occllem - Hilbert Space Explorer

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Theorem occllem 28502

Description: Lemma for occl 28503. (Contributed by NM, 7-Aug-2000.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
occl.1	⊢ (𝜑 → 𝐴 ⊆ ℋ)
occl.2	⊢ (𝜑 → 𝐹 ∈ Cauchy)
occl.3	⊢ (𝜑 → 𝐹:ℕ⟶(⊥‘𝐴))
occl.4	⊢ (𝜑 → 𝐵 ∈ 𝐴)

**Assertion**
Ref	Expression
occllem	⊢ (𝜑 → (( ⇝_𝑣 ‘𝐹) ·_ih 𝐵) = 0)

Proof of Theorem occllem Dummy variables 𝑥 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2771	. . . 4 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
2	1	cnfldhaus 22808	. . 3 ⊢ (TopOpen‘ℂ_fld) ∈ Haus
3	2	a1i 11	. 2 ⊢ (𝜑 → (TopOpen‘ℂ_fld) ∈ Haus)
4		occl.2	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ Cauchy)
5		ax-hcompl 28399	. . . . . . 7 ⊢ (𝐹 ∈ Cauchy → ∃𝑥 ∈ ℋ 𝐹 ⇝_𝑣 𝑥)
6		hlimf 28434	. . . . . . . . . 10 ⊢ ⇝_𝑣 :dom ⇝_𝑣 ⟶ ℋ
7		ffn 6184	. . . . . . . . . 10 ⊢ ( ⇝_𝑣 :dom ⇝_𝑣 ⟶ ℋ → ⇝_𝑣 Fn dom ⇝_𝑣 )
8	6, 7	ax-mp 5	. . . . . . . . 9 ⊢ ⇝_𝑣 Fn dom ⇝_𝑣
9		fnbr 6132	. . . . . . . . 9 ⊢ (( ⇝_𝑣 Fn dom ⇝_𝑣 ∧ 𝐹 ⇝_𝑣 𝑥) → 𝐹 ∈ dom ⇝_𝑣 )
10	8, 9	mpan 670	. . . . . . . 8 ⊢ (𝐹 ⇝_𝑣 𝑥 → 𝐹 ∈ dom ⇝_𝑣 )
11	10	rexlimivw 3177	. . . . . . 7 ⊢ (∃𝑥 ∈ ℋ 𝐹 ⇝_𝑣 𝑥 → 𝐹 ∈ dom ⇝_𝑣 )
12	4, 5, 11	3syl 18	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ dom ⇝_𝑣 )
13		ffun 6187	. . . . . . 7 ⊢ ( ⇝_𝑣 :dom ⇝_𝑣 ⟶ ℋ → Fun ⇝_𝑣 )
14		funfvbrb 6475	. . . . . . 7 ⊢ (Fun ⇝_𝑣 → (𝐹 ∈ dom ⇝_𝑣 ↔ 𝐹 ⇝_𝑣 ( ⇝_𝑣 ‘𝐹)))
15	6, 13, 14	mp2b 10	. . . . . 6 ⊢ (𝐹 ∈ dom ⇝_𝑣 ↔ 𝐹 ⇝_𝑣 ( ⇝_𝑣 ‘𝐹))
16	12, 15	sylib 208	. . . . 5 ⊢ (𝜑 → 𝐹 ⇝_𝑣 ( ⇝_𝑣 ‘𝐹))
17		eqid 2771	. . . . . . . 8 ⊢ ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩ = ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩
18		eqid 2771	. . . . . . . . 9 ⊢ (norm_ℎ ∘ −_ℎ ) = (norm_ℎ ∘ −_ℎ )
19	17, 18	hhims 28369	. . . . . . . 8 ⊢ (norm_ℎ ∘ −_ℎ ) = (IndMet‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
20		eqid 2771	. . . . . . . 8 ⊢ (MetOpen‘(norm_ℎ ∘ −_ℎ )) = (MetOpen‘(norm_ℎ ∘ −_ℎ ))
21	17, 19, 20	hhlm 28396	. . . . . . 7 ⊢ ⇝_𝑣 = ((⇝_𝑡‘(MetOpen‘(norm_ℎ ∘ −_ℎ ))) ↾ ( ℋ ↑_𝑚 ℕ))
22		resss 5562	. . . . . . 7 ⊢ ((⇝_𝑡‘(MetOpen‘(norm_ℎ ∘ −_ℎ ))) ↾ ( ℋ ↑_𝑚 ℕ)) ⊆ (⇝_𝑡‘(MetOpen‘(norm_ℎ ∘ −_ℎ )))
23	21, 22	eqsstri 3784	. . . . . 6 ⊢ ⇝_𝑣 ⊆ (⇝_𝑡‘(MetOpen‘(norm_ℎ ∘ −_ℎ )))
24	23	ssbri 4832	. . . . 5 ⊢ (𝐹 ⇝_𝑣 ( ⇝_𝑣 ‘𝐹) → 𝐹(⇝_𝑡‘(MetOpen‘(norm_ℎ ∘ −_ℎ )))( ⇝_𝑣 ‘𝐹))
25	16, 24	syl 17	. . . 4 ⊢ (𝜑 → 𝐹(⇝_𝑡‘(MetOpen‘(norm_ℎ ∘ −_ℎ )))( ⇝_𝑣 ‘𝐹))
26	18	hilxmet 28392	. . . . . 6 ⊢ (norm_ℎ ∘ −_ℎ ) ∈ (∞Met‘ ℋ)
27	20	mopntopon 22464	. . . . . 6 ⊢ ((norm_ℎ ∘ −_ℎ ) ∈ (∞Met‘ ℋ) → (MetOpen‘(norm_ℎ ∘ −_ℎ )) ∈ (TopOn‘ ℋ))
28	26, 27	mp1i 13	. . . . 5 ⊢ (𝜑 → (MetOpen‘(norm_ℎ ∘ −_ℎ )) ∈ (TopOn‘ ℋ))
29	28	cnmptid 21685	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ℋ ↦ 𝑥) ∈ ((MetOpen‘(norm_ℎ ∘ −_ℎ )) Cn (MetOpen‘(norm_ℎ ∘ −_ℎ ))))
30		occl.1	. . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ ℋ)
31		occl.4	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ 𝐴)
32	30, 31	sseldd 3753	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℋ)
33	28, 28, 32	cnmptc 21686	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ℋ ↦ 𝐵) ∈ ((MetOpen‘(norm_ℎ ∘ −_ℎ )) Cn (MetOpen‘(norm_ℎ ∘ −_ℎ ))))
34	17	hhnv 28362	. . . . . 6 ⊢ ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩ ∈ NrmCVec
35	17	hhip 28374	. . . . . . 7 ⊢ ·_ih = (·_𝑖OLD‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
36	35, 19, 20, 1	dipcn 27915	. . . . . 6 ⊢ (⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩ ∈ NrmCVec → ·_ih ∈ (((MetOpen‘(norm_ℎ ∘ −_ℎ )) ×_t (MetOpen‘(norm_ℎ ∘ −_ℎ ))) Cn (TopOpen‘ℂ_fld)))
37	34, 36	mp1i 13	. . . . 5 ⊢ (𝜑 → ·_ih ∈ (((MetOpen‘(norm_ℎ ∘ −_ℎ )) ×_t (MetOpen‘(norm_ℎ ∘ −_ℎ ))) Cn (TopOpen‘ℂ_fld)))
38	28, 29, 33, 37	cnmpt12f 21690	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵)) ∈ ((MetOpen‘(norm_ℎ ∘ −_ℎ )) Cn (TopOpen‘ℂ_fld)))
39	25, 38	lmcn 21330	. . 3 ⊢ (𝜑 → ((𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵)) ∘ 𝐹)(⇝_𝑡‘(TopOpen‘ℂ_fld))((𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵))‘( ⇝_𝑣 ‘𝐹)))
40		occl.3	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:ℕ⟶(⊥‘𝐴))
41	40	ffvelrnda 6504	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ (⊥‘𝐴))
42		ocel 28480	. . . . . . . . . . . 12 ⊢ (𝐴 ⊆ ℋ → ((𝐹‘𝑘) ∈ (⊥‘𝐴) ↔ ((𝐹‘𝑘) ∈ ℋ ∧ ∀𝑥 ∈ 𝐴 ((𝐹‘𝑘) ·_ih 𝑥) = 0)))
43	30, 42	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐹‘𝑘) ∈ (⊥‘𝐴) ↔ ((𝐹‘𝑘) ∈ ℋ ∧ ∀𝑥 ∈ 𝐴 ((𝐹‘𝑘) ·_ih 𝑥) = 0)))
44	43	adantr 466	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝑘) ∈ (⊥‘𝐴) ↔ ((𝐹‘𝑘) ∈ ℋ ∧ ∀𝑥 ∈ 𝐴 ((𝐹‘𝑘) ·_ih 𝑥) = 0)))
45	41, 44	mpbid 222	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝑘) ∈ ℋ ∧ ∀𝑥 ∈ 𝐴 ((𝐹‘𝑘) ·_ih 𝑥) = 0))
46	45	simpld 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℋ)
47		oveq1 6803	. . . . . . . . 9 ⊢ (𝑥 = (𝐹‘𝑘) → (𝑥 ·_ih 𝐵) = ((𝐹‘𝑘) ·_ih 𝐵))
48		eqid 2771	. . . . . . . . 9 ⊢ (𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵)) = (𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵))
49		ovex 6827	. . . . . . . . 9 ⊢ ((𝐹‘𝑘) ·_ih 𝐵) ∈ V
50	47, 48, 49	fvmpt 6426	. . . . . . . 8 ⊢ ((𝐹‘𝑘) ∈ ℋ → ((𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵))‘(𝐹‘𝑘)) = ((𝐹‘𝑘) ·_ih 𝐵))
51	46, 50	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵))‘(𝐹‘𝑘)) = ((𝐹‘𝑘) ·_ih 𝐵))
52		oveq2 6804	. . . . . . . . 9 ⊢ (𝑥 = 𝐵 → ((𝐹‘𝑘) ·_ih 𝑥) = ((𝐹‘𝑘) ·_ih 𝐵))
53	52	eqeq1d 2773	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → (((𝐹‘𝑘) ·_ih 𝑥) = 0 ↔ ((𝐹‘𝑘) ·_ih 𝐵) = 0))
54	45	simprd 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ∀𝑥 ∈ 𝐴 ((𝐹‘𝑘) ·_ih 𝑥) = 0)
55	31	adantr 466	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐵 ∈ 𝐴)
56	53, 54, 55	rspcdva 3466	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝑘) ·_ih 𝐵) = 0)
57	51, 56	eqtrd 2805	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵))‘(𝐹‘𝑘)) = 0)
58		ocss 28484	. . . . . . . . 9 ⊢ (𝐴 ⊆ ℋ → (⊥‘𝐴) ⊆ ℋ)
59	30, 58	syl 17	. . . . . . . 8 ⊢ (𝜑 → (⊥‘𝐴) ⊆ ℋ)
60	40, 59	fssd 6198	. . . . . . 7 ⊢ (𝜑 → 𝐹:ℕ⟶ ℋ)
61		fvco3 6419	. . . . . . 7 ⊢ ((𝐹:ℕ⟶ ℋ ∧ 𝑘 ∈ ℕ) → (((𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵)) ∘ 𝐹)‘𝑘) = ((𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵))‘(𝐹‘𝑘)))
62	60, 61	sylan 569	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (((𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵)) ∘ 𝐹)‘𝑘) = ((𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵))‘(𝐹‘𝑘)))
63		c0ex 10240	. . . . . . . 8 ⊢ 0 ∈ V
64	63	fvconst2 6616	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → ((ℕ × {0})‘𝑘) = 0)
65	64	adantl 467	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((ℕ × {0})‘𝑘) = 0)
66	57, 62, 65	3eqtr4d 2815	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (((𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵)) ∘ 𝐹)‘𝑘) = ((ℕ × {0})‘𝑘))
67	66	ralrimiva 3115	. . . 4 ⊢ (𝜑 → ∀𝑘 ∈ ℕ (((𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵)) ∘ 𝐹)‘𝑘) = ((ℕ × {0})‘𝑘))
68		ovex 6827	. . . . . . 7 ⊢ (𝑥 ·_ih 𝐵) ∈ V
69	68, 48	fnmpti 6161	. . . . . 6 ⊢ (𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵)) Fn ℋ
70		fnfco 6210	. . . . . 6 ⊢ (((𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵)) Fn ℋ ∧ 𝐹:ℕ⟶ ℋ) → ((𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵)) ∘ 𝐹) Fn ℕ)
71	69, 60, 70	sylancr 575	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵)) ∘ 𝐹) Fn ℕ)
72	63	fconst 6232	. . . . . 6 ⊢ (ℕ × {0}):ℕ⟶{0}
73		ffn 6184	. . . . . 6 ⊢ ((ℕ × {0}):ℕ⟶{0} → (ℕ × {0}) Fn ℕ)
74	72, 73	ax-mp 5	. . . . 5 ⊢ (ℕ × {0}) Fn ℕ
75		eqfnfv 6456	. . . . 5 ⊢ ((((𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵)) ∘ 𝐹) Fn ℕ ∧ (ℕ × {0}) Fn ℕ) → (((𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵)) ∘ 𝐹) = (ℕ × {0}) ↔ ∀𝑘 ∈ ℕ (((𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵)) ∘ 𝐹)‘𝑘) = ((ℕ × {0})‘𝑘)))
76	71, 74, 75	sylancl 574	. . . 4 ⊢ (𝜑 → (((𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵)) ∘ 𝐹) = (ℕ × {0}) ↔ ∀𝑘 ∈ ℕ (((𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵)) ∘ 𝐹)‘𝑘) = ((ℕ × {0})‘𝑘)))
77	67, 76	mpbird 247	. . 3 ⊢ (𝜑 → ((𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵)) ∘ 𝐹) = (ℕ × {0}))
78		fvex 6344	. . . . 5 ⊢ ( ⇝_𝑣 ‘𝐹) ∈ V
79	78	hlimveci 28387	. . . 4 ⊢ (𝐹 ⇝_𝑣 ( ⇝_𝑣 ‘𝐹) → ( ⇝_𝑣 ‘𝐹) ∈ ℋ)
80		oveq1 6803	. . . . 5 ⊢ (𝑥 = ( ⇝_𝑣 ‘𝐹) → (𝑥 ·_ih 𝐵) = (( ⇝_𝑣 ‘𝐹) ·_ih 𝐵))
81		ovex 6827	. . . . 5 ⊢ (( ⇝_𝑣 ‘𝐹) ·_ih 𝐵) ∈ V
82	80, 48, 81	fvmpt 6426	. . . 4 ⊢ (( ⇝_𝑣 ‘𝐹) ∈ ℋ → ((𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵))‘( ⇝_𝑣 ‘𝐹)) = (( ⇝_𝑣 ‘𝐹) ·_ih 𝐵))
83	16, 79, 82	3syl 18	. . 3 ⊢ (𝜑 → ((𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵))‘( ⇝_𝑣 ‘𝐹)) = (( ⇝_𝑣 ‘𝐹) ·_ih 𝐵))
84	39, 77, 83	3brtr3d 4818	. 2 ⊢ (𝜑 → (ℕ × {0})(⇝_𝑡‘(TopOpen‘ℂ_fld))(( ⇝_𝑣 ‘𝐹) ·_ih 𝐵))
85	1	cnfldtopon 22806	. . . 4 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
86	85	a1i 11	. . 3 ⊢ (𝜑 → (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ))
87		0cnd 10239	. . 3 ⊢ (𝜑 → 0 ∈ ℂ)
88		1zzd 11615	. . 3 ⊢ (𝜑 → 1 ∈ ℤ)
89		nnuz 11930	. . . 4 ⊢ ℕ = (ℤ_≥‘1)
90	89	lmconst 21286	. . 3 ⊢ (((TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ) ∧ 0 ∈ ℂ ∧ 1 ∈ ℤ) → (ℕ × {0})(⇝_𝑡‘(TopOpen‘ℂ_fld))0)
91	86, 87, 88, 90	syl3anc 1476	. 2 ⊢ (𝜑 → (ℕ × {0})(⇝_𝑡‘(TopOpen‘ℂ_fld))0)
92	3, 84, 91	lmmo 21405	1 ⊢ (𝜑 → (( ⇝_𝑣 ‘𝐹) ·_ih 𝐵) = 0)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 382 = wceq 1631 ∈ wcel 2145 ∀wral 3061 ∃wrex 3062 ⊆ wss 3723 {csn 4317 ⟨cop 4323 class class class wbr 4787 ↦ cmpt 4864 × cxp 5248 dom cdm 5250 ↾ cres 5252 ∘ ccom 5254 Fun wfun 6024 Fn wfn 6025 ⟶wf 6026 ‘cfv 6030 (class class class)co 6796 ↑_𝑚 cmap 8013 ℂcc 10140 0cc0 10142 1c1 10143 ℕcn 11226 ℤcz 11584 TopOpenctopn 16290 ∞Metcxmt 19946 MetOpencmopn 19951 ℂ_fldccnfld 19961 TopOnctopon 20935 Cn ccn 21249 ⇝_𝑡clm 21251 Hauscha 21333 ×_t ctx 21584 NrmCVeccnv 27779 ℋchil 28116 +_ℎ cva 28117 ·_ℎ csm 28118 ·_ih csp 28119 norm_ℎcno 28120 −_ℎ cmv 28122 Cauchyccau 28123 ⇝_𝑣 chli 28124 ⊥cort 28127

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4905 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7100 ax-inf2 8706 ax-cnex 10198 ax-resscn 10199 ax-1cn 10200 ax-icn 10201 ax-addcl 10202 ax-addrcl 10203 ax-mulcl 10204 ax-mulrcl 10205 ax-mulcom 10206 ax-addass 10207 ax-mulass 10208 ax-distr 10209 ax-i2m1 10210 ax-1ne0 10211 ax-1rid 10212 ax-rnegex 10213 ax-rrecex 10214 ax-cnre 10215 ax-pre-lttri 10216 ax-pre-lttrn 10217 ax-pre-ltadd 10218 ax-pre-mulgt0 10219 ax-pre-sup 10220 ax-addf 10221 ax-mulf 10222 ax-hilex 28196 ax-hfvadd 28197 ax-hvcom 28198 ax-hvass 28199 ax-hv0cl 28200 ax-hvaddid 28201 ax-hfvmul 28202 ax-hvmulid 28203 ax-hvmulass 28204 ax-hvdistr1 28205 ax-hvdistr2 28206 ax-hvmul0 28207 ax-hfi 28276 ax-his1 28279 ax-his2 28280 ax-his3 28281 ax-his4 28282 ax-hcompl 28399

This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-fal 1637 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-tp 4322 df-op 4324 df-uni 4576 df-int 4613 df-iun 4657 df-iin 4658 df-br 4788 df-opab 4848 df-mpt 4865 df-tr 4888 df-id 5158 df-eprel 5163 df-po 5171 df-so 5172 df-fr 5209 df-se 5210 df-we 5211 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-pred 5822 df-ord 5868 df-on 5869 df-lim 5870 df-suc 5871 df-iota 5993 df-fun 6032 df-fn 6033 df-f 6034 df-f1 6035 df-fo 6036 df-f1o 6037 df-fv 6038 df-isom 6039 df-riota 6757 df-ov 6799 df-oprab 6800 df-mpt2 6801 df-of 7048 df-om 7217 df-1st 7319 df-2nd 7320 df-supp 7451 df-wrecs 7563 df-recs 7625 df-rdg 7663 df-1o 7717 df-2o 7718 df-oadd 7721 df-er 7900 df-map 8015 df-pm 8016 df-ixp 8067 df-en 8114 df-dom 8115 df-sdom 8116 df-fin 8117 df-fsupp 8436 df-fi 8477 df-sup 8508 df-inf 8509 df-oi 8575 df-card 8969 df-cda 9196 df-pnf 10282 df-mnf 10283 df-xr 10284 df-ltxr 10285 df-le 10286 df-sub 10474 df-neg 10475 df-div 10891 df-nn 11227 df-2 11285 df-3 11286 df-4 11287 df-5 11288 df-6 11289 df-7 11290 df-8 11291 df-9 11292 df-n0 11500 df-z 11585 df-dec 11701 df-uz 11894 df-q 11997 df-rp 12036 df-xneg 12151 df-xadd 12152 df-xmul 12153 df-ioo 12384 df-icc 12387 df-fz 12534 df-fzo 12674 df-seq 13009 df-exp 13068 df-hash 13322 df-cj 14047 df-re 14048 df-im 14049 df-sqrt 14183 df-abs 14184 df-clim 14427 df-sum 14625 df-struct 16066 df-ndx 16067 df-slot 16068 df-base 16070 df-sets 16071 df-ress 16072 df-plusg 16162 df-mulr 16163 df-starv 16164 df-sca 16165 df-vsca 16166 df-ip 16167 df-tset 16168 df-ple 16169 df-ds 16172 df-unif 16173 df-hom 16174 df-cco 16175 df-rest 16291 df-topn 16292 df-0g 16310 df-gsum 16311 df-topgen 16312 df-pt 16313 df-prds 16316 df-xrs 16370 df-qtop 16375 df-imas 16376 df-xps 16378 df-mre 16454 df-mrc 16455 df-acs 16457 df-mgm 17450 df-sgrp 17492 df-mnd 17503 df-submnd 17544 df-mulg 17749 df-cntz 17957 df-cmn 18402 df-psmet 19953 df-xmet 19954 df-met 19955 df-bl 19956 df-mopn 19957 df-cnfld 19962 df-top 20919 df-topon 20936 df-topsp 20958 df-bases 20971 df-cn 21252 df-cnp 21253 df-lm 21254 df-haus 21340 df-tx 21586 df-hmeo 21779 df-xms 22345 df-ms 22346 df-tms 22347 df-grpo 27687 df-gid 27688 df-ginv 27689 df-gdiv 27690 df-ablo 27739 df-vc 27754 df-nv 27787 df-va 27790 df-ba 27791 df-sm 27792 df-0v 27793 df-vs 27794 df-nmcv 27795 df-ims 27796 df-dip 27896 df-hnorm 28165 df-hvsub 28168 df-hlim 28169 df-sh 28404 df-oc 28449

This theorem is referenced by: occl 28503

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