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Theorem bcthlem1 22816
Description: Lemma for bcth 22821. 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 5010 . . . . . . 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 6013 . . . . . . . . 9 (𝐷 ∈ (CMet‘𝑋) → 𝑋 ∈ dom CMet)
42, 3syl 17 . . . . . . . 8 (𝜑𝑋 ∈ dom CMet)
5 reex 9780 . . . . . . . . 9 ℝ ∈ V
6 rpssre 11581 . . . . . . . . 9 + ⊆ ℝ
75, 6ssexi 4630 . . . . . . . 8 + ∈ V
8 xpexg 6732 . . . . . . . 8 ((𝑋 ∈ dom CMet ∧ ℝ+ ∈ V) → (𝑋 × ℝ+) ∈ V)
94, 7, 8sylancl 692 . . . . . . 7 (𝜑 → (𝑋 × ℝ+) ∈ V)
10 ssexg 4631 . . . . . . 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 693 . . . . . 6 (𝜑 → {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝑋𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝐴) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀𝐴))))} ∈ V)
12 oveq2 6433 . . . . . . . . . . 11 (𝑘 = 𝐴 → (1 / 𝑘) = (1 / 𝐴))
1312breq2d 4493 . . . . . . . . . 10 (𝑘 = 𝐴 → (𝑟 < (1 / 𝑘) ↔ 𝑟 < (1 / 𝐴)))
14 fveq2 5986 . . . . . . . . . . . 12 (𝑘 = 𝐴 → (𝑀𝑘) = (𝑀𝐴))
1514difeq2d 3594 . . . . . . . . . . 11 (𝑘 = 𝐴 → (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘)) = (((ball‘𝐷)‘𝑧) ∖ (𝑀𝐴)))
1615sseq2d 3500 . . . . . . . . . 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 742 . . . . . . . . 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 735 . . . . . . . 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 4546 . . . . . . 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 5986 . . . . . . . . . . . 12 (𝑧 = 𝐵 → ((ball‘𝐷)‘𝑧) = ((ball‘𝐷)‘𝐵))
2120difeq1d 3593 . . . . . . . . . . 11 (𝑧 = 𝐵 → (((ball‘𝐷)‘𝑧) ∖ (𝑀𝐴)) = (((ball‘𝐷)‘𝐵) ∖ (𝑀𝐴)))
2221sseq2d 3500 . . . . . . . . . 10 (𝑧 = 𝐵 → (((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝐴)) ↔ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀𝐴))))
2322anbi2d 735 . . . . . . . . 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 735 . . . . . . . 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 4546 . . . . . . 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, 26ovmpt2g 6568 . . . . . 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 1352 . . . . 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 1256 . . . 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 467 . . 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 2577 . 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 3468 . . 3 (𝐶 ∈ {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝑋𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝐴) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀𝐴))))} → 𝐶 ∈ (𝑋 × ℝ+))
33 simp1 1053 . . 3 ((𝐶 ∈ (𝑋 × ℝ+) ∧ (2nd𝐶) < (1 / 𝐴) ∧ ((cls‘𝐽)‘((ball‘𝐷)‘𝐶)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀𝐴))) → 𝐶 ∈ (𝑋 × ℝ+))
34 1st2nd2 6969 . . . . . 6 (𝐶 ∈ (𝑋 × ℝ+) → 𝐶 = ⟨(1st𝐶), (2nd𝐶)⟩)
3534eleq1d 2576 . . . . 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 5996 . . . . . 6 (1st𝐶) ∈ V
37 fvex 5996 . . . . . 6 (2nd𝐶) ∈ V
38 eleq1 2580 . . . . . . . 8 (𝑥 = (1st𝐶) → (𝑥𝑋 ↔ (1st𝐶) ∈ 𝑋))
39 eleq1 2580 . . . . . . . 8 (𝑟 = (2nd𝐶) → (𝑟 ∈ ℝ+ ↔ (2nd𝐶) ∈ ℝ+))
4038, 39bi2anan9 912 . . . . . . 7 ((𝑥 = (1st𝐶) ∧ 𝑟 = (2nd𝐶)) → ((𝑥𝑋𝑟 ∈ ℝ+) ↔ ((1st𝐶) ∈ 𝑋 ∧ (2nd𝐶) ∈ ℝ+)))
41 simpr 475 . . . . . . . . 9 ((𝑥 = (1st𝐶) ∧ 𝑟 = (2nd𝐶)) → 𝑟 = (2nd𝐶))
4241breq1d 4491 . . . . . . . 8 ((𝑥 = (1st𝐶) ∧ 𝑟 = (2nd𝐶)) → (𝑟 < (1 / 𝐴) ↔ (2nd𝐶) < (1 / 𝐴)))
43 oveq12 6434 . . . . . . . . . 10 ((𝑥 = (1st𝐶) ∧ 𝑟 = (2nd𝐶)) → (𝑥(ball‘𝐷)𝑟) = ((1st𝐶)(ball‘𝐷)(2nd𝐶)))
4443fveq2d 5990 . . . . . . . . 9 ((𝑥 = (1st𝐶) ∧ 𝑟 = (2nd𝐶)) → ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) = ((cls‘𝐽)‘((1st𝐶)(ball‘𝐷)(2nd𝐶))))
4544sseq1d 3499 . . . . . . . 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 742 . . . . . . 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 742 . . . . . 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 4810 . . . . 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 274 . . . 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 2576 . . . . . . 7 (𝐶 ∈ (𝑋 × ℝ+) → (𝐶 ∈ (𝑋 × ℝ+) ↔ ⟨(1st𝐶), (2nd𝐶)⟩ ∈ (𝑋 × ℝ+)))
51 opelxp 4964 . . . . . . 7 (⟨(1st𝐶), (2nd𝐶)⟩ ∈ (𝑋 × ℝ+) ↔ ((1st𝐶) ∈ 𝑋 ∧ (2nd𝐶) ∈ ℝ+))
5250, 51syl6rbb 275 . . . . . 6 (𝐶 ∈ (𝑋 × ℝ+) → (((1st𝐶) ∈ 𝑋 ∧ (2nd𝐶) ∈ ℝ+) ↔ 𝐶 ∈ (𝑋 × ℝ+)))
5334fveq2d 5990 . . . . . . . . . 10 (𝐶 ∈ (𝑋 × ℝ+) → ((ball‘𝐷)‘𝐶) = ((ball‘𝐷)‘⟨(1st𝐶), (2nd𝐶)⟩))
54 df-ov 6428 . . . . . . . . . 10 ((1st𝐶)(ball‘𝐷)(2nd𝐶)) = ((ball‘𝐷)‘⟨(1st𝐶), (2nd𝐶)⟩)
5553, 54syl6reqr 2567 . . . . . . . . 9 (𝐶 ∈ (𝑋 × ℝ+) → ((1st𝐶)(ball‘𝐷)(2nd𝐶)) = ((ball‘𝐷)‘𝐶))
5655fveq2d 5990 . . . . . . . 8 (𝐶 ∈ (𝑋 × ℝ+) → ((cls‘𝐽)‘((1st𝐶)(ball‘𝐷)(2nd𝐶))) = ((cls‘𝐽)‘((ball‘𝐷)‘𝐶)))
5756sseq1d 3499 . . . . . . 7 (𝐶 ∈ (𝑋 × ℝ+) → (((cls‘𝐽)‘((1st𝐶)(ball‘𝐷)(2nd𝐶))) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀𝐴)) ↔ ((cls‘𝐽)‘((ball‘𝐷)‘𝐶)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀𝐴))))
5857anbi2d 735 . . . . . 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 742 . . . . 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 1034 . . . . 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 276 . . . 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 266 . . 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 366 . 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 274 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 194  wa 382  w3a 1030   = wceq 1474  wcel 1938  Vcvv 3077  cdif 3441  wss 3444  cop 4034   class class class wbr 4481  {copab 4540   × cxp 4930  dom cdm 4932  cfv 5689  (class class class)co 6425  cmpt2 6427  1st c1st 6930  2nd c2nd 6931  cr 9688  1c1 9690   < clt 9827   / cdiv 10431  cn 10773  +crp 11570  ballcbl 19479  MetOpencmopn 19482  clsccl 20556  CMetcms 22748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-sep 4607  ax-nul 4616  ax-pow 4668  ax-pr 4732  ax-un 6721  ax-cnex 9745  ax-resscn 9746
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-ral 2805  df-rex 2806  df-rab 2809  df-v 3079  df-sbc 3307  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-nul 3778  df-if 3940  df-pw 4013  df-sn 4029  df-pr 4031  df-op 4035  df-uni 4271  df-br 4482  df-opab 4542  df-mpt 4543  df-id 4847  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-iota 5653  df-fun 5691  df-fv 5697  df-ov 6428  df-oprab 6429  df-mpt2 6430  df-1st 6932  df-2nd 6933  df-rp 11571
This theorem is referenced by:  bcthlem2  22817  bcthlem3  22818  bcthlem4  22819
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