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Theorem bcthlem1 23530

Description: Lemma for bcth 23535. 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 5441	. . . . . . 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 6478	. . . . . . . . 9 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝑋 ∈ dom CMet)
4	2, 3	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ dom CMet)
5		reex 10363	. . . . . . . . 9 ⊢ ℝ ∈ V
6		rpssre 12144	. . . . . . . . 9 ⊢ ℝ⁺ ⊆ ℝ
7	5, 6	ssexi 5040	. . . . . . . 8 ⊢ ℝ⁺ ∈ V
8		xpexg 7237	. . . . . . . 8 ⊢ ((𝑋 ∈ dom CMet ∧ ℝ⁺ ∈ V) → (𝑋 × ℝ⁺) ∈ V)
9	4, 7, 8	sylancl 580	. . . . . . 7 ⊢ (𝜑 → (𝑋 × ℝ⁺) ∈ V)
10		ssexg 5041	. . . . . . 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 581	. . . . . 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 6930	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐴 → (1 / 𝑘) = (1 / 𝐴))
13	12	breq2d 4898	. . . . . . . . . 10 ⊢ (𝑘 = 𝐴 → (𝑟 < (1 / 𝑘) ↔ 𝑟 < (1 / 𝐴)))
14		fveq2 6446	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐴 → (𝑀‘𝑘) = (𝑀‘𝐴))
15	14	difeq2d 3951	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐴 → (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘)) = (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝐴)))
16	15	sseq2d 3852	. . . . . . . . . 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 624	. . . . . . . . 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 622	. . . . . . . 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 4952	. . . . . . 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 6446	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝐵 → ((ball‘𝐷)‘𝑧) = ((ball‘𝐷)‘𝐵))
21	20	difeq1d 3950	. . . . . . . . . . 11 ⊢ (𝑧 = 𝐵 → (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝐴)) = (((ball‘𝐷)‘𝐵) ∖ (𝑀‘𝐴)))
22	21	sseq2d 3852	. . . . . . . . . 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 622	. . . . . . . . 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 622	. . . . . . . 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 4952	. . . . . . 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	ovmpt2g 7072	. . . . . 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 1166	. . . . 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 1108	. . . 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 452	. . 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 2845	. 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 3817	. . 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 1127	. . 3 ⊢ ((𝐶 ∈ (𝑋 × ℝ⁺) ∧ (2^nd ‘𝐶) < (1 / 𝐴) ∧ ((cls‘𝐽)‘((ball‘𝐷)‘𝐶)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀‘𝐴))) → 𝐶 ∈ (𝑋 × ℝ⁺))
34		1st2nd2 7484	. . . . . 6 ⊢ (𝐶 ∈ (𝑋 × ℝ⁺) → 𝐶 = ⟨(1^st ‘𝐶), (2^nd ‘𝐶)⟩)
35	34	eleq1d 2844	. . . . 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 6459	. . . . . 6 ⊢ (1^st ‘𝐶) ∈ V
37		fvex 6459	. . . . . 6 ⊢ (2^nd ‘𝐶) ∈ V
38		eleq1 2847	. . . . . . . 8 ⊢ (𝑥 = (1^st ‘𝐶) → (𝑥 ∈ 𝑋 ↔ (1^st ‘𝐶) ∈ 𝑋))
39		eleq1 2847	. . . . . . . 8 ⊢ (𝑟 = (2^nd ‘𝐶) → (𝑟 ∈ ℝ⁺ ↔ (2^nd ‘𝐶) ∈ ℝ⁺))
40	38, 39	bi2anan9 629	. . . . . . 7 ⊢ ((𝑥 = (1^st ‘𝐶) ∧ 𝑟 = (2^nd ‘𝐶)) → ((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺) ↔ ((1^st ‘𝐶) ∈ 𝑋 ∧ (2^nd ‘𝐶) ∈ ℝ⁺)))
41		simpr 479	. . . . . . . . 9 ⊢ ((𝑥 = (1^st ‘𝐶) ∧ 𝑟 = (2^nd ‘𝐶)) → 𝑟 = (2^nd ‘𝐶))
42	41	breq1d 4896	. . . . . . . 8 ⊢ ((𝑥 = (1^st ‘𝐶) ∧ 𝑟 = (2^nd ‘𝐶)) → (𝑟 < (1 / 𝐴) ↔ (2^nd ‘𝐶) < (1 / 𝐴)))
43		oveq12 6931	. . . . . . . . . 10 ⊢ ((𝑥 = (1^st ‘𝐶) ∧ 𝑟 = (2^nd ‘𝐶)) → (𝑥(ball‘𝐷)𝑟) = ((1^st ‘𝐶)(ball‘𝐷)(2^nd ‘𝐶)))
44	43	fveq2d 6450	. . . . . . . . 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 3851	. . . . . . . 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 624	. . . . . . 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 624	. . . . . 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 5228	. . . . 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 279	. . . 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 2844	. . . . . . 7 ⊢ (𝐶 ∈ (𝑋 × ℝ⁺) → (𝐶 ∈ (𝑋 × ℝ⁺) ↔ ⟨(1^st ‘𝐶), (2^nd ‘𝐶)⟩ ∈ (𝑋 × ℝ⁺)))
51		opelxp 5391	. . . . . . 7 ⊢ (⟨(1^st ‘𝐶), (2^nd ‘𝐶)⟩ ∈ (𝑋 × ℝ⁺) ↔ ((1^st ‘𝐶) ∈ 𝑋 ∧ (2^nd ‘𝐶) ∈ ℝ⁺))
52	50, 51	syl6rbb 280	. . . . . 6 ⊢ (𝐶 ∈ (𝑋 × ℝ⁺) → (((1^st ‘𝐶) ∈ 𝑋 ∧ (2^nd ‘𝐶) ∈ ℝ⁺) ↔ 𝐶 ∈ (𝑋 × ℝ⁺)))
53	34	fveq2d 6450	. . . . . . . . . 10 ⊢ (𝐶 ∈ (𝑋 × ℝ⁺) → ((ball‘𝐷)‘𝐶) = ((ball‘𝐷)‘⟨(1^st ‘𝐶), (2^nd ‘𝐶)⟩))
54		df-ov 6925	. . . . . . . . . 10 ⊢ ((1^st ‘𝐶)(ball‘𝐷)(2^nd ‘𝐶)) = ((ball‘𝐷)‘⟨(1^st ‘𝐶), (2^nd ‘𝐶)⟩)
55	53, 54	syl6reqr 2833	. . . . . . . . 9 ⊢ (𝐶 ∈ (𝑋 × ℝ⁺) → ((1^st ‘𝐶)(ball‘𝐷)(2^nd ‘𝐶)) = ((ball‘𝐷)‘𝐶))
56	55	fveq2d 6450	. . . . . . . 8 ⊢ (𝐶 ∈ (𝑋 × ℝ⁺) → ((cls‘𝐽)‘((1^st ‘𝐶)(ball‘𝐷)(2^nd ‘𝐶))) = ((cls‘𝐽)‘((ball‘𝐷)‘𝐶)))
57	56	sseq1d 3851	. . . . . . 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 622	. . . . . 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 624	. . . . 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 1079	. . . . 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 281	. . . 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 271	. . 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 370	. 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 279	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 198 ∧ wa 386 ∧ w3a 1071 = wceq 1601 ∈ wcel 2107 Vcvv 3398 ∖ cdif 3789 ⊆ wss 3792 ⟨cop 4404 class class class wbr 4886 {copab 4948 × cxp 5353 dom cdm 5355 ‘cfv 6135 (class class class)co 6922 ↦ cmpt2 6924 1^st c1st 7443 2^nd c2nd 7444 ℝcr 10271 1c1 10273 < clt 10411 / cdiv 11032 ℕcn 11374 ℝ⁺crp 12137 ballcbl 20129 MetOpencmopn 20132 clsccl 21230 CMetccmet 23460

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329

This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-iota 6099 df-fun 6137 df-fv 6143 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-1st 7445 df-2nd 7446 df-rp 12138

This theorem is referenced by: bcthlem2 23531 bcthlem3 23532 bcthlem4 23533

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