hmopidmchi - Hilbert Space Explorer

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Theorem hmopidmchi 28394

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 28185	. . . 4 ⊢ (𝑇 ∈ HrmOp → 𝑇 ∈ LinOp)
3	1, 2	ax-mp 5	. . 3 ⊢ 𝑇 ∈ LinOp
4	3	rnelshi 28302	. 2 ⊢ ran 𝑇 ∈ S_ℋ
5		eqid 2610	. . . . . . . 8 ⊢ (norm_ℎ ∘ −_ℎ ) = (norm_ℎ ∘ −_ℎ )
6	5	hilxmet 27436	. . . . . . 7 ⊢ (norm_ℎ ∘ −_ℎ ) ∈ (∞Met‘ ℋ)
7		eqid 2610	. . . . . . . 8 ⊢ (MetOpen‘(norm_ℎ ∘ −_ℎ )) = (MetOpen‘(norm_ℎ ∘ −_ℎ ))
8	7	methaus 22135	. . . . . . 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 2610	. . . . . . . . . . . 12 ⊢ ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩ = ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩
11	10, 5	hhims 27413	. . . . . . . . . . . 12 ⊢ (norm_ℎ ∘ −_ℎ ) = (IndMet‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
12	10, 11, 7	hhlm 27440	. . . . . . . . . . 11 ⊢ ⇝_𝑣 = ((⇝_𝑡‘(MetOpen‘(norm_ℎ ∘ −_ℎ ))) ↾ ( ℋ ↑_𝑚 ℕ))
13		resss 5342	. . . . . . . . . . 11 ⊢ ((⇝_𝑡‘(MetOpen‘(norm_ℎ ∘ −_ℎ ))) ↾ ( ℋ ↑_𝑚 ℕ)) ⊆ (⇝_𝑡‘(MetOpen‘(norm_ℎ ∘ −_ℎ )))
14	12, 13	eqsstri 3598	. . . . . . . . . 10 ⊢ ⇝_𝑣 ⊆ (⇝_𝑡‘(MetOpen‘(norm_ℎ ∘ −_ℎ )))
15	14	ssbri 4627	. . . . . . . . 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 22054	. . . . . . . . . 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 28212	. . . . . . . . . . . 12 ⊢ 𝑇: ℋ⟶ ℋ
20	19	a1i 11	. . . . . . . . . . 11 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → 𝑇: ℋ⟶ ℋ)
21	20	feqmptd 6159	. . . . . . . . . 10 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → 𝑇 = (𝑦 ∈ ℋ ↦ (𝑇‘𝑦)))
22		hmopbdoptHIL 28231	. . . . . . . . . . . . 13 ⊢ (𝑇 ∈ HrmOp → 𝑇 ∈ BndLinOp)
23	1, 22	ax-mp 5	. . . . . . . . . . . 12 ⊢ 𝑇 ∈ BndLinOp
24		lnopcnbd 28279	. . . . . . . . . . . . 13 ⊢ (𝑇 ∈ LinOp → (𝑇 ∈ ConOp ↔ 𝑇 ∈ BndLinOp))
25	3, 24	ax-mp 5	. . . . . . . . . . . 12 ⊢ (𝑇 ∈ ConOp ↔ 𝑇 ∈ BndLinOp)
26	23, 25	mpbir 220	. . . . . . . . . . 11 ⊢ 𝑇 ∈ ConOp
27	5, 7	hhcno 28147	. . . . . . . . . . 11 ⊢ ConOp = ((MetOpen‘(norm_ℎ ∘ −_ℎ )) Cn (MetOpen‘(norm_ℎ ∘ −_ℎ )))
28	26, 27	eleqtri 2686	. . . . . . . . . 10 ⊢ 𝑇 ∈ ((MetOpen‘(norm_ℎ ∘ −_ℎ )) Cn (MetOpen‘(norm_ℎ ∘ −_ℎ )))
29	21, 28	syl6eqelr 2697	. . . . . . . . 9 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → (𝑦 ∈ ℋ ↦ (𝑇‘𝑦)) ∈ ((MetOpen‘(norm_ℎ ∘ −_ℎ )) Cn (MetOpen‘(norm_ℎ ∘ −_ℎ ))))
30	18	cnmptid 21274	. . . . . . . . 9 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → (𝑦 ∈ ℋ ↦ 𝑦) ∈ ((MetOpen‘(norm_ℎ ∘ −_ℎ )) Cn (MetOpen‘(norm_ℎ ∘ −_ℎ ))))
31	10	hhnv 27406	. . . . . . . . . 10 ⊢ ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩ ∈ NrmCVec
32	10	hhvs 27411	. . . . . . . . . . 11 ⊢ −_ℎ = ( −_𝑣 ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
33	11, 7, 32	vmcn 26938	. . . . . . . . . 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 21279	. . . . . . . 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 20919	. . . . . . 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 472	. . . . . . . . . . . . . 14 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → 𝑓:ℕ⟶ran 𝑇)
38	4	shssii 27454	. . . . . . . . . . . . . 14 ⊢ ran 𝑇 ⊆ ℋ
39		fss 5969	. . . . . . . . . . . . . 14 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ ran 𝑇 ⊆ ℋ) → 𝑓:ℕ⟶ ℋ)
40	37, 38, 39	sylancl 693	. . . . . . . . . . . . 13 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → 𝑓:ℕ⟶ ℋ)
41	40	ffvelrnda 6267	. . . . . . . . . . . 12 ⊢ (((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) ∧ 𝑘 ∈ ℕ) → (𝑓‘𝑘) ∈ ℋ)
42		fveq2 6103	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑓‘𝑘) → (𝑇‘𝑦) = (𝑇‘(𝑓‘𝑘)))
43		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑓‘𝑘) → 𝑦 = (𝑓‘𝑘))
44	42, 43	oveq12d 6567	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑓‘𝑘) → ((𝑇‘𝑦) −_ℎ 𝑦) = ((𝑇‘(𝑓‘𝑘)) −_ℎ (𝑓‘𝑘)))
45		eqid 2610	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℋ ↦ ((𝑇‘𝑦) −_ℎ 𝑦)) = (𝑦 ∈ ℋ ↦ ((𝑇‘𝑦) −_ℎ 𝑦))
46		ovex 6577	. . . . . . . . . . . . 13 ⊢ ((𝑇‘(𝑓‘𝑘)) −_ℎ (𝑓‘𝑘)) ∈ V
47	44, 45, 46	fvmpt 6191	. . . . . . . . . . . 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 5958	. . . . . . . . . . . . . . . 16 ⊢ (𝑇: ℋ⟶ ℋ → 𝑇 Fn ℋ)
50	19, 49	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ 𝑇 Fn ℋ
51		fveq2 6103	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = (𝑇‘𝑥) → (𝑇‘𝑦) = (𝑇‘(𝑇‘𝑥)))
52		id 22	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = (𝑇‘𝑥) → 𝑦 = (𝑇‘𝑥))
53	51, 52	eqeq12d 2625	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = (𝑇‘𝑥) → ((𝑇‘𝑦) = 𝑦 ↔ (𝑇‘(𝑇‘𝑥)) = (𝑇‘𝑥)))
54	53	ralrn 6270	. . . . . . . . . . . . . . 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 6104	. . . . . . . . . . . . . . 15 ⊢ ((𝑇 ∘ 𝑇)‘𝑥) = (𝑇‘𝑥)
58	19, 19	hocoi 28007	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℋ → ((𝑇 ∘ 𝑇)‘𝑥) = (𝑇‘(𝑇‘𝑥)))
59	57, 58	syl5reqr 2659	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℋ → (𝑇‘(𝑇‘𝑥)) = (𝑇‘𝑥))
60	55, 59	mprgbir 2911	. . . . . . . . . . . . 13 ⊢ ∀𝑦 ∈ ran 𝑇(𝑇‘𝑦) = 𝑦
61		ffvelrn 6265	. . . . . . . . . . . . . 14 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑘 ∈ ℕ) → (𝑓‘𝑘) ∈ ran 𝑇)
62	61	adantlr 747	. . . . . . . . . . . . 13 ⊢ (((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) ∧ 𝑘 ∈ ℕ) → (𝑓‘𝑘) ∈ ran 𝑇)
63	42, 43	eqeq12d 2625	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑓‘𝑘) → ((𝑇‘𝑦) = 𝑦 ↔ (𝑇‘(𝑓‘𝑘)) = (𝑓‘𝑘)))
64	63	rspccv 3279	. . . . . . . . . . . . 13 ⊢ (∀𝑦 ∈ ran 𝑇(𝑇‘𝑦) = 𝑦 → ((𝑓‘𝑘) ∈ ran 𝑇 → (𝑇‘(𝑓‘𝑘)) = (𝑓‘𝑘)))
65	60, 62, 64	mpsyl 66	. . . . . . . . . . . 12 ⊢ (((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) ∧ 𝑘 ∈ ℕ) → (𝑇‘(𝑓‘𝑘)) = (𝑓‘𝑘))
66	65, 41	eqeltrd 2688	. . . . . . . . . . . . 13 ⊢ (((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) ∧ 𝑘 ∈ ℕ) → (𝑇‘(𝑓‘𝑘)) ∈ ℋ)
67		hvsubeq0 27309	. . . . . . . . . . . . 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 691	. . . . . . . . . . . 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 246	. . . . . . . . . . 11 ⊢ (((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) ∧ 𝑘 ∈ ℕ) → ((𝑇‘(𝑓‘𝑘)) −_ℎ (𝑓‘𝑘)) = 0_ℎ)
70	48, 69	eqtrd 2644	. . . . . . . . . 10 ⊢ (((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) ∧ 𝑘 ∈ ℕ) → ((𝑦 ∈ ℋ ↦ ((𝑇‘𝑦) −_ℎ 𝑦))‘(𝑓‘𝑘)) = 0_ℎ)
71		fvco3 6185	. . . . . . . . . . 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 747	. . . . . . . . . 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 27244	. . . . . . . . . . . . 13 ⊢ 0_ℎ ∈ ℋ
74	73	elexi 3186	. . . . . . . . . . . 12 ⊢ 0_ℎ ∈ V
75	74	fvconst2 6374	. . . . . . . . . . 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 2654	. . . . . . . . 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 2949	. . . . . . . 8 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → ∀𝑘 ∈ ℕ (((𝑦 ∈ ℋ ↦ ((𝑇‘𝑦) −_ℎ 𝑦)) ∘ 𝑓)‘𝑘) = ((ℕ × {0_ℎ})‘𝑘))
79		ovex 6577	. . . . . . . . . . 11 ⊢ ((𝑇‘𝑦) −_ℎ 𝑦) ∈ V
80	79, 45	fnmpti 5935	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℋ ↦ ((𝑇‘𝑦) −_ℎ 𝑦)) Fn ℋ
81		fnfco 5982	. . . . . . . . . 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 694	. . . . . . . . 9 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → ((𝑦 ∈ ℋ ↦ ((𝑇‘𝑦) −_ℎ 𝑦)) ∘ 𝑓) Fn ℕ)
83	74	fconst 6004	. . . . . . . . . 10 ⊢ (ℕ × {0_ℎ}):ℕ⟶{0_ℎ}
84		ffn 5958	. . . . . . . . . 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 6219	. . . . . . . . 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 693	. . . . . . . 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 246	. . . . . . 7 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → ((𝑦 ∈ ℋ ↦ ((𝑇‘𝑦) −_ℎ 𝑦)) ∘ 𝑓) = (ℕ × {0_ℎ}))
89		vex 3176	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
90	89	hlimveci 27431	. . . . . . . . 9 ⊢ (𝑓 ⇝_𝑣 𝑥 → 𝑥 ∈ ℋ)
91	90	adantl 481	. . . . . . . 8 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → 𝑥 ∈ ℋ)
92		fveq2 6103	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (𝑇‘𝑦) = (𝑇‘𝑥))
93		id 22	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → 𝑦 = 𝑥)
94	92, 93	oveq12d 6567	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → ((𝑇‘𝑦) −_ℎ 𝑦) = ((𝑇‘𝑥) −_ℎ 𝑥))
95		ovex 6577	. . . . . . . . 9 ⊢ ((𝑇‘𝑥) −_ℎ 𝑥) ∈ V
96	94, 45, 95	fvmpt 6191	. . . . . . . 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 4614	. . . . . 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 11285	. . . . . . 7 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → 1 ∈ ℤ)
101		nnuz 11599	. . . . . . . 8 ⊢ ℕ = (ℤ_≥‘1)
102	101	lmconst 20875	. . . . . . 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 1318	. . . . . 6 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → (ℕ × {0_ℎ})(⇝_𝑡‘(MetOpen‘(norm_ℎ ∘ −_ℎ )))0_ℎ)
104	9, 98, 103	lmmo 20994	. . . . 5 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → ((𝑇‘𝑥) −_ℎ 𝑥) = 0_ℎ)
105	19	ffvelrni 6266	. . . . . . 7 ⊢ (𝑥 ∈ ℋ → (𝑇‘𝑥) ∈ ℋ)
106	91, 105	syl 17	. . . . . 6 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → (𝑇‘𝑥) ∈ ℋ)
107		hvsubeq0 27309	. . . . . 6 ⊢ (((𝑇‘𝑥) ∈ ℋ ∧ 𝑥 ∈ ℋ) → (((𝑇‘𝑥) −_ℎ 𝑥) = 0_ℎ ↔ (𝑇‘𝑥) = 𝑥))
108	106, 91, 107	syl2anc 691	. . . . 5 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → (((𝑇‘𝑥) −_ℎ 𝑥) = 0_ℎ ↔ (𝑇‘𝑥) = 𝑥))
109	104, 108	mpbid 221	. . . 4 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → (𝑇‘𝑥) = 𝑥)
110		fnfvelrn 6264	. . . . 5 ⊢ ((𝑇 Fn ℋ ∧ 𝑥 ∈ ℋ) → (𝑇‘𝑥) ∈ ran 𝑇)
111	50, 91, 110	sylancr 694	. . . 4 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → (𝑇‘𝑥) ∈ ran 𝑇)
112	109, 111	eqeltrrd 2689	. . 3 ⊢ ((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → 𝑥 ∈ ran 𝑇)
113	112	gen2 1714	. 2 ⊢ ∀𝑓∀𝑥((𝑓:ℕ⟶ran 𝑇 ∧ 𝑓 ⇝_𝑣 𝑥) → 𝑥 ∈ ran 𝑇)
114		isch2 27464	. 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 957	1 ⊢ ran 𝑇 ∈ C_ℋ

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∀wal 1473 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ⊆ wss 3540 {csn 4125 ⟨cop 4131 class class class wbr 4583 ↦ cmpt 4643 × cxp 5036 ran crn 5039 ↾ cres 5040 ∘ ccom 5042 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ↑_𝑚 cmap 7744 1c1 9816 ℕcn 10897 ℤcz 11254 ∞Metcxmt 19552 MetOpencmopn 19557 TopOnctopon 20518 Cn ccn 20838 ⇝_𝑡clm 20840 Hauscha 20922 ×_t ctx 21173 NrmCVeccnv 26823 ℋchil 27160 +_ℎ cva 27161 ·_ℎ csm 27162 norm_ℎcno 27164 0_ℎc0v 27165 −_ℎ cmv 27166 ⇝_𝑣 chli 27168 S_ℋ csh 27169 C_ℋ cch 27170 ConOpccop 27187 LinOpclo 27188 BndLinOpcbo 27189 HrmOpcho 27191

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cc 9140 ax-dc 9151 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895 ax-hilex 27240 ax-hfvadd 27241 ax-hvcom 27242 ax-hvass 27243 ax-hv0cl 27244 ax-hvaddid 27245 ax-hfvmul 27246 ax-hvmulid 27247 ax-hvmulass 27248 ax-hvdistr1 27249 ax-hvdistr2 27250 ax-hvmul0 27251 ax-hfi 27320 ax-his1 27323 ax-his2 27324 ax-his3 27325 ax-his4 27326 ax-hcompl 27443

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-omul 7452 df-er 7629 df-map 7746 df-pm 7747 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-fi 8200 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-acn 8651 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-ioo 12050 df-ico 12052 df-icc 12053 df-fz 12198 df-fzo 12335 df-fl 12455 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-rlim 14068 df-sum 14265 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-hom 15793 df-cco 15794 df-rest 15906 df-topn 15907 df-0g 15925 df-gsum 15926 df-topgen 15927 df-pt 15928 df-prds 15931 df-xrs 15985 df-qtop 15990 df-imas 15991 df-xps 15993 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-mulg 17364 df-cntz 17573 df-cmn 18018 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-fbas 19564 df-fg 19565 df-cnfld 19568 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-cld 20633 df-ntr 20634 df-cls 20635 df-nei 20712 df-cn 20841 df-cnp 20842 df-lm 20843 df-t1 20928 df-haus 20929 df-cmp 21000 df-tx 21175 df-hmeo 21368 df-fil 21460 df-fm 21552 df-flim 21553 df-flf 21554 df-fcls 21555 df-xms 21935 df-ms 21936 df-tms 21937 df-cncf 22489 df-cfil 22861 df-cau 22862 df-cmet 22863 df-grpo 26731 df-gid 26732 df-ginv 26733 df-gdiv 26734 df-ablo 26783 df-vc 26798 df-nv 26831 df-va 26834 df-ba 26835 df-sm 26836 df-0v 26837 df-vs 26838 df-nmcv 26839 df-ims 26840 df-dip 26940 df-ssp 26961 df-lno 26983 df-nmoo 26984 df-blo 26985 df-0o 26986 df-ph 27052 df-cbn 27103 df-hlo 27126 df-hnorm 27209 df-hba 27210 df-hvsub 27212 df-hlim 27213 df-hcau 27214 df-sh 27448 df-ch 27462 df-oc 27493 df-ch0 27494 df-shs 27551 df-pjh 27638 df-h0op 27991 df-nmop 28082 df-cnop 28083 df-lnop 28084 df-bdop 28085 df-unop 28086 df-hmop 28087

This theorem is referenced by: hmopidmpji 28395 hmopidmch 28396

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