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Theorem occllem 27546

Description: Lemma for occl 27547. (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 2610	. . . 4 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
2	1	cnfldhaus 22398	. . 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 27443	. . . . . . 7 ⊢ (𝐹 ∈ Cauchy → ∃𝑥 ∈ ℋ 𝐹 ⇝_𝑣 𝑥)
6		hlimf 27478	. . . . . . . . . 10 ⊢ ⇝_𝑣 :dom ⇝_𝑣 ⟶ ℋ
7		ffn 5958	. . . . . . . . . 10 ⊢ ( ⇝_𝑣 :dom ⇝_𝑣 ⟶ ℋ → ⇝_𝑣 Fn dom ⇝_𝑣 )
8	6, 7	ax-mp 5	. . . . . . . . 9 ⊢ ⇝_𝑣 Fn dom ⇝_𝑣
9		fnbr 5907	. . . . . . . . 9 ⊢ (( ⇝_𝑣 Fn dom ⇝_𝑣 ∧ 𝐹 ⇝_𝑣 𝑥) → 𝐹 ∈ dom ⇝_𝑣 )
10	8, 9	mpan 702	. . . . . . . 8 ⊢ (𝐹 ⇝_𝑣 𝑥 → 𝐹 ∈ dom ⇝_𝑣 )
11	10	rexlimivw 3011	. . . . . . 7 ⊢ (∃𝑥 ∈ ℋ 𝐹 ⇝_𝑣 𝑥 → 𝐹 ∈ dom ⇝_𝑣 )
12	4, 5, 11	3syl 18	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ dom ⇝_𝑣 )
13		ffun 5961	. . . . . . 7 ⊢ ( ⇝_𝑣 :dom ⇝_𝑣 ⟶ ℋ → Fun ⇝_𝑣 )
14		funfvbrb 6238	. . . . . . 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 207	. . . . 5 ⊢ (𝜑 → 𝐹 ⇝_𝑣 ( ⇝_𝑣 ‘𝐹))
17		eqid 2610	. . . . . . . 8 ⊢ ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩ = ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩
18		eqid 2610	. . . . . . . . 9 ⊢ (norm_ℎ ∘ −_ℎ ) = (norm_ℎ ∘ −_ℎ )
19	17, 18	hhims 27413	. . . . . . . 8 ⊢ (norm_ℎ ∘ −_ℎ ) = (IndMet‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
20		eqid 2610	. . . . . . . 8 ⊢ (MetOpen‘(norm_ℎ ∘ −_ℎ )) = (MetOpen‘(norm_ℎ ∘ −_ℎ ))
21	17, 19, 20	hhlm 27440	. . . . . . 7 ⊢ ⇝_𝑣 = ((⇝_𝑡‘(MetOpen‘(norm_ℎ ∘ −_ℎ ))) ↾ ( ℋ ↑_𝑚 ℕ))
22		resss 5342	. . . . . . 7 ⊢ ((⇝_𝑡‘(MetOpen‘(norm_ℎ ∘ −_ℎ ))) ↾ ( ℋ ↑_𝑚 ℕ)) ⊆ (⇝_𝑡‘(MetOpen‘(norm_ℎ ∘ −_ℎ )))
23	21, 22	eqsstri 3598	. . . . . 6 ⊢ ⇝_𝑣 ⊆ (⇝_𝑡‘(MetOpen‘(norm_ℎ ∘ −_ℎ )))
24	23	ssbri 4627	. . . . 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 27436	. . . . . 6 ⊢ (norm_ℎ ∘ −_ℎ ) ∈ (∞Met‘ ℋ)
27	20	mopntopon 22054	. . . . . 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 21274	. . . . 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 3569	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℋ)
33	28, 28, 32	cnmptc 21275	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ℋ ↦ 𝐵) ∈ ((MetOpen‘(norm_ℎ ∘ −_ℎ )) Cn (MetOpen‘(norm_ℎ ∘ −_ℎ ))))
34	17	hhnv 27406	. . . . . 6 ⊢ ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩ ∈ NrmCVec
35	17	hhip 27418	. . . . . . 7 ⊢ ·_ih = (·_𝑖OLD‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
36	35, 19, 20, 1	dipcn 26959	. . . . . 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 21279	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵)) ∈ ((MetOpen‘(norm_ℎ ∘ −_ℎ )) Cn (TopOpen‘ℂ_fld)))
39	25, 38	lmcn 20919	. . 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 6267	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ (⊥‘𝐴))
42		ocel 27524	. . . . . . . . . . . 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 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝑘) ∈ (⊥‘𝐴) ↔ ((𝐹‘𝑘) ∈ ℋ ∧ ∀𝑥 ∈ 𝐴 ((𝐹‘𝑘) ·_ih 𝑥) = 0)))
45	41, 44	mpbid 221	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝑘) ∈ ℋ ∧ ∀𝑥 ∈ 𝐴 ((𝐹‘𝑘) ·_ih 𝑥) = 0))
46	45	simpld 474	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℋ)
47		oveq1 6556	. . . . . . . . 9 ⊢ (𝑥 = (𝐹‘𝑘) → (𝑥 ·_ih 𝐵) = ((𝐹‘𝑘) ·_ih 𝐵))
48		eqid 2610	. . . . . . . . 9 ⊢ (𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵)) = (𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵))
49		ovex 6577	. . . . . . . . 9 ⊢ ((𝐹‘𝑘) ·_ih 𝐵) ∈ V
50	47, 48, 49	fvmpt 6191	. . . . . . . 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 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐵 ∈ 𝐴)
53	45	simprd 478	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ∀𝑥 ∈ 𝐴 ((𝐹‘𝑘) ·_ih 𝑥) = 0)
54		oveq2 6557	. . . . . . . . . 10 ⊢ (𝑥 = 𝐵 → ((𝐹‘𝑘) ·_ih 𝑥) = ((𝐹‘𝑘) ·_ih 𝐵))
55	54	eqeq1d 2612	. . . . . . . . 9 ⊢ (𝑥 = 𝐵 → (((𝐹‘𝑘) ·_ih 𝑥) = 0 ↔ ((𝐹‘𝑘) ·_ih 𝐵) = 0))
56	55	rspcv 3278	. . . . . . . 8 ⊢ (𝐵 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 ((𝐹‘𝑘) ·_ih 𝑥) = 0 → ((𝐹‘𝑘) ·_ih 𝐵) = 0))
57	52, 53, 56	sylc 63	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝑘) ·_ih 𝐵) = 0)
58	51, 57	eqtrd 2644	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵))‘(𝐹‘𝑘)) = 0)
59		ocss 27528	. . . . . . . . 9 ⊢ (𝐴 ⊆ ℋ → (⊥‘𝐴) ⊆ ℋ)
60	30, 59	syl 17	. . . . . . . 8 ⊢ (𝜑 → (⊥‘𝐴) ⊆ ℋ)
61	40, 60	fssd 5970	. . . . . . 7 ⊢ (𝜑 → 𝐹:ℕ⟶ ℋ)
62		fvco3 6185	. . . . . . 7 ⊢ ((𝐹:ℕ⟶ ℋ ∧ 𝑘 ∈ ℕ) → (((𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵)) ∘ 𝐹)‘𝑘) = ((𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵))‘(𝐹‘𝑘)))
63	61, 62	sylan 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (((𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵)) ∘ 𝐹)‘𝑘) = ((𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵))‘(𝐹‘𝑘)))
64		c0ex 9913	. . . . . . . 8 ⊢ 0 ∈ V
65	64	fvconst2 6374	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → ((ℕ × {0})‘𝑘) = 0)
66	65	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((ℕ × {0})‘𝑘) = 0)
67	58, 63, 66	3eqtr4d 2654	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (((𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵)) ∘ 𝐹)‘𝑘) = ((ℕ × {0})‘𝑘))
68	67	ralrimiva 2949	. . . 4 ⊢ (𝜑 → ∀𝑘 ∈ ℕ (((𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵)) ∘ 𝐹)‘𝑘) = ((ℕ × {0})‘𝑘))
69		ovex 6577	. . . . . . 7 ⊢ (𝑥 ·_ih 𝐵) ∈ V
70	69, 48	fnmpti 5935	. . . . . 6 ⊢ (𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵)) Fn ℋ
71		fnfco 5982	. . . . . 6 ⊢ (((𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵)) Fn ℋ ∧ 𝐹:ℕ⟶ ℋ) → ((𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵)) ∘ 𝐹) Fn ℕ)
72	70, 61, 71	sylancr 694	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵)) ∘ 𝐹) Fn ℕ)
73	64	fconst 6004	. . . . . 6 ⊢ (ℕ × {0}):ℕ⟶{0}
74		ffn 5958	. . . . . 6 ⊢ ((ℕ × {0}):ℕ⟶{0} → (ℕ × {0}) Fn ℕ)
75	73, 74	ax-mp 5	. . . . 5 ⊢ (ℕ × {0}) Fn ℕ
76		eqfnfv 6219	. . . . 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 693	. . . 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 246	. . 3 ⊢ (𝜑 → ((𝑥 ∈ ℋ ↦ (𝑥 ·_ih 𝐵)) ∘ 𝐹) = (ℕ × {0}))
79		fvex 6113	. . . . 5 ⊢ ( ⇝_𝑣 ‘𝐹) ∈ V
80	79	hlimveci 27431	. . . 4 ⊢ (𝐹 ⇝_𝑣 ( ⇝_𝑣 ‘𝐹) → ( ⇝_𝑣 ‘𝐹) ∈ ℋ)
81		oveq1 6556	. . . . 5 ⊢ (𝑥 = ( ⇝_𝑣 ‘𝐹) → (𝑥 ·_ih 𝐵) = (( ⇝_𝑣 ‘𝐹) ·_ih 𝐵))
82		ovex 6577	. . . . 5 ⊢ (( ⇝_𝑣 ‘𝐹) ·_ih 𝐵) ∈ V
83	81, 48, 82	fvmpt 6191	. . . 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 4614	. 2 ⊢ (𝜑 → (ℕ × {0})(⇝_𝑡‘(TopOpen‘ℂ_fld))(( ⇝_𝑣 ‘𝐹) ·_ih 𝐵))
86	1	cnfldtopon 22396	. . . 4 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
87	86	a1i 11	. . 3 ⊢ (𝜑 → (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ))
88		0cnd 9912	. . 3 ⊢ (𝜑 → 0 ∈ ℂ)
89		1zzd 11285	. . 3 ⊢ (𝜑 → 1 ∈ ℤ)
90		nnuz 11599	. . . 4 ⊢ ℕ = (ℤ_≥‘1)
91	90	lmconst 20875	. . 3 ⊢ (((TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ) ∧ 0 ∈ ℂ ∧ 1 ∈ ℤ) → (ℕ × {0})(⇝_𝑡‘(TopOpen‘ℂ_fld))0)
92	87, 88, 89, 91	syl3anc 1318	. 2 ⊢ (𝜑 → (ℕ × {0})(⇝_𝑡‘(TopOpen‘ℂ_fld))0)
93	3, 85, 92	lmmo 20994	1 ⊢ (𝜑 → (( ⇝_𝑣 ‘𝐹) ·_ih 𝐵) = 0)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 ⊆ wss 3540 {csn 4125 ⟨cop 4131 class class class wbr 4583 ↦ cmpt 4643 × cxp 5036 dom cdm 5038 ↾ cres 5040 ∘ ccom 5042 Fun wfun 5798 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ↑_𝑚 cmap 7744 ℂcc 9813 0cc0 9815 1c1 9816 ℕcn 10897 ℤcz 11254 TopOpenctopn 15905 ∞Metcxmt 19552 MetOpencmopn 19557 ℂ_fldccnfld 19567 TopOnctopon 20518 Cn ccn 20838 ⇝_𝑡clm 20840 Hauscha 20922 ×_t ctx 21173 NrmCVeccnv 26823 ℋchil 27160 +_ℎ cva 27161 ·_ℎ csm 27162 ·_ih csp 27163 norm_ℎcno 27164 −_ℎ cmv 27166 Cauchyccau 27167 ⇝_𝑣 chli 27168 ⊥cort 27171

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-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-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-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-icc 12053 df-fz 12198 df-fzo 12335 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-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-cnfld 19568 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-cn 20841 df-cnp 20842 df-lm 20843 df-haus 20929 df-tx 21175 df-hmeo 21368 df-xms 21935 df-ms 21936 df-tms 21937 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-hnorm 27209 df-hvsub 27212 df-hlim 27213 df-sh 27448 df-oc 27493

This theorem is referenced by: occl 27547

W3C validator