hmopidmchi - Hilbert Space Explorer

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Theorem hmopidmchi 32080

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 31871	. . . 4 ⊢ (𝑇 ∈ HrmOp → 𝑇 ∈ LinOp)
3	1, 2	ax-mp 5	. . 3 ⊢ 𝑇 ∈ LinOp
4	3	rnelshi 31988	. 2 ⊢ ran 𝑇 ∈ S_ℋ
5		eqid 2729	. . . . . . . 8 ⊢ (norm_ℎ ∘ −_ℎ ) = (norm_ℎ ∘ −_ℎ )
6	5	hilxmet 31124	. . . . . . 7 ⊢ (norm_ℎ ∘ −_ℎ ) ∈ (∞Met‘ ℋ)
7		eqid 2729	. . . . . . . 8 ⊢ (MetOpen‘(norm_ℎ ∘ −_ℎ )) = (MetOpen‘(norm_ℎ ∘ −_ℎ ))
8	7	methaus 24408	. . . . . . 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 2729	. . . . . . . . . . . 12 ⊢ ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩ = ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩
11	10, 5	hhims 31101	. . . . . . . . . . . 12 ⊢ (norm_ℎ ∘ −_ℎ ) = (IndMet‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
12	10, 11, 7	hhlm 31128	. . . . . . . . . . 11 ⊢ ⇝_𝑣 = ((⇝_𝑡‘(MetOpen‘(norm_ℎ ∘ −_ℎ ))) ↾ ( ℋ ↑_m ℕ))
13		resss 5972	. . . . . . . . . . 11 ⊢ ((⇝_𝑡‘(MetOpen‘(norm_ℎ ∘ −_ℎ ))) ↾ ( ℋ ↑_m ℕ)) ⊆ (⇝_𝑡‘(MetOpen‘(norm_ℎ ∘ −_ℎ )))
14	12, 13	eqsstri 3993	. . . . . . . . . 10 ⊢ ⇝_𝑣 ⊆ (⇝_𝑡‘(MetOpen‘(norm_ℎ ∘ −_ℎ )))
15	14	ssbri 5152	. . . . . . . . 9 ⊢ (𝑓 ⇝_𝑣 𝑥 → 𝑓(⇝_𝑡‘(MetOpen‘(norm_ℎ ∘ −_ℎ )))𝑥)
16	15	adantl 481	. . . . . . . 8 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → 𝑓(⇝_𝑡‘(MetOpen‘(norm_ℎ ∘ −_ℎ )))𝑥)
17	7	mopntopon 24327	. . . . . . . . . 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 31898	. . . . . . . . . . . 12 ⊢ 𝑇: ℋ⟶ ℋ
20	19	a1i 11	. . . . . . . . . . 11 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → 𝑇: ℋ⟶ ℋ)
21	20	feqmptd 6929	. . . . . . . . . 10 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → 𝑇 = (𝑦 ∈ ℋ ↦ (𝑇‘𝑦)))
22		hmopbdoptHIL 31917	. . . . . . . . . . . . 13 ⊢ (𝑇 ∈ HrmOp → 𝑇 ∈ BndLinOp)
23	1, 22	ax-mp 5	. . . . . . . . . . . 12 ⊢ 𝑇 ∈ BndLinOp
24		lnopcnbd 31965	. . . . . . . . . . . . 13 ⊢ (𝑇 ∈ LinOp → (𝑇 ∈ ContOp ↔ 𝑇 ∈ BndLinOp))
25	3, 24	ax-mp 5	. . . . . . . . . . . 12 ⊢ (𝑇 ∈ ContOp ↔ 𝑇 ∈ BndLinOp)
26	23, 25	mpbir 231	. . . . . . . . . . 11 ⊢ 𝑇 ∈ ContOp
27	5, 7	hhcno 31833	. . . . . . . . . . 11 ⊢ ContOp = ((MetOpen‘(norm_ℎ ∘ −_ℎ )) Cn (MetOpen‘(norm_ℎ ∘ −_ℎ )))
28	26, 27	eleqtri 2826	. . . . . . . . . 10 ⊢ 𝑇 ∈ ((MetOpen‘(norm_ℎ ∘ −_ℎ )) Cn (MetOpen‘(norm_ℎ ∘ −_ℎ )))
29	21, 28	eqeltrrdi 2837	. . . . . . . . 9 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → (𝑦 ∈ ℋ ↦ (𝑇‘𝑦)) ∈ ((MetOpen‘(norm_ℎ ∘ −_ℎ )) Cn (MetOpen‘(norm_ℎ ∘ −_ℎ ))))
30	18	cnmptid 23548	. . . . . . . . 9 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → (𝑦 ∈ ℋ ↦ 𝑦) ∈ ((MetOpen‘(norm_ℎ ∘ −_ℎ )) Cn (MetOpen‘(norm_ℎ ∘ −_ℎ ))))
31	10	hhnv 31094	. . . . . . . . . 10 ⊢ ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩ ∈ NrmCVec
32	10	hhvs 31099	. . . . . . . . . . 11 ⊢ −_ℎ = ( −_𝑣 ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
33	11, 7, 32	vmcn 30628	. . . . . . . . . 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 23553	. . . . . . . 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 23192	. . . . . . 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 482	. . . . . . . . . . . . . 14 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → 𝑓:ℕ⟶ran 𝑇)
38	4	shssii 31142	. . . . . . . . . . . . . 14 ⊢ ran 𝑇 ⊆ ℋ
39		fss 6704	. . . . . . . . . . . . . 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 7056	. . . . . . . . . . . 12 ⊢ (((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) ∧ 𝑘 ∈ ℕ) → (𝑓‘𝑘) ∈ ℋ)
42		fveq2 6858	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑓‘𝑘) → (𝑇‘𝑦) = (𝑇‘(𝑓‘𝑘)))
43		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑓‘𝑘) → 𝑦 = (𝑓‘𝑘))
44	42, 43	oveq12d 7405	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑓‘𝑘) → ((𝑇‘𝑦) −_ℎ 𝑦) = ((𝑇‘(𝑓‘𝑘)) −_ℎ (𝑓‘𝑘)))
45		eqid 2729	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℋ ↦ ((𝑇‘𝑦) −_ℎ 𝑦)) = (𝑦 ∈ ℋ ↦ ((𝑇‘𝑦) −_ℎ 𝑦))
46		ovex 7420	. . . . . . . . . . . . 13 ⊢ ((𝑇‘(𝑓‘𝑘)) −_ℎ (𝑓‘𝑘)) ∈ V
47	44, 45, 46	fvmpt 6968	. . . . . . . . . . . 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 6688	. . . . . . . . . . . . . . . 16 ⊢ (𝑇: ℋ⟶ ℋ → 𝑇 Fn ℋ)
50	19, 49	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ 𝑇 Fn ℋ
51		fveq2 6858	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = (𝑇‘𝑥) → (𝑇‘𝑦) = (𝑇‘(𝑇‘𝑥)))
52		id 22	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = (𝑇‘𝑥) → 𝑦 = (𝑇‘𝑥))
53	51, 52	eqeq12d 2745	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = (𝑇‘𝑥) → ((𝑇‘𝑦) = 𝑦 ↔ (𝑇‘(𝑇‘𝑥)) = (𝑇‘𝑥)))
54	53	ralrn 7060	. . . . . . . . . . . . . . 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 31693	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℋ → ((𝑇 ∘ 𝑇)‘𝑥) = (𝑇‘(𝑇‘𝑥)))
57		hmopidmch.2	. . . . . . . . . . . . . . . 16 ⊢ (𝑇 ∘ 𝑇) = 𝑇
58	57	fveq1i 6859	. . . . . . . . . . . . . . 15 ⊢ ((𝑇 ∘ 𝑇)‘𝑥) = (𝑇‘𝑥)
59	56, 58	eqtr3di 2779	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℋ → (𝑇‘(𝑇‘𝑥)) = (𝑇‘𝑥))
60	55, 59	mprgbir 3051	. . . . . . . . . . . . 13 ⊢ ∀𝑦 ∈ ran 𝑇(𝑇‘𝑦) = 𝑦
61		ffvelcdm 7053	. . . . . . . . . . . . . 14 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑘 ∈ ℕ) → (𝑓‘𝑘) ∈ ran 𝑇)
62	61	adantlr 715	. . . . . . . . . . . . 13 ⊢ (((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) ∧ 𝑘 ∈ ℕ) → (𝑓‘𝑘) ∈ ran 𝑇)
63	42, 43	eqeq12d 2745	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑓‘𝑘) → ((𝑇‘𝑦) = 𝑦 ↔ (𝑇‘(𝑓‘𝑘)) = (𝑓‘𝑘)))
64	63	rspccv 3585	. . . . . . . . . . . . 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 2828	. . . . . . . . . . . . 13 ⊢ (((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) ∧ 𝑘 ∈ ℕ) → (𝑇‘(𝑓‘𝑘)) ∈ ℋ)
67		hvsubeq0 30997	. . . . . . . . . . . . 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 257	. . . . . . . . . . 11 ⊢ (((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) ∧ 𝑘 ∈ ℕ) → ((𝑇‘(𝑓‘𝑘)) −_ℎ (𝑓‘𝑘)) = 0_ℎ)
70	48, 69	eqtrd 2764	. . . . . . . . . 10 ⊢ (((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) ∧ 𝑘 ∈ ℕ) → ((𝑦 ∈ ℋ ↦ ((𝑇‘𝑦) −_ℎ 𝑦))‘(𝑓‘𝑘)) = 0_ℎ)
71		fvco3 6960	. . . . . . . . . . 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 715	. . . . . . . . . 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 30932	. . . . . . . . . . . . 13 ⊢ 0_ℎ ∈ ℋ
74	73	elexi 3470	. . . . . . . . . . . 12 ⊢ 0_ℎ ∈ V
75	74	fvconst2 7178	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → ((ℕ × {0_ℎ})‘𝑘) = 0_ℎ)
76	75	adantl 481	. . . . . . . . . 10 ⊢ (((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) ∧ 𝑘 ∈ ℕ) → ((ℕ × {0_ℎ})‘𝑘) = 0_ℎ)
77	70, 72, 76	3eqtr4d 2774	. . . . . . . . 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 3125	. . . . . . . 8 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → ∀𝑘 ∈ ℕ (((𝑦 ∈ ℋ ↦ ((𝑇‘𝑦) −_ℎ 𝑦)) ∘ 𝑓)‘𝑘) = ((ℕ × {0_ℎ})‘𝑘))
79		ovex 7420	. . . . . . . . . . 11 ⊢ ((𝑇‘𝑦) −_ℎ 𝑦) ∈ V
80	79, 45	fnmpti 6661	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℋ ↦ ((𝑇‘𝑦) −_ℎ 𝑦)) Fn ℋ
81		fnfco 6725	. . . . . . . . . 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 6746	. . . . . . . . . 10 ⊢ (ℕ × {0_ℎ}):ℕ⟶{0_ℎ}
84		ffn 6688	. . . . . . . . . 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 7003	. . . . . . . . 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 257	. . . . . . 7 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → ((𝑦 ∈ ℋ ↦ ((𝑇‘𝑦) −_ℎ 𝑦)) ∘ 𝑓) = (ℕ × {0_ℎ}))
89		vex 3451	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
90	89	hlimveci 31119	. . . . . . . . 9 ⊢ (𝑓 ⇝_𝑣 𝑥 → 𝑥 ∈ ℋ)
91	90	adantl 481	. . . . . . . 8 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → 𝑥 ∈ ℋ)
92		fveq2 6858	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (𝑇‘𝑦) = (𝑇‘𝑥))
93		id 22	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → 𝑦 = 𝑥)
94	92, 93	oveq12d 7405	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → ((𝑇‘𝑦) −_ℎ 𝑦) = ((𝑇‘𝑥) −_ℎ 𝑥))
95		ovex 7420	. . . . . . . . 9 ⊢ ((𝑇‘𝑥) −_ℎ 𝑥) ∈ V
96	94, 45, 95	fvmpt 6968	. . . . . . . 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 5138	. . . . . 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 12564	. . . . . . 7 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → 1 ∈ ℤ)
101		nnuz 12836	. . . . . . . 8 ⊢ ℕ = (ℤ_≥‘1)
102	101	lmconst 23148	. . . . . . 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 1373	. . . . . 6 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → (ℕ × {0_ℎ})(⇝_𝑡‘(MetOpen‘(norm_ℎ ∘ −_ℎ )))0_ℎ)
104	9, 98, 103	lmmo 23267	. . . . 5 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → ((𝑇‘𝑥) −_ℎ 𝑥) = 0_ℎ)
105	19	ffvelcdmi 7055	. . . . . . 7 ⊢ (𝑥 ∈ ℋ → (𝑇‘𝑥) ∈ ℋ)
106	91, 105	syl 17	. . . . . 6 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → (𝑇‘𝑥) ∈ ℋ)
107		hvsubeq0 30997	. . . . . 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 232	. . . 4 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → (𝑇‘𝑥) = 𝑥)
110		fnfvelrn 7052	. . . . 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 2829	. . 3 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → 𝑥 ∈ ran 𝑇)
113	112	gen2 1796	. 2 ⊢ ∀𝑓∀𝑥((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → 𝑥 ∈ ran 𝑇)
114		isch2 31152	. 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 711	1 ⊢ ran 𝑇 ∈ C_ℋ

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∀wal 1538 = wceq 1540 ∈ wcel 2109 ∀wral 3044 ⊆ wss 3914 {csn 4589 ⟨cop 4595 class class class wbr 5107 ↦ cmpt 5188 × cxp 5636 ran crn 5639 ↾ cres 5640 ∘ ccom 5642 Fn wfn 6506 ⟶wf 6507 ‘cfv 6511 (class class class)co 7387 ↑_m cmap 8799 1c1 11069 ℕcn 12186 ℤcz 12529 ∞Metcxmet 21249 MetOpencmopn 21254 TopOnctopon 22797 Cn ccn 23111 ⇝_𝑡clm 23113 Hauscha 23195 ×_t ctx 23447 NrmCVeccnv 30513 ℋchba 30848 +_ℎ cva 30849 ·_ℎ csm 30850 norm_ℎcno 30852 0_ℎc0v 30853 −_ℎ cmv 30854 ⇝_𝑣 chli 30856 S_ℋ csh 30857 C_ℋ cch 30858 ContOpccop 30875 LinOpclo 30876 BndLinOpcbo 30877 HrmOpcho 30879

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-inf2 9594 ax-cc 10388 ax-dc 10399 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146 ax-addf 11147 ax-mulf 11148 ax-hilex 30928 ax-hfvadd 30929 ax-hvcom 30930 ax-hvass 30931 ax-hv0cl 30932 ax-hvaddid 30933 ax-hfvmul 30934 ax-hvmulid 30935 ax-hvmulass 30936 ax-hvdistr1 30937 ax-hvdistr2 30938 ax-hvmul0 30939 ax-hfi 31008 ax-his1 31011 ax-his2 31012 ax-his3 31013 ax-his4 31014 ax-hcompl 31131

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-iin 4958 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-se 5592 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-isom 6520 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-of 7653 df-om 7843 df-1st 7968 df-2nd 7969 df-supp 8140 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-2o 8435 df-oadd 8438 df-omul 8439 df-er 8671 df-map 8801 df-pm 8802 df-ixp 8871 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-fsupp 9313 df-fi 9362 df-sup 9393 df-inf 9394 df-oi 9463 df-card 9892 df-acn 9895 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-7 12254 df-8 12255 df-9 12256 df-n0 12443 df-z 12530 df-dec 12650 df-uz 12794 df-q 12908 df-rp 12952 df-xneg 13072 df-xadd 13073 df-xmul 13074 df-ioo 13310 df-ico 13312 df-icc 13313 df-fz 13469 df-fzo 13616 df-fl 13754 df-seq 13967 df-exp 14027 df-hash 14296 df-cj 15065 df-re 15066 df-im 15067 df-sqrt 15201 df-abs 15202 df-clim 15454 df-rlim 15455 df-sum 15653 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-rest 17385 df-topn 17386 df-0g 17404 df-gsum 17405 df-topgen 17406 df-pt 17407 df-prds 17410 df-xrs 17465 df-qtop 17470 df-imas 17471 df-xps 17473 df-mre 17547 df-mrc 17548 df-acs 17550 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-submnd 18711 df-mulg 19000 df-cntz 19249 df-cmn 19712 df-psmet 21256 df-xmet 21257 df-met 21258 df-bl 21259 df-mopn 21260 df-fbas 21261 df-fg 21262 df-cnfld 21265 df-top 22781 df-topon 22798 df-topsp 22820 df-bases 22833 df-cld 22906 df-ntr 22907 df-cls 22908 df-nei 22985 df-cn 23114 df-cnp 23115 df-lm 23116 df-t1 23201 df-haus 23202 df-cmp 23274 df-tx 23449 df-hmeo 23642 df-fil 23733 df-fm 23825 df-flim 23826 df-flf 23827 df-fcls 23828 df-xms 24208 df-ms 24209 df-tms 24210 df-cncf 24771 df-cfil 25155 df-cau 25156 df-cmet 25157 df-grpo 30422 df-gid 30423 df-ginv 30424 df-gdiv 30425 df-ablo 30474 df-vc 30488 df-nv 30521 df-va 30524 df-ba 30525 df-sm 30526 df-0v 30527 df-vs 30528 df-nmcv 30529 df-ims 30530 df-dip 30630 df-ssp 30651 df-lno 30673 df-nmoo 30674 df-blo 30675 df-0o 30676 df-ph 30742 df-cbn 30792 df-hlo 30815 df-hnorm 30897 df-hba 30898 df-hvsub 30900 df-hlim 30901 df-hcau 30902 df-sh 31136 df-ch 31150 df-oc 31181 df-ch0 31182 df-shs 31237 df-pjh 31324 df-h0op 31677 df-nmop 31768 df-cnop 31769 df-lnop 31770 df-bdop 31771 df-unop 31772 df-hmop 31773

This theorem is referenced by: hmopidmpji 32081 hmopidmch 32082

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