Theorem clsocv 24414

Description: The orthogonal complement of the closure of a subset is the same as the orthogonal complement of the subset itself. (Contributed by Mario Carneiro, 13-Oct-2015.)

Hypotheses

Ref	Expression
clsocv.v	⊢ 𝑉 = (Base‘𝑊)
clsocv.o	⊢ 𝑂 = (ocv‘𝑊)
clsocv.j	⊢ 𝐽 = (TopOpen‘𝑊)

Assertion

Ref	Expression
clsocv	⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑆 ⊆ 𝑉) → (𝑂‘((cls‘𝐽)‘𝑆)) = (𝑂‘𝑆))

Proof of Theorem clsocv

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cphngp 24337	. . . . . . . 8 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmGrp)
2		ngptps 23758	. . . . . . . 8 ⊢ (𝑊 ∈ NrmGrp → 𝑊 ∈ TopSp)
3	1, 2	syl 17	. . . . . . 7 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ TopSp)
4	3	adantr 481	. . . . . 6 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑆 ⊆ 𝑉) → 𝑊 ∈ TopSp)
5		clsocv.v	. . . . . . 7 ⊢ 𝑉 = (Base‘𝑊)
6		clsocv.j	. . . . . . 7 ⊢ 𝐽 = (TopOpen‘𝑊)
7	5, 6	istps 22083	. . . . . 6 ⊢ (𝑊 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝑉))
8	4, 7	sylib 217	. . . . 5 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑆 ⊆ 𝑉) → 𝐽 ∈ (TopOn‘𝑉))
9		topontop 22062	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑉) → 𝐽 ∈ Top)
10	8, 9	syl 17	. . . 4 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑆 ⊆ 𝑉) → 𝐽 ∈ Top)
11		simpr 485	. . . . 5 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑆 ⊆ 𝑉) → 𝑆 ⊆ 𝑉)
12		toponuni 22063	. . . . . 6 ⊢ (𝐽 ∈ (TopOn‘𝑉) → 𝑉 = ∪ 𝐽)
13	8, 12	syl 17	. . . . 5 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑆 ⊆ 𝑉) → 𝑉 = ∪ 𝐽)
14	11, 13	sseqtrd 3961	. . . 4 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑆 ⊆ 𝑉) → 𝑆 ⊆ ∪ 𝐽)
15		eqid 2738	. . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽
16	15	sscls 22207	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
17	10, 14, 16	syl2anc 584	. . 3 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑆 ⊆ 𝑉) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
18		clsocv.o	. . . 4 ⊢ 𝑂 = (ocv‘𝑊)
19	18	ocv2ss 20878	. . 3 ⊢ (𝑆 ⊆ ((cls‘𝐽)‘𝑆) → (𝑂‘((cls‘𝐽)‘𝑆)) ⊆ (𝑂‘𝑆))
20	17, 19	syl 17	. 2 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑆 ⊆ 𝑉) → (𝑂‘((cls‘𝐽)‘𝑆)) ⊆ (𝑂‘𝑆))
21	15	clsss3 22210	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → ((cls‘𝐽)‘𝑆) ⊆ ∪ 𝐽)
22	10, 14, 21	syl2anc 584	. . . . 5 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑆 ⊆ 𝑉) → ((cls‘𝐽)‘𝑆) ⊆ ∪ 𝐽)
23	22, 13	sseqtrrd 3962	. . . 4 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑆 ⊆ 𝑉) → ((cls‘𝐽)‘𝑆) ⊆ 𝑉)
24	23	adantr 481	. . 3 ⊢ (((𝑊 ∈ ℂPreHil ∧ 𝑆 ⊆ 𝑉) ∧ 𝑥 ∈ (𝑂‘𝑆)) → ((cls‘𝐽)‘𝑆) ⊆ 𝑉)
25	5, 18	ocvss 20875	. . . . 5 ⊢ (𝑂‘𝑆) ⊆ 𝑉
26	25	a1i 11	. . . 4 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑆 ⊆ 𝑉) → (𝑂‘𝑆) ⊆ 𝑉)
27	26	sselda 3921	. . 3 ⊢ (((𝑊 ∈ ℂPreHil ∧ 𝑆 ⊆ 𝑉) ∧ 𝑥 ∈ (𝑂‘𝑆)) → 𝑥 ∈ 𝑉)
28		df-ss 3904	. . . . . . . . . 10 ⊢ (((cls‘𝐽)‘𝑆) ⊆ 𝑉 ↔ (((cls‘𝐽)‘𝑆) ∩ 𝑉) = ((cls‘𝐽)‘𝑆))
29	24, 28	sylib 217	. . . . . . . . 9 ⊢ (((𝑊 ∈ ℂPreHil ∧ 𝑆 ⊆ 𝑉) ∧ 𝑥 ∈ (𝑂‘𝑆)) → (((cls‘𝐽)‘𝑆) ∩ 𝑉) = ((cls‘𝐽)‘𝑆))
30	29	ineq1d 4145	. . . . . . . 8 ⊢ (((𝑊 ∈ ℂPreHil ∧ 𝑆 ⊆ 𝑉) ∧ 𝑥 ∈ (𝑂‘𝑆)) → ((((cls‘𝐽)‘𝑆) ∩ 𝑉) ∩ {𝑦 ∣ (𝑥(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))}) = (((cls‘𝐽)‘𝑆) ∩ {𝑦 ∣ (𝑥(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))}))
31		dfrab3 4243	. . . . . . . . . 10 ⊢ {𝑦 ∈ 𝑉 ∣ (𝑥(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))} = (𝑉 ∩ {𝑦 ∣ (𝑥(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))})
32	31	ineq2i 4143	. . . . . . . . 9 ⊢ (((cls‘𝐽)‘𝑆) ∩ {𝑦 ∈ 𝑉 ∣ (𝑥(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))}) = (((cls‘𝐽)‘𝑆) ∩ (𝑉 ∩ {𝑦 ∣ (𝑥(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))}))
33		inass 4153	. . . . . . . . 9 ⊢ ((((cls‘𝐽)‘𝑆) ∩ 𝑉) ∩ {𝑦 ∣ (𝑥(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))}) = (((cls‘𝐽)‘𝑆) ∩ (𝑉 ∩ {𝑦 ∣ (𝑥(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))}))
34	32, 33	eqtr4i 2769	. . . . . . . 8 ⊢ (((cls‘𝐽)‘𝑆) ∩ {𝑦 ∈ 𝑉 ∣ (𝑥(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))}) = ((((cls‘𝐽)‘𝑆) ∩ 𝑉) ∩ {𝑦 ∣ (𝑥(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))})
35		dfrab3 4243	. . . . . . . 8 ⊢ {𝑦 ∈ ((cls‘𝐽)‘𝑆) ∣ (𝑥(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))} = (((cls‘𝐽)‘𝑆) ∩ {𝑦 ∣ (𝑥(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))})
36	30, 34, 35	3eqtr4g 2803	. . . . . . 7 ⊢ (((𝑊 ∈ ℂPreHil ∧ 𝑆 ⊆ 𝑉) ∧ 𝑥 ∈ (𝑂‘𝑆)) → (((cls‘𝐽)‘𝑆) ∩ {𝑦 ∈ 𝑉 ∣ (𝑥(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))}) = {𝑦 ∈ ((cls‘𝐽)‘𝑆) ∣ (𝑥(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))})
37	15	clscld 22198	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → ((cls‘𝐽)‘𝑆) ∈ (Clsd‘𝐽))
38	10, 14, 37	syl2anc 584	. . . . . . . . 9 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑆 ⊆ 𝑉) → ((cls‘𝐽)‘𝑆) ∈ (Clsd‘𝐽))
39	38	adantr 481	. . . . . . . 8 ⊢ (((𝑊 ∈ ℂPreHil ∧ 𝑆 ⊆ 𝑉) ∧ 𝑥 ∈ (𝑂‘𝑆)) → ((cls‘𝐽)‘𝑆) ∈ (Clsd‘𝐽))
40		fvex 6787	. . . . . . . . . 10 ⊢ (0g‘(Scalar‘𝑊)) ∈ V
41		eqid 2738	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝑉 ↦ (𝑥(·𝑖‘𝑊)𝑦)) = (𝑦 ∈ 𝑉 ↦ (𝑥(·𝑖‘𝑊)𝑦))
42	41	mptiniseg 6142	. . . . . . . . . 10 ⊢ ((0g‘(Scalar‘𝑊)) ∈ V → (◡(𝑦 ∈ 𝑉 ↦ (𝑥(·𝑖‘𝑊)𝑦)) “ {(0g‘(Scalar‘𝑊))}) = {𝑦 ∈ 𝑉 ∣ (𝑥(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))})
43	40, 42	ax-mp 5	. . . . . . . . 9 ⊢ (◡(𝑦 ∈ 𝑉 ↦ (𝑥(·𝑖‘𝑊)𝑦)) “ {(0g‘(Scalar‘𝑊))}) = {𝑦 ∈ 𝑉 ∣ (𝑥(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))}
44		eqid 2738	. . . . . . . . . . 11 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
45		eqid 2738	. . . . . . . . . . 11 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊)
46		simpll 764	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ ℂPreHil ∧ 𝑆 ⊆ 𝑉) ∧ 𝑥 ∈ (𝑂‘𝑆)) → 𝑊 ∈ ℂPreHil)
47	8	adantr 481	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ ℂPreHil ∧ 𝑆 ⊆ 𝑉) ∧ 𝑥 ∈ (𝑂‘𝑆)) → 𝐽 ∈ (TopOn‘𝑉))
48	47, 47, 27	cnmptc 22813	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ ℂPreHil ∧ 𝑆 ⊆ 𝑉) ∧ 𝑥 ∈ (𝑂‘𝑆)) → (𝑦 ∈ 𝑉 ↦ 𝑥) ∈ (𝐽 Cn 𝐽))
49	47	cnmptid 22812	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ ℂPreHil ∧ 𝑆 ⊆ 𝑉) ∧ 𝑥 ∈ (𝑂‘𝑆)) → (𝑦 ∈ 𝑉 ↦ 𝑦) ∈ (𝐽 Cn 𝐽))
50	6, 44, 45, 46, 47, 48, 49	cnmpt1ip 24411	. . . . . . . . . 10 ⊢ (((𝑊 ∈ ℂPreHil ∧ 𝑆 ⊆ 𝑉) ∧ 𝑥 ∈ (𝑂‘𝑆)) → (𝑦 ∈ 𝑉 ↦ (𝑥(·𝑖‘𝑊)𝑦)) ∈ (𝐽 Cn (TopOpen‘ℂfld)))
51	44	cnfldhaus 23948	. . . . . . . . . . 11 ⊢ (TopOpen‘ℂfld) ∈ Haus
52		cphclm 24353	. . . . . . . . . . . . . 14 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ ℂMod)
53		eqid 2738	. . . . . . . . . . . . . . 15 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
54	53	clm0 24235	. . . . . . . . . . . . . 14 ⊢ (𝑊 ∈ ℂMod → 0 = (0g‘(Scalar‘𝑊)))
55	52, 54	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑊 ∈ ℂPreHil → 0 = (0g‘(Scalar‘𝑊)))
56	55	ad2antrr 723	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ ℂPreHil ∧ 𝑆 ⊆ 𝑉) ∧ 𝑥 ∈ (𝑂‘𝑆)) → 0 = (0g‘(Scalar‘𝑊)))
57		0cn 10967	. . . . . . . . . . . 12 ⊢ 0 ∈ ℂ
58	56, 57	eqeltrrdi 2848	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ ℂPreHil ∧ 𝑆 ⊆ 𝑉) ∧ 𝑥 ∈ (𝑂‘𝑆)) → (0g‘(Scalar‘𝑊)) ∈ ℂ)
59		unicntop 23949	. . . . . . . . . . . 12 ⊢ ℂ = ∪ (TopOpen‘ℂfld)
60	59	sncld 22522	. . . . . . . . . . 11 ⊢ (((TopOpen‘ℂfld) ∈ Haus ∧ (0g‘(Scalar‘𝑊)) ∈ ℂ) → {(0g‘(Scalar‘𝑊))} ∈ (Clsd‘(TopOpen‘ℂfld)))
61	51, 58, 60	sylancr 587	. . . . . . . . . 10 ⊢ (((𝑊 ∈ ℂPreHil ∧ 𝑆 ⊆ 𝑉) ∧ 𝑥 ∈ (𝑂‘𝑆)) → {(0g‘(Scalar‘𝑊))} ∈ (Clsd‘(TopOpen‘ℂfld)))
62		cnclima 22419	. . . . . . . . . 10 ⊢ (((𝑦 ∈ 𝑉 ↦ (𝑥(·𝑖‘𝑊)𝑦)) ∈ (𝐽 Cn (TopOpen‘ℂfld)) ∧ {(0g‘(Scalar‘𝑊))} ∈ (Clsd‘(TopOpen‘ℂfld))) → (◡(𝑦 ∈ 𝑉 ↦ (𝑥(·𝑖‘𝑊)𝑦)) “ {(0g‘(Scalar‘𝑊))}) ∈ (Clsd‘𝐽))
63	50, 61, 62	syl2anc 584	. . . . . . . . 9 ⊢ (((𝑊 ∈ ℂPreHil ∧ 𝑆 ⊆ 𝑉) ∧ 𝑥 ∈ (𝑂‘𝑆)) → (◡(𝑦 ∈ 𝑉 ↦ (𝑥(·𝑖‘𝑊)𝑦)) “ {(0g‘(Scalar‘𝑊))}) ∈ (Clsd‘𝐽))
64	43, 63	eqeltrrid 2844	. . . . . . . 8 ⊢ (((𝑊 ∈ ℂPreHil ∧ 𝑆 ⊆ 𝑉) ∧ 𝑥 ∈ (𝑂‘𝑆)) → {𝑦 ∈ 𝑉 ∣ (𝑥(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))} ∈ (Clsd‘𝐽))
65		incld 22194	. . . . . . . 8 ⊢ ((((cls‘𝐽)‘𝑆) ∈ (Clsd‘𝐽) ∧ {𝑦 ∈ 𝑉 ∣ (𝑥(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))} ∈ (Clsd‘𝐽)) → (((cls‘𝐽)‘𝑆) ∩ {𝑦 ∈ 𝑉 ∣ (𝑥(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))}) ∈ (Clsd‘𝐽))
66	39, 64, 65	syl2anc 584	. . . . . . 7 ⊢ (((𝑊 ∈ ℂPreHil ∧ 𝑆 ⊆ 𝑉) ∧ 𝑥 ∈ (𝑂‘𝑆)) → (((cls‘𝐽)‘𝑆) ∩ {𝑦 ∈ 𝑉 ∣ (𝑥(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))}) ∈ (Clsd‘𝐽))
67	36, 66	eqeltrrd 2840	. . . . . 6 ⊢ (((𝑊 ∈ ℂPreHil ∧ 𝑆 ⊆ 𝑉) ∧ 𝑥 ∈ (𝑂‘𝑆)) → {𝑦 ∈ ((cls‘𝐽)‘𝑆) ∣ (𝑥(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))} ∈ (Clsd‘𝐽))
68	17	adantr 481	. . . . . . 7 ⊢ (((𝑊 ∈ ℂPreHil ∧ 𝑆 ⊆ 𝑉) ∧ 𝑥 ∈ (𝑂‘𝑆)) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
69		eqid 2738	. . . . . . . . . 10 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
70	5, 45, 53, 69, 18	ocvi 20874	. . . . . . . . 9 ⊢ ((𝑥 ∈ (𝑂‘𝑆) ∧ 𝑦 ∈ 𝑆) → (𝑥(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊)))
71	70	ralrimiva 3103	. . . . . . . 8 ⊢ (𝑥 ∈ (𝑂‘𝑆) → ∀𝑦 ∈ 𝑆 (𝑥(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊)))
72	71	adantl 482	. . . . . . 7 ⊢ (((𝑊 ∈ ℂPreHil ∧ 𝑆 ⊆ 𝑉) ∧ 𝑥 ∈ (𝑂‘𝑆)) → ∀𝑦 ∈ 𝑆 (𝑥(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊)))
73		ssrab 4006	. . . . . . 7 ⊢ (𝑆 ⊆ {𝑦 ∈ ((cls‘𝐽)‘𝑆) ∣ (𝑥(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))} ↔ (𝑆 ⊆ ((cls‘𝐽)‘𝑆) ∧ ∀𝑦 ∈ 𝑆 (𝑥(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))))
74	68, 72, 73	sylanbrc 583	. . . . . 6 ⊢ (((𝑊 ∈ ℂPreHil ∧ 𝑆 ⊆ 𝑉) ∧ 𝑥 ∈ (𝑂‘𝑆)) → 𝑆 ⊆ {𝑦 ∈ ((cls‘𝐽)‘𝑆) ∣ (𝑥(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))})
75	15	clsss2 22223	. . . . . 6 ⊢ (({𝑦 ∈ ((cls‘𝐽)‘𝑆) ∣ (𝑥(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))} ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ {𝑦 ∈ ((cls‘𝐽)‘𝑆) ∣ (𝑥(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))}) → ((cls‘𝐽)‘𝑆) ⊆ {𝑦 ∈ ((cls‘𝐽)‘𝑆) ∣ (𝑥(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))})
76	67, 74, 75	syl2anc 584	. . . . 5 ⊢ (((𝑊 ∈ ℂPreHil ∧ 𝑆 ⊆ 𝑉) ∧ 𝑥 ∈ (𝑂‘𝑆)) → ((cls‘𝐽)‘𝑆) ⊆ {𝑦 ∈ ((cls‘𝐽)‘𝑆) ∣ (𝑥(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))})
77		ssrab2 4013	. . . . . 6 ⊢ {𝑦 ∈ ((cls‘𝐽)‘𝑆) ∣ (𝑥(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))} ⊆ ((cls‘𝐽)‘𝑆)
78	77	a1i 11	. . . . 5 ⊢ (((𝑊 ∈ ℂPreHil ∧ 𝑆 ⊆ 𝑉) ∧ 𝑥 ∈ (𝑂‘𝑆)) → {𝑦 ∈ ((cls‘𝐽)‘𝑆) ∣ (𝑥(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))} ⊆ ((cls‘𝐽)‘𝑆))
79	76, 78	eqssd 3938	. . . 4 ⊢ (((𝑊 ∈ ℂPreHil ∧ 𝑆 ⊆ 𝑉) ∧ 𝑥 ∈ (𝑂‘𝑆)) → ((cls‘𝐽)‘𝑆) = {𝑦 ∈ ((cls‘𝐽)‘𝑆) ∣ (𝑥(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))})
80		rabid2 3314	. . . 4 ⊢ (((cls‘𝐽)‘𝑆) = {𝑦 ∈ ((cls‘𝐽)‘𝑆) ∣ (𝑥(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))} ↔ ∀𝑦 ∈ ((cls‘𝐽)‘𝑆)(𝑥(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊)))
81	79, 80	sylib 217	. . 3 ⊢ (((𝑊 ∈ ℂPreHil ∧ 𝑆 ⊆ 𝑉) ∧ 𝑥 ∈ (𝑂‘𝑆)) → ∀𝑦 ∈ ((cls‘𝐽)‘𝑆)(𝑥(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊)))
82	5, 45, 53, 69, 18	elocv 20873	. . 3 ⊢ (𝑥 ∈ (𝑂‘((cls‘𝐽)‘𝑆)) ↔ (((cls‘𝐽)‘𝑆) ⊆ 𝑉 ∧ 𝑥 ∈ 𝑉 ∧ ∀𝑦 ∈ ((cls‘𝐽)‘𝑆)(𝑥(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))))
83	24, 27, 81, 82	syl3anbrc 1342	. 2 ⊢ (((𝑊 ∈ ℂPreHil ∧ 𝑆 ⊆ 𝑉) ∧ 𝑥 ∈ (𝑂‘𝑆)) → 𝑥 ∈ (𝑂‘((cls‘𝐽)‘𝑆)))
84	20, 83	eqelssd 3942	1 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑆 ⊆ 𝑉) → (𝑂‘((cls‘𝐽)‘𝑆)) = (𝑂‘𝑆))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  {cab 2715  ∀wral 3064  {crab 3068  Vcvv 3432   ∩ cin 3886   ⊆ wss 3887  {csn 4561  ∪ cuni 4839   ↦ cmpt 5157  ^◡ccnv 5588   “ cima 5592  ‘cfv 6433  (class class class)co 7275  ℂcc 10869  0cc0 10871  Basecbs 16912  Scalarcsca 16965  ·_𝑖cip 16967  TopOpenctopn 17132  0_gc0g 17150  ℂ_fldccnfld 20597  ocvcocv 20865  Topctop 22042  TopOnctopon 22059  TopSpctps 22081  Clsdccld 22167  clsccl 22169   Cn ccn 22375  Hauscha 22459  NrmGrpcngp 23733  ℂModcclm 24225  ℂPreHilccph 24330

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948  ax-pre-sup 10949  ax-addf 10950  ax-mulf 10951

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-iin 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-of 7533  df-om 7713  df-1st 7831  df-2nd 7832  df-supp 7978  df-tpos 8042  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-2o 8298  df-er 8498  df-map 8617  df-ixp 8686  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-fsupp 9129  df-fi 9170  df-sup 9201  df-inf 9202  df-oi 9269  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-div 11633  df-nn 11974  df-2 12036  df-3 12037  df-4 12038  df-5 12039  df-6 12040  df-7 12041  df-8 12042  df-9 12043  df-n0 12234  df-z 12320  df-dec 12438  df-uz 12583  df-q 12689  df-rp 12731  df-xneg 12848  df-xadd 12849  df-xmul 12850  df-ico 13085  df-icc 13086  df-fz 13240  df-fzo 13383  df-seq 13722  df-exp 13783  df-hash 14045  df-cj 14810  df-re 14811  df-im 14812  df-sqrt 14946  df-abs 14947  df-struct 16848  df-sets 16865  df-slot 16883  df-ndx 16895  df-base 16913  df-ress 16942  df-plusg 16975  df-mulr 16976  df-starv 16977  df-sca 16978  df-vsca 16979  df-ip 16980  df-tset 16981  df-ple 16982  df-ds 16984  df-unif 16985  df-hom 16986  df-cco 16987  df-rest 17133  df-topn 17134  df-0g 17152  df-gsum 17153  df-topgen 17154  df-pt 17155  df-prds 17158  df-xrs 17213  df-qtop 17218  df-imas 17219  df-xps 17221  df-mre 17295  df-mrc 17296  df-acs 17298  df-mgm 18326  df-sgrp 18375  df-mnd 18386  df-mhm 18430  df-submnd 18431  df-grp 18580  df-minusg 18581  df-sbg 18582  df-mulg 18701  df-subg 18752  df-ghm 18832  df-cntz 18923  df-cmn 19388  df-abl 19389  df-mgp 19721  df-ur 19738  df-ring 19785  df-cring 19786  df-oppr 19862  df-dvdsr 19883  df-unit 19884  df-invr 19914  df-dvr 19925  df-rnghom 19959  df-drng 19993  df-subrg 20022  df-staf 20105  df-srng 20106  df-lmod 20125  df-lmhm 20284  df-lvec 20365  df-sra 20434  df-rgmod 20435  df-psmet 20589  df-xmet 20590  df-met 20591  df-bl 20592  df-mopn 20593  df-cnfld 20598  df-phl 20831  df-ipf 20832  df-ocv 20868  df-top 22043  df-topon 22060  df-topsp 22082  df-bases 22096  df-cld 22170  df-cls 22172  df-cn 22378  df-cnp 22379  df-t1 22465  df-haus 22466  df-tx 22713  df-hmeo 22906  df-xms 23473  df-ms 23474  df-tms 23475  df-nm 23738  df-ngp 23739  df-tng 23740  df-nlm 23742  df-clm 24226  df-cph 24332  df-tcph 24333

This theorem is referenced by: (None)