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Theorem occllem 31898

Description: Lemma for occl 31899. (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 2761	. . . 4 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
2	1	cnfldhaus 25096	. . 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 31797	. . . . . . 7 ⊢ (𝐹 ∈ Cauchy → ∃𝑥 ∈ ℋ 𝐹 ⇝_𝑣 𝑥)
6		hlimf 31832	. . . . . . . . . 10 ⊢ ⇝_𝑣 :dom ⇝_𝑣 ⟶ ℋ
7		ffn 6707	. . . . . . . . . 10 ⊢ ( ⇝_𝑣 :dom ⇝_𝑣 ⟶ ℋ → ⇝_𝑣 Fn dom ⇝_𝑣 )
8	6, 7	ax-mp 5	. . . . . . . . 9 ⊢ ⇝_𝑣 Fn dom ⇝_𝑣
9		fnbr 6645	. . . . . . . . 9 ⊢ (( ⇝_𝑣 Fn dom ⇝_𝑣 ∧ 𝐹 ⇝_𝑣 𝑥) → 𝐹 ∈ dom ⇝_𝑣 )
10	8, 9	mpan 703	. . . . . . . 8 ⊢ (𝐹 ⇝_𝑣 𝑥 → 𝐹 ∈ dom ⇝_𝑣 )
11	10	rexlimivw 3160	. . . . . . 7 ⊢ (∃𝑥 ∈ ℋ 𝐹 ⇝_𝑣 𝑥 → 𝐹 ∈ dom ⇝_𝑣 )
12	4, 5, 11	3syl 19	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ dom ⇝_𝑣 )
13		ffun 6710	. . . . . . 7 ⊢ ( ⇝_𝑣 :dom ⇝_𝑣 ⟶ ℋ → Fun ⇝_𝑣 )
14		funfvbrb 7048	. . . . . . 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 221	. . . . 5 ⊢ (𝜑 → 𝐹 ⇝_𝑣 ( ⇝_𝑣 ‘𝐹))
17		eqid 2761	. . . . . . . 8 ⊢ ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩ = ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩
18		eqid 2761	. . . . . . . . 9 ⊢ (norm_ℎ ∘ −_ℎ ) = (norm_ℎ ∘ −_ℎ )
19	17, 18	hhims 31767	. . . . . . . 8 ⊢ (norm_ℎ ∘ −_ℎ ) = (IndMet‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
20		eqid 2761	. . . . . . . 8 ⊢ (MetOpen‘(norm_ℎ ∘ −_ℎ )) = (MetOpen‘(norm_ℎ ∘ −_ℎ ))
21	17, 19, 20	hhlm 31794	. . . . . . 7 ⊢ ⇝_𝑣 = ((⇝_𝑡‘(MetOpen‘(norm_ℎ ∘ −_ℎ ))) ↾ ( ℋ ↑_m ℕ))
22		resss 5992	. . . . . . 7 ⊢ ((⇝_𝑡‘(MetOpen‘(norm_ℎ ∘ −_ℎ ))) ↾ ( ℋ ↑_m ℕ)) ⊆ (⇝_𝑡‘(MetOpen‘(norm_ℎ ∘ −_ℎ )))
23	21, 22	eqsstri 3977	. . . . . 6 ⊢ ⇝_𝑣 ⊆ (⇝_𝑡‘(MetOpen‘(norm_ℎ ∘ −_ℎ )))
24	23	ssbri 5150	. . . . 5 ⊢ (𝐹 ⇝_𝑣 ( ⇝_𝑣 ‘𝐹) → 𝐹(⇝_𝑡‘(MetOpen‘(norm_ℎ ∘ −_ℎ )))( ⇝_𝑣 ‘𝐹))
25	16, 24	syl 18	. . . 4 ⊢ (𝜑 → 𝐹(⇝_𝑡‘(MetOpen‘(norm_ℎ ∘ −_ℎ )))( ⇝_𝑣 ‘𝐹))
26	18	hilxmet 31790	. . . . . 6 ⊢ (norm_ℎ ∘ −_ℎ ) ∈ (∞Met‘ ℋ)
27	20	mopntopon 24751	. . . . . 6 ⊢ ((norm_ℎ ∘ −_ℎ ) ∈ (∞Met‘ ℋ) → (MetOpen‘(norm_ℎ ∘ −_ℎ )) ∈ (TopOn‘ ℋ))
28	26, 27	mp1i 14	. . . . 5 ⊢ (𝜑 → (MetOpen‘(norm_ℎ ∘ −_ℎ )) ∈ (TopOn‘ ℋ))
29	28	cnmptid 23973	. . . . 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 3932	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℋ)
33	28, 28, 32	cnmptc 23974	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ℋ ↦ 𝐵) ∈ ((MetOpen‘(norm_ℎ ∘ −_ℎ )) Cn (MetOpen‘(norm_ℎ ∘ −_ℎ ))))
34	17	hhnv 31760	. . . . . 6 ⊢ ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩ ∈ NrmCVec
35	17	hhip 31772	. . . . . . 7 ⊢ ·_ih = (·_𝑖OLD‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
36	35, 19, 20, 1	dipcn 31315	. . . . . 6 ⊢ (⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩ ∈ NrmCVec → ·_ih ∈ (((MetOpen‘(norm_ℎ ∘ −_ℎ )) ×_t (MetOpen‘(norm_ℎ ∘ −_ℎ ))) Cn (TopOpen‘ℂ_fld)))
37	34, 36	mp1i 14	. . . . 5 ⊢ (𝜑 → ·_ih ∈ (((MetOpen‘(norm_ℎ ∘ −_ℎ )) ×_t (MetOpen‘(norm_ℎ ∘ −_ℎ ))) Cn (TopOpen‘ℂ_fld)))
38	28, 29, 33, 37	cnmpt12f 23978	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵)) ∈ ((MetOpen‘(norm_ℎ ∘ −_ℎ )) Cn (TopOpen‘ℂ_fld)))
39	25, 38	lmcn 23616	. . 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	ffvelcdmda 7082	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ (⊥‘𝐴))
42		ocel 31876	. . . . . . . . . . . 12 ⊢ (𝐴 ⊆ ℋ → ((𝐹‘𝑘) ∈ (⊥‘𝐴) ↔ ((𝐹‘𝑘) ∈ ℋ ∧ ∀𝑥 ∈ 𝐴 ((𝐹‘𝑘) ·_ih 𝑥) = 0)))
43	30, 42	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐹‘𝑘) ∈ (⊥‘𝐴) ↔ ((𝐹‘𝑘) ∈ ℋ ∧ ∀𝑥 ∈ 𝐴 ((𝐹‘𝑘) ·_ih 𝑥) = 0)))
44	43	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝑘) ∈ (⊥‘𝐴) ↔ ((𝐹‘𝑘) ∈ ℋ ∧ ∀𝑥 ∈ 𝐴 ((𝐹‘𝑘) ·_ih 𝑥) = 0)))
45	41, 44	mpbid 235	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝑘) ∈ ℋ ∧ ∀𝑥 ∈ 𝐴 ((𝐹‘𝑘) ·_ih 𝑥) = 0))
46	45	simpld 500	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℋ)
47		oveq1 7425	. . . . . . . . 9 ⊢ (𝑥 = (𝐹‘𝑘) → (𝑥 ·_ih 𝐵) = ((𝐹‘𝑘) ·_ih 𝐵))
48		eqid 2761	. . . . . . . . 9 ⊢ (𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵)) = (𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵))
49		ovex 7451	. . . . . . . . 9 ⊢ ((𝐹‘𝑘) ·_ih 𝐵) ∈ V
50	47, 48, 49	fvmpt 6991	. . . . . . . 8 ⊢ ((𝐹‘𝑘) ∈ ℋ → ((𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵))‘(𝐹‘𝑘)) = ((𝐹‘𝑘) ·_ih 𝐵))
51	46, 50	syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵))‘(𝐹‘𝑘)) = ((𝐹‘𝑘) ·_ih 𝐵))
52		oveq2 7426	. . . . . . . . 9 ⊢ (𝑥 = 𝐵 → ((𝐹‘𝑘) ·_ih 𝑥) = ((𝐹‘𝑘) ·_ih 𝐵))
53	52	eqeq1d 2763	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → (((𝐹‘𝑘) ·_ih 𝑥) = 0 ↔ ((𝐹‘𝑘) ·_ih 𝐵) = 0))
54	45	simprd 501	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ∀𝑥 ∈ 𝐴 ((𝐹‘𝑘) ·_ih 𝑥) = 0)
55	31	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐵 ∈ 𝐴)
56	53, 54, 55	rspcdva 3578	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝑘) ·_ih 𝐵) = 0)
57	51, 56	eqtrd 2796	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵))‘(𝐹‘𝑘)) = 0)
58		ocss 31880	. . . . . . . . 9 ⊢ (𝐴 ⊆ ℋ → (⊥‘𝐴) ⊆ ℋ)
59	30, 58	syl 18	. . . . . . . 8 ⊢ (𝜑 → (⊥‘𝐴) ⊆ ℋ)
60	40, 59	fssd 6725	. . . . . . 7 ⊢ (𝜑 → 𝐹:ℕ⟶ ℋ)
61		fvco3 6983	. . . . . . 7 ⊢ ((𝐹:ℕ⟶ ℋ ∧ 𝑘 ∈ ℕ) → (((𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵)) ∘ 𝐹)‘𝑘) = ((𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵))‘(𝐹‘𝑘)))
62	60, 61	sylan 592	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (((𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵)) ∘ 𝐹)‘𝑘) = ((𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵))‘(𝐹‘𝑘)))
63		c0ex 11293	. . . . . . . 8 ⊢ 0 ∈ V
64	63	fvconst2 7208	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → ((ℕ × {0})‘𝑘) = 0)
65	64	adantl 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((ℕ × {0})‘𝑘) = 0)
66	57, 62, 65	3eqtr4d 2806	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (((𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵)) ∘ 𝐹)‘𝑘) = ((ℕ × {0})‘𝑘))
67	66	ralrimiva 3155	. . . 4 ⊢ (𝜑 → ∀𝑘 ∈ ℕ (((𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵)) ∘ 𝐹)‘𝑘) = ((ℕ × {0})‘𝑘))
68		ovex 7451	. . . . . . 7 ⊢ (𝑥 ·_ih 𝐵) ∈ V
69	68, 48	fnmpti 6680	. . . . . 6 ⊢ (𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵)) Fn ℋ
70		fnfco 6745	. . . . . 6 ⊢ (((𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵)) Fn ℋ ∧ 𝐹:ℕ⟶ ℋ) → ((𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵)) ∘ 𝐹) Fn ℕ)
71	69, 60, 70	sylancr 599	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵)) ∘ 𝐹) Fn ℕ)
72	63	fconst 6766	. . . . . 6 ⊢ (ℕ × {0}):ℕ⟶{0}
73		ffn 6707	. . . . . 6 ⊢ ((ℕ × {0}):ℕ⟶{0} → (ℕ × {0}) Fn ℕ)
74	72, 73	ax-mp 5	. . . . 5 ⊢ (ℕ × {0}) Fn ℕ
75		eqfnfv 7027	. . . . 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 598	. . . 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 260	. . 3 ⊢ (𝜑 → ((𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵)) ∘ 𝐹) = (ℕ × {0}))
78		fvex 6896	. . . . 5 ⊢ ( ⇝_𝑣 ‘𝐹) ∈ V
79	78	hlimveci 31785	. . . 4 ⊢ (𝐹 ⇝_𝑣 ( ⇝_𝑣 ‘𝐹) → ( ⇝_𝑣 ‘𝐹) ∈ ℋ)
80		oveq1 7425	. . . . 5 ⊢ (𝑥 = ( ⇝_𝑣 ‘𝐹) → (𝑥 ·_ih 𝐵) = (( ⇝_𝑣 ‘𝐹) ·_ih 𝐵))
81		ovex 7451	. . . . 5 ⊢ (( ⇝_𝑣 ‘𝐹) ·_ih 𝐵) ∈ V
82	80, 48, 81	fvmpt 6991	. . . 4 ⊢ (( ⇝_𝑣 ‘𝐹) ∈ ℋ → ((𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵))‘( ⇝_𝑣 ‘𝐹)) = (( ⇝_𝑣 ‘𝐹) ·_ih 𝐵))
83	16, 79, 82	3syl 19	. . 3 ⊢ (𝜑 → ((𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵))‘( ⇝_𝑣 ‘𝐹)) = (( ⇝_𝑣 ‘𝐹) ·_ih 𝐵))
84	39, 77, 83	3brtr3d 5136	. 2 ⊢ (𝜑 → (ℕ × {0})(⇝_𝑡‘(TopOpen‘ℂ_fld))(( ⇝_𝑣 ‘𝐹) ·_ih 𝐵))
85	1	cnfldtopon 25094	. . . 4 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
86	85	a1i 11	. . 3 ⊢ (𝜑 → (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ))
87		0cnd 11292	. . 3 ⊢ (𝜑 → 0 ∈ ℂ)
88		1zzd 12720	. . 3 ⊢ (𝜑 → 1 ∈ ℤ)
89		nnuz 12997	. . . 4 ⊢ ℕ = (ℤ_≥‘1)
90	89	lmconst 23572	. . 3 ⊢ (((TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ) ∧ 0 ∈ ℂ ∧ 1 ∈ ℤ) → (ℕ × {0})(⇝_𝑡‘(TopOpen‘ℂ_fld))0)
91	86, 87, 88, 90	syl3anc 1398	. 2 ⊢ (𝜑 → (ℕ × {0})(⇝_𝑡‘(TopOpen‘ℂ_fld))0)
92	3, 84, 91	lmmo 23691	1 ⊢ (𝜑 → (( ⇝_𝑣 ‘𝐹) ·_ih 𝐵) = 0)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ∀wral 3077 ∃wrex 3087 ⊆ wss 3899 {csn 4584 ⟨cop 4590 class class class wbr 5103 ↦ cmpt 5186 × cxp 5649 dom cdm 5651 ↾ cres 5653 ∘ ccom 5655 Fun wfun 6531 Fn wfn 6532 ⟶wf 6533 ‘cfv 6537 (class class class)co 7418 ↑_m cmap 8840 ℂcc 11191 0cc0 11193 1c1 11194 ℕcn 12328 ℤcz 12686 TopOpenctopn 17585 ∞Metcxmet 21656 MetOpencmopn 21661 ℂ_fldccnfld 21671 TopOnctopon 23221 Cn ccn 23535 ⇝_𝑡clm 23537 Hauscha 23619 ×_t ctx 23872 NrmCVeccnv 31179 ℋchba 31514 +_ℎ cva 31515 ·_ℎ csm 31516 ·_ih csp 31517 norm_ℎcno 31518 −_ℎ cmv 31520 Cauchyccauold 31521 ⇝_𝑣 chli 31522 ⊥cort 31525

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7749 ax-inf2 9635 ax-cnex 11249 ax-resscn 11250 ax-1cn 11251 ax-icn 11252 ax-addcl 11253 ax-addrcl 11254 ax-mulcl 11255 ax-mulrcl 11256 ax-mulcom 11257 ax-addass 11258 ax-mulass 11259 ax-distr 11260 ax-i2m1 11261 ax-1ne0 11262 ax-1rid 11263 ax-rnegex 11264 ax-rrecex 11265 ax-cnre 11266 ax-pre-lttri 11267 ax-pre-lttrn 11268 ax-pre-ltadd 11269 ax-pre-mulgt0 11270 ax-pre-sup 11271 ax-addf 11272 ax-mulf 11273 ax-hilex 31594 ax-hfvadd 31595 ax-hvcom 31596 ax-hvass 31597 ax-hv0cl 31598 ax-hvaddid 31599 ax-hfvmul 31600 ax-hvmulid 31601 ax-hvmulass 31602 ax-hvdistr1 31603 ax-hvdistr2 31604 ax-hvmul0 31605 ax-hfi 31674 ax-his1 31677 ax-his2 31678 ax-his3 31679 ax-his4 31680 ax-hcompl 31797

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7375 df-ov 7421 df-oprab 7422 df-mpo 7423 df-of 7691 df-om 7876 df-1st 7999 df-2nd 8000 df-supp 8171 df-frecs 8292 df-wrecs 8323 df-recs 8372 df-rdg 8411 df-1o 8469 df-2o 8470 df-er 8710 df-map 8842 df-pm 8843 df-ixp 8919 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-fsupp 9347 df-fi 9396 df-sup 9427 df-inf 9428 df-oi 9497 df-card 10013 df-pnf 11338 df-mnf 11339 df-xr 11340 df-ltxr 11341 df-le 11342 df-sub 11536 df-neg 11537 df-div 11967 df-nn 12329 df-2 12398 df-3 12399 df-4 12400 df-5 12401 df-6 12402 df-7 12403 df-8 12404 df-9 12405 df-n0 12600 df-z 12687 df-dec 12808 df-uz 12959 df-q 13069 df-rp 13114 df-xneg 13234 df-xadd 13235 df-xmul 13236 df-ioo 13473 df-icc 13476 df-fz 13633 df-fzo 13782 df-seq 14138 df-exp 14198 df-hash 14468 df-cj 15259 df-re 15260 df-im 15261 df-sqrt 15395 df-abs 15396 df-clim 15648 df-sum 15847 df-struct 17318 df-sets 17335 df-slot 17353 df-ndx 17365 df-base 17381 df-ress 17402 df-plusg 17434 df-mulr 17435 df-starv 17436 df-sca 17437 df-vsca 17438 df-ip 17439 df-tset 17440 df-ple 17441 df-ds 17443 df-unif 17444 df-hom 17445 df-cco 17446 df-rest 17586 df-topn 17587 df-0g 17605 df-gsum 17606 df-topgen 17607 df-pt 17608 df-prds 17611 df-xrs 17667 df-qtop 17672 df-imas 17673 df-xps 17675 df-mre 17749 df-mrc 17750 df-acs 17752 df-mgm 18809 df-sgrp 18901 df-mnd 18917 df-submnd 18972 df-mulg 19271 df-cntz 19524 df-cmn 19989 df-psmet 21663 df-xmet 21664 df-met 21665 df-bl 21666 df-mopn 21667 df-cnfld 21672 df-top 23205 df-topon 23222 df-topsp 23244 df-bases 23257 df-cn 23538 df-cnp 23539 df-lm 23540 df-haus 23626 df-tx 23874 df-hmeo 24067 df-xms 24632 df-ms 24633 df-tms 24634 df-grpo 31088 df-gid 31089 df-ginv 31090 df-gdiv 31091 df-ablo 31140 df-vc 31154 df-nv 31187 df-va 31190 df-ba 31191 df-sm 31192 df-0v 31193 df-vs 31194 df-nmcv 31195 df-ims 31196 df-dip 31296 df-hnorm 31563 df-hvsub 31566 df-hlim 31567 df-sh 31802 df-oc 31847

This theorem is used by: occl 31899

W3C validator