occllem - Hilbert Space Explorer

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Theorem occllem 28771

Description: Lemma for occl 28772. (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 2795	. . . 4 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
2	1	cnfldhaus 23076	. . 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 28670	. . . . . . 7 ⊢ (𝐹 ∈ Cauchy → ∃𝑥 ∈ ℋ 𝐹 ⇝_𝑣 𝑥)
6		hlimf 28705	. . . . . . . . . 10 ⊢ ⇝_𝑣 :dom ⇝_𝑣 ⟶ ℋ
7		ffn 6382	. . . . . . . . . 10 ⊢ ( ⇝_𝑣 :dom ⇝_𝑣 ⟶ ℋ → ⇝_𝑣 Fn dom ⇝_𝑣 )
8	6, 7	ax-mp 5	. . . . . . . . 9 ⊢ ⇝_𝑣 Fn dom ⇝_𝑣
9		fnbr 6329	. . . . . . . . 9 ⊢ (( ⇝_𝑣 Fn dom ⇝_𝑣 ∧ 𝐹 ⇝_𝑣 𝑥) → 𝐹 ∈ dom ⇝_𝑣 )
10	8, 9	mpan 686	. . . . . . . 8 ⊢ (𝐹 ⇝_𝑣 𝑥 → 𝐹 ∈ dom ⇝_𝑣 )
11	10	rexlimivw 3245	. . . . . . 7 ⊢ (∃𝑥 ∈ ℋ 𝐹 ⇝_𝑣 𝑥 → 𝐹 ∈ dom ⇝_𝑣 )
12	4, 5, 11	3syl 18	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ dom ⇝_𝑣 )
13		ffun 6385	. . . . . . 7 ⊢ ( ⇝_𝑣 :dom ⇝_𝑣 ⟶ ℋ → Fun ⇝_𝑣 )
14		funfvbrb 6686	. . . . . . 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 219	. . . . 5 ⊢ (𝜑 → 𝐹 ⇝_𝑣 ( ⇝_𝑣 ‘𝐹))
17		eqid 2795	. . . . . . . 8 ⊢ ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩ = ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩
18		eqid 2795	. . . . . . . . 9 ⊢ (norm_ℎ ∘ −_ℎ ) = (norm_ℎ ∘ −_ℎ )
19	17, 18	hhims 28640	. . . . . . . 8 ⊢ (norm_ℎ ∘ −_ℎ ) = (IndMet‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
20		eqid 2795	. . . . . . . 8 ⊢ (MetOpen‘(norm_ℎ ∘ −_ℎ )) = (MetOpen‘(norm_ℎ ∘ −_ℎ ))
21	17, 19, 20	hhlm 28667	. . . . . . 7 ⊢ ⇝_𝑣 = ((⇝_𝑡‘(MetOpen‘(norm_ℎ ∘ −_ℎ ))) ↾ ( ℋ ↑_𝑚 ℕ))
22		resss 5759	. . . . . . 7 ⊢ ((⇝_𝑡‘(MetOpen‘(norm_ℎ ∘ −_ℎ ))) ↾ ( ℋ ↑_𝑚 ℕ)) ⊆ (⇝_𝑡‘(MetOpen‘(norm_ℎ ∘ −_ℎ )))
23	21, 22	eqsstri 3922	. . . . . 6 ⊢ ⇝_𝑣 ⊆ (⇝_𝑡‘(MetOpen‘(norm_ℎ ∘ −_ℎ )))
24	23	ssbri 5007	. . . . 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 28663	. . . . . 6 ⊢ (norm_ℎ ∘ −_ℎ ) ∈ (∞Met‘ ℋ)
27	20	mopntopon 22732	. . . . . 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 21953	. . . . 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 3890	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℋ)
33	28, 28, 32	cnmptc 21954	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ℋ ↦ 𝐵) ∈ ((MetOpen‘(norm_ℎ ∘ −_ℎ )) Cn (MetOpen‘(norm_ℎ ∘ −_ℎ ))))
34	17	hhnv 28633	. . . . . 6 ⊢ ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩ ∈ NrmCVec
35	17	hhip 28645	. . . . . . 7 ⊢ ·_ih = (·_𝑖OLD‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
36	35, 19, 20, 1	dipcn 28188	. . . . . 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 21958	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵)) ∈ ((MetOpen‘(norm_ℎ ∘ −_ℎ )) Cn (TopOpen‘ℂ_fld)))
39	25, 38	lmcn 21597	. . 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 6716	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ (⊥‘𝐴))
42		ocel 28749	. . . . . . . . . . . 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 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝑘) ∈ (⊥‘𝐴) ↔ ((𝐹‘𝑘) ∈ ℋ ∧ ∀𝑥 ∈ 𝐴 ((𝐹‘𝑘) ·_ih 𝑥) = 0)))
45	41, 44	mpbid 233	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝑘) ∈ ℋ ∧ ∀𝑥 ∈ 𝐴 ((𝐹‘𝑘) ·_ih 𝑥) = 0))
46	45	simpld 495	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℋ)
47		oveq1 7023	. . . . . . . . 9 ⊢ (𝑥 = (𝐹‘𝑘) → (𝑥 ·_ih 𝐵) = ((𝐹‘𝑘) ·_ih 𝐵))
48		eqid 2795	. . . . . . . . 9 ⊢ (𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵)) = (𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵))
49		ovex 7048	. . . . . . . . 9 ⊢ ((𝐹‘𝑘) ·_ih 𝐵) ∈ V
50	47, 48, 49	fvmpt 6635	. . . . . . . 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 7024	. . . . . . . . 9 ⊢ (𝑥 = 𝐵 → ((𝐹‘𝑘) ·_ih 𝑥) = ((𝐹‘𝑘) ·_ih 𝐵))
53	52	eqeq1d 2797	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → (((𝐹‘𝑘) ·_ih 𝑥) = 0 ↔ ((𝐹‘𝑘) ·_ih 𝐵) = 0))
54	45	simprd 496	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ∀𝑥 ∈ 𝐴 ((𝐹‘𝑘) ·_ih 𝑥) = 0)
55	31	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐵 ∈ 𝐴)
56	53, 54, 55	rspcdva 3565	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝑘) ·_ih 𝐵) = 0)
57	51, 56	eqtrd 2831	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵))‘(𝐹‘𝑘)) = 0)
58		ocss 28753	. . . . . . . . 9 ⊢ (𝐴 ⊆ ℋ → (⊥‘𝐴) ⊆ ℋ)
59	30, 58	syl 17	. . . . . . . 8 ⊢ (𝜑 → (⊥‘𝐴) ⊆ ℋ)
60	40, 59	fssd 6396	. . . . . . 7 ⊢ (𝜑 → 𝐹:ℕ⟶ ℋ)
61		fvco3 6627	. . . . . . 7 ⊢ ((𝐹:ℕ⟶ ℋ ∧ 𝑘 ∈ ℕ) → (((𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵)) ∘ 𝐹)‘𝑘) = ((𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵))‘(𝐹‘𝑘)))
62	60, 61	sylan 580	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (((𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵)) ∘ 𝐹)‘𝑘) = ((𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵))‘(𝐹‘𝑘)))
63		c0ex 10481	. . . . . . . 8 ⊢ 0 ∈ V
64	63	fvconst2 6833	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → ((ℕ × {0})‘𝑘) = 0)
65	64	adantl 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((ℕ × {0})‘𝑘) = 0)
66	57, 62, 65	3eqtr4d 2841	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (((𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵)) ∘ 𝐹)‘𝑘) = ((ℕ × {0})‘𝑘))
67	66	ralrimiva 3149	. . . 4 ⊢ (𝜑 → ∀𝑘 ∈ ℕ (((𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵)) ∘ 𝐹)‘𝑘) = ((ℕ × {0})‘𝑘))
68		ovex 7048	. . . . . . 7 ⊢ (𝑥 ·_ih 𝐵) ∈ V
69	68, 48	fnmpti 6359	. . . . . 6 ⊢ (𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵)) Fn ℋ
70		fnfco 6411	. . . . . 6 ⊢ (((𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵)) Fn ℋ ∧ 𝐹:ℕ⟶ ℋ) → ((𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵)) ∘ 𝐹) Fn ℕ)
71	69, 60, 70	sylancr 587	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵)) ∘ 𝐹) Fn ℕ)
72	63	fconst 6433	. . . . . 6 ⊢ (ℕ × {0}):ℕ⟶{0}
73		ffn 6382	. . . . . 6 ⊢ ((ℕ × {0}):ℕ⟶{0} → (ℕ × {0}) Fn ℕ)
74	72, 73	ax-mp 5	. . . . 5 ⊢ (ℕ × {0}) Fn ℕ
75		eqfnfv 6667	. . . . 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 586	. . . 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 258	. . 3 ⊢ (𝜑 → ((𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵)) ∘ 𝐹) = (ℕ × {0}))
78		fvex 6551	. . . . 5 ⊢ ( ⇝_𝑣 ‘𝐹) ∈ V
79	78	hlimveci 28658	. . . 4 ⊢ (𝐹 ⇝_𝑣 ( ⇝_𝑣 ‘𝐹) → ( ⇝_𝑣 ‘𝐹) ∈ ℋ)
80		oveq1 7023	. . . . 5 ⊢ (𝑥 = ( ⇝_𝑣 ‘𝐹) → (𝑥 ·_ih 𝐵) = (( ⇝_𝑣 ‘𝐹) ·_ih 𝐵))
81		ovex 7048	. . . . 5 ⊢ (( ⇝_𝑣 ‘𝐹) ·_ih 𝐵) ∈ V
82	80, 48, 81	fvmpt 6635	. . . 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 4993	. 2 ⊢ (𝜑 → (ℕ × {0})(⇝_𝑡‘(TopOpen‘ℂ_fld))(( ⇝_𝑣 ‘𝐹) ·_ih 𝐵))
85	1	cnfldtopon 23074	. . . 4 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
86	85	a1i 11	. . 3 ⊢ (𝜑 → (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ))
87		0cnd 10480	. . 3 ⊢ (𝜑 → 0 ∈ ℂ)
88		1zzd 11862	. . 3 ⊢ (𝜑 → 1 ∈ ℤ)
89		nnuz 12130	. . . 4 ⊢ ℕ = (ℤ_≥‘1)
90	89	lmconst 21553	. . 3 ⊢ (((TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ) ∧ 0 ∈ ℂ ∧ 1 ∈ ℤ) → (ℕ × {0})(⇝_𝑡‘(TopOpen‘ℂ_fld))0)
91	86, 87, 88, 90	syl3anc 1364	. 2 ⊢ (𝜑 → (ℕ × {0})(⇝_𝑡‘(TopOpen‘ℂ_fld))0)
92	3, 84, 91	lmmo 21672	1 ⊢ (𝜑 → (( ⇝_𝑣 ‘𝐹) ·_ih 𝐵) = 0)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1522 ∈ wcel 2081 ∀wral 3105 ∃wrex 3106 ⊆ wss 3859 {csn 4472 ⟨cop 4478 class class class wbr 4962 ↦ cmpt 5041 × cxp 5441 dom cdm 5443 ↾ cres 5445 ∘ ccom 5447 Fun wfun 6219 Fn wfn 6220 ⟶wf 6221 ‘cfv 6225 (class class class)co 7016 ↑_𝑚 cmap 8256 ℂcc 10381 0cc0 10383 1c1 10384 ℕcn 11486 ℤcz 11829 TopOpenctopn 16524 ∞Metcxmet 20212 MetOpencmopn 20217 ℂ_fldccnfld 20227 TopOnctopon 21202 Cn ccn 21516 ⇝_𝑡clm 21518 Hauscha 21600 ×_t ctx 21852 NrmCVeccnv 28052 ℋchba 28387 +_ℎ cva 28388 ·_ℎ csm 28389 ·_ih csp 28390 norm_ℎcno 28391 −_ℎ cmv 28393 Cauchyccauold 28394 ⇝_𝑣 chli 28395 ⊥cort 28398

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5081 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-inf2 8950 ax-cnex 10439 ax-resscn 10440 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-mulcom 10447 ax-addass 10448 ax-mulass 10449 ax-distr 10450 ax-i2m1 10451 ax-1ne0 10452 ax-1rid 10453 ax-rnegex 10454 ax-rrecex 10455 ax-cnre 10456 ax-pre-lttri 10457 ax-pre-lttrn 10458 ax-pre-ltadd 10459 ax-pre-mulgt0 10460 ax-pre-sup 10461 ax-addf 10462 ax-mulf 10463 ax-hilex 28467 ax-hfvadd 28468 ax-hvcom 28469 ax-hvass 28470 ax-hv0cl 28471 ax-hvaddid 28472 ax-hfvmul 28473 ax-hvmulid 28474 ax-hvmulass 28475 ax-hvdistr1 28476 ax-hvdistr2 28477 ax-hvmul0 28478 ax-hfi 28547 ax-his1 28550 ax-his2 28551 ax-his3 28552 ax-his4 28553 ax-hcompl 28670

This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-fal 1535 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-int 4783 df-iun 4827 df-iin 4828 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-se 5403 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-isom 6234 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-of 7267 df-om 7437 df-1st 7545 df-2nd 7546 df-supp 7682 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-1o 7953 df-2o 7954 df-oadd 7957 df-er 8139 df-map 8258 df-pm 8259 df-ixp 8311 df-en 8358 df-dom 8359 df-sdom 8360 df-fin 8361 df-fsupp 8680 df-fi 8721 df-sup 8752 df-inf 8753 df-oi 8820 df-card 9214 df-pnf 10523 df-mnf 10524 df-xr 10525 df-ltxr 10526 df-le 10527 df-sub 10719 df-neg 10720 df-div 11146 df-nn 11487 df-2 11548 df-3 11549 df-4 11550 df-5 11551 df-6 11552 df-7 11553 df-8 11554 df-9 11555 df-n0 11746 df-z 11830 df-dec 11948 df-uz 12094 df-q 12198 df-rp 12240 df-xneg 12357 df-xadd 12358 df-xmul 12359 df-ioo 12592 df-icc 12595 df-fz 12743 df-fzo 12884 df-seq 13220 df-exp 13280 df-hash 13541 df-cj 14292 df-re 14293 df-im 14294 df-sqrt 14428 df-abs 14429 df-clim 14679 df-sum 14877 df-struct 16314 df-ndx 16315 df-slot 16316 df-base 16318 df-sets 16319 df-ress 16320 df-plusg 16407 df-mulr 16408 df-starv 16409 df-sca 16410 df-vsca 16411 df-ip 16412 df-tset 16413 df-ple 16414 df-ds 16416 df-unif 16417 df-hom 16418 df-cco 16419 df-rest 16525 df-topn 16526 df-0g 16544 df-gsum 16545 df-topgen 16546 df-pt 16547 df-prds 16550 df-xrs 16604 df-qtop 16609 df-imas 16610 df-xps 16612 df-mre 16686 df-mrc 16687 df-acs 16689 df-mgm 17681 df-sgrp 17723 df-mnd 17734 df-submnd 17775 df-mulg 17982 df-cntz 18188 df-cmn 18635 df-psmet 20219 df-xmet 20220 df-met 20221 df-bl 20222 df-mopn 20223 df-cnfld 20228 df-top 21186 df-topon 21203 df-topsp 21225 df-bases 21238 df-cn 21519 df-cnp 21520 df-lm 21521 df-haus 21607 df-tx 21854 df-hmeo 22047 df-xms 22613 df-ms 22614 df-tms 22615 df-grpo 27961 df-gid 27962 df-ginv 27963 df-gdiv 27964 df-ablo 28013 df-vc 28027 df-nv 28060 df-va 28063 df-ba 28064 df-sm 28065 df-0v 28066 df-vs 28067 df-nmcv 28068 df-ims 28069 df-dip 28169 df-hnorm 28436 df-hvsub 28439 df-hlim 28440 df-sh 28675 df-oc 28720

This theorem is referenced by: occl 28772

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