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Theorem bcthlem1 23928

Description: Lemma for bcth 23933. Substitutions for the function 𝐹. (Contributed by Mario Carneiro, 9-Jan-2014.)

**Hypotheses**
Ref	Expression
bcth.2	⊢ 𝐽 = (MetOpen‘𝐷)
bcthlem.4	⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋))
bcthlem.5	⊢ 𝐹 = (𝑘 ∈ ℕ, 𝑧 ∈ (𝑋 × ℝ⁺) ↦ {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))))})

**Assertion**
Ref	Expression
bcthlem1	⊢ ((𝜑 ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ (𝑋 × ℝ⁺))) → (𝐶 ∈ (𝐴𝐹𝐵) ↔ (𝐶 ∈ (𝑋 × ℝ⁺) ∧ (2^nd ‘𝐶) < (1 / 𝐴) ∧ ((cls‘𝐽)‘((ball‘𝐷)‘𝐶)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀‘𝐴)))))

Distinct variable groups: 𝑘,𝑟,𝑥,𝑧,𝐴 𝐵,𝑘,𝑟,𝑥,𝑧 𝐶,𝑟,𝑥 𝐷,𝑘,𝑟,𝑥,𝑧 𝑘,𝐹,𝑟,𝑥,𝑧 𝑘,𝐽,𝑟,𝑥,𝑧 𝑘,𝑀,𝑟,𝑥,𝑧 𝜑,𝑘,𝑟,𝑥,𝑧 𝑘,𝑋,𝑟,𝑥,𝑧 Allowed substitution hints: 𝐶(𝑧,𝑘)

**Proof of Theorem bcthlem1**
Step	Hyp	Ref	Expression
1		opabssxp 5607	. . . . . . 7 ⊢ {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑟 < (1 / 𝐴) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀‘𝐴))))} ⊆ (𝑋 × ℝ⁺)
2		bcthlem.4	. . . . . . . . 9 ⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋))
3		elfvdm 6677	. . . . . . . . 9 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝑋 ∈ dom CMet)
4	2, 3	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ dom CMet)
5		reex 10617	. . . . . . . . 9 ⊢ ℝ ∈ V
6		rpssre 12384	. . . . . . . . 9 ⊢ ℝ⁺ ⊆ ℝ
7	5, 6	ssexi 5190	. . . . . . . 8 ⊢ ℝ⁺ ∈ V
8		xpexg 7453	. . . . . . . 8 ⊢ ((𝑋 ∈ dom CMet ∧ ℝ⁺ ∈ V) → (𝑋 × ℝ⁺) ∈ V)
9	4, 7, 8	sylancl 589	. . . . . . 7 ⊢ (𝜑 → (𝑋 × ℝ⁺) ∈ V)
10		ssexg 5191	. . . . . . 7 ⊢ (({⟨𝑥, 𝑟⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑟 < (1 / 𝐴) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀‘𝐴))))} ⊆ (𝑋 × ℝ⁺) ∧ (𝑋 × ℝ⁺) ∈ V) → {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑟 < (1 / 𝐴) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀‘𝐴))))} ∈ V)
11	1, 9, 10	sylancr 590	. . . . . 6 ⊢ (𝜑 → {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑟 < (1 / 𝐴) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀‘𝐴))))} ∈ V)
12		oveq2 7143	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐴 → (1 / 𝑘) = (1 / 𝐴))
13	12	breq2d 5042	. . . . . . . . . 10 ⊢ (𝑘 = 𝐴 → (𝑟 < (1 / 𝑘) ↔ 𝑟 < (1 / 𝐴)))
14		fveq2 6645	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐴 → (𝑀‘𝑘) = (𝑀‘𝐴))
15	14	difeq2d 4050	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐴 → (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘)) = (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝐴)))
16	15	sseq2d 3947	. . . . . . . . . 10 ⊢ (𝑘 = 𝐴 → (((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘)) ↔ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝐴))))
17	13, 16	anbi12d 633	. . . . . . . . 9 ⊢ (𝑘 = 𝐴 → ((𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))) ↔ (𝑟 < (1 / 𝐴) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝐴)))))
18	17	anbi2d 631	. . . . . . . 8 ⊢ (𝑘 = 𝐴 → (((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘)))) ↔ ((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑟 < (1 / 𝐴) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝐴))))))
19	18	opabbidv 5096	. . . . . . 7 ⊢ (𝑘 = 𝐴 → {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))))} = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑟 < (1 / 𝐴) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝐴))))})
20		fveq2 6645	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝐵 → ((ball‘𝐷)‘𝑧) = ((ball‘𝐷)‘𝐵))
21	20	difeq1d 4049	. . . . . . . . . . 11 ⊢ (𝑧 = 𝐵 → (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝐴)) = (((ball‘𝐷)‘𝐵) ∖ (𝑀‘𝐴)))
22	21	sseq2d 3947	. . . . . . . . . 10 ⊢ (𝑧 = 𝐵 → (((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝐴)) ↔ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀‘𝐴))))
23	22	anbi2d 631	. . . . . . . . 9 ⊢ (𝑧 = 𝐵 → ((𝑟 < (1 / 𝐴) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝐴))) ↔ (𝑟 < (1 / 𝐴) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀‘𝐴)))))
24	23	anbi2d 631	. . . . . . . 8 ⊢ (𝑧 = 𝐵 → (((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑟 < (1 / 𝐴) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝐴)))) ↔ ((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑟 < (1 / 𝐴) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀‘𝐴))))))
25	24	opabbidv 5096	. . . . . . 7 ⊢ (𝑧 = 𝐵 → {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑟 < (1 / 𝐴) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝐴))))} = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑟 < (1 / 𝐴) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀‘𝐴))))})
26		bcthlem.5	. . . . . . 7 ⊢ 𝐹 = (𝑘 ∈ ℕ, 𝑧 ∈ (𝑋 × ℝ⁺) ↦ {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))))})
27	19, 25, 26	ovmpog 7288	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (𝑋 × ℝ⁺) ∧ {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑟 < (1 / 𝐴) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀‘𝐴))))} ∈ V) → (𝐴𝐹𝐵) = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑟 < (1 / 𝐴) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀‘𝐴))))})
28	11, 27	syl3an3 1162	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (𝑋 × ℝ⁺) ∧ 𝜑) → (𝐴𝐹𝐵) = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑟 < (1 / 𝐴) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀‘𝐴))))})
29	28	3expa 1115	. . . 4 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ (𝑋 × ℝ⁺)) ∧ 𝜑) → (𝐴𝐹𝐵) = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑟 < (1 / 𝐴) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀‘𝐴))))})
30	29	ancoms 462	. . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ (𝑋 × ℝ⁺))) → (𝐴𝐹𝐵) = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑟 < (1 / 𝐴) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀‘𝐴))))})
31	30	eleq2d 2875	. 2 ⊢ ((𝜑 ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ (𝑋 × ℝ⁺))) → (𝐶 ∈ (𝐴𝐹𝐵) ↔ 𝐶 ∈ {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑟 < (1 / 𝐴) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀‘𝐴))))}))
32	1	sseli 3911	. . 3 ⊢ (𝐶 ∈ {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑟 < (1 / 𝐴) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀‘𝐴))))} → 𝐶 ∈ (𝑋 × ℝ⁺))
33		simp1 1133	. . 3 ⊢ ((𝐶 ∈ (𝑋 × ℝ⁺) ∧ (2^nd ‘𝐶) < (1 / 𝐴) ∧ ((cls‘𝐽)‘((ball‘𝐷)‘𝐶)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀‘𝐴))) → 𝐶 ∈ (𝑋 × ℝ⁺))
34		1st2nd2 7710	. . . . . 6 ⊢ (𝐶 ∈ (𝑋 × ℝ⁺) → 𝐶 = ⟨(1^st ‘𝐶), (2^nd ‘𝐶)⟩)
35	34	eleq1d 2874	. . . . 5 ⊢ (𝐶 ∈ (𝑋 × ℝ⁺) → (𝐶 ∈ {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑟 < (1 / 𝐴) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀‘𝐴))))} ↔ ⟨(1^st ‘𝐶), (2^nd ‘𝐶)⟩ ∈ {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑟 < (1 / 𝐴) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀‘𝐴))))}))
36		fvex 6658	. . . . . 6 ⊢ (1^st ‘𝐶) ∈ V
37		fvex 6658	. . . . . 6 ⊢ (2^nd ‘𝐶) ∈ V
38		eleq1 2877	. . . . . . . 8 ⊢ (𝑥 = (1^st ‘𝐶) → (𝑥 ∈ 𝑋 ↔ (1^st ‘𝐶) ∈ 𝑋))
39		eleq1 2877	. . . . . . . 8 ⊢ (𝑟 = (2^nd ‘𝐶) → (𝑟 ∈ ℝ⁺ ↔ (2^nd ‘𝐶) ∈ ℝ⁺))
40	38, 39	bi2anan9 638	. . . . . . 7 ⊢ ((𝑥 = (1^st ‘𝐶) ∧ 𝑟 = (2^nd ‘𝐶)) → ((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺) ↔ ((1^st ‘𝐶) ∈ 𝑋 ∧ (2^nd ‘𝐶) ∈ ℝ⁺)))
41		simpr 488	. . . . . . . . 9 ⊢ ((𝑥 = (1^st ‘𝐶) ∧ 𝑟 = (2^nd ‘𝐶)) → 𝑟 = (2^nd ‘𝐶))
42	41	breq1d 5040	. . . . . . . 8 ⊢ ((𝑥 = (1^st ‘𝐶) ∧ 𝑟 = (2^nd ‘𝐶)) → (𝑟 < (1 / 𝐴) ↔ (2^nd ‘𝐶) < (1 / 𝐴)))
43		oveq12 7144	. . . . . . . . . 10 ⊢ ((𝑥 = (1^st ‘𝐶) ∧ 𝑟 = (2^nd ‘𝐶)) → (𝑥(ball‘𝐷)𝑟) = ((1^st ‘𝐶)(ball‘𝐷)(2^nd ‘𝐶)))
44	43	fveq2d 6649	. . . . . . . . 9 ⊢ ((𝑥 = (1^st ‘𝐶) ∧ 𝑟 = (2^nd ‘𝐶)) → ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) = ((cls‘𝐽)‘((1^st ‘𝐶)(ball‘𝐷)(2^nd ‘𝐶))))
45	44	sseq1d 3946	. . . . . . . 8 ⊢ ((𝑥 = (1^st ‘𝐶) ∧ 𝑟 = (2^nd ‘𝐶)) → (((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀‘𝐴)) ↔ ((cls‘𝐽)‘((1^st ‘𝐶)(ball‘𝐷)(2^nd ‘𝐶))) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀‘𝐴))))
46	42, 45	anbi12d 633	. . . . . . 7 ⊢ ((𝑥 = (1^st ‘𝐶) ∧ 𝑟 = (2^nd ‘𝐶)) → ((𝑟 < (1 / 𝐴) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀‘𝐴))) ↔ ((2^nd ‘𝐶) < (1 / 𝐴) ∧ ((cls‘𝐽)‘((1^st ‘𝐶)(ball‘𝐷)(2^nd ‘𝐶))) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀‘𝐴)))))
47	40, 46	anbi12d 633	. . . . . 6 ⊢ ((𝑥 = (1^st ‘𝐶) ∧ 𝑟 = (2^nd ‘𝐶)) → (((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑟 < (1 / 𝐴) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀‘𝐴)))) ↔ (((1^st ‘𝐶) ∈ 𝑋 ∧ (2^nd ‘𝐶) ∈ ℝ⁺) ∧ ((2^nd ‘𝐶) < (1 / 𝐴) ∧ ((cls‘𝐽)‘((1^st ‘𝐶)(ball‘𝐷)(2^nd ‘𝐶))) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀‘𝐴))))))
48	36, 37, 47	opelopaba 5388	. . . . 5 ⊢ (⟨(1^st ‘𝐶), (2^nd ‘𝐶)⟩ ∈ {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑟 < (1 / 𝐴) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀‘𝐴))))} ↔ (((1^st ‘𝐶) ∈ 𝑋 ∧ (2^nd ‘𝐶) ∈ ℝ⁺) ∧ ((2^nd ‘𝐶) < (1 / 𝐴) ∧ ((cls‘𝐽)‘((1^st ‘𝐶)(ball‘𝐷)(2^nd ‘𝐶))) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀‘𝐴)))))
49	35, 48	syl6bb 290	. . . 4 ⊢ (𝐶 ∈ (𝑋 × ℝ⁺) → (𝐶 ∈ {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑟 < (1 / 𝐴) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀‘𝐴))))} ↔ (((1^st ‘𝐶) ∈ 𝑋 ∧ (2^nd ‘𝐶) ∈ ℝ⁺) ∧ ((2^nd ‘𝐶) < (1 / 𝐴) ∧ ((cls‘𝐽)‘((1^st ‘𝐶)(ball‘𝐷)(2^nd ‘𝐶))) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀‘𝐴))))))
50	34	eleq1d 2874	. . . . . . 7 ⊢ (𝐶 ∈ (𝑋 × ℝ⁺) → (𝐶 ∈ (𝑋 × ℝ⁺) ↔ ⟨(1^st ‘𝐶), (2^nd ‘𝐶)⟩ ∈ (𝑋 × ℝ⁺)))
51		opelxp 5555	. . . . . . 7 ⊢ (⟨(1^st ‘𝐶), (2^nd ‘𝐶)⟩ ∈ (𝑋 × ℝ⁺) ↔ ((1^st ‘𝐶) ∈ 𝑋 ∧ (2^nd ‘𝐶) ∈ ℝ⁺))
52	50, 51	syl6rbb 291	. . . . . 6 ⊢ (𝐶 ∈ (𝑋 × ℝ⁺) → (((1^st ‘𝐶) ∈ 𝑋 ∧ (2^nd ‘𝐶) ∈ ℝ⁺) ↔ 𝐶 ∈ (𝑋 × ℝ⁺)))
53		df-ov 7138	. . . . . . . . . 10 ⊢ ((1^st ‘𝐶)(ball‘𝐷)(2^nd ‘𝐶)) = ((ball‘𝐷)‘⟨(1^st ‘𝐶), (2^nd ‘𝐶)⟩)
54	34	fveq2d 6649	. . . . . . . . . 10 ⊢ (𝐶 ∈ (𝑋 × ℝ⁺) → ((ball‘𝐷)‘𝐶) = ((ball‘𝐷)‘⟨(1^st ‘𝐶), (2^nd ‘𝐶)⟩))
55	53, 54	eqtr4id 2852	. . . . . . . . 9 ⊢ (𝐶 ∈ (𝑋 × ℝ⁺) → ((1^st ‘𝐶)(ball‘𝐷)(2^nd ‘𝐶)) = ((ball‘𝐷)‘𝐶))
56	55	fveq2d 6649	. . . . . . . 8 ⊢ (𝐶 ∈ (𝑋 × ℝ⁺) → ((cls‘𝐽)‘((1^st ‘𝐶)(ball‘𝐷)(2^nd ‘𝐶))) = ((cls‘𝐽)‘((ball‘𝐷)‘𝐶)))
57	56	sseq1d 3946	. . . . . . 7 ⊢ (𝐶 ∈ (𝑋 × ℝ⁺) → (((cls‘𝐽)‘((1^st ‘𝐶)(ball‘𝐷)(2^nd ‘𝐶))) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀‘𝐴)) ↔ ((cls‘𝐽)‘((ball‘𝐷)‘𝐶)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀‘𝐴))))
58	57	anbi2d 631	. . . . . 6 ⊢ (𝐶 ∈ (𝑋 × ℝ⁺) → (((2^nd ‘𝐶) < (1 / 𝐴) ∧ ((cls‘𝐽)‘((1^st ‘𝐶)(ball‘𝐷)(2^nd ‘𝐶))) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀‘𝐴))) ↔ ((2^nd ‘𝐶) < (1 / 𝐴) ∧ ((cls‘𝐽)‘((ball‘𝐷)‘𝐶)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀‘𝐴)))))
59	52, 58	anbi12d 633	. . . . 5 ⊢ (𝐶 ∈ (𝑋 × ℝ⁺) → ((((1^st ‘𝐶) ∈ 𝑋 ∧ (2^nd ‘𝐶) ∈ ℝ⁺) ∧ ((2^nd ‘𝐶) < (1 / 𝐴) ∧ ((cls‘𝐽)‘((1^st ‘𝐶)(ball‘𝐷)(2^nd ‘𝐶))) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀‘𝐴)))) ↔ (𝐶 ∈ (𝑋 × ℝ⁺) ∧ ((2^nd ‘𝐶) < (1 / 𝐴) ∧ ((cls‘𝐽)‘((ball‘𝐷)‘𝐶)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀‘𝐴))))))
60		3anass 1092	. . . . 5 ⊢ ((𝐶 ∈ (𝑋 × ℝ⁺) ∧ (2^nd ‘𝐶) < (1 / 𝐴) ∧ ((cls‘𝐽)‘((ball‘𝐷)‘𝐶)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀‘𝐴))) ↔ (𝐶 ∈ (𝑋 × ℝ⁺) ∧ ((2^nd ‘𝐶) < (1 / 𝐴) ∧ ((cls‘𝐽)‘((ball‘𝐷)‘𝐶)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀‘𝐴)))))
61	59, 60	syl6bbr 292	. . . 4 ⊢ (𝐶 ∈ (𝑋 × ℝ⁺) → ((((1^st ‘𝐶) ∈ 𝑋 ∧ (2^nd ‘𝐶) ∈ ℝ⁺) ∧ ((2^nd ‘𝐶) < (1 / 𝐴) ∧ ((cls‘𝐽)‘((1^st ‘𝐶)(ball‘𝐷)(2^nd ‘𝐶))) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀‘𝐴)))) ↔ (𝐶 ∈ (𝑋 × ℝ⁺) ∧ (2^nd ‘𝐶) < (1 / 𝐴) ∧ ((cls‘𝐽)‘((ball‘𝐷)‘𝐶)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀‘𝐴)))))
62	49, 61	bitrd 282	. . 3 ⊢ (𝐶 ∈ (𝑋 × ℝ⁺) → (𝐶 ∈ {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑟 < (1 / 𝐴) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀‘𝐴))))} ↔ (𝐶 ∈ (𝑋 × ℝ⁺) ∧ (2^nd ‘𝐶) < (1 / 𝐴) ∧ ((cls‘𝐽)‘((ball‘𝐷)‘𝐶)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀‘𝐴)))))
63	32, 33, 62	pm5.21nii 383	. 2 ⊢ (𝐶 ∈ {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑟 < (1 / 𝐴) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀‘𝐴))))} ↔ (𝐶 ∈ (𝑋 × ℝ⁺) ∧ (2^nd ‘𝐶) < (1 / 𝐴) ∧ ((cls‘𝐽)‘((ball‘𝐷)‘𝐶)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀‘𝐴))))
64	31, 63	syl6bb 290	1 ⊢ ((𝜑 ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ (𝑋 × ℝ⁺))) → (𝐶 ∈ (𝐴𝐹𝐵) ↔ (𝐶 ∈ (𝑋 × ℝ⁺) ∧ (2^nd ‘𝐶) < (1 / 𝐴) ∧ ((cls‘𝐽)‘((ball‘𝐷)‘𝐶)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀‘𝐴)))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ∖ cdif 3878 ⊆ wss 3881 ⟨cop 4531 class class class wbr 5030 {copab 5092 × cxp 5517 dom cdm 5519 ‘cfv 6324 (class class class)co 7135 ∈ cmpo 7137 1^st c1st 7669 2^nd c2nd 7670 ℝcr 10525 1c1 10527 < clt 10664 / cdiv 11286 ℕcn 11625 ℝ⁺crp 12377 ballcbl 20078 MetOpencmopn 20081 clsccl 21623 CMetccmet 23858

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-iota 6283 df-fun 6326 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-rp 12378

This theorem is referenced by: bcthlem2 23929 bcthlem3 23930 bcthlem4 23931

W3C validator