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Theorem bcthlem1 24393

Description: Lemma for bcth 24398. 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 5669	. . . . . . 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 6788	. . . . . . . . 9 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝑋 ∈ dom CMet)
4	2, 3	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ dom CMet)
5		reex 10893	. . . . . . . . 9 ⊢ ℝ ∈ V
6		rpssre 12666	. . . . . . . . 9 ⊢ ℝ⁺ ⊆ ℝ
7	5, 6	ssexi 5241	. . . . . . . 8 ⊢ ℝ⁺ ∈ V
8		xpexg 7578	. . . . . . . 8 ⊢ ((𝑋 ∈ dom CMet ∧ ℝ⁺ ∈ V) → (𝑋 × ℝ⁺) ∈ V)
9	4, 7, 8	sylancl 585	. . . . . . 7 ⊢ (𝜑 → (𝑋 × ℝ⁺) ∈ V)
10		ssexg 5242	. . . . . . 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 586	. . . . . 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 7263	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐴 → (1 / 𝑘) = (1 / 𝐴))
13	12	breq2d 5082	. . . . . . . . . 10 ⊢ (𝑘 = 𝐴 → (𝑟 < (1 / 𝑘) ↔ 𝑟 < (1 / 𝐴)))
14		fveq2 6756	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐴 → (𝑀‘𝑘) = (𝑀‘𝐴))
15	14	difeq2d 4053	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐴 → (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘)) = (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝐴)))
16	15	sseq2d 3949	. . . . . . . . . 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 630	. . . . . . . . 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 628	. . . . . . . 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 5136	. . . . . . 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 6756	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝐵 → ((ball‘𝐷)‘𝑧) = ((ball‘𝐷)‘𝐵))
21	20	difeq1d 4052	. . . . . . . . . . 11 ⊢ (𝑧 = 𝐵 → (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝐴)) = (((ball‘𝐷)‘𝐵) ∖ (𝑀‘𝐴)))
22	21	sseq2d 3949	. . . . . . . . . 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 628	. . . . . . . . 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 628	. . . . . . . 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 5136	. . . . . . 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 7410	. . . . . 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 1163	. . . . 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 1116	. . . 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 458	. . 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 2824	. 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 3913	. . 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 1134	. . 3 ⊢ ((𝐶 ∈ (𝑋 × ℝ⁺) ∧ (2^nd ‘𝐶) < (1 / 𝐴) ∧ ((cls‘𝐽)‘((ball‘𝐷)‘𝐶)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀‘𝐴))) → 𝐶 ∈ (𝑋 × ℝ⁺))
34		1st2nd2 7843	. . . . . 6 ⊢ (𝐶 ∈ (𝑋 × ℝ⁺) → 𝐶 = ⟨(1^st ‘𝐶), (2^nd ‘𝐶)⟩)
35	34	eleq1d 2823	. . . . 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 6769	. . . . . 6 ⊢ (1^st ‘𝐶) ∈ V
37		fvex 6769	. . . . . 6 ⊢ (2^nd ‘𝐶) ∈ V
38		eleq1 2826	. . . . . . . 8 ⊢ (𝑥 = (1^st ‘𝐶) → (𝑥 ∈ 𝑋 ↔ (1^st ‘𝐶) ∈ 𝑋))
39		eleq1 2826	. . . . . . . 8 ⊢ (𝑟 = (2^nd ‘𝐶) → (𝑟 ∈ ℝ⁺ ↔ (2^nd ‘𝐶) ∈ ℝ⁺))
40	38, 39	bi2anan9 635	. . . . . . 7 ⊢ ((𝑥 = (1^st ‘𝐶) ∧ 𝑟 = (2^nd ‘𝐶)) → ((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺) ↔ ((1^st ‘𝐶) ∈ 𝑋 ∧ (2^nd ‘𝐶) ∈ ℝ⁺)))
41		simpr 484	. . . . . . . . 9 ⊢ ((𝑥 = (1^st ‘𝐶) ∧ 𝑟 = (2^nd ‘𝐶)) → 𝑟 = (2^nd ‘𝐶))
42	41	breq1d 5080	. . . . . . . 8 ⊢ ((𝑥 = (1^st ‘𝐶) ∧ 𝑟 = (2^nd ‘𝐶)) → (𝑟 < (1 / 𝐴) ↔ (2^nd ‘𝐶) < (1 / 𝐴)))
43		oveq12 7264	. . . . . . . . . 10 ⊢ ((𝑥 = (1^st ‘𝐶) ∧ 𝑟 = (2^nd ‘𝐶)) → (𝑥(ball‘𝐷)𝑟) = ((1^st ‘𝐶)(ball‘𝐷)(2^nd ‘𝐶)))
44	43	fveq2d 6760	. . . . . . . . 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 3948	. . . . . . . 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 630	. . . . . . 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 630	. . . . . 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 5442	. . . . 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	bitrdi 286	. . . 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 2823	. . . . . . 7 ⊢ (𝐶 ∈ (𝑋 × ℝ⁺) → (𝐶 ∈ (𝑋 × ℝ⁺) ↔ ⟨(1^st ‘𝐶), (2^nd ‘𝐶)⟩ ∈ (𝑋 × ℝ⁺)))
51		opelxp 5616	. . . . . . 7 ⊢ (⟨(1^st ‘𝐶), (2^nd ‘𝐶)⟩ ∈ (𝑋 × ℝ⁺) ↔ ((1^st ‘𝐶) ∈ 𝑋 ∧ (2^nd ‘𝐶) ∈ ℝ⁺))
52	50, 51	bitr2di 287	. . . . . 6 ⊢ (𝐶 ∈ (𝑋 × ℝ⁺) → (((1^st ‘𝐶) ∈ 𝑋 ∧ (2^nd ‘𝐶) ∈ ℝ⁺) ↔ 𝐶 ∈ (𝑋 × ℝ⁺)))
53		df-ov 7258	. . . . . . . . . 10 ⊢ ((1^st ‘𝐶)(ball‘𝐷)(2^nd ‘𝐶)) = ((ball‘𝐷)‘⟨(1^st ‘𝐶), (2^nd ‘𝐶)⟩)
54	34	fveq2d 6760	. . . . . . . . . 10 ⊢ (𝐶 ∈ (𝑋 × ℝ⁺) → ((ball‘𝐷)‘𝐶) = ((ball‘𝐷)‘⟨(1^st ‘𝐶), (2^nd ‘𝐶)⟩))
55	53, 54	eqtr4id 2798	. . . . . . . . 9 ⊢ (𝐶 ∈ (𝑋 × ℝ⁺) → ((1^st ‘𝐶)(ball‘𝐷)(2^nd ‘𝐶)) = ((ball‘𝐷)‘𝐶))
56	55	fveq2d 6760	. . . . . . . 8 ⊢ (𝐶 ∈ (𝑋 × ℝ⁺) → ((cls‘𝐽)‘((1^st ‘𝐶)(ball‘𝐷)(2^nd ‘𝐶))) = ((cls‘𝐽)‘((ball‘𝐷)‘𝐶)))
57	56	sseq1d 3948	. . . . . . 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 628	. . . . . 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 630	. . . . 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 1093	. . . . 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	bitr4di 288	. . . 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 278	. . 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 379	. 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	bitrdi 286	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 205 ∧ wa 395 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 Vcvv 3422 ∖ cdif 3880 ⊆ wss 3883 ⟨cop 4564 class class class wbr 5070 {copab 5132 × cxp 5578 dom cdm 5580 ‘cfv 6418 (class class class)co 7255 ∈ cmpo 7257 1^st c1st 7802 2^nd c2nd 7803 ℝcr 10801 1c1 10803 < clt 10940 / cdiv 11562 ℕcn 11903 ℝ⁺crp 12659 ballcbl 20497 MetOpencmopn 20500 clsccl 22077 CMetccmet 24323

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-ral 3068 df-rex 3069 df-rab 3072 df-v 3424 df-sbc 3712 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4837 df-br 5071 df-opab 5133 df-mpt 5154 df-id 5480 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-iota 6376 df-fun 6420 df-fv 6426 df-ov 7258 df-oprab 7259 df-mpo 7260 df-1st 7804 df-2nd 7805 df-rp 12660

This theorem is referenced by: bcthlem2 24394 bcthlem3 24395 bcthlem4 24396

W3C validator