Theorem clsocv 25626

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 25549	. . . . . . . 8 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmGrp)
2		ngptps 24976	. . . . . . . 8 ⊢ (𝑊 ∈ NrmGrp → 𝑊 ∈ TopSp)
3	1, 2	syl 18	. . . . . . 7 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ TopSp)
4	3	adantr 486	. . . . . 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 23307	. . . . . 6 ⊢ (𝑊 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝑉))
8	4, 7	sylib 221	. . . . 5 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑆 ⊆ 𝑉) → 𝐽 ∈ (TopOn‘𝑉))
9		topontop 23286	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑉) → 𝐽 ∈ Top)
10	8, 9	syl 18	. . . 4 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑆 ⊆ 𝑉) → 𝐽 ∈ Top)
11		simpr 490	. . . . 5 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑆 ⊆ 𝑉) → 𝑆 ⊆ 𝑉)
12		toponuni 23287	. . . . . 6 ⊢ (𝐽 ∈ (TopOn‘𝑉) → 𝑉 = ∪ 𝐽)
13	8, 12	syl 18	. . . . 5 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑆 ⊆ 𝑉) → 𝑉 = ∪ 𝐽)
14	11, 13	sseqtrd 3949	. . . 4 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑆 ⊆ 𝑉) → 𝑆 ⊆ ∪ 𝐽)
15		eqid 2763	. . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽
16	15	sscls 23429	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
17	10, 14, 16	syl2anc 596	. . 3 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑆 ⊆ 𝑉) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
18		clsocv.o	. . . 4 ⊢ 𝑂 = (ocv‘𝑊)
19	18	ocv2ss 22034	. . 3 ⊢ (𝑆 ⊆ ((cls‘𝐽)‘𝑆) → (𝑂‘((cls‘𝐽)‘𝑆)) ⊆ (𝑂‘𝑆))
20	17, 19	syl 18	. 2 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑆 ⊆ 𝑉) → (𝑂‘((cls‘𝐽)‘𝑆)) ⊆ (𝑂‘𝑆))
21	15	clsss3 23432	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → ((cls‘𝐽)‘𝑆) ⊆ ∪ 𝐽)
22	10, 14, 21	syl2anc 596	. . . . 5 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑆 ⊆ 𝑉) → ((cls‘𝐽)‘𝑆) ⊆ ∪ 𝐽)
23	22, 13	sseqtrrd 3950	. . . 4 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑆 ⊆ 𝑉) → ((cls‘𝐽)‘𝑆) ⊆ 𝑉)
24	23	adantr 486	. . 3 ⊢ (((𝑊 ∈ ℂPreHil ∧ 𝑆 ⊆ 𝑉) ∧ 𝑥 ∈ (𝑂‘𝑆)) → ((cls‘𝐽)‘𝑆) ⊆ 𝑉)
25	5, 18	ocvss 22031	. . . . 5 ⊢ (𝑂‘𝑆) ⊆ 𝑉
26	25	a1i 11	. . . 4 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑆 ⊆ 𝑉) → (𝑂‘𝑆) ⊆ 𝑉)
27	26	sselda 3913	. . 3 ⊢ (((𝑊 ∈ ℂPreHil ∧ 𝑆 ⊆ 𝑉) ∧ 𝑥 ∈ (𝑂‘𝑆)) → 𝑥 ∈ 𝑉)
28		dfss2 4097	. . . . . . . . . 10 ⊢ (((cls‘𝐽)‘𝑆) ⊆ 𝑉 ↔ (((cls‘𝐽)‘𝑆) ∩ 𝑉) = ((cls‘𝐽)‘𝑆))
29	24, 28	sylib 221	. . . . . . . . 9 ⊢ (((𝑊 ∈ ℂPreHil ∧ 𝑆 ⊆ 𝑉) ∧ 𝑥 ∈ (𝑂‘𝑆)) → (((cls‘𝐽)‘𝑆) ∩ 𝑉) = ((cls‘𝐽)‘𝑆))
30	29	ineq1d 4119	. . . . . . . 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 4267	. . . . . . . . . 10 ⊢ {𝑦 ∈ 𝑉 ∣ (𝑥(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))} = (𝑉 ∩ {𝑦 ∣ (𝑥(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))})
32	31	ineq2i 4117	. . . . . . . . 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 4127	. . . . . . . . 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 2789	. . . . . . . 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 4267	. . . . . . . 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 2823	. . . . . . 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 23420	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → ((cls‘𝐽)‘𝑆) ∈ (Clsd‘𝐽))
38	10, 14, 37	syl2anc 596	. . . . . . . . 9 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑆 ⊆ 𝑉) → ((cls‘𝐽)‘𝑆) ∈ (Clsd‘𝐽))
39	38	adantr 486	. . . . . . . 8 ⊢ (((𝑊 ∈ ℂPreHil ∧ 𝑆 ⊆ 𝑉) ∧ 𝑥 ∈ (𝑂‘𝑆)) → ((cls‘𝐽)‘𝑆) ∈ (Clsd‘𝐽))
40		fvex 6929	. . . . . . . . . 10 ⊢ (0g‘(Scalar‘𝑊)) ∈ V
41		eqid 2763	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝑉 ↦ (𝑥(·𝑖‘𝑊)𝑦)) = (𝑦 ∈ 𝑉 ↦ (𝑥(·𝑖‘𝑊)𝑦))
42	41	mptiniseg 6259	. . . . . . . . . 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 2763	. . . . . . . . . . 11 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
45		eqid 2763	. . . . . . . . . . 11 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊)
46		simpll 779	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ ℂPreHil ∧ 𝑆 ⊆ 𝑉) ∧ 𝑥 ∈ (𝑂‘𝑆)) → 𝑊 ∈ ℂPreHil)
47	8	adantr 486	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ ℂPreHil ∧ 𝑆 ⊆ 𝑉) ∧ 𝑥 ∈ (𝑂‘𝑆)) → 𝐽 ∈ (TopOn‘𝑉))
48	47, 47, 27	cnmptc 24036	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ ℂPreHil ∧ 𝑆 ⊆ 𝑉) ∧ 𝑥 ∈ (𝑂‘𝑆)) → (𝑦 ∈ 𝑉 ↦ 𝑥) ∈ (𝐽 Cn 𝐽))
49	47	cnmptid 24035	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ ℂPreHil ∧ 𝑆 ⊆ 𝑉) ∧ 𝑥 ∈ (𝑂‘𝑆)) → (𝑦 ∈ 𝑉 ↦ 𝑦) ∈ (𝐽 Cn 𝐽))
50	6, 44, 45, 46, 47, 48, 49	cnmpt1ip 25623	. . . . . . . . . 10 ⊢ (((𝑊 ∈ ℂPreHil ∧ 𝑆 ⊆ 𝑉) ∧ 𝑥 ∈ (𝑂‘𝑆)) → (𝑦 ∈ 𝑉 ↦ (𝑥(·𝑖‘𝑊)𝑦)) ∈ (𝐽 Cn (TopOpen‘ℂfld)))
51	44	cnfldhaus 25158	. . . . . . . . . . 11 ⊢ (TopOpen‘ℂfld) ∈ Haus
52		cphclm 25565	. . . . . . . . . . . . . 14 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ ℂMod)
53		eqid 2763	. . . . . . . . . . . . . . 15 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
54	53	clm0 25448	. . . . . . . . . . . . . 14 ⊢ (𝑊 ∈ ℂMod → 0 = (0g‘(Scalar‘𝑊)))
55	52, 54	syl 18	. . . . . . . . . . . . 13 ⊢ (𝑊 ∈ ℂPreHil → 0 = (0g‘(Scalar‘𝑊)))
56	55	ad2antrr 739	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ ℂPreHil ∧ 𝑆 ⊆ 𝑉) ∧ 𝑥 ∈ (𝑂‘𝑆)) → 0 = (0g‘(Scalar‘𝑊)))
57		0cn 11337	. . . . . . . . . . . 12 ⊢ 0 ∈ ℂ
58	56, 57	eqeltrrdi 2873	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ ℂPreHil ∧ 𝑆 ⊆ 𝑉) ∧ 𝑥 ∈ (𝑂‘𝑆)) → (0g‘(Scalar‘𝑊)) ∈ ℂ)
59		unicntop 25159	. . . . . . . . . . . 12 ⊢ ℂ = ∪ (TopOpen‘ℂfld)
60	59	sncld 23744	. . . . . . . . . . 11 ⊢ (((TopOpen‘ℂfld) ∈ Haus ∧ (0g‘(Scalar‘𝑊)) ∈ ℂ) → {(0g‘(Scalar‘𝑊))} ∈ (Clsd‘(TopOpen‘ℂfld)))
61	51, 58, 60	sylancr 599	. . . . . . . . . 10 ⊢ (((𝑊 ∈ ℂPreHil ∧ 𝑆 ⊆ 𝑉) ∧ 𝑥 ∈ (𝑂‘𝑆)) → {(0g‘(Scalar‘𝑊))} ∈ (Clsd‘(TopOpen‘ℂfld)))
62		cnclima 23641	. . . . . . . . . 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 596	. . . . . . . . 9 ⊢ (((𝑊 ∈ ℂPreHil ∧ 𝑆 ⊆ 𝑉) ∧ 𝑥 ∈ (𝑂‘𝑆)) → (◡(𝑦 ∈ 𝑉 ↦ (𝑥(·𝑖‘𝑊)𝑦)) “ {(0g‘(Scalar‘𝑊))}) ∈ (Clsd‘𝐽))
64	43, 63	eqeltrrid 2869	. . . . . . . 8 ⊢ (((𝑊 ∈ ℂPreHil ∧ 𝑆 ⊆ 𝑉) ∧ 𝑥 ∈ (𝑂‘𝑆)) → {𝑦 ∈ 𝑉 ∣ (𝑥(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))} ∈ (Clsd‘𝐽))
65		incld 23416	. . . . . . . 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 596	. . . . . . 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 2865	. . . . . 6 ⊢ (((𝑊 ∈ ℂPreHil ∧ 𝑆 ⊆ 𝑉) ∧ 𝑥 ∈ (𝑂‘𝑆)) → {𝑦 ∈ ((cls‘𝐽)‘𝑆) ∣ (𝑥(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))} ∈ (Clsd‘𝐽))
68	17	adantr 486	. . . . . . 7 ⊢ (((𝑊 ∈ ℂPreHil ∧ 𝑆 ⊆ 𝑉) ∧ 𝑥 ∈ (𝑂‘𝑆)) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
69		eqid 2763	. . . . . . . . . 10 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
70	5, 45, 53, 69, 18	ocvi 22030	. . . . . . . . 9 ⊢ ((𝑥 ∈ (𝑂‘𝑆) ∧ 𝑦 ∈ 𝑆) → (𝑥(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊)))
71	70	ralrimiva 3158	. . . . . . . 8 ⊢ (𝑥 ∈ (𝑂‘𝑆) → ∀𝑦 ∈ 𝑆 (𝑥(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊)))
72	71	adantl 487	. . . . . . 7 ⊢ (((𝑊 ∈ ℂPreHil ∧ 𝑆 ⊆ 𝑉) ∧ 𝑥 ∈ (𝑂‘𝑆)) → ∀𝑦 ∈ 𝑆 (𝑥(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊)))
73		ssrab 3999	. . . . . . 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 595	. . . . . 6 ⊢ (((𝑊 ∈ ℂPreHil ∧ 𝑆 ⊆ 𝑉) ∧ 𝑥 ∈ (𝑂‘𝑆)) → 𝑆 ⊆ {𝑦 ∈ ((cls‘𝐽)‘𝑆) ∣ (𝑥(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))})
75	15	clsss2 23445	. . . . . 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 596	. . . . 5 ⊢ (((𝑊 ∈ ℂPreHil ∧ 𝑆 ⊆ 𝑉) ∧ 𝑥 ∈ (𝑂‘𝑆)) → ((cls‘𝐽)‘𝑆) ⊆ {𝑦 ∈ ((cls‘𝐽)‘𝑆) ∣ (𝑥(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))})
77		ssrab2 4008	. . . . . 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 3930	. . . 4 ⊢ (((𝑊 ∈ ℂPreHil ∧ 𝑆 ⊆ 𝑉) ∧ 𝑥 ∈ (𝑂‘𝑆)) → ((cls‘𝐽)‘𝑆) = {𝑦 ∈ ((cls‘𝐽)‘𝑆) ∣ (𝑥(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))})
80		rabid2 3448	. . . 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 221	. . 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 22029	. . 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 1362	. 2 ⊢ (((𝑊 ∈ ℂPreHil ∧ 𝑆 ⊆ 𝑉) ∧ 𝑥 ∈ (𝑂‘𝑆)) → 𝑥 ∈ (𝑂‘((cls‘𝐽)‘𝑆)))
84	20, 83	eqelssd 3934	1 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑆 ⊆ 𝑉) → (𝑂‘((cls‘𝐽)‘𝑆)) = (𝑂‘𝑆))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  {cab 2741  ∀wral 3080  {crab 3416  Vcvv 3454   ⊆ wss 3900   ∩ cin 4092  {csn 4588  ∪ cuni 4872   ↦ cmpt 5192  ^◡ccnv 5657   “ cima 5661  ‘cfv 6566  (class class class)co 7451  ℂcc 11237  0cc0 11239  Basecbs 17431  Scalarcsca 17475  ·_𝑖cip 17477  TopOpenctopn 17642  0_gc0g 17660  ℂ_fldccnfld 21729  ocvcocv 22021  Topctop 23266  TopOnctopon 23283  TopSpctps 23305  Clsdccld 23389  clsccl 23391   Cn ccn 23597  Hauscha 23681  NrmGrpcngp 24951  ℂModcclm 25438  ℂPreHilccph 25542

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-cnex 11295  ax-resscn 11296  ax-1cn 11297  ax-icn 11298  ax-addcl 11299  ax-addrcl 11300  ax-mulcl 11301  ax-mulrcl 11302  ax-mulcom 11303  ax-addass 11304  ax-mulass 11305  ax-distr 11306  ax-i2m1 11307  ax-1ne0 11308  ax-1rid 11309  ax-rnegex 11310  ax-rrecex 11311  ax-cnre 11312  ax-pre-lttri 11313  ax-pre-lttrn 11314  ax-pre-ltadd 11315  ax-pre-mulgt0 11316  ax-pre-sup 11317  ax-addf 11318  ax-mulf 11319

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4841  df-uni 4873  df-int 4913  df-iun 4958  df-iin 4959  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-isom 6575  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-of 7725  df-om 7912  df-1st 8035  df-2nd 8036  df-supp 8215  df-tpos 8281  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-1o 8514  df-2o 8515  df-er 8755  df-map 8887  df-ixp 8964  df-en 9012  df-dom 9013  df-sdom 9014  df-fin 9015  df-fsupp 9392  df-fi 9440  df-sup 9471  df-inf 9472  df-oi 9541  df-card 10057  df-pnf 11384  df-mnf 11385  df-xr 11386  df-ltxr 11387  df-le 11388  df-sub 11582  df-neg 11583  df-div 12014  df-nn 12376  df-2 12445  df-3 12446  df-4 12447  df-5 12448  df-6 12449  df-7 12450  df-8 12451  df-9 12452  df-n0 12647  df-z 12734  df-dec 12855  df-uz 13006  df-q 13116  df-rp 13161  df-xneg 13281  df-xadd 13282  df-xmul 13283  df-ico 13522  df-icc 13523  df-fz 13680  df-fzo 13829  df-seq 14185  df-exp 14245  df-hash 14516  df-cj 15309  df-re 15310  df-im 15311  df-sqrt 15445  df-abs 15446  df-struct 17369  df-sets 17386  df-slot 17404  df-ndx 17416  df-base 17432  df-ress 17453  df-plusg 17486  df-mulr 17487  df-starv 17488  df-sca 17489  df-vsca 17490  df-ip 17491  df-tset 17492  df-ple 17493  df-ds 17495  df-unif 17496  df-hom 17497  df-cco 17498  df-euf 17499  df-rest 17643  df-topn 17644  df-0g 17662  df-gsum 17663  df-topgen 17664  df-pt 17665  df-prds 17668  df-xrs 17724  df-qtop 17729  df-imas 17730  df-xps 17732  df-mre 17806  df-mrc 17807  df-acs 17809  df-mgm 18866  df-sgrp 18958  df-mnd 18974  df-mhm 19028  df-submnd 19029  df-grp 19197  df-minusg 19198  df-sbg 19199  df-mulg 19328  df-subg 19383  df-ghm 19478  df-cntz 19582  df-cmn 20047  df-abl 20048  df-mgp 20412  df-rng 20426  df-ur 20459  df-ring 20512  df-cring 20513  df-oppr 20618  df-dvdsr 20638  df-unit 20639  df-invr 20669  df-dvr 20682  df-rhm 20753  df-subrng 20849  df-subrg 20873  df-drng 21033  df-staf 21147  df-srng 21148  df-lmod 21188  df-lmhm 21348  df-lvec 21429  df-sra 21499  df-rgmod 21500  df-psmet 21721  df-xmet 21722  df-met 21723  df-bl 21724  df-mopn 21725  df-cnfld 21730  df-phl 21987  df-ipf 21988  df-ocv 22024  df-top 23267  df-topon 23284  df-topsp 23306  df-bases 23319  df-cld 23392  df-cls 23394  df-cn 23600  df-cnp 23601  df-t1 23687  df-haus 23688  df-tx 23936  df-hmeo 24129  df-xms 24694  df-ms 24695  df-tms 24696  df-nm 24956  df-ngp 24957  df-tng 24958  df-nlm 24960  df-clm 25439  df-cph 25544  df-tcph 25545

This theorem is used by: (None)