hmopidmchi - Hilbert Space Explorer

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Theorem hmopidmchi 29930

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 29721	. . . 4 ⊢ (𝑇 ∈ HrmOp → 𝑇 ∈ LinOp)
3	1, 2	ax-mp 5	. . 3 ⊢ 𝑇 ∈ LinOp
4	3	rnelshi 29838	. 2 ⊢ ran 𝑇 ∈ S_ℋ
5		eqid 2823	. . . . . . . 8 ⊢ (norm_ℎ ∘ −_ℎ ) = (norm_ℎ ∘ −_ℎ )
6	5	hilxmet 28974	. . . . . . 7 ⊢ (norm_ℎ ∘ −_ℎ ) ∈ (∞Met‘ ℋ)
7		eqid 2823	. . . . . . . 8 ⊢ (MetOpen‘(norm_ℎ ∘ −_ℎ )) = (MetOpen‘(norm_ℎ ∘ −_ℎ ))
8	7	methaus 23132	. . . . . . 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 2823	. . . . . . . . . . . 12 ⊢ ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩ = ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩
11	10, 5	hhims 28951	. . . . . . . . . . . 12 ⊢ (norm_ℎ ∘ −_ℎ ) = (IndMet‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
12	10, 11, 7	hhlm 28978	. . . . . . . . . . 11 ⊢ ⇝_𝑣 = ((⇝_𝑡‘(MetOpen‘(norm_ℎ ∘ −_ℎ ))) ↾ ( ℋ ↑_m ℕ))
13		resss 5880	. . . . . . . . . . 11 ⊢ ((⇝_𝑡‘(MetOpen‘(norm_ℎ ∘ −_ℎ ))) ↾ ( ℋ ↑_m ℕ)) ⊆ (⇝_𝑡‘(MetOpen‘(norm_ℎ ∘ −_ℎ )))
14	12, 13	eqsstri 4003	. . . . . . . . . 10 ⊢ ⇝_𝑣 ⊆ (⇝_𝑡‘(MetOpen‘(norm_ℎ ∘ −_ℎ )))
15	14	ssbri 5113	. . . . . . . . 9 ⊢ (𝑓 ⇝_𝑣 𝑥 → 𝑓(⇝_𝑡‘(MetOpen‘(norm_ℎ ∘ −_ℎ )))𝑥)
16	15	adantl 484	. . . . . . . 8 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → 𝑓(⇝_𝑡‘(MetOpen‘(norm_ℎ ∘ −_ℎ )))𝑥)
17	7	mopntopon 23051	. . . . . . . . . 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 29748	. . . . . . . . . . . 12 ⊢ 𝑇: ℋ⟶ ℋ
20	19	a1i 11	. . . . . . . . . . 11 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → 𝑇: ℋ⟶ ℋ)
21	20	feqmptd 6735	. . . . . . . . . 10 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → 𝑇 = (𝑦 ∈ ℋ ↦ (𝑇‘𝑦)))
22		hmopbdoptHIL 29767	. . . . . . . . . . . . 13 ⊢ (𝑇 ∈ HrmOp → 𝑇 ∈ BndLinOp)
23	1, 22	ax-mp 5	. . . . . . . . . . . 12 ⊢ 𝑇 ∈ BndLinOp
24		lnopcnbd 29815	. . . . . . . . . . . . 13 ⊢ (𝑇 ∈ LinOp → (𝑇 ∈ ContOp ↔ 𝑇 ∈ BndLinOp))
25	3, 24	ax-mp 5	. . . . . . . . . . . 12 ⊢ (𝑇 ∈ ContOp ↔ 𝑇 ∈ BndLinOp)
26	23, 25	mpbir 233	. . . . . . . . . . 11 ⊢ 𝑇 ∈ ContOp
27	5, 7	hhcno 29683	. . . . . . . . . . 11 ⊢ ContOp = ((MetOpen‘(norm_ℎ ∘ −_ℎ )) Cn (MetOpen‘(norm_ℎ ∘ −_ℎ )))
28	26, 27	eleqtri 2913	. . . . . . . . . 10 ⊢ 𝑇 ∈ ((MetOpen‘(norm_ℎ ∘ −_ℎ )) Cn (MetOpen‘(norm_ℎ ∘ −_ℎ )))
29	21, 28	eqeltrrdi 2924	. . . . . . . . 9 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → (𝑦 ∈ ℋ ↦ (𝑇‘𝑦)) ∈ ((MetOpen‘(norm_ℎ ∘ −_ℎ )) Cn (MetOpen‘(norm_ℎ ∘ −_ℎ ))))
30	18	cnmptid 22271	. . . . . . . . 9 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → (𝑦 ∈ ℋ ↦ 𝑦) ∈ ((MetOpen‘(norm_ℎ ∘ −_ℎ )) Cn (MetOpen‘(norm_ℎ ∘ −_ℎ ))))
31	10	hhnv 28944	. . . . . . . . . 10 ⊢ ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩ ∈ NrmCVec
32	10	hhvs 28949	. . . . . . . . . . 11 ⊢ −_ℎ = ( −_𝑣 ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
33	11, 7, 32	vmcn 28478	. . . . . . . . . 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 22276	. . . . . . . 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 21915	. . . . . . 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 485	. . . . . . . . . . . . . 14 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → 𝑓:ℕ⟶ran 𝑇)
38	4	shssii 28992	. . . . . . . . . . . . . 14 ⊢ ran 𝑇 ⊆ ℋ
39		fss 6529	. . . . . . . . . . . . . 14 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ ran 𝑇 ⊆ ℋ) → 𝑓:ℕ⟶ ℋ)
40	37, 38, 39	sylancl 588	. . . . . . . . . . . . 13 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → 𝑓:ℕ⟶ ℋ)
41	40	ffvelrnda 6853	. . . . . . . . . . . 12 ⊢ (((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) ∧ 𝑘 ∈ ℕ) → (𝑓‘𝑘) ∈ ℋ)
42		fveq2 6672	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑓‘𝑘) → (𝑇‘𝑦) = (𝑇‘(𝑓‘𝑘)))
43		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑓‘𝑘) → 𝑦 = (𝑓‘𝑘))
44	42, 43	oveq12d 7176	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑓‘𝑘) → ((𝑇‘𝑦) −_ℎ 𝑦) = ((𝑇‘(𝑓‘𝑘)) −_ℎ (𝑓‘𝑘)))
45		eqid 2823	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℋ ↦ ((𝑇‘𝑦) −_ℎ 𝑦)) = (𝑦 ∈ ℋ ↦ ((𝑇‘𝑦) −_ℎ 𝑦))
46		ovex 7191	. . . . . . . . . . . . 13 ⊢ ((𝑇‘(𝑓‘𝑘)) −_ℎ (𝑓‘𝑘)) ∈ V
47	44, 45, 46	fvmpt 6770	. . . . . . . . . . . 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 6516	. . . . . . . . . . . . . . . 16 ⊢ (𝑇: ℋ⟶ ℋ → 𝑇 Fn ℋ)
50	19, 49	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ 𝑇 Fn ℋ
51		fveq2 6672	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = (𝑇‘𝑥) → (𝑇‘𝑦) = (𝑇‘(𝑇‘𝑥)))
52		id 22	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = (𝑇‘𝑥) → 𝑦 = (𝑇‘𝑥))
53	51, 52	eqeq12d 2839	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = (𝑇‘𝑥) → ((𝑇‘𝑦) = 𝑦 ↔ (𝑇‘(𝑇‘𝑥)) = (𝑇‘𝑥)))
54	53	ralrn 6856	. . . . . . . . . . . . . . 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		hmopidmch.2	. . . . . . . . . . . . . . . 16 ⊢ (𝑇 ∘ 𝑇) = 𝑇
57	56	fveq1i 6673	. . . . . . . . . . . . . . 15 ⊢ ((𝑇 ∘ 𝑇)‘𝑥) = (𝑇‘𝑥)
58	19, 19	hocoi 29543	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℋ → ((𝑇 ∘ 𝑇)‘𝑥) = (𝑇‘(𝑇‘𝑥)))
59	57, 58	syl5reqr 2873	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℋ → (𝑇‘(𝑇‘𝑥)) = (𝑇‘𝑥))
60	55, 59	mprgbir 3155	. . . . . . . . . . . . 13 ⊢ ∀𝑦 ∈ ran 𝑇(𝑇‘𝑦) = 𝑦
61		ffvelrn 6851	. . . . . . . . . . . . . 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 2839	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑓‘𝑘) → ((𝑇‘𝑦) = 𝑦 ↔ (𝑇‘(𝑓‘𝑘)) = (𝑓‘𝑘)))
64	63	rspccv 3622	. . . . . . . . . . . . 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 2915	. . . . . . . . . . . . 13 ⊢ (((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) ∧ 𝑘 ∈ ℕ) → (𝑇‘(𝑓‘𝑘)) ∈ ℋ)
67		hvsubeq0 28847	. . . . . . . . . . . . 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 586	. . . . . . . . . . . 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 259	. . . . . . . . . . 11 ⊢ (((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) ∧ 𝑘 ∈ ℕ) → ((𝑇‘(𝑓‘𝑘)) −_ℎ (𝑓‘𝑘)) = 0_ℎ)
70	48, 69	eqtrd 2858	. . . . . . . . . 10 ⊢ (((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) ∧ 𝑘 ∈ ℕ) → ((𝑦 ∈ ℋ ↦ ((𝑇‘𝑦) −_ℎ 𝑦))‘(𝑓‘𝑘)) = 0_ℎ)
71		fvco3 6762	. . . . . . . . . . 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 28782	. . . . . . . . . . . . 13 ⊢ 0_ℎ ∈ ℋ
74	73	elexi 3515	. . . . . . . . . . . 12 ⊢ 0_ℎ ∈ V
75	74	fvconst2 6968	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → ((ℕ × {0_ℎ})‘𝑘) = 0_ℎ)
76	75	adantl 484	. . . . . . . . . 10 ⊢ (((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) ∧ 𝑘 ∈ ℕ) → ((ℕ × {0_ℎ})‘𝑘) = 0_ℎ)
77	70, 72, 76	3eqtr4d 2868	. . . . . . . . 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 3184	. . . . . . . 8 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → ∀𝑘 ∈ ℕ (((𝑦 ∈ ℋ ↦ ((𝑇‘𝑦) −_ℎ 𝑦)) ∘ 𝑓)‘𝑘) = ((ℕ × {0_ℎ})‘𝑘))
79		ovex 7191	. . . . . . . . . . 11 ⊢ ((𝑇‘𝑦) −_ℎ 𝑦) ∈ V
80	79, 45	fnmpti 6493	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℋ ↦ ((𝑇‘𝑦) −_ℎ 𝑦)) Fn ℋ
81		fnfco 6545	. . . . . . . . . 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 589	. . . . . . . . 9 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → ((𝑦 ∈ ℋ ↦ ((𝑇‘𝑦) −_ℎ 𝑦)) ∘ 𝑓) Fn ℕ)
83	74	fconst 6567	. . . . . . . . . 10 ⊢ (ℕ × {0_ℎ}):ℕ⟶{0_ℎ}
84		ffn 6516	. . . . . . . . . 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 6804	. . . . . . . . 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 588	. . . . . . . 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 259	. . . . . . 7 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → ((𝑦 ∈ ℋ ↦ ((𝑇‘𝑦) −_ℎ 𝑦)) ∘ 𝑓) = (ℕ × {0_ℎ}))
89		vex 3499	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
90	89	hlimveci 28969	. . . . . . . . 9 ⊢ (𝑓 ⇝_𝑣 𝑥 → 𝑥 ∈ ℋ)
91	90	adantl 484	. . . . . . . 8 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → 𝑥 ∈ ℋ)
92		fveq2 6672	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (𝑇‘𝑦) = (𝑇‘𝑥))
93		id 22	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → 𝑦 = 𝑥)
94	92, 93	oveq12d 7176	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → ((𝑇‘𝑦) −_ℎ 𝑦) = ((𝑇‘𝑥) −_ℎ 𝑥))
95		ovex 7191	. . . . . . . . 9 ⊢ ((𝑇‘𝑥) −_ℎ 𝑥) ∈ V
96	94, 45, 95	fvmpt 6770	. . . . . . . 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 5099	. . . . . 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 12016	. . . . . . 7 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → 1 ∈ ℤ)
101		nnuz 12284	. . . . . . . 8 ⊢ ℕ = (ℤ_≥‘1)
102	101	lmconst 21871	. . . . . . 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 1367	. . . . . 6 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → (ℕ × {0_ℎ})(⇝_𝑡‘(MetOpen‘(norm_ℎ ∘ −_ℎ )))0_ℎ)
104	9, 98, 103	lmmo 21990	. . . . 5 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → ((𝑇‘𝑥) −_ℎ 𝑥) = 0_ℎ)
105	19	ffvelrni 6852	. . . . . . 7 ⊢ (𝑥 ∈ ℋ → (𝑇‘𝑥) ∈ ℋ)
106	91, 105	syl 17	. . . . . 6 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → (𝑇‘𝑥) ∈ ℋ)
107		hvsubeq0 28847	. . . . . 6 ⊢ (((𝑇‘𝑥) ∈ ℋ ∧ 𝑥 ∈ ℋ) → (((𝑇‘𝑥) −_ℎ 𝑥) = 0_ℎ ↔ (𝑇‘𝑥) = 𝑥))
108	106, 91, 107	syl2anc 586	. . . . 5 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → (((𝑇‘𝑥) −_ℎ 𝑥) = 0_ℎ ↔ (𝑇‘𝑥) = 𝑥))
109	104, 108	mpbid 234	. . . 4 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → (𝑇‘𝑥) = 𝑥)
110		fnfvelrn 6850	. . . . 5 ⊢ ((𝑇 Fn ℋ ∧ 𝑥 ∈ ℋ) → (𝑇‘𝑥) ∈ ran 𝑇)
111	50, 91, 110	sylancr 589	. . . 4 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → (𝑇‘𝑥) ∈ ran 𝑇)
112	109, 111	eqeltrrd 2916	. . 3 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → 𝑥 ∈ ran 𝑇)
113	112	gen2 1797	. 2 ⊢ ∀𝑓∀𝑥((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → 𝑥 ∈ ran 𝑇)
114		isch2 29002	. 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 208 ∧ wa 398 ∀wal 1535 = wceq 1537 ∈ wcel 2114 ∀wral 3140 ⊆ wss 3938 {csn 4569 ⟨cop 4575 class class class wbr 5068 ↦ cmpt 5148 × cxp 5555 ran crn 5558 ↾ cres 5559 ∘ ccom 5561 Fn wfn 6352 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 ↑_m cmap 8408 1c1 10540 ℕcn 11640 ℤcz 11984 ∞Metcxmet 20532 MetOpencmopn 20537 TopOnctopon 21520 Cn ccn 21834 ⇝_𝑡clm 21836 Hauscha 21918 ×_t ctx 22170 NrmCVeccnv 28363 ℋchba 28698 +_ℎ cva 28699 ·_ℎ csm 28700 norm_ℎcno 28702 0_ℎc0v 28703 −_ℎ cmv 28704 ⇝_𝑣 chli 28706 S_ℋ csh 28707 C_ℋ cch 28708 ContOpccop 28725 LinOpclo 28726 BndLinOpcbo 28727 HrmOpcho 28729

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cc 9859 ax-dc 9870 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 ax-addf 10618 ax-mulf 10619 ax-hilex 28778 ax-hfvadd 28779 ax-hvcom 28780 ax-hvass 28781 ax-hv0cl 28782 ax-hvaddid 28783 ax-hfvmul 28784 ax-hvmulid 28785 ax-hvmulass 28786 ax-hvdistr1 28787 ax-hvdistr2 28788 ax-hvmul0 28789 ax-hfi 28858 ax-his1 28861 ax-his2 28862 ax-his3 28863 ax-his4 28864 ax-hcompl 28981

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-omul 8109 df-er 8291 df-map 8410 df-pm 8411 df-ixp 8464 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-fi 8877 df-sup 8908 df-inf 8909 df-oi 8976 df-card 9370 df-acn 9373 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-ioo 12745 df-ico 12747 df-icc 12748 df-fz 12896 df-fzo 13037 df-fl 13165 df-seq 13373 df-exp 13433 df-hash 13694 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-clim 14847 df-rlim 14848 df-sum 15045 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-starv 16582 df-sca 16583 df-vsca 16584 df-ip 16585 df-tset 16586 df-ple 16587 df-ds 16589 df-unif 16590 df-hom 16591 df-cco 16592 df-rest 16698 df-topn 16699 df-0g 16717 df-gsum 16718 df-topgen 16719 df-pt 16720 df-prds 16723 df-xrs 16777 df-qtop 16782 df-imas 16783 df-xps 16785 df-mre 16859 df-mrc 16860 df-acs 16862 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-submnd 17959 df-mulg 18227 df-cntz 18449 df-cmn 18910 df-psmet 20539 df-xmet 20540 df-met 20541 df-bl 20542 df-mopn 20543 df-fbas 20544 df-fg 20545 df-cnfld 20548 df-top 21504 df-topon 21521 df-topsp 21543 df-bases 21556 df-cld 21629 df-ntr 21630 df-cls 21631 df-nei 21708 df-cn 21837 df-cnp 21838 df-lm 21839 df-t1 21924 df-haus 21925 df-cmp 21997 df-tx 22172 df-hmeo 22365 df-fil 22456 df-fm 22548 df-flim 22549 df-flf 22550 df-fcls 22551 df-xms 22932 df-ms 22933 df-tms 22934 df-cncf 23488 df-cfil 23860 df-cau 23861 df-cmet 23862 df-grpo 28272 df-gid 28273 df-ginv 28274 df-gdiv 28275 df-ablo 28324 df-vc 28338 df-nv 28371 df-va 28374 df-ba 28375 df-sm 28376 df-0v 28377 df-vs 28378 df-nmcv 28379 df-ims 28380 df-dip 28480 df-ssp 28501 df-lno 28523 df-nmoo 28524 df-blo 28525 df-0o 28526 df-ph 28592 df-cbn 28642 df-hlo 28665 df-hnorm 28747 df-hba 28748 df-hvsub 28750 df-hlim 28751 df-hcau 28752 df-sh 28986 df-ch 29000 df-oc 29031 df-ch0 29032 df-shs 29087 df-pjh 29174 df-h0op 29527 df-nmop 29618 df-cnop 29619 df-lnop 29620 df-bdop 29621 df-unop 29622 df-hmop 29623

This theorem is referenced by: hmopidmpji 29931 hmopidmch 29932

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