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Theorem occllem 28162

Description: Lemma for occl 28163. (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 2622	. . . 4 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
2	1	cnfldhaus 22588	. . 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 28059	. . . . . . 7 ⊢ (𝐹 ∈ Cauchy → ∃𝑥 ∈ ℋ 𝐹 ⇝_𝑣 𝑥)
6		hlimf 28094	. . . . . . . . . 10 ⊢ ⇝_𝑣 :dom ⇝_𝑣 ⟶ ℋ
7		ffn 6045	. . . . . . . . . 10 ⊢ ( ⇝_𝑣 :dom ⇝_𝑣 ⟶ ℋ → ⇝_𝑣 Fn dom ⇝_𝑣 )
8	6, 7	ax-mp 5	. . . . . . . . 9 ⊢ ⇝_𝑣 Fn dom ⇝_𝑣
9		fnbr 5993	. . . . . . . . 9 ⊢ (( ⇝_𝑣 Fn dom ⇝_𝑣 ∧ 𝐹 ⇝_𝑣 𝑥) → 𝐹 ∈ dom ⇝_𝑣 )
10	8, 9	mpan 706	. . . . . . . 8 ⊢ (𝐹 ⇝_𝑣 𝑥 → 𝐹 ∈ dom ⇝_𝑣 )
11	10	rexlimivw 3029	. . . . . . 7 ⊢ (∃𝑥 ∈ ℋ 𝐹 ⇝_𝑣 𝑥 → 𝐹 ∈ dom ⇝_𝑣 )
12	4, 5, 11	3syl 18	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ dom ⇝_𝑣 )
13		ffun 6048	. . . . . . 7 ⊢ ( ⇝_𝑣 :dom ⇝_𝑣 ⟶ ℋ → Fun ⇝_𝑣 )
14		funfvbrb 6330	. . . . . . 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 2622	. . . . . . . 8 ⊢ ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩ = ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩
18		eqid 2622	. . . . . . . . 9 ⊢ (norm_ℎ ∘ −_ℎ ) = (norm_ℎ ∘ −_ℎ )
19	17, 18	hhims 28029	. . . . . . . 8 ⊢ (norm_ℎ ∘ −_ℎ ) = (IndMet‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
20		eqid 2622	. . . . . . . 8 ⊢ (MetOpen‘(norm_ℎ ∘ −_ℎ )) = (MetOpen‘(norm_ℎ ∘ −_ℎ ))
21	17, 19, 20	hhlm 28056	. . . . . . 7 ⊢ ⇝_𝑣 = ((⇝_𝑡‘(MetOpen‘(norm_ℎ ∘ −_ℎ ))) ↾ ( ℋ ↑_𝑚 ℕ))
22		resss 5422	. . . . . . 7 ⊢ ((⇝_𝑡‘(MetOpen‘(norm_ℎ ∘ −_ℎ ))) ↾ ( ℋ ↑_𝑚 ℕ)) ⊆ (⇝_𝑡‘(MetOpen‘(norm_ℎ ∘ −_ℎ )))
23	21, 22	eqsstri 3635	. . . . . 6 ⊢ ⇝_𝑣 ⊆ (⇝_𝑡‘(MetOpen‘(norm_ℎ ∘ −_ℎ )))
24	23	ssbri 4697	. . . . 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 28052	. . . . . 6 ⊢ (norm_ℎ ∘ −_ℎ ) ∈ (∞Met‘ ℋ)
27	20	mopntopon 22244	. . . . . 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 21464	. . . . 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 3604	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℋ)
33	28, 28, 32	cnmptc 21465	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ℋ ↦ 𝐵) ∈ ((MetOpen‘(norm_ℎ ∘ −_ℎ )) Cn (MetOpen‘(norm_ℎ ∘ −_ℎ ))))
34	17	hhnv 28022	. . . . . 6 ⊢ ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩ ∈ NrmCVec
35	17	hhip 28034	. . . . . . 7 ⊢ ·_ih = (·_𝑖OLD‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
36	35, 19, 20, 1	dipcn 27575	. . . . . 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 21469	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵)) ∈ ((MetOpen‘(norm_ℎ ∘ −_ℎ )) Cn (TopOpen‘ℂ_fld)))
39	25, 38	lmcn 21109	. . 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 6359	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ (⊥‘𝐴))
42		ocel 28140	. . . . . . . . . . . 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 222	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝑘) ∈ ℋ ∧ ∀𝑥 ∈ 𝐴 ((𝐹‘𝑘) ·_ih 𝑥) = 0))
46	45	simpld 475	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℋ)
47		oveq1 6657	. . . . . . . . 9 ⊢ (𝑥 = (𝐹‘𝑘) → (𝑥 ·_ih 𝐵) = ((𝐹‘𝑘) ·_ih 𝐵))
48		eqid 2622	. . . . . . . . 9 ⊢ (𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵)) = (𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵))
49		ovex 6678	. . . . . . . . 9 ⊢ ((𝐹‘𝑘) ·_ih 𝐵) ∈ V
50	47, 48, 49	fvmpt 6282	. . . . . . . 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	31	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐵 ∈ 𝐴)
53	45	simprd 479	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ∀𝑥 ∈ 𝐴 ((𝐹‘𝑘) ·_ih 𝑥) = 0)
54		oveq2 6658	. . . . . . . . . 10 ⊢ (𝑥 = 𝐵 → ((𝐹‘𝑘) ·_ih 𝑥) = ((𝐹‘𝑘) ·_ih 𝐵))
55	54	eqeq1d 2624	. . . . . . . . 9 ⊢ (𝑥 = 𝐵 → (((𝐹‘𝑘) ·_ih 𝑥) = 0 ↔ ((𝐹‘𝑘) ·_ih 𝐵) = 0))
56	55	rspcv 3305	. . . . . . . 8 ⊢ (𝐵 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 ((𝐹‘𝑘) ·_ih 𝑥) = 0 → ((𝐹‘𝑘) ·_ih 𝐵) = 0))
57	52, 53, 56	sylc 65	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝑘) ·_ih 𝐵) = 0)
58	51, 57	eqtrd 2656	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵))‘(𝐹‘𝑘)) = 0)
59		ocss 28144	. . . . . . . . 9 ⊢ (𝐴 ⊆ ℋ → (⊥‘𝐴) ⊆ ℋ)
60	30, 59	syl 17	. . . . . . . 8 ⊢ (𝜑 → (⊥‘𝐴) ⊆ ℋ)
61	40, 60	fssd 6057	. . . . . . 7 ⊢ (𝜑 → 𝐹:ℕ⟶ ℋ)
62		fvco3 6275	. . . . . . 7 ⊢ ((𝐹:ℕ⟶ ℋ ∧ 𝑘 ∈ ℕ) → (((𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵)) ∘ 𝐹)‘𝑘) = ((𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵))‘(𝐹‘𝑘)))
63	61, 62	sylan 488	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (((𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵)) ∘ 𝐹)‘𝑘) = ((𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵))‘(𝐹‘𝑘)))
64		c0ex 10034	. . . . . . . 8 ⊢ 0 ∈ V
65	64	fvconst2 6469	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → ((ℕ × {0})‘𝑘) = 0)
66	65	adantl 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((ℕ × {0})‘𝑘) = 0)
67	58, 63, 66	3eqtr4d 2666	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (((𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵)) ∘ 𝐹)‘𝑘) = ((ℕ × {0})‘𝑘))
68	67	ralrimiva 2966	. . . 4 ⊢ (𝜑 → ∀𝑘 ∈ ℕ (((𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵)) ∘ 𝐹)‘𝑘) = ((ℕ × {0})‘𝑘))
69		ovex 6678	. . . . . . 7 ⊢ (𝑥 ·_ih 𝐵) ∈ V
70	69, 48	fnmpti 6022	. . . . . 6 ⊢ (𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵)) Fn ℋ
71		fnfco 6069	. . . . . 6 ⊢ (((𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵)) Fn ℋ ∧ 𝐹:ℕ⟶ ℋ) → ((𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵)) ∘ 𝐹) Fn ℕ)
72	70, 61, 71	sylancr 695	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵)) ∘ 𝐹) Fn ℕ)
73	64	fconst 6091	. . . . . 6 ⊢ (ℕ × {0}):ℕ⟶{0}
74		ffn 6045	. . . . . 6 ⊢ ((ℕ × {0}):ℕ⟶{0} → (ℕ × {0}) Fn ℕ)
75	73, 74	ax-mp 5	. . . . 5 ⊢ (ℕ × {0}) Fn ℕ
76		eqfnfv 6311	. . . . 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)))
77	72, 75, 76	sylancl 694	. . . 4 ⊢ (𝜑 → (((𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵)) ∘ 𝐹) = (ℕ × {0}) ↔ ∀𝑘 ∈ ℕ (((𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵)) ∘ 𝐹)‘𝑘) = ((ℕ × {0})‘𝑘)))
78	68, 77	mpbird 247	. . 3 ⊢ (𝜑 → ((𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵)) ∘ 𝐹) = (ℕ × {0}))
79		fvex 6201	. . . . 5 ⊢ ( ⇝_𝑣 ‘𝐹) ∈ V
80	79	hlimveci 28047	. . . 4 ⊢ (𝐹 ⇝_𝑣 ( ⇝_𝑣 ‘𝐹) → ( ⇝_𝑣 ‘𝐹) ∈ ℋ)
81		oveq1 6657	. . . . 5 ⊢ (𝑥 = ( ⇝_𝑣 ‘𝐹) → (𝑥 ·_ih 𝐵) = (( ⇝_𝑣 ‘𝐹) ·_ih 𝐵))
82		ovex 6678	. . . . 5 ⊢ (( ⇝_𝑣 ‘𝐹) ·_ih 𝐵) ∈ V
83	81, 48, 82	fvmpt 6282	. . . 4 ⊢ (( ⇝_𝑣 ‘𝐹) ∈ ℋ → ((𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵))‘( ⇝_𝑣 ‘𝐹)) = (( ⇝_𝑣 ‘𝐹) ·_ih 𝐵))
84	16, 80, 83	3syl 18	. . 3 ⊢ (𝜑 → ((𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵))‘( ⇝_𝑣 ‘𝐹)) = (( ⇝_𝑣 ‘𝐹) ·_ih 𝐵))
85	39, 78, 84	3brtr3d 4684	. 2 ⊢ (𝜑 → (ℕ × {0})(⇝_𝑡‘(TopOpen‘ℂ_fld))(( ⇝_𝑣 ‘𝐹) ·_ih 𝐵))
86	1	cnfldtopon 22586	. . . 4 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
87	86	a1i 11	. . 3 ⊢ (𝜑 → (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ))
88		0cnd 10033	. . 3 ⊢ (𝜑 → 0 ∈ ℂ)
89		1zzd 11408	. . 3 ⊢ (𝜑 → 1 ∈ ℤ)
90		nnuz 11723	. . . 4 ⊢ ℕ = (ℤ_≥‘1)
91	90	lmconst 21065	. . 3 ⊢ (((TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ) ∧ 0 ∈ ℂ ∧ 1 ∈ ℤ) → (ℕ × {0})(⇝_𝑡‘(TopOpen‘ℂ_fld))0)
92	87, 88, 89, 91	syl3anc 1326	. 2 ⊢ (𝜑 → (ℕ × {0})(⇝_𝑡‘(TopOpen‘ℂ_fld))0)
93	3, 85, 92	lmmo 21184	1 ⊢ (𝜑 → (( ⇝_𝑣 ‘𝐹) ·_ih 𝐵) = 0)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ∃wrex 2913 ⊆ wss 3574 {csn 4177 ⟨cop 4183 class class class wbr 4653 ↦ cmpt 4729 × cxp 5112 dom cdm 5114 ↾ cres 5116 ∘ ccom 5118 Fun wfun 5882 Fn wfn 5883 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ↑_𝑚 cmap 7857 ℂcc 9934 0cc0 9936 1c1 9937 ℕcn 11020 ℤcz 11377 TopOpenctopn 16082 ∞Metcxmt 19731 MetOpencmopn 19736 ℂ_fldccnfld 19746 TopOnctopon 20715 Cn ccn 21028 ⇝_𝑡clm 21030 Hauscha 21112 ×_t ctx 21363 NrmCVeccnv 27439 ℋchil 27776 +_ℎ cva 27777 ·_ℎ csm 27778 ·_ih csp 27779 norm_ℎcno 27780 −_ℎ cmv 27782 Cauchyccau 27783 ⇝_𝑣 chli 27784 ⊥cort 27787

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016 ax-hilex 27856 ax-hfvadd 27857 ax-hvcom 27858 ax-hvass 27859 ax-hv0cl 27860 ax-hvaddid 27861 ax-hfvmul 27862 ax-hvmulid 27863 ax-hvmulass 27864 ax-hvdistr1 27865 ax-hvdistr2 27866 ax-hvmul0 27867 ax-hfi 27936 ax-his1 27939 ax-his2 27940 ax-his3 27941 ax-his4 27942 ax-hcompl 28059

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-icc 12182 df-fz 12327 df-fzo 12466 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-sum 14417 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-pt 16105 df-prds 16108 df-xrs 16162 df-qtop 16167 df-imas 16168 df-xps 16170 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-mulg 17541 df-cntz 17750 df-cmn 18195 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-cnfld 19747 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cn 21031 df-cnp 21032 df-lm 21033 df-haus 21119 df-tx 21365 df-hmeo 21558 df-xms 22125 df-ms 22126 df-tms 22127 df-grpo 27347 df-gid 27348 df-ginv 27349 df-gdiv 27350 df-ablo 27399 df-vc 27414 df-nv 27447 df-va 27450 df-ba 27451 df-sm 27452 df-0v 27453 df-vs 27454 df-nmcv 27455 df-ims 27456 df-dip 27556 df-hnorm 27825 df-hvsub 27828 df-hlim 27829 df-sh 28064 df-oc 28109

This theorem is referenced by: occl 28163

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