hmopidmchi - Hilbert Space Explorer

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Theorem hmopidmchi 29934

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 29725	. . . 4 ⊢ (𝑇 ∈ HrmOp → 𝑇 ∈ LinOp)
3	1, 2	ax-mp 5	. . 3 ⊢ 𝑇 ∈ LinOp
4	3	rnelshi 29842	. 2 ⊢ ran 𝑇 ∈ S_ℋ
5		eqid 2798	. . . . . . . 8 ⊢ (norm_ℎ ∘ −_ℎ ) = (norm_ℎ ∘ −_ℎ )
6	5	hilxmet 28978	. . . . . . 7 ⊢ (norm_ℎ ∘ −_ℎ ) ∈ (∞Met‘ ℋ)
7		eqid 2798	. . . . . . . 8 ⊢ (MetOpen‘(norm_ℎ ∘ −_ℎ )) = (MetOpen‘(norm_ℎ ∘ −_ℎ ))
8	7	methaus 23127	. . . . . . 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 2798	. . . . . . . . . . . 12 ⊢ ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩ = ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩
11	10, 5	hhims 28955	. . . . . . . . . . . 12 ⊢ (norm_ℎ ∘ −_ℎ ) = (IndMet‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
12	10, 11, 7	hhlm 28982	. . . . . . . . . . 11 ⊢ ⇝_𝑣 = ((⇝_𝑡‘(MetOpen‘(norm_ℎ ∘ −_ℎ ))) ↾ ( ℋ ↑_m ℕ))
13		resss 5843	. . . . . . . . . . 11 ⊢ ((⇝_𝑡‘(MetOpen‘(norm_ℎ ∘ −_ℎ ))) ↾ ( ℋ ↑_m ℕ)) ⊆ (⇝_𝑡‘(MetOpen‘(norm_ℎ ∘ −_ℎ )))
14	12, 13	eqsstri 3949	. . . . . . . . . 10 ⊢ ⇝_𝑣 ⊆ (⇝_𝑡‘(MetOpen‘(norm_ℎ ∘ −_ℎ )))
15	14	ssbri 5075	. . . . . . . . 9 ⊢ (𝑓 ⇝_𝑣 𝑥 → 𝑓(⇝_𝑡‘(MetOpen‘(norm_ℎ ∘ −_ℎ )))𝑥)
16	15	adantl 485	. . . . . . . 8 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → 𝑓(⇝_𝑡‘(MetOpen‘(norm_ℎ ∘ −_ℎ )))𝑥)
17	7	mopntopon 23046	. . . . . . . . . 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 29752	. . . . . . . . . . . 12 ⊢ 𝑇: ℋ⟶ ℋ
20	19	a1i 11	. . . . . . . . . . 11 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → 𝑇: ℋ⟶ ℋ)
21	20	feqmptd 6708	. . . . . . . . . 10 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → 𝑇 = (𝑦 ∈ ℋ ↦ (𝑇‘𝑦)))
22		hmopbdoptHIL 29771	. . . . . . . . . . . . 13 ⊢ (𝑇 ∈ HrmOp → 𝑇 ∈ BndLinOp)
23	1, 22	ax-mp 5	. . . . . . . . . . . 12 ⊢ 𝑇 ∈ BndLinOp
24		lnopcnbd 29819	. . . . . . . . . . . . 13 ⊢ (𝑇 ∈ LinOp → (𝑇 ∈ ContOp ↔ 𝑇 ∈ BndLinOp))
25	3, 24	ax-mp 5	. . . . . . . . . . . 12 ⊢ (𝑇 ∈ ContOp ↔ 𝑇 ∈ BndLinOp)
26	23, 25	mpbir 234	. . . . . . . . . . 11 ⊢ 𝑇 ∈ ContOp
27	5, 7	hhcno 29687	. . . . . . . . . . 11 ⊢ ContOp = ((MetOpen‘(norm_ℎ ∘ −_ℎ )) Cn (MetOpen‘(norm_ℎ ∘ −_ℎ )))
28	26, 27	eleqtri 2888	. . . . . . . . . 10 ⊢ 𝑇 ∈ ((MetOpen‘(norm_ℎ ∘ −_ℎ )) Cn (MetOpen‘(norm_ℎ ∘ −_ℎ )))
29	21, 28	eqeltrrdi 2899	. . . . . . . . 9 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → (𝑦 ∈ ℋ ↦ (𝑇‘𝑦)) ∈ ((MetOpen‘(norm_ℎ ∘ −_ℎ )) Cn (MetOpen‘(norm_ℎ ∘ −_ℎ ))))
30	18	cnmptid 22266	. . . . . . . . 9 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → (𝑦 ∈ ℋ ↦ 𝑦) ∈ ((MetOpen‘(norm_ℎ ∘ −_ℎ )) Cn (MetOpen‘(norm_ℎ ∘ −_ℎ ))))
31	10	hhnv 28948	. . . . . . . . . 10 ⊢ ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩ ∈ NrmCVec
32	10	hhvs 28953	. . . . . . . . . . 11 ⊢ −_ℎ = ( −_𝑣 ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
33	11, 7, 32	vmcn 28482	. . . . . . . . . 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 22271	. . . . . . . 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 21910	. . . . . . 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 486	. . . . . . . . . . . . . 14 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → 𝑓:ℕ⟶ran 𝑇)
38	4	shssii 28996	. . . . . . . . . . . . . 14 ⊢ ran 𝑇 ⊆ ℋ
39		fss 6501	. . . . . . . . . . . . . 14 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ ran 𝑇 ⊆ ℋ) → 𝑓:ℕ⟶ ℋ)
40	37, 38, 39	sylancl 589	. . . . . . . . . . . . 13 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → 𝑓:ℕ⟶ ℋ)
41	40	ffvelrnda 6828	. . . . . . . . . . . 12 ⊢ (((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) ∧ 𝑘 ∈ ℕ) → (𝑓‘𝑘) ∈ ℋ)
42		fveq2 6645	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑓‘𝑘) → (𝑇‘𝑦) = (𝑇‘(𝑓‘𝑘)))
43		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑓‘𝑘) → 𝑦 = (𝑓‘𝑘))
44	42, 43	oveq12d 7153	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑓‘𝑘) → ((𝑇‘𝑦) −_ℎ 𝑦) = ((𝑇‘(𝑓‘𝑘)) −_ℎ (𝑓‘𝑘)))
45		eqid 2798	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℋ ↦ ((𝑇‘𝑦) −_ℎ 𝑦)) = (𝑦 ∈ ℋ ↦ ((𝑇‘𝑦) −_ℎ 𝑦))
46		ovex 7168	. . . . . . . . . . . . 13 ⊢ ((𝑇‘(𝑓‘𝑘)) −_ℎ (𝑓‘𝑘)) ∈ V
47	44, 45, 46	fvmpt 6745	. . . . . . . . . . . 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 6487	. . . . . . . . . . . . . . . 16 ⊢ (𝑇: ℋ⟶ ℋ → 𝑇 Fn ℋ)
50	19, 49	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ 𝑇 Fn ℋ
51		fveq2 6645	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = (𝑇‘𝑥) → (𝑇‘𝑦) = (𝑇‘(𝑇‘𝑥)))
52		id 22	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = (𝑇‘𝑥) → 𝑦 = (𝑇‘𝑥))
53	51, 52	eqeq12d 2814	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = (𝑇‘𝑥) → ((𝑇‘𝑦) = 𝑦 ↔ (𝑇‘(𝑇‘𝑥)) = (𝑇‘𝑥)))
54	53	ralrn 6831	. . . . . . . . . . . . . . 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 6646	. . . . . . . . . . . . . . 15 ⊢ ((𝑇 ∘ 𝑇)‘𝑥) = (𝑇‘𝑥)
58	19, 19	hocoi 29547	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℋ → ((𝑇 ∘ 𝑇)‘𝑥) = (𝑇‘(𝑇‘𝑥)))
59	57, 58	syl5reqr 2848	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℋ → (𝑇‘(𝑇‘𝑥)) = (𝑇‘𝑥))
60	55, 59	mprgbir 3121	. . . . . . . . . . . . 13 ⊢ ∀𝑦 ∈ ran 𝑇(𝑇‘𝑦) = 𝑦
61		ffvelrn 6826	. . . . . . . . . . . . . 14 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑘 ∈ ℕ) → (𝑓‘𝑘) ∈ ran 𝑇)
62	61	adantlr 714	. . . . . . . . . . . . 13 ⊢ (((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) ∧ 𝑘 ∈ ℕ) → (𝑓‘𝑘) ∈ ran 𝑇)
63	42, 43	eqeq12d 2814	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑓‘𝑘) → ((𝑇‘𝑦) = 𝑦 ↔ (𝑇‘(𝑓‘𝑘)) = (𝑓‘𝑘)))
64	63	rspccv 3568	. . . . . . . . . . . . 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 2890	. . . . . . . . . . . . 13 ⊢ (((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) ∧ 𝑘 ∈ ℕ) → (𝑇‘(𝑓‘𝑘)) ∈ ℋ)
67		hvsubeq0 28851	. . . . . . . . . . . . 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 587	. . . . . . . . . . . 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 260	. . . . . . . . . . 11 ⊢ (((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) ∧ 𝑘 ∈ ℕ) → ((𝑇‘(𝑓‘𝑘)) −_ℎ (𝑓‘𝑘)) = 0_ℎ)
70	48, 69	eqtrd 2833	. . . . . . . . . 10 ⊢ (((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) ∧ 𝑘 ∈ ℕ) → ((𝑦 ∈ ℋ ↦ ((𝑇‘𝑦) −_ℎ 𝑦))‘(𝑓‘𝑘)) = 0_ℎ)
71		fvco3 6737	. . . . . . . . . . 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 714	. . . . . . . . . 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 28786	. . . . . . . . . . . . 13 ⊢ 0_ℎ ∈ ℋ
74	73	elexi 3460	. . . . . . . . . . . 12 ⊢ 0_ℎ ∈ V
75	74	fvconst2 6943	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → ((ℕ × {0_ℎ})‘𝑘) = 0_ℎ)
76	75	adantl 485	. . . . . . . . . 10 ⊢ (((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) ∧ 𝑘 ∈ ℕ) → ((ℕ × {0_ℎ})‘𝑘) = 0_ℎ)
77	70, 72, 76	3eqtr4d 2843	. . . . . . . . 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 3149	. . . . . . . 8 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → ∀𝑘 ∈ ℕ (((𝑦 ∈ ℋ ↦ ((𝑇‘𝑦) −_ℎ 𝑦)) ∘ 𝑓)‘𝑘) = ((ℕ × {0_ℎ})‘𝑘))
79		ovex 7168	. . . . . . . . . . 11 ⊢ ((𝑇‘𝑦) −_ℎ 𝑦) ∈ V
80	79, 45	fnmpti 6463	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℋ ↦ ((𝑇‘𝑦) −_ℎ 𝑦)) Fn ℋ
81		fnfco 6517	. . . . . . . . . 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 590	. . . . . . . . 9 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → ((𝑦 ∈ ℋ ↦ ((𝑇‘𝑦) −_ℎ 𝑦)) ∘ 𝑓) Fn ℕ)
83	74	fconst 6539	. . . . . . . . . 10 ⊢ (ℕ × {0_ℎ}):ℕ⟶{0_ℎ}
84		ffn 6487	. . . . . . . . . 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 6779	. . . . . . . . 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 589	. . . . . . . 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 260	. . . . . . 7 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → ((𝑦 ∈ ℋ ↦ ((𝑇‘𝑦) −_ℎ 𝑦)) ∘ 𝑓) = (ℕ × {0_ℎ}))
89		vex 3444	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
90	89	hlimveci 28973	. . . . . . . . 9 ⊢ (𝑓 ⇝_𝑣 𝑥 → 𝑥 ∈ ℋ)
91	90	adantl 485	. . . . . . . 8 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → 𝑥 ∈ ℋ)
92		fveq2 6645	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (𝑇‘𝑦) = (𝑇‘𝑥))
93		id 22	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → 𝑦 = 𝑥)
94	92, 93	oveq12d 7153	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → ((𝑇‘𝑦) −_ℎ 𝑦) = ((𝑇‘𝑥) −_ℎ 𝑥))
95		ovex 7168	. . . . . . . . 9 ⊢ ((𝑇‘𝑥) −_ℎ 𝑥) ∈ V
96	94, 45, 95	fvmpt 6745	. . . . . . . 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 5061	. . . . . 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 12001	. . . . . . 7 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → 1 ∈ ℤ)
101		nnuz 12269	. . . . . . . 8 ⊢ ℕ = (ℤ_≥‘1)
102	101	lmconst 21866	. . . . . . 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 1368	. . . . . 6 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → (ℕ × {0_ℎ})(⇝_𝑡‘(MetOpen‘(norm_ℎ ∘ −_ℎ )))0_ℎ)
104	9, 98, 103	lmmo 21985	. . . . 5 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → ((𝑇‘𝑥) −_ℎ 𝑥) = 0_ℎ)
105	19	ffvelrni 6827	. . . . . . 7 ⊢ (𝑥 ∈ ℋ → (𝑇‘𝑥) ∈ ℋ)
106	91, 105	syl 17	. . . . . 6 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → (𝑇‘𝑥) ∈ ℋ)
107		hvsubeq0 28851	. . . . . 6 ⊢ (((𝑇‘𝑥) ∈ ℋ ∧ 𝑥 ∈ ℋ) → (((𝑇‘𝑥) −_ℎ 𝑥) = 0_ℎ ↔ (𝑇‘𝑥) = 𝑥))
108	106, 91, 107	syl2anc 587	. . . . 5 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → (((𝑇‘𝑥) −_ℎ 𝑥) = 0_ℎ ↔ (𝑇‘𝑥) = 𝑥))
109	104, 108	mpbid 235	. . . 4 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → (𝑇‘𝑥) = 𝑥)
110		fnfvelrn 6825	. . . . 5 ⊢ ((𝑇 Fn ℋ ∧ 𝑥 ∈ ℋ) → (𝑇‘𝑥) ∈ ran 𝑇)
111	50, 91, 110	sylancr 590	. . . 4 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → (𝑇‘𝑥) ∈ ran 𝑇)
112	109, 111	eqeltrrd 2891	. . 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 29006	. 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 710	1 ⊢ ran 𝑇 ∈ C_ℋ

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∀wal 1536 = wceq 1538 ∈ wcel 2111 ∀wral 3106 ⊆ wss 3881 {csn 4525 ⟨cop 4531 class class class wbr 5030 ↦ cmpt 5110 × cxp 5517 ran crn 5520 ↾ cres 5521 ∘ ccom 5523 Fn wfn 6319 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ↑_m cmap 8389 1c1 10527 ℕcn 11625 ℤcz 11969 ∞Metcxmet 20076 MetOpencmopn 20081 TopOnctopon 21515 Cn ccn 21829 ⇝_𝑡clm 21831 Hauscha 21913 ×_t ctx 22165 NrmCVeccnv 28367 ℋchba 28702 +_ℎ cva 28703 ·_ℎ csm 28704 norm_ℎcno 28706 0_ℎc0v 28707 −_ℎ cmv 28708 ⇝_𝑣 chli 28710 S_ℋ csh 28711 C_ℋ cch 28712 ContOpccop 28729 LinOpclo 28730 BndLinOpcbo 28731 HrmOpcho 28733

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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cc 9846 ax-dc 9857 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606 ax-hilex 28782 ax-hfvadd 28783 ax-hvcom 28784 ax-hvass 28785 ax-hv0cl 28786 ax-hvaddid 28787 ax-hfvmul 28788 ax-hvmulid 28789 ax-hvmulass 28790 ax-hvdistr1 28791 ax-hvdistr2 28792 ax-hvmul0 28793 ax-hfi 28862 ax-his1 28865 ax-his2 28866 ax-his3 28867 ax-his4 28868 ax-hcompl 28985

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-omul 8090 df-er 8272 df-map 8391 df-pm 8392 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-fi 8859 df-sup 8890 df-inf 8891 df-oi 8958 df-card 9352 df-acn 9355 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ioo 12730 df-ico 12732 df-icc 12733 df-fz 12886 df-fzo 13029 df-fl 13157 df-seq 13365 df-exp 13426 df-hash 13687 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-clim 14837 df-rlim 14838 df-sum 15035 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-hom 16581 df-cco 16582 df-rest 16688 df-topn 16689 df-0g 16707 df-gsum 16708 df-topgen 16709 df-pt 16710 df-prds 16713 df-xrs 16767 df-qtop 16772 df-imas 16773 df-xps 16775 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-mulg 18217 df-cntz 18439 df-cmn 18900 df-psmet 20083 df-xmet 20084 df-met 20085 df-bl 20086 df-mopn 20087 df-fbas 20088 df-fg 20089 df-cnfld 20092 df-top 21499 df-topon 21516 df-topsp 21538 df-bases 21551 df-cld 21624 df-ntr 21625 df-cls 21626 df-nei 21703 df-cn 21832 df-cnp 21833 df-lm 21834 df-t1 21919 df-haus 21920 df-cmp 21992 df-tx 22167 df-hmeo 22360 df-fil 22451 df-fm 22543 df-flim 22544 df-flf 22545 df-fcls 22546 df-xms 22927 df-ms 22928 df-tms 22929 df-cncf 23483 df-cfil 23859 df-cau 23860 df-cmet 23861 df-grpo 28276 df-gid 28277 df-ginv 28278 df-gdiv 28279 df-ablo 28328 df-vc 28342 df-nv 28375 df-va 28378 df-ba 28379 df-sm 28380 df-0v 28381 df-vs 28382 df-nmcv 28383 df-ims 28384 df-dip 28484 df-ssp 28505 df-lno 28527 df-nmoo 28528 df-blo 28529 df-0o 28530 df-ph 28596 df-cbn 28646 df-hlo 28669 df-hnorm 28751 df-hba 28752 df-hvsub 28754 df-hlim 28755 df-hcau 28756 df-sh 28990 df-ch 29004 df-oc 29035 df-ch0 29036 df-shs 29091 df-pjh 29178 df-h0op 29531 df-nmop 29622 df-cnop 29623 df-lnop 29624 df-bdop 29625 df-unop 29626 df-hmop 29627

This theorem is referenced by: hmopidmpji 29935 hmopidmch 29936

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