ((𝑊 ∈ CMetSp ∧ 𝐾 ∈ {ℝ, ℂ}) ∧ 𝑊 ∈ ℂPreHil))
3		3ancomb 1099	. . 3 ⊢ ((𝑊 ∈ CMetSp ∧ 𝑊 ∈ ℂPreHil ∧ 𝐾 ∈ {ℝ, ℂ}) ↔ (𝑊 ∈ CMetSp ∧ 𝐾 ∈ {ℝ, ℂ} ∧ 𝑊 ∈ ℂPreHil))
4		cphnvc 25210	. . . . . 6 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmVec)
5		hlress.f	. . . . . . . . 9 ⊢ 𝐹 = (Scalar‘𝑊)
6	5	isbn 25372	. . . . . . . 8 ⊢ (𝑊 ∈ Ban ↔ (𝑊 ∈ NrmVec ∧ 𝑊 ∈ CMetSp ∧ 𝐹 ∈ CMetSp))
7		3anass 1095	. . . . . . . 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 25213	. . . . . . . 8 ⊢ (𝑊 ∈ ℂPreHil → 𝐹 = (ℂ_fld ↾_s 𝐾))
13	12	eleq1d 2826	. . . . . . 7 ⊢ (𝑊 ∈ ℂPreHil → (𝐹 ∈ CMetSp ↔ (ℂ_fld ↾_s 𝐾) ∈ CMetSp))
14	5, 11	cphsubrg 25214	. . . . . . . . 9 ⊢ (𝑊 ∈ ℂPreHil → 𝐾 ∈ (SubRing‘ℂ_fld))
15		cphlvec 25209	. . . . . . . . . . 11 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ LVec)
16	5	lvecdrng 21104	. . . . . . . . . . 11 ⊢ (𝑊 ∈ LVec → 𝐹 ∈ DivRing)
17	15, 16	syl 17	. . . . . . . . . 10 ⊢ (𝑊 ∈ ℂPreHil → 𝐹 ∈ DivRing)
18	12, 17	eqeltrrd 2842	. . . . . . . . 9 ⊢ (𝑊 ∈ ℂPreHil → (ℂ_fld ↾_s 𝐾) ∈ DivRing)
19		eqid 2737	. . . . . . . . . . 11 ⊢ (ℂ_fld ↾_s 𝐾) = (ℂ_fld ↾_s 𝐾)
20	19	cncdrg 25393	. . . . . . . . . 10 ⊢ ((𝐾 ∈ (SubRing‘ℂ_fld) ∧ (ℂ_fld ↾_s 𝐾) ∈ DivRing ∧ (ℂ_fld ↾_s 𝐾) ∈ CMetSp) → 𝐾 ∈ {ℝ, ℂ})
21	20	3expia 1122	. . . . . . . . 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 4649	. . . . . . . . 9 ⊢ (𝐾 ∈ {ℝ, ℂ} → (𝐾 = ℝ ∨ 𝐾 = ℂ))
24		oveq2 7439	. . . . . . . . . . 11 ⊢ (𝐾 = ℝ → (ℂ_fld ↾_s 𝐾) = (ℂ_fld ↾_s ℝ))
25		eqid 2737	. . . . . . . . . . . . 13 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
26	25	recld2 24836	. . . . . . . . . . . 12 ⊢ ℝ ∈ (Clsd‘(TopOpen‘ℂ_fld))
27		cncms 25389	. . . . . . . . . . . . 13 ⊢ ℂ_fld ∈ CMetSp
28		ax-resscn 11212	. . . . . . . . . . . . 13 ⊢ ℝ ⊆ ℂ
29		eqid 2737	. . . . . . . . . . . . . 14 ⊢ (ℂ_fld ↾_s ℝ) = (ℂ_fld ↾_s ℝ)
30		cnfldbas 21368	. . . . . . . . . . . . . 14 ⊢ ℂ = (Base‘ℂ_fld)
31	29, 30, 25	cmsss 25385	. . . . . . . . . . . . 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 2849	. . . . . . . . . 10 ⊢ (𝐾 = ℝ → (ℂ_fld ↾_s 𝐾) ∈ CMetSp)
35		oveq2 7439	. . . . . . . . . . 11 ⊢ (𝐾 = ℂ → (ℂ_fld ↾_s 𝐾) = (ℂ_fld ↾_s ℂ))
36	30	ressid 17290	. . . . . . . . . . . . 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 2837	. . . . . . . . . . 11 ⊢ (ℂ_fld ↾_s ℂ) ∈ CMetSp
39	35, 38	eqeltrdi 2849	. . . . . . . . . 10 ⊢ (𝐾 = ℂ → (ℂ_fld ↾_s 𝐾) ∈ CMetSp)
40	34, 39	jaoi 858	. . . . . . . . 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 848 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ⊆ wss 3951 {cpr 4628 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 ℝcr 11154 Basecbs 17247 ↾_s cress 17274 Scalarcsca 17300 TopOpenctopn 17466 SubRingcsubrg 20569 DivRingcdr 20729 LVecclvec 21101 ℂ_fldccnfld 21364 Clsdccld 23024 NrmVeccnvc 24594 ℂPreHilccph 25200 CMetSpccms 25366 Bancbn 25367 ℂHilchl 25368

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234 ax-mulf 11235

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-tpos 8251 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-fi 9451 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ioo 13391 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17467 df-topn 17468 df-0g 17486 df-gsum 17487 df-topgen 17488 df-pt 17489 df-prds 17492 df-xrs 17547 df-qtop 17552 df-imas 17553 df-xps 17555 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-grp 18954 df-minusg 18955 df-mulg 19086 df-subg 19141 df-cntz 19335 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-cring 20233 df-oppr 20334 df-dvdsr 20357 df-unit 20358 df-invr 20388 df-dvr 20401 df-subrng 20546 df-subrg 20570 df-drng 20731 df-lvec 21102 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-fbas 21361 df-fg 21362 df-cnfld 21365 df-phl 21644 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-cld 23027 df-ntr 23028 df-cls 23029 df-nei 23106 df-cn 23235 df-cnp 23236 df-haus 23323 df-cmp 23395 df-tx 23570 df-hmeo 23763 df-fil 23854 df-flim 23947 df-fcls 23949 df-xms 24330 df-ms 24331 df-tms 24332 df-nvc 24600 df-cncf 24904 df-cph 25202 df-cfil 25289 df-cmet 25291 df-cms 25369 df-bn 25370 df-hl 25371

This theorem is referenced by: (None)

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