hmopidmchi - Hilbert Space Explorer

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Theorem hmopidmchi 31093

Description: An idempotent Hermitian operator generates a closed subspace. Part of proof of Theorem of [AkhiezerGlazman] p. 64. (Contributed by NM, 21-Apr-2006.) (Proof shortened by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
hmopidmch.1	⊢ 𝑇 ∈ HrmOp
hmopidmch.2	⊢ (𝑇 ∘ 𝑇) = 𝑇

**Assertion**
Ref	Expression
hmopidmchi	⊢ ran 𝑇 ∈ C_ℋ

Proof of Theorem hmopidmchi Dummy variables 𝑓 𝑘 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		hmopidmch.1	. . . 4 ⊢ 𝑇 ∈ HrmOp
2		hmoplin 30884	. . . 4 ⊢ (𝑇 ∈ HrmOp → 𝑇 ∈ LinOp)
3	1, 2	ax-mp 5	. . 3 ⊢ 𝑇 ∈ LinOp
4	3	rnelshi 31001	. 2 ⊢ ran 𝑇 ∈ S_ℋ
5		eqid 2736	. . . . . . . 8 ⊢ (norm_ℎ ∘ −_ℎ ) = (norm_ℎ ∘ −_ℎ )
6	5	hilxmet 30137	. . . . . . 7 ⊢ (norm_ℎ ∘ −_ℎ ) ∈ (∞Met‘ ℋ)
7		eqid 2736	. . . . . . . 8 ⊢ (MetOpen‘(norm_ℎ ∘ −_ℎ )) = (MetOpen‘(norm_ℎ ∘ −_ℎ ))
8	7	methaus 23876	. . . . . . 7 ⊢ ((norm_ℎ ∘ −_ℎ ) ∈ (∞Met‘ ℋ) → (MetOpen‘(norm_ℎ ∘ −_ℎ )) ∈ Haus)
9	6, 8	mp1i 13	. . . . . 6 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → (MetOpen‘(norm_ℎ ∘ −_ℎ )) ∈ Haus)
10		eqid 2736	. . . . . . . . . . . 12 ⊢ ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩ = ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩
11	10, 5	hhims 30114	. . . . . . . . . . . 12 ⊢ (norm_ℎ ∘ −_ℎ ) = (IndMet‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
12	10, 11, 7	hhlm 30141	. . . . . . . . . . 11 ⊢ ⇝_𝑣 = ((⇝_𝑡‘(MetOpen‘(norm_ℎ ∘ −_ℎ ))) ↾ ( ℋ ↑_m ℕ))
13		resss 5962	. . . . . . . . . . 11 ⊢ ((⇝_𝑡‘(MetOpen‘(norm_ℎ ∘ −_ℎ ))) ↾ ( ℋ ↑_m ℕ)) ⊆ (⇝_𝑡‘(MetOpen‘(norm_ℎ ∘ −_ℎ )))
14	12, 13	eqsstri 3978	. . . . . . . . . 10 ⊢ ⇝_𝑣 ⊆ (⇝_𝑡‘(MetOpen‘(norm_ℎ ∘ −_ℎ )))
15	14	ssbri 5150	. . . . . . . . 9 ⊢ (𝑓 ⇝_𝑣 𝑥 → 𝑓(⇝_𝑡‘(MetOpen‘(norm_ℎ ∘ −_ℎ )))𝑥)
16	15	adantl 482	. . . . . . . 8 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → 𝑓(⇝_𝑡‘(MetOpen‘(norm_ℎ ∘ −_ℎ )))𝑥)
17	7	mopntopon 23792	. . . . . . . . . 10 ⊢ ((norm_ℎ ∘ −_ℎ ) ∈ (∞Met‘ ℋ) → (MetOpen‘(norm_ℎ ∘ −_ℎ )) ∈ (TopOn‘ ℋ))
18	6, 17	mp1i 13	. . . . . . . . 9 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → (MetOpen‘(norm_ℎ ∘ −_ℎ )) ∈ (TopOn‘ ℋ))
19	3	lnopfi 30911	. . . . . . . . . . . 12 ⊢ 𝑇: ℋ⟶ ℋ
20	19	a1i 11	. . . . . . . . . . 11 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → 𝑇: ℋ⟶ ℋ)
21	20	feqmptd 6910	. . . . . . . . . 10 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → 𝑇 = (𝑦 ∈ ℋ ↦ (𝑇‘𝑦)))
22		hmopbdoptHIL 30930	. . . . . . . . . . . . 13 ⊢ (𝑇 ∈ HrmOp → 𝑇 ∈ BndLinOp)
23	1, 22	ax-mp 5	. . . . . . . . . . . 12 ⊢ 𝑇 ∈ BndLinOp
24		lnopcnbd 30978	. . . . . . . . . . . . 13 ⊢ (𝑇 ∈ LinOp → (𝑇 ∈ ContOp ↔ 𝑇 ∈ BndLinOp))
25	3, 24	ax-mp 5	. . . . . . . . . . . 12 ⊢ (𝑇 ∈ ContOp ↔ 𝑇 ∈ BndLinOp)
26	23, 25	mpbir 230	. . . . . . . . . . 11 ⊢ 𝑇 ∈ ContOp
27	5, 7	hhcno 30846	. . . . . . . . . . 11 ⊢ ContOp = ((MetOpen‘(norm_ℎ ∘ −_ℎ )) Cn (MetOpen‘(norm_ℎ ∘ −_ℎ )))
28	26, 27	eleqtri 2836	. . . . . . . . . 10 ⊢ 𝑇 ∈ ((MetOpen‘(norm_ℎ ∘ −_ℎ )) Cn (MetOpen‘(norm_ℎ ∘ −_ℎ )))
29	21, 28	eqeltrrdi 2847	. . . . . . . . 9 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → (𝑦 ∈ ℋ ↦ (𝑇‘𝑦)) ∈ ((MetOpen‘(norm_ℎ ∘ −_ℎ )) Cn (MetOpen‘(norm_ℎ ∘ −_ℎ ))))
30	18	cnmptid 23012	. . . . . . . . 9 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → (𝑦 ∈ ℋ ↦ 𝑦) ∈ ((MetOpen‘(norm_ℎ ∘ −_ℎ )) Cn (MetOpen‘(norm_ℎ ∘ −_ℎ ))))
31	10	hhnv 30107	. . . . . . . . . 10 ⊢ ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩ ∈ NrmCVec
32	10	hhvs 30112	. . . . . . . . . . 11 ⊢ −_ℎ = ( −_𝑣 ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
33	11, 7, 32	vmcn 29641	. . . . . . . . . 10 ⊢ (⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩ ∈ NrmCVec → −_ℎ ∈ (((MetOpen‘(norm_ℎ ∘ −_ℎ )) ×_t (MetOpen‘(norm_ℎ ∘ −_ℎ ))) Cn (MetOpen‘(norm_ℎ ∘ −_ℎ ))))
34	31, 33	mp1i 13	. . . . . . . . 9 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → −_ℎ ∈ (((MetOpen‘(norm_ℎ ∘ −_ℎ )) ×_t (MetOpen‘(norm_ℎ ∘ −_ℎ ))) Cn (MetOpen‘(norm_ℎ ∘ −_ℎ ))))
35	18, 29, 30, 34	cnmpt12f 23017	. . . . . . . 8 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → (𝑦 ∈ ℋ ↦ ((𝑇‘𝑦) −_ℎ 𝑦)) ∈ ((MetOpen‘(norm_ℎ ∘ −_ℎ )) Cn (MetOpen‘(norm_ℎ ∘ −_ℎ ))))
36	16, 35	lmcn 22656	. . . . . . 7 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → ((𝑦 ∈ ℋ ↦ ((𝑇‘𝑦) −_ℎ 𝑦)) ∘ 𝑓)(⇝_𝑡‘(MetOpen‘(norm_ℎ ∘ −_ℎ )))((𝑦 ∈ ℋ ↦ ((𝑇‘𝑦) −_ℎ 𝑦))‘𝑥))
37		simpl 483	. . . . . . . . . . . . . 14 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → 𝑓:ℕ⟶ran 𝑇)
38	4	shssii 30155	. . . . . . . . . . . . . 14 ⊢ ran 𝑇 ⊆ ℋ
39		fss 6685	. . . . . . . . . . . . . 14 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ ran 𝑇 ⊆ ℋ) → 𝑓:ℕ⟶ ℋ)
40	37, 38, 39	sylancl 586	. . . . . . . . . . . . 13 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → 𝑓:ℕ⟶ ℋ)
41	40	ffvelcdmda 7035	. . . . . . . . . . . 12 ⊢ (((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) ∧ 𝑘 ∈ ℕ) → (𝑓‘𝑘) ∈ ℋ)
42		fveq2 6842	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑓‘𝑘) → (𝑇‘𝑦) = (𝑇‘(𝑓‘𝑘)))
43		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑓‘𝑘) → 𝑦 = (𝑓‘𝑘))
44	42, 43	oveq12d 7375	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑓‘𝑘) → ((𝑇‘𝑦) −_ℎ 𝑦) = ((𝑇‘(𝑓‘𝑘)) −_ℎ (𝑓‘𝑘)))
45		eqid 2736	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℋ ↦ ((𝑇‘𝑦) −_ℎ 𝑦)) = (𝑦 ∈ ℋ ↦ ((𝑇‘𝑦) −_ℎ 𝑦))
46		ovex 7390	. . . . . . . . . . . . 13 ⊢ ((𝑇‘(𝑓‘𝑘)) −_ℎ (𝑓‘𝑘)) ∈ V
47	44, 45, 46	fvmpt 6948	. . . . . . . . . . . 12 ⊢ ((𝑓‘𝑘) ∈ ℋ → ((𝑦 ∈ ℋ ↦ ((𝑇‘𝑦) −_ℎ 𝑦))‘(𝑓‘𝑘)) = ((𝑇‘(𝑓‘𝑘)) −_ℎ (𝑓‘𝑘)))
48	41, 47	syl 17	. . . . . . . . . . 11 ⊢ (((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) ∧ 𝑘 ∈ ℕ) → ((𝑦 ∈ ℋ ↦ ((𝑇‘𝑦) −_ℎ 𝑦))‘(𝑓‘𝑘)) = ((𝑇‘(𝑓‘𝑘)) −_ℎ (𝑓‘𝑘)))
49		ffn 6668	. . . . . . . . . . . . . . . 16 ⊢ (𝑇: ℋ⟶ ℋ → 𝑇 Fn ℋ)
50	19, 49	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ 𝑇 Fn ℋ
51		fveq2 6842	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = (𝑇‘𝑥) → (𝑇‘𝑦) = (𝑇‘(𝑇‘𝑥)))
52		id 22	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = (𝑇‘𝑥) → 𝑦 = (𝑇‘𝑥))
53	51, 52	eqeq12d 2752	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = (𝑇‘𝑥) → ((𝑇‘𝑦) = 𝑦 ↔ (𝑇‘(𝑇‘𝑥)) = (𝑇‘𝑥)))
54	53	ralrn 7038	. . . . . . . . . . . . . . 15 ⊢ (𝑇 Fn ℋ → (∀𝑦 ∈ ran 𝑇(𝑇‘𝑦) = 𝑦 ↔ ∀𝑥 ∈ ℋ (𝑇‘(𝑇‘𝑥)) = (𝑇‘𝑥)))
55	50, 54	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (∀𝑦 ∈ ran 𝑇(𝑇‘𝑦) = 𝑦 ↔ ∀𝑥 ∈ ℋ (𝑇‘(𝑇‘𝑥)) = (𝑇‘𝑥))
56	19, 19	hocoi 30706	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℋ → ((𝑇 ∘ 𝑇)‘𝑥) = (𝑇‘(𝑇‘𝑥)))
57		hmopidmch.2	. . . . . . . . . . . . . . . 16 ⊢ (𝑇 ∘ 𝑇) = 𝑇
58	57	fveq1i 6843	. . . . . . . . . . . . . . 15 ⊢ ((𝑇 ∘ 𝑇)‘𝑥) = (𝑇‘𝑥)
59	56, 58	eqtr3di 2791	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℋ → (𝑇‘(𝑇‘𝑥)) = (𝑇‘𝑥))
60	55, 59	mprgbir 3071	. . . . . . . . . . . . 13 ⊢ ∀𝑦 ∈ ran 𝑇(𝑇‘𝑦) = 𝑦
61		ffvelcdm 7032	. . . . . . . . . . . . . 14 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑘 ∈ ℕ) → (𝑓‘𝑘) ∈ ran 𝑇)
62	61	adantlr 713	. . . . . . . . . . . . 13 ⊢ (((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) ∧ 𝑘 ∈ ℕ) → (𝑓‘𝑘) ∈ ran 𝑇)
63	42, 43	eqeq12d 2752	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑓‘𝑘) → ((𝑇‘𝑦) = 𝑦 ↔ (𝑇‘(𝑓‘𝑘)) = (𝑓‘𝑘)))
64	63	rspccv 3578	. . . . . . . . . . . . 13 ⊢ (∀𝑦 ∈ ran 𝑇(𝑇‘𝑦) = 𝑦 → ((𝑓‘𝑘) ∈ ran 𝑇 → (𝑇‘(𝑓‘𝑘)) = (𝑓‘𝑘)))
65	60, 62, 64	mpsyl 68	. . . . . . . . . . . 12 ⊢ (((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) ∧ 𝑘 ∈ ℕ) → (𝑇‘(𝑓‘𝑘)) = (𝑓‘𝑘))
66	65, 41	eqeltrd 2838	. . . . . . . . . . . . 13 ⊢ (((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) ∧ 𝑘 ∈ ℕ) → (𝑇‘(𝑓‘𝑘)) ∈ ℋ)
67		hvsubeq0 30010	. . . . . . . . . . . . 13 ⊢ (((𝑇‘(𝑓‘𝑘)) ∈ ℋ ∧ (𝑓‘𝑘) ∈ ℋ) → (((𝑇‘(𝑓‘𝑘)) −_ℎ (𝑓‘𝑘)) = 0_ℎ ↔ (𝑇‘(𝑓‘𝑘)) = (𝑓‘𝑘)))
68	66, 41, 67	syl2anc 584	. . . . . . . . . . . 12 ⊢ (((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) ∧ 𝑘 ∈ ℕ) → (((𝑇‘(𝑓‘𝑘)) −_ℎ (𝑓‘𝑘)) = 0_ℎ ↔ (𝑇‘(𝑓‘𝑘)) = (𝑓‘𝑘)))
69	65, 68	mpbird 256	. . . . . . . . . . 11 ⊢ (((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) ∧ 𝑘 ∈ ℕ) → ((𝑇‘(𝑓‘𝑘)) −_ℎ (𝑓‘𝑘)) = 0_ℎ)
70	48, 69	eqtrd 2776	. . . . . . . . . 10 ⊢ (((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) ∧ 𝑘 ∈ ℕ) → ((𝑦 ∈ ℋ ↦ ((𝑇‘𝑦) −_ℎ 𝑦))‘(𝑓‘𝑘)) = 0_ℎ)
71		fvco3 6940	. . . . . . . . . . 11 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑘 ∈ ℕ) → (((𝑦 ∈ ℋ ↦ ((𝑇‘𝑦) −_ℎ 𝑦)) ∘ 𝑓)‘𝑘) = ((𝑦 ∈ ℋ ↦ ((𝑇‘𝑦) −_ℎ 𝑦))‘(𝑓‘𝑘)))
72	71	adantlr 713	. . . . . . . . . 10 ⊢ (((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) ∧ 𝑘 ∈ ℕ) → (((𝑦 ∈ ℋ ↦ ((𝑇‘𝑦) −_ℎ 𝑦)) ∘ 𝑓)‘𝑘) = ((𝑦 ∈ ℋ ↦ ((𝑇‘𝑦) −_ℎ 𝑦))‘(𝑓‘𝑘)))
73		ax-hv0cl 29945	. . . . . . . . . . . . 13 ⊢ 0_ℎ ∈ ℋ
74	73	elexi 3464	. . . . . . . . . . . 12 ⊢ 0_ℎ ∈ V
75	74	fvconst2 7153	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → ((ℕ × {0_ℎ})‘𝑘) = 0_ℎ)
76	75	adantl 482	. . . . . . . . . 10 ⊢ (((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) ∧ 𝑘 ∈ ℕ) → ((ℕ × {0_ℎ})‘𝑘) = 0_ℎ)
77	70, 72, 76	3eqtr4d 2786	. . . . . . . . 9 ⊢ (((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) ∧ 𝑘 ∈ ℕ) → (((𝑦 ∈ ℋ ↦ ((𝑇‘𝑦) −_ℎ 𝑦)) ∘ 𝑓)‘𝑘) = ((ℕ × {0_ℎ})‘𝑘))
78	77	ralrimiva 3143	. . . . . . . 8 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → ∀𝑘 ∈ ℕ (((𝑦 ∈ ℋ ↦ ((𝑇‘𝑦) −_ℎ 𝑦)) ∘ 𝑓)‘𝑘) = ((ℕ × {0_ℎ})‘𝑘))
79		ovex 7390	. . . . . . . . . . 11 ⊢ ((𝑇‘𝑦) −_ℎ 𝑦) ∈ V
80	79, 45	fnmpti 6644	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℋ ↦ ((𝑇‘𝑦) −_ℎ 𝑦)) Fn ℋ
81		fnfco 6707	. . . . . . . . . 10 ⊢ (((𝑦 ∈ ℋ ↦ ((𝑇‘𝑦) −_ℎ 𝑦)) Fn ℋ ∧ 𝑓:ℕ⟶ ℋ) → ((𝑦 ∈ ℋ ↦ ((𝑇‘𝑦) −_ℎ 𝑦)) ∘ 𝑓) Fn ℕ)
82	80, 40, 81	sylancr 587	. . . . . . . . 9 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → ((𝑦 ∈ ℋ ↦ ((𝑇‘𝑦) −_ℎ 𝑦)) ∘ 𝑓) Fn ℕ)
83	74	fconst 6728	. . . . . . . . . 10 ⊢ (ℕ × {0_ℎ}):ℕ⟶{0_ℎ}
84		ffn 6668	. . . . . . . . . 10 ⊢ ((ℕ × {0_ℎ}):ℕ⟶{0_ℎ} → (ℕ × {0_ℎ}) Fn ℕ)
85	83, 84	ax-mp 5	. . . . . . . . 9 ⊢ (ℕ × {0_ℎ}) Fn ℕ
86		eqfnfv 6982	. . . . . . . . 9 ⊢ ((((𝑦 ∈ ℋ ↦ ((𝑇‘𝑦) −_ℎ 𝑦)) ∘ 𝑓) Fn ℕ ∧ (ℕ × {0_ℎ}) Fn ℕ) → (((𝑦 ∈ ℋ ↦ ((𝑇‘𝑦) −_ℎ 𝑦)) ∘ 𝑓) = (ℕ × {0_ℎ}) ↔ ∀𝑘 ∈ ℕ (((𝑦 ∈ ℋ ↦ ((𝑇‘𝑦) −_ℎ 𝑦)) ∘ 𝑓)‘𝑘) = ((ℕ × {0_ℎ})‘𝑘)))
87	82, 85, 86	sylancl 586	. . . . . . . 8 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → (((𝑦 ∈ ℋ ↦ ((𝑇‘𝑦) −_ℎ 𝑦)) ∘ 𝑓) = (ℕ × {0_ℎ}) ↔ ∀𝑘 ∈ ℕ (((𝑦 ∈ ℋ ↦ ((𝑇‘𝑦) −_ℎ 𝑦)) ∘ 𝑓)‘𝑘) = ((ℕ × {0_ℎ})‘𝑘)))
88	78, 87	mpbird 256	. . . . . . 7 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → ((𝑦 ∈ ℋ ↦ ((𝑇‘𝑦) −_ℎ 𝑦)) ∘ 𝑓) = (ℕ × {0_ℎ}))
89		vex 3449	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
90	89	hlimveci 30132	. . . . . . . . 9 ⊢ (𝑓 ⇝_𝑣 𝑥 → 𝑥 ∈ ℋ)
91	90	adantl 482	. . . . . . . 8 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → 𝑥 ∈ ℋ)
92		fveq2 6842	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (𝑇‘𝑦) = (𝑇‘𝑥))
93		id 22	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → 𝑦 = 𝑥)
94	92, 93	oveq12d 7375	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → ((𝑇‘𝑦) −_ℎ 𝑦) = ((𝑇‘𝑥) −_ℎ 𝑥))
95		ovex 7390	. . . . . . . . 9 ⊢ ((𝑇‘𝑥) −_ℎ 𝑥) ∈ V
96	94, 45, 95	fvmpt 6948	. . . . . . . 8 ⊢ (𝑥 ∈ ℋ → ((𝑦 ∈ ℋ ↦ ((𝑇‘𝑦) −_ℎ 𝑦))‘𝑥) = ((𝑇‘𝑥) −_ℎ 𝑥))
97	91, 96	syl 17	. . . . . . 7 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → ((𝑦 ∈ ℋ ↦ ((𝑇‘𝑦) −_ℎ 𝑦))‘𝑥) = ((𝑇‘𝑥) −_ℎ 𝑥))
98	36, 88, 97	3brtr3d 5136	. . . . . 6 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → (ℕ × {0_ℎ})(⇝_𝑡‘(MetOpen‘(norm_ℎ ∘ −_ℎ )))((𝑇‘𝑥) −_ℎ 𝑥))
99	73	a1i 11	. . . . . . 7 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → 0_ℎ ∈ ℋ)
100		1zzd 12534	. . . . . . 7 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → 1 ∈ ℤ)
101		nnuz 12806	. . . . . . . 8 ⊢ ℕ = (ℤ_≥‘1)
102	101	lmconst 22612	. . . . . . 7 ⊢ (((MetOpen‘(norm_ℎ ∘ −_ℎ )) ∈ (TopOn‘ ℋ) ∧ 0_ℎ ∈ ℋ ∧ 1 ∈ ℤ) → (ℕ × {0_ℎ})(⇝_𝑡‘(MetOpen‘(norm_ℎ ∘ −_ℎ )))0_ℎ)
103	18, 99, 100, 102	syl3anc 1371	. . . . . 6 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → (ℕ × {0_ℎ})(⇝_𝑡‘(MetOpen‘(norm_ℎ ∘ −_ℎ )))0_ℎ)
104	9, 98, 103	lmmo 22731	. . . . 5 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → ((𝑇‘𝑥) −_ℎ 𝑥) = 0_ℎ)
105	19	ffvelcdmi 7034	. . . . . . 7 ⊢ (𝑥 ∈ ℋ → (𝑇‘𝑥) ∈ ℋ)
106	91, 105	syl 17	. . . . . 6 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → (𝑇‘𝑥) ∈ ℋ)
107		hvsubeq0 30010	. . . . . 6 ⊢ (((𝑇‘𝑥) ∈ ℋ ∧ 𝑥 ∈ ℋ) → (((𝑇‘𝑥) −_ℎ 𝑥) = 0_ℎ ↔ (𝑇‘𝑥) = 𝑥))
108	106, 91, 107	syl2anc 584	. . . . 5 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → (((𝑇‘𝑥) −_ℎ 𝑥) = 0_ℎ ↔ (𝑇‘𝑥) = 𝑥))
109	104, 108	mpbid 231	. . . 4 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → (𝑇‘𝑥) = 𝑥)
110		fnfvelrn 7031	. . . . 5 ⊢ ((𝑇 Fn ℋ ∧ 𝑥 ∈ ℋ) → (𝑇‘𝑥) ∈ ran 𝑇)
111	50, 91, 110	sylancr 587	. . . 4 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → (𝑇‘𝑥) ∈ ran 𝑇)
112	109, 111	eqeltrrd 2839	. . 3 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → 𝑥 ∈ ran 𝑇)
113	112	gen2 1798	. 2 ⊢ ∀𝑓∀𝑥((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → 𝑥 ∈ ran 𝑇)
114		isch2 30165	. 2 ⊢ (ran 𝑇 ∈ C_ℋ ↔ (ran 𝑇 ∈ S_ℋ ∧ ∀𝑓∀𝑥((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → 𝑥 ∈ ran 𝑇)))
115	4, 113, 114	mpbir2an 709	1 ⊢ ran 𝑇 ∈ C_ℋ

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∀wal 1539 = wceq 1541 ∈ wcel 2106 ∀wral 3064 ⊆ wss 3910 {csn 4586 ⟨cop 4592 class class class wbr 5105 ↦ cmpt 5188 × cxp 5631 ran crn 5634 ↾ cres 5635 ∘ ccom 5637 Fn wfn 6491 ⟶wf 6492 ‘cfv 6496 (class class class)co 7357 ↑_m cmap 8765 1c1 11052 ℕcn 12153 ℤcz 12499 ∞Metcxmet 20781 MetOpencmopn 20786 TopOnctopon 22259 Cn ccn 22575 ⇝_𝑡clm 22577 Hauscha 22659 ×_t ctx 22911 NrmCVeccnv 29526 ℋchba 29861 +_ℎ cva 29862 ·_ℎ csm 29863 norm_ℎcno 29865 0_ℎc0v 29866 −_ℎ cmv 29867 ⇝_𝑣 chli 29869 S_ℋ csh 29870 C_ℋ cch 29871 ContOpccop 29888 LinOpclo 29889 BndLinOpcbo 29890 HrmOpcho 29892

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 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 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-inf2 9577 ax-cc 10371 ax-dc 10382 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 ax-addf 11130 ax-mulf 11131 ax-hilex 29941 ax-hfvadd 29942 ax-hvcom 29943 ax-hvass 29944 ax-hv0cl 29945 ax-hvaddid 29946 ax-hfvmul 29947 ax-hvmulid 29948 ax-hvmulass 29949 ax-hvdistr1 29950 ax-hvdistr2 29951 ax-hvmul0 29952 ax-hfi 30021 ax-his1 30024 ax-his2 30025 ax-his3 30026 ax-his4 30027 ax-hcompl 30144

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-om 7803 df-1st 7921 df-2nd 7922 df-supp 8093 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-2o 8413 df-oadd 8416 df-omul 8417 df-er 8648 df-map 8767 df-pm 8768 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9306 df-fi 9347 df-sup 9378 df-inf 9379 df-oi 9446 df-card 9875 df-acn 9878 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-dec 12619 df-uz 12764 df-q 12874 df-rp 12916 df-xneg 13033 df-xadd 13034 df-xmul 13035 df-ioo 13268 df-ico 13270 df-icc 13271 df-fz 13425 df-fzo 13568 df-fl 13697 df-seq 13907 df-exp 13968 df-hash 14231 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-clim 15370 df-rlim 15371 df-sum 15571 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-starv 17148 df-sca 17149 df-vsca 17150 df-ip 17151 df-tset 17152 df-ple 17153 df-ds 17155 df-unif 17156 df-hom 17157 df-cco 17158 df-rest 17304 df-topn 17305 df-0g 17323 df-gsum 17324 df-topgen 17325 df-pt 17326 df-prds 17329 df-xrs 17384 df-qtop 17389 df-imas 17390 df-xps 17392 df-mre 17466 df-mrc 17467 df-acs 17469 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-submnd 18602 df-mulg 18873 df-cntz 19097 df-cmn 19564 df-psmet 20788 df-xmet 20789 df-met 20790 df-bl 20791 df-mopn 20792 df-fbas 20793 df-fg 20794 df-cnfld 20797 df-top 22243 df-topon 22260 df-topsp 22282 df-bases 22296 df-cld 22370 df-ntr 22371 df-cls 22372 df-nei 22449 df-cn 22578 df-cnp 22579 df-lm 22580 df-t1 22665 df-haus 22666 df-cmp 22738 df-tx 22913 df-hmeo 23106 df-fil 23197 df-fm 23289 df-flim 23290 df-flf 23291 df-fcls 23292 df-xms 23673 df-ms 23674 df-tms 23675 df-cncf 24241 df-cfil 24619 df-cau 24620 df-cmet 24621 df-grpo 29435 df-gid 29436 df-ginv 29437 df-gdiv 29438 df-ablo 29487 df-vc 29501 df-nv 29534 df-va 29537 df-ba 29538 df-sm 29539 df-0v 29540 df-vs 29541 df-nmcv 29542 df-ims 29543 df-dip 29643 df-ssp 29664 df-lno 29686 df-nmoo 29687 df-blo 29688 df-0o 29689 df-ph 29755 df-cbn 29805 df-hlo 29828 df-hnorm 29910 df-hba 29911 df-hvsub 29913 df-hlim 29914 df-hcau 29915 df-sh 30149 df-ch 30163 df-oc 30194 df-ch0 30195 df-shs 30250 df-pjh 30337 df-h0op 30690 df-nmop 30781 df-cnop 30782 df-lnop 30783 df-bdop 30784 df-unop 30785 df-hmop 30786

This theorem is referenced by: hmopidmpji 31094 hmopidmch 31095

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