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Theorem bcthlem1 24498
Description: Lemma for bcth 24503. Substitutions for the function 𝐹. (Contributed by Mario Carneiro, 9-Jan-2014.)
Hypotheses
Ref Expression
bcth.2 𝐽 = (MetOpen‘𝐷)
bcthlem.4 (𝜑𝐷 ∈ (CMet‘𝑋))
bcthlem.5 𝐹 = (𝑘 ∈ ℕ, 𝑧 ∈ (𝑋 × ℝ+) ↦ {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝑋𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘))))})
Assertion
Ref Expression
bcthlem1 ((𝜑 ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ (𝑋 × ℝ+))) → (𝐶 ∈ (𝐴𝐹𝐵) ↔ (𝐶 ∈ (𝑋 × ℝ+) ∧ (2nd𝐶) < (1 / 𝐴) ∧ ((cls‘𝐽)‘((ball‘𝐷)‘𝐶)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀𝐴)))))
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
StepHypRef Expression
1 opabssxp 5673 . . . . . . 7 {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝑋𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝐴) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀𝐴))))} ⊆ (𝑋 × ℝ+)
2 bcthlem.4 . . . . . . . . 9 (𝜑𝐷 ∈ (CMet‘𝑋))
3 elfvdm 6798 . . . . . . . . 9 (𝐷 ∈ (CMet‘𝑋) → 𝑋 ∈ dom CMet)
42, 3syl 17 . . . . . . . 8 (𝜑𝑋 ∈ dom CMet)
5 reex 10972 . . . . . . . . 9 ℝ ∈ V
6 rpssre 12747 . . . . . . . . 9 + ⊆ ℝ
75, 6ssexi 5244 . . . . . . . 8 + ∈ V
8 xpexg 7590 . . . . . . . 8 ((𝑋 ∈ dom CMet ∧ ℝ+ ∈ V) → (𝑋 × ℝ+) ∈ V)
94, 7, 8sylancl 586 . . . . . . 7 (𝜑 → (𝑋 × ℝ+) ∈ V)
10 ssexg 5245 . . . . . . 7 (({⟨𝑥, 𝑟⟩ ∣ ((𝑥𝑋𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝐴) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀𝐴))))} ⊆ (𝑋 × ℝ+) ∧ (𝑋 × ℝ+) ∈ V) → {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝑋𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝐴) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀𝐴))))} ∈ V)
111, 9, 10sylancr 587 . . . . . 6 (𝜑 → {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝑋𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝐴) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀𝐴))))} ∈ V)
12 oveq2 7275 . . . . . . . . . . 11 (𝑘 = 𝐴 → (1 / 𝑘) = (1 / 𝐴))
1312breq2d 5085 . . . . . . . . . 10 (𝑘 = 𝐴 → (𝑟 < (1 / 𝑘) ↔ 𝑟 < (1 / 𝐴)))
14 fveq2 6766 . . . . . . . . . . . 12 (𝑘 = 𝐴 → (𝑀𝑘) = (𝑀𝐴))
1514difeq2d 4056 . . . . . . . . . . 11 (𝑘 = 𝐴 → (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘)) = (((ball‘𝐷)‘𝑧) ∖ (𝑀𝐴)))
1615sseq2d 3952 . . . . . . . . . 10 (𝑘 = 𝐴 → (((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘)) ↔ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝐴))))
1713, 16anbi12d 631 . . . . . . . . 9 (𝑘 = 𝐴 → ((𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘))) ↔ (𝑟 < (1 / 𝐴) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝐴)))))
1817anbi2d 629 . . . . . . . 8 (𝑘 = 𝐴 → (((𝑥𝑋𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘)))) ↔ ((𝑥𝑋𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝐴) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝐴))))))
1918opabbidv 5139 . . . . . . 7 (𝑘 = 𝐴 → {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝑋𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘))))} = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝑋𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝐴) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝐴))))})
20 fveq2 6766 . . . . . . . . . . . 12 (𝑧 = 𝐵 → ((ball‘𝐷)‘𝑧) = ((ball‘𝐷)‘𝐵))
2120difeq1d 4055 . . . . . . . . . . 11 (𝑧 = 𝐵 → (((ball‘𝐷)‘𝑧) ∖ (𝑀𝐴)) = (((ball‘𝐷)‘𝐵) ∖ (𝑀𝐴)))
2221sseq2d 3952 . . . . . . . . . 10 (𝑧 = 𝐵 → (((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝐴)) ↔ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀𝐴))))
2322anbi2d 629 . . . . . . . . 9 (𝑧 = 𝐵 → ((𝑟 < (1 / 𝐴) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝐴))) ↔ (𝑟 < (1 / 𝐴) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀𝐴)))))
2423anbi2d 629 . . . . . . . 8 (𝑧 = 𝐵 → (((𝑥𝑋𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝐴) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝐴)))) ↔ ((𝑥𝑋𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝐴) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀𝐴))))))
2524opabbidv 5139 . . . . . . 7 (𝑧 = 𝐵 → {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝑋𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝐴) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝐴))))} = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝑋𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝐴) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀𝐴))))})
26 bcthlem.5 . . . . . . 7 𝐹 = (𝑘 ∈ ℕ, 𝑧 ∈ (𝑋 × ℝ+) ↦ {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝑋𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘))))})
2719, 25, 26ovmpog 7422 . . . . . 6 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (𝑋 × ℝ+) ∧ {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝑋𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝐴) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀𝐴))))} ∈ V) → (𝐴𝐹𝐵) = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝑋𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝐴) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀𝐴))))})
2811, 27syl3an3 1164 . . . . 5 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (𝑋 × ℝ+) ∧ 𝜑) → (𝐴𝐹𝐵) = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝑋𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝐴) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀𝐴))))})
29283expa 1117 . . . 4 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ (𝑋 × ℝ+)) ∧ 𝜑) → (𝐴𝐹𝐵) = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝑋𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝐴) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀𝐴))))})
3029ancoms 459 . . 3 ((𝜑 ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ (𝑋 × ℝ+))) → (𝐴𝐹𝐵) = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝑋𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝐴) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀𝐴))))})
3130eleq2d 2824 . 2 ((𝜑 ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ (𝑋 × ℝ+))) → (𝐶 ∈ (𝐴𝐹𝐵) ↔ 𝐶 ∈ {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝑋𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝐴) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀𝐴))))}))
321sseli 3916 . . 3 (𝐶 ∈ {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝑋𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝐴) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀𝐴))))} → 𝐶 ∈ (𝑋 × ℝ+))
33 simp1 1135 . . 3 ((𝐶 ∈ (𝑋 × ℝ+) ∧ (2nd𝐶) < (1 / 𝐴) ∧ ((cls‘𝐽)‘((ball‘𝐷)‘𝐶)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀𝐴))) → 𝐶 ∈ (𝑋 × ℝ+))
34 1st2nd2 7859 . . . . . 6 (𝐶 ∈ (𝑋 × ℝ+) → 𝐶 = ⟨(1st𝐶), (2nd𝐶)⟩)
3534eleq1d 2823 . . . . 5 (𝐶 ∈ (𝑋 × ℝ+) → (𝐶 ∈ {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝑋𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝐴) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀𝐴))))} ↔ ⟨(1st𝐶), (2nd𝐶)⟩ ∈ {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝑋𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝐴) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀𝐴))))}))
36 fvex 6779 . . . . . 6 (1st𝐶) ∈ V
37 fvex 6779 . . . . . 6 (2nd𝐶) ∈ V
38 eleq1 2826 . . . . . . . 8 (𝑥 = (1st𝐶) → (𝑥𝑋 ↔ (1st𝐶) ∈ 𝑋))
39 eleq1 2826 . . . . . . . 8 (𝑟 = (2nd𝐶) → (𝑟 ∈ ℝ+ ↔ (2nd𝐶) ∈ ℝ+))
4038, 39bi2anan9 636 . . . . . . 7 ((𝑥 = (1st𝐶) ∧ 𝑟 = (2nd𝐶)) → ((𝑥𝑋𝑟 ∈ ℝ+) ↔ ((1st𝐶) ∈ 𝑋 ∧ (2nd𝐶) ∈ ℝ+)))
41 simpr 485 . . . . . . . . 9 ((𝑥 = (1st𝐶) ∧ 𝑟 = (2nd𝐶)) → 𝑟 = (2nd𝐶))
4241breq1d 5083 . . . . . . . 8 ((𝑥 = (1st𝐶) ∧ 𝑟 = (2nd𝐶)) → (𝑟 < (1 / 𝐴) ↔ (2nd𝐶) < (1 / 𝐴)))
43 oveq12 7276 . . . . . . . . . 10 ((𝑥 = (1st𝐶) ∧ 𝑟 = (2nd𝐶)) → (𝑥(ball‘𝐷)𝑟) = ((1st𝐶)(ball‘𝐷)(2nd𝐶)))
4443fveq2d 6770 . . . . . . . . 9 ((𝑥 = (1st𝐶) ∧ 𝑟 = (2nd𝐶)) → ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) = ((cls‘𝐽)‘((1st𝐶)(ball‘𝐷)(2nd𝐶))))
4544sseq1d 3951 . . . . . . . 8 ((𝑥 = (1st𝐶) ∧ 𝑟 = (2nd𝐶)) → (((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀𝐴)) ↔ ((cls‘𝐽)‘((1st𝐶)(ball‘𝐷)(2nd𝐶))) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀𝐴))))
4642, 45anbi12d 631 . . . . . . 7 ((𝑥 = (1st𝐶) ∧ 𝑟 = (2nd𝐶)) → ((𝑟 < (1 / 𝐴) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀𝐴))) ↔ ((2nd𝐶) < (1 / 𝐴) ∧ ((cls‘𝐽)‘((1st𝐶)(ball‘𝐷)(2nd𝐶))) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀𝐴)))))
4740, 46anbi12d 631 . . . . . 6 ((𝑥 = (1st𝐶) ∧ 𝑟 = (2nd𝐶)) → (((𝑥𝑋𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝐴) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀𝐴)))) ↔ (((1st𝐶) ∈ 𝑋 ∧ (2nd𝐶) ∈ ℝ+) ∧ ((2nd𝐶) < (1 / 𝐴) ∧ ((cls‘𝐽)‘((1st𝐶)(ball‘𝐷)(2nd𝐶))) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀𝐴))))))
4836, 37, 47opelopaba 5446 . . . . 5 (⟨(1st𝐶), (2nd𝐶)⟩ ∈ {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝑋𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝐴) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀𝐴))))} ↔ (((1st𝐶) ∈ 𝑋 ∧ (2nd𝐶) ∈ ℝ+) ∧ ((2nd𝐶) < (1 / 𝐴) ∧ ((cls‘𝐽)‘((1st𝐶)(ball‘𝐷)(2nd𝐶))) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀𝐴)))))
4935, 48bitrdi 287 . . . 4 (𝐶 ∈ (𝑋 × ℝ+) → (𝐶 ∈ {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝑋𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝐴) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀𝐴))))} ↔ (((1st𝐶) ∈ 𝑋 ∧ (2nd𝐶) ∈ ℝ+) ∧ ((2nd𝐶) < (1 / 𝐴) ∧ ((cls‘𝐽)‘((1st𝐶)(ball‘𝐷)(2nd𝐶))) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀𝐴))))))
5034eleq1d 2823 . . . . . . 7 (𝐶 ∈ (𝑋 × ℝ+) → (𝐶 ∈ (𝑋 × ℝ+) ↔ ⟨(1st𝐶), (2nd𝐶)⟩ ∈ (𝑋 × ℝ+)))
51 opelxp 5620 . . . . . . 7 (⟨(1st𝐶), (2nd𝐶)⟩ ∈ (𝑋 × ℝ+) ↔ ((1st𝐶) ∈ 𝑋 ∧ (2nd𝐶) ∈ ℝ+))
5250, 51bitr2di 288 . . . . . 6 (𝐶 ∈ (𝑋 × ℝ+) → (((1st𝐶) ∈ 𝑋 ∧ (2nd𝐶) ∈ ℝ+) ↔ 𝐶 ∈ (𝑋 × ℝ+)))
53 df-ov 7270 . . . . . . . . . 10 ((1st𝐶)(ball‘𝐷)(2nd𝐶)) = ((ball‘𝐷)‘⟨(1st𝐶), (2nd𝐶)⟩)
5434fveq2d 6770 . . . . . . . . . 10 (𝐶 ∈ (𝑋 × ℝ+) → ((ball‘𝐷)‘𝐶) = ((ball‘𝐷)‘⟨(1st𝐶), (2nd𝐶)⟩))
5553, 54eqtr4id 2797 . . . . . . . . 9 (𝐶 ∈ (𝑋 × ℝ+) → ((1st𝐶)(ball‘𝐷)(2nd𝐶)) = ((ball‘𝐷)‘𝐶))
5655fveq2d 6770 . . . . . . . 8 (𝐶 ∈ (𝑋 × ℝ+) → ((cls‘𝐽)‘((1st𝐶)(ball‘𝐷)(2nd𝐶))) = ((cls‘𝐽)‘((ball‘𝐷)‘𝐶)))
5756sseq1d 3951 . . . . . . 7 (𝐶 ∈ (𝑋 × ℝ+) → (((cls‘𝐽)‘((1st𝐶)(ball‘𝐷)(2nd𝐶))) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀𝐴)) ↔ ((cls‘𝐽)‘((ball‘𝐷)‘𝐶)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀𝐴))))
5857anbi2d 629 . . . . . 6 (𝐶 ∈ (𝑋 × ℝ+) → (((2nd𝐶) < (1 / 𝐴) ∧ ((cls‘𝐽)‘((1st𝐶)(ball‘𝐷)(2nd𝐶))) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀𝐴))) ↔ ((2nd𝐶) < (1 / 𝐴) ∧ ((cls‘𝐽)‘((ball‘𝐷)‘𝐶)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀𝐴)))))
5952, 58anbi12d 631 . . . . 5 (𝐶 ∈ (𝑋 × ℝ+) → ((((1st𝐶) ∈ 𝑋 ∧ (2nd𝐶) ∈ ℝ+) ∧ ((2nd𝐶) < (1 / 𝐴) ∧ ((cls‘𝐽)‘((1st𝐶)(ball‘𝐷)(2nd𝐶))) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀𝐴)))) ↔ (𝐶 ∈ (𝑋 × ℝ+) ∧ ((2nd𝐶) < (1 / 𝐴) ∧ ((cls‘𝐽)‘((ball‘𝐷)‘𝐶)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀𝐴))))))
60 3anass 1094 . . . . 5 ((𝐶 ∈ (𝑋 × ℝ+) ∧ (2nd𝐶) < (1 / 𝐴) ∧ ((cls‘𝐽)‘((ball‘𝐷)‘𝐶)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀𝐴))) ↔ (𝐶 ∈ (𝑋 × ℝ+) ∧ ((2nd𝐶) < (1 / 𝐴) ∧ ((cls‘𝐽)‘((ball‘𝐷)‘𝐶)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀𝐴)))))
6159, 60bitr4di 289 . . . 4 (𝐶 ∈ (𝑋 × ℝ+) → ((((1st𝐶) ∈ 𝑋 ∧ (2nd𝐶) ∈ ℝ+) ∧ ((2nd𝐶) < (1 / 𝐴) ∧ ((cls‘𝐽)‘((1st𝐶)(ball‘𝐷)(2nd𝐶))) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀𝐴)))) ↔ (𝐶 ∈ (𝑋 × ℝ+) ∧ (2nd𝐶) < (1 / 𝐴) ∧ ((cls‘𝐽)‘((ball‘𝐷)‘𝐶)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀𝐴)))))
6249, 61bitrd 278 . . 3 (𝐶 ∈ (𝑋 × ℝ+) → (𝐶 ∈ {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝑋𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝐴) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀𝐴))))} ↔ (𝐶 ∈ (𝑋 × ℝ+) ∧ (2nd𝐶) < (1 / 𝐴) ∧ ((cls‘𝐽)‘((ball‘𝐷)‘𝐶)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀𝐴)))))
6332, 33, 62pm5.21nii 380 . 2 (𝐶 ∈ {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝑋𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝐴) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀𝐴))))} ↔ (𝐶 ∈ (𝑋 × ℝ+) ∧ (2nd𝐶) < (1 / 𝐴) ∧ ((cls‘𝐽)‘((ball‘𝐷)‘𝐶)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀𝐴))))
6431, 63bitrdi 287 1 ((𝜑 ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ (𝑋 × ℝ+))) → (𝐶 ∈ (𝐴𝐹𝐵) ↔ (𝐶 ∈ (𝑋 × ℝ+) ∧ (2nd𝐶) < (1 / 𝐴) ∧ ((cls‘𝐽)‘((ball‘𝐷)‘𝐶)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀𝐴)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1086   = wceq 1539  wcel 2106  Vcvv 3429  cdif 3883  wss 3886  cop 4567   class class class wbr 5073  {copab 5135   × cxp 5582  dom cdm 5584  cfv 6426  (class class class)co 7267  cmpo 7269  1st c1st 7818  2nd c2nd 7819  cr 10880  1c1 10882   < clt 11019   / cdiv 11642  cn 11983  +crp 12740  ballcbl 20594  MetOpencmopn 20597  clsccl 22179  CMetccmet 24428
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5221  ax-nul 5228  ax-pow 5286  ax-pr 5350  ax-un 7578  ax-cnex 10937  ax-resscn 10938
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3431  df-sbc 3716  df-dif 3889  df-un 3891  df-in 3893  df-ss 3903  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5074  df-opab 5136  df-mpt 5157  df-id 5484  df-xp 5590  df-rel 5591  df-cnv 5592  df-co 5593  df-dm 5594  df-rn 5595  df-iota 6384  df-fun 6428  df-fv 6434  df-ov 7270  df-oprab 7271  df-mpo 7272  df-1st 7820  df-2nd 7821  df-rp 12741
This theorem is referenced by:  bcthlem2  24499  bcthlem3  24500  bcthlem4  24501
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