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Theorem bcthlem1 23934
Description: Lemma for bcth 23939. 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 5630 . . . . . . 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 6693 . . . . . . . . 9 (𝐷 ∈ (CMet‘𝑋) → 𝑋 ∈ dom CMet)
42, 3syl 17 . . . . . . . 8 (𝜑𝑋 ∈ dom CMet)
5 reex 10626 . . . . . . . . 9 ℝ ∈ V
6 rpssre 12393 . . . . . . . . 9 + ⊆ ℝ
75, 6ssexi 5212 . . . . . . . 8 + ∈ V
8 xpexg 7467 . . . . . . . 8 ((𝑋 ∈ dom CMet ∧ ℝ+ ∈ V) → (𝑋 × ℝ+) ∈ V)
94, 7, 8sylancl 589 . . . . . . 7 (𝜑 → (𝑋 × ℝ+) ∈ V)
10 ssexg 5213 . . . . . . 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 590 . . . . . 6 (𝜑 → {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝑋𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝐴) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀𝐴))))} ∈ V)
12 oveq2 7157 . . . . . . . . . . 11 (𝑘 = 𝐴 → (1 / 𝑘) = (1 / 𝐴))
1312breq2d 5064 . . . . . . . . . 10 (𝑘 = 𝐴 → (𝑟 < (1 / 𝑘) ↔ 𝑟 < (1 / 𝐴)))
14 fveq2 6661 . . . . . . . . . . . 12 (𝑘 = 𝐴 → (𝑀𝑘) = (𝑀𝐴))
1514difeq2d 4085 . . . . . . . . . . 11 (𝑘 = 𝐴 → (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘)) = (((ball‘𝐷)‘𝑧) ∖ (𝑀𝐴)))
1615sseq2d 3985 . . . . . . . . . 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 633 . . . . . . . . 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 631 . . . . . . . 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 5118 . . . . . . 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 6661 . . . . . . . . . . . 12 (𝑧 = 𝐵 → ((ball‘𝐷)‘𝑧) = ((ball‘𝐷)‘𝐵))
2120difeq1d 4084 . . . . . . . . . . 11 (𝑧 = 𝐵 → (((ball‘𝐷)‘𝑧) ∖ (𝑀𝐴)) = (((ball‘𝐷)‘𝐵) ∖ (𝑀𝐴)))
2221sseq2d 3985 . . . . . . . . . 10 (𝑧 = 𝐵 → (((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝐴)) ↔ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀𝐴))))
2322anbi2d 631 . . . . . . . . 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 631 . . . . . . . 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 5118 . . . . . . 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 7302 . . . . . 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 1162 . . . . 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 1115 . . . 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 462 . . 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 2901 . 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 3949 . . 3 (𝐶 ∈ {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝑋𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝐴) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀𝐴))))} → 𝐶 ∈ (𝑋 × ℝ+))
33 simp1 1133 . . 3 ((𝐶 ∈ (𝑋 × ℝ+) ∧ (2nd𝐶) < (1 / 𝐴) ∧ ((cls‘𝐽)‘((ball‘𝐷)‘𝐶)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀𝐴))) → 𝐶 ∈ (𝑋 × ℝ+))
34 1st2nd2 7723 . . . . . 6 (𝐶 ∈ (𝑋 × ℝ+) → 𝐶 = ⟨(1st𝐶), (2nd𝐶)⟩)
3534eleq1d 2900 . . . . 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 6674 . . . . . 6 (1st𝐶) ∈ V
37 fvex 6674 . . . . . 6 (2nd𝐶) ∈ V
38 eleq1 2903 . . . . . . . 8 (𝑥 = (1st𝐶) → (𝑥𝑋 ↔ (1st𝐶) ∈ 𝑋))
39 eleq1 2903 . . . . . . . 8 (𝑟 = (2nd𝐶) → (𝑟 ∈ ℝ+ ↔ (2nd𝐶) ∈ ℝ+))
4038, 39bi2anan9 638 . . . . . . 7 ((𝑥 = (1st𝐶) ∧ 𝑟 = (2nd𝐶)) → ((𝑥𝑋𝑟 ∈ ℝ+) ↔ ((1st𝐶) ∈ 𝑋 ∧ (2nd𝐶) ∈ ℝ+)))
41 simpr 488 . . . . . . . . 9 ((𝑥 = (1st𝐶) ∧ 𝑟 = (2nd𝐶)) → 𝑟 = (2nd𝐶))
4241breq1d 5062 . . . . . . . 8 ((𝑥 = (1st𝐶) ∧ 𝑟 = (2nd𝐶)) → (𝑟 < (1 / 𝐴) ↔ (2nd𝐶) < (1 / 𝐴)))
43 oveq12 7158 . . . . . . . . . 10 ((𝑥 = (1st𝐶) ∧ 𝑟 = (2nd𝐶)) → (𝑥(ball‘𝐷)𝑟) = ((1st𝐶)(ball‘𝐷)(2nd𝐶)))
4443fveq2d 6665 . . . . . . . . 9 ((𝑥 = (1st𝐶) ∧ 𝑟 = (2nd𝐶)) → ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) = ((cls‘𝐽)‘((1st𝐶)(ball‘𝐷)(2nd𝐶))))
4544sseq1d 3984 . . . . . . . 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 633 . . . . . . 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 633 . . . . . 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 5410 . . . . 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, 48syl6bb 290 . . . 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 2900 . . . . . . 7 (𝐶 ∈ (𝑋 × ℝ+) → (𝐶 ∈ (𝑋 × ℝ+) ↔ ⟨(1st𝐶), (2nd𝐶)⟩ ∈ (𝑋 × ℝ+)))
51 opelxp 5578 . . . . . . 7 (⟨(1st𝐶), (2nd𝐶)⟩ ∈ (𝑋 × ℝ+) ↔ ((1st𝐶) ∈ 𝑋 ∧ (2nd𝐶) ∈ ℝ+))
5250, 51syl6rbb 291 . . . . . 6 (𝐶 ∈ (𝑋 × ℝ+) → (((1st𝐶) ∈ 𝑋 ∧ (2nd𝐶) ∈ ℝ+) ↔ 𝐶 ∈ (𝑋 × ℝ+)))
5334fveq2d 6665 . . . . . . . . . 10 (𝐶 ∈ (𝑋 × ℝ+) → ((ball‘𝐷)‘𝐶) = ((ball‘𝐷)‘⟨(1st𝐶), (2nd𝐶)⟩))
54 df-ov 7152 . . . . . . . . . 10 ((1st𝐶)(ball‘𝐷)(2nd𝐶)) = ((ball‘𝐷)‘⟨(1st𝐶), (2nd𝐶)⟩)
5553, 54syl6reqr 2878 . . . . . . . . 9 (𝐶 ∈ (𝑋 × ℝ+) → ((1st𝐶)(ball‘𝐷)(2nd𝐶)) = ((ball‘𝐷)‘𝐶))
5655fveq2d 6665 . . . . . . . 8 (𝐶 ∈ (𝑋 × ℝ+) → ((cls‘𝐽)‘((1st𝐶)(ball‘𝐷)(2nd𝐶))) = ((cls‘𝐽)‘((ball‘𝐷)‘𝐶)))
5756sseq1d 3984 . . . . . . 7 (𝐶 ∈ (𝑋 × ℝ+) → (((cls‘𝐽)‘((1st𝐶)(ball‘𝐷)(2nd𝐶))) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀𝐴)) ↔ ((cls‘𝐽)‘((ball‘𝐷)‘𝐶)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀𝐴))))
5857anbi2d 631 . . . . . 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 633 . . . . 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 1092 . . . . 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, 60syl6bbr 292 . . . 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 282 . . 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 383 . 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, 63syl6bb 290 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 209  wa 399  w3a 1084   = wceq 1538  wcel 2115  Vcvv 3480  cdif 3916  wss 3919  cop 4556   class class class wbr 5052  {copab 5114   × cxp 5540  dom cdm 5542  cfv 6343  (class class class)co 7149  cmpo 7151  1st c1st 7682  2nd c2nd 7683  cr 10534  1c1 10536   < clt 10673   / cdiv 11295  cn 11634  +crp 12386  ballcbl 20085  MetOpencmopn 20088  clsccl 21629  CMetccmet 23864
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-cnex 10591  ax-resscn 10592
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-iota 6302  df-fun 6345  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154  df-1st 7684  df-2nd 7685  df-rp 12387
This theorem is referenced by:  bcthlem2  23935  bcthlem3  23936  bcthlem4  23937
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