| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | eqid 2737 | . 2
⊢
(ℤ≥‘𝐴) = (ℤ≥‘𝐴) | 
| 2 |  | caubl.2 | . . . 4
⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋)) | 
| 3 | 2 | 3ad2ant1 1134 | . . 3
⊢ ((𝜑 ∧ (1st ∘
𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) → 𝐷 ∈ (∞Met‘𝑋)) | 
| 4 |  | caublcls.6 | . . . 4
⊢ 𝐽 = (MetOpen‘𝐷) | 
| 5 | 4 | mopntopon 24449 | . . 3
⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ (TopOn‘𝑋)) | 
| 6 | 3, 5 | syl 17 | . 2
⊢ ((𝜑 ∧ (1st ∘
𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) → 𝐽 ∈ (TopOn‘𝑋)) | 
| 7 |  | simp3 1139 | . . 3
⊢ ((𝜑 ∧ (1st ∘
𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) → 𝐴 ∈ ℕ) | 
| 8 | 7 | nnzd 12640 | . 2
⊢ ((𝜑 ∧ (1st ∘
𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) → 𝐴 ∈ ℤ) | 
| 9 |  | simp2 1138 | . 2
⊢ ((𝜑 ∧ (1st ∘
𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) → (1st
∘ 𝐹)(⇝𝑡‘𝐽)𝑃) | 
| 10 |  | 2fveq3 6911 | . . . . . . . 8
⊢ (𝑟 = 𝐴 → ((ball‘𝐷)‘(𝐹‘𝑟)) = ((ball‘𝐷)‘(𝐹‘𝐴))) | 
| 11 | 10 | sseq1d 4015 | . . . . . . 7
⊢ (𝑟 = 𝐴 → (((ball‘𝐷)‘(𝐹‘𝑟)) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴)) ↔ ((ball‘𝐷)‘(𝐹‘𝐴)) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴)))) | 
| 12 | 11 | imbi2d 340 | . . . . . 6
⊢ (𝑟 = 𝐴 → (((𝜑 ∧ 𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘𝑟)) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴))) ↔ ((𝜑 ∧ 𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘𝐴)) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴))))) | 
| 13 |  | 2fveq3 6911 | . . . . . . . 8
⊢ (𝑟 = 𝑘 → ((ball‘𝐷)‘(𝐹‘𝑟)) = ((ball‘𝐷)‘(𝐹‘𝑘))) | 
| 14 | 13 | sseq1d 4015 | . . . . . . 7
⊢ (𝑟 = 𝑘 → (((ball‘𝐷)‘(𝐹‘𝑟)) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴)) ↔ ((ball‘𝐷)‘(𝐹‘𝑘)) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴)))) | 
| 15 | 14 | imbi2d 340 | . . . . . 6
⊢ (𝑟 = 𝑘 → (((𝜑 ∧ 𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘𝑟)) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴))) ↔ ((𝜑 ∧ 𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘𝑘)) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴))))) | 
| 16 |  | 2fveq3 6911 | . . . . . . . 8
⊢ (𝑟 = (𝑘 + 1) → ((ball‘𝐷)‘(𝐹‘𝑟)) = ((ball‘𝐷)‘(𝐹‘(𝑘 + 1)))) | 
| 17 | 16 | sseq1d 4015 | . . . . . . 7
⊢ (𝑟 = (𝑘 + 1) → (((ball‘𝐷)‘(𝐹‘𝑟)) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴)) ↔ ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴)))) | 
| 18 | 17 | imbi2d 340 | . . . . . 6
⊢ (𝑟 = (𝑘 + 1) → (((𝜑 ∧ 𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘𝑟)) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴))) ↔ ((𝜑 ∧ 𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴))))) | 
| 19 |  | ssid 4006 | . . . . . . 7
⊢
((ball‘𝐷)‘(𝐹‘𝐴)) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴)) | 
| 20 | 19 | 2a1i 12 | . . . . . 6
⊢ (𝐴 ∈ ℤ → ((𝜑 ∧ 𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘𝐴)) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴)))) | 
| 21 |  | caubl.4 | . . . . . . . . . . 11
⊢ (𝜑 → ∀𝑛 ∈ ℕ ((ball‘𝐷)‘(𝐹‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝑛))) | 
| 22 |  | eluznn 12960 | . . . . . . . . . . 11
⊢ ((𝐴 ∈ ℕ ∧ 𝑘 ∈
(ℤ≥‘𝐴)) → 𝑘 ∈ ℕ) | 
| 23 |  | fvoveq1 7454 | . . . . . . . . . . . . . 14
⊢ (𝑛 = 𝑘 → (𝐹‘(𝑛 + 1)) = (𝐹‘(𝑘 + 1))) | 
| 24 | 23 | fveq2d 6910 | . . . . . . . . . . . . 13
⊢ (𝑛 = 𝑘 → ((ball‘𝐷)‘(𝐹‘(𝑛 + 1))) = ((ball‘𝐷)‘(𝐹‘(𝑘 + 1)))) | 
| 25 |  | 2fveq3 6911 | . . . . . . . . . . . . 13
⊢ (𝑛 = 𝑘 → ((ball‘𝐷)‘(𝐹‘𝑛)) = ((ball‘𝐷)‘(𝐹‘𝑘))) | 
| 26 | 24, 25 | sseq12d 4017 | . . . . . . . . . . . 12
⊢ (𝑛 = 𝑘 → (((ball‘𝐷)‘(𝐹‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝑛)) ↔ ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝑘)))) | 
| 27 | 26 | rspccva 3621 | . . . . . . . . . . 11
⊢
((∀𝑛 ∈
ℕ ((ball‘𝐷)‘(𝐹‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝑛)) ∧ 𝑘 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝑘))) | 
| 28 | 21, 22, 27 | syl2an 596 | . . . . . . . . . 10
⊢ ((𝜑 ∧ (𝐴 ∈ ℕ ∧ 𝑘 ∈ (ℤ≥‘𝐴))) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝑘))) | 
| 29 | 28 | anassrs 467 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝐴)) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝑘))) | 
| 30 |  | sstr2 3990 | . . . . . . . . 9
⊢
(((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝑘)) → (((ball‘𝐷)‘(𝐹‘𝑘)) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴)) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴)))) | 
| 31 | 29, 30 | syl 17 | . . . . . . . 8
⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝐴)) → (((ball‘𝐷)‘(𝐹‘𝑘)) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴)) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴)))) | 
| 32 | 31 | expcom 413 | . . . . . . 7
⊢ (𝑘 ∈
(ℤ≥‘𝐴) → ((𝜑 ∧ 𝐴 ∈ ℕ) → (((ball‘𝐷)‘(𝐹‘𝑘)) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴)) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴))))) | 
| 33 | 32 | a2d 29 | . . . . . 6
⊢ (𝑘 ∈
(ℤ≥‘𝐴) → (((𝜑 ∧ 𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘𝑘)) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴))) → ((𝜑 ∧ 𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴))))) | 
| 34 | 12, 15, 18, 15, 20, 33 | uzind4 12948 | . . . . 5
⊢ (𝑘 ∈
(ℤ≥‘𝐴) → ((𝜑 ∧ 𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘𝑘)) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴)))) | 
| 35 | 34 | impcom 407 | . . . 4
⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝐴)) → ((ball‘𝐷)‘(𝐹‘𝑘)) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴))) | 
| 36 | 35 | 3adantl2 1168 | . . 3
⊢ (((𝜑 ∧ (1st ∘
𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝐴)) → ((ball‘𝐷)‘(𝐹‘𝑘)) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴))) | 
| 37 | 3 | adantr 480 | . . . . 5
⊢ (((𝜑 ∧ (1st ∘
𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝐴)) → 𝐷 ∈ (∞Met‘𝑋)) | 
| 38 |  | simpl1 1192 | . . . . . . . 8
⊢ (((𝜑 ∧ (1st ∘
𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝐴)) → 𝜑) | 
| 39 |  | caubl.3 | . . . . . . . 8
⊢ (𝜑 → 𝐹:ℕ⟶(𝑋 ×
ℝ+)) | 
| 40 | 38, 39 | syl 17 | . . . . . . 7
⊢ (((𝜑 ∧ (1st ∘
𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝐴)) → 𝐹:ℕ⟶(𝑋 ×
ℝ+)) | 
| 41 | 22 | 3ad2antl3 1188 | . . . . . . 7
⊢ (((𝜑 ∧ (1st ∘
𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝐴)) → 𝑘 ∈ ℕ) | 
| 42 | 40, 41 | ffvelcdmd 7105 | . . . . . 6
⊢ (((𝜑 ∧ (1st ∘
𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝐴)) → (𝐹‘𝑘) ∈ (𝑋 ×
ℝ+)) | 
| 43 |  | xp1st 8046 | . . . . . 6
⊢ ((𝐹‘𝑘) ∈ (𝑋 × ℝ+) →
(1st ‘(𝐹‘𝑘)) ∈ 𝑋) | 
| 44 | 42, 43 | syl 17 | . . . . 5
⊢ (((𝜑 ∧ (1st ∘
𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝐴)) → (1st
‘(𝐹‘𝑘)) ∈ 𝑋) | 
| 45 |  | xp2nd 8047 | . . . . . 6
⊢ ((𝐹‘𝑘) ∈ (𝑋 × ℝ+) →
(2nd ‘(𝐹‘𝑘)) ∈
ℝ+) | 
| 46 | 42, 45 | syl 17 | . . . . 5
⊢ (((𝜑 ∧ (1st ∘
𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝐴)) → (2nd
‘(𝐹‘𝑘)) ∈
ℝ+) | 
| 47 |  | blcntr 24423 | . . . . 5
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (1st
‘(𝐹‘𝑘)) ∈ 𝑋 ∧ (2nd ‘(𝐹‘𝑘)) ∈ ℝ+) →
(1st ‘(𝐹‘𝑘)) ∈ ((1st ‘(𝐹‘𝑘))(ball‘𝐷)(2nd ‘(𝐹‘𝑘)))) | 
| 48 | 37, 44, 46, 47 | syl3anc 1373 | . . . 4
⊢ (((𝜑 ∧ (1st ∘
𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝐴)) → (1st
‘(𝐹‘𝑘)) ∈ ((1st
‘(𝐹‘𝑘))(ball‘𝐷)(2nd ‘(𝐹‘𝑘)))) | 
| 49 |  | fvco3 7008 | . . . . 5
⊢ ((𝐹:ℕ⟶(𝑋 × ℝ+)
∧ 𝑘 ∈ ℕ)
→ ((1st ∘ 𝐹)‘𝑘) = (1st ‘(𝐹‘𝑘))) | 
| 50 | 40, 41, 49 | syl2anc 584 | . . . 4
⊢ (((𝜑 ∧ (1st ∘
𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝐴)) → ((1st
∘ 𝐹)‘𝑘) = (1st
‘(𝐹‘𝑘))) | 
| 51 |  | 1st2nd2 8053 | . . . . . . 7
⊢ ((𝐹‘𝑘) ∈ (𝑋 × ℝ+) → (𝐹‘𝑘) = 〈(1st ‘(𝐹‘𝑘)), (2nd ‘(𝐹‘𝑘))〉) | 
| 52 | 42, 51 | syl 17 | . . . . . 6
⊢ (((𝜑 ∧ (1st ∘
𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝐴)) → (𝐹‘𝑘) = 〈(1st ‘(𝐹‘𝑘)), (2nd ‘(𝐹‘𝑘))〉) | 
| 53 | 52 | fveq2d 6910 | . . . . 5
⊢ (((𝜑 ∧ (1st ∘
𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝐴)) → ((ball‘𝐷)‘(𝐹‘𝑘)) = ((ball‘𝐷)‘〈(1st ‘(𝐹‘𝑘)), (2nd ‘(𝐹‘𝑘))〉)) | 
| 54 |  | df-ov 7434 | . . . . 5
⊢
((1st ‘(𝐹‘𝑘))(ball‘𝐷)(2nd ‘(𝐹‘𝑘))) = ((ball‘𝐷)‘〈(1st ‘(𝐹‘𝑘)), (2nd ‘(𝐹‘𝑘))〉) | 
| 55 | 53, 54 | eqtr4di 2795 | . . . 4
⊢ (((𝜑 ∧ (1st ∘
𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝐴)) → ((ball‘𝐷)‘(𝐹‘𝑘)) = ((1st ‘(𝐹‘𝑘))(ball‘𝐷)(2nd ‘(𝐹‘𝑘)))) | 
| 56 | 48, 50, 55 | 3eltr4d 2856 | . . 3
⊢ (((𝜑 ∧ (1st ∘
𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝐴)) → ((1st
∘ 𝐹)‘𝑘) ∈ ((ball‘𝐷)‘(𝐹‘𝑘))) | 
| 57 | 36, 56 | sseldd 3984 | . 2
⊢ (((𝜑 ∧ (1st ∘
𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝐴)) → ((1st
∘ 𝐹)‘𝑘) ∈ ((ball‘𝐷)‘(𝐹‘𝐴))) | 
| 58 | 39 | ffvelcdmda 7104 | . . . . . . 7
⊢ ((𝜑 ∧ 𝐴 ∈ ℕ) → (𝐹‘𝐴) ∈ (𝑋 ×
ℝ+)) | 
| 59 | 58 | 3adant2 1132 | . . . . . 6
⊢ ((𝜑 ∧ (1st ∘
𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) → (𝐹‘𝐴) ∈ (𝑋 ×
ℝ+)) | 
| 60 |  | 1st2nd2 8053 | . . . . . 6
⊢ ((𝐹‘𝐴) ∈ (𝑋 × ℝ+) → (𝐹‘𝐴) = 〈(1st ‘(𝐹‘𝐴)), (2nd ‘(𝐹‘𝐴))〉) | 
| 61 | 59, 60 | syl 17 | . . . . 5
⊢ ((𝜑 ∧ (1st ∘
𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) → (𝐹‘𝐴) = 〈(1st ‘(𝐹‘𝐴)), (2nd ‘(𝐹‘𝐴))〉) | 
| 62 | 61 | fveq2d 6910 | . . . 4
⊢ ((𝜑 ∧ (1st ∘
𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘𝐴)) = ((ball‘𝐷)‘〈(1st ‘(𝐹‘𝐴)), (2nd ‘(𝐹‘𝐴))〉)) | 
| 63 |  | df-ov 7434 | . . . 4
⊢
((1st ‘(𝐹‘𝐴))(ball‘𝐷)(2nd ‘(𝐹‘𝐴))) = ((ball‘𝐷)‘〈(1st ‘(𝐹‘𝐴)), (2nd ‘(𝐹‘𝐴))〉) | 
| 64 | 62, 63 | eqtr4di 2795 | . . 3
⊢ ((𝜑 ∧ (1st ∘
𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘𝐴)) = ((1st ‘(𝐹‘𝐴))(ball‘𝐷)(2nd ‘(𝐹‘𝐴)))) | 
| 65 |  | xp1st 8046 | . . . . 5
⊢ ((𝐹‘𝐴) ∈ (𝑋 × ℝ+) →
(1st ‘(𝐹‘𝐴)) ∈ 𝑋) | 
| 66 | 59, 65 | syl 17 | . . . 4
⊢ ((𝜑 ∧ (1st ∘
𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) → (1st
‘(𝐹‘𝐴)) ∈ 𝑋) | 
| 67 |  | xp2nd 8047 | . . . . . 6
⊢ ((𝐹‘𝐴) ∈ (𝑋 × ℝ+) →
(2nd ‘(𝐹‘𝐴)) ∈
ℝ+) | 
| 68 | 59, 67 | syl 17 | . . . . 5
⊢ ((𝜑 ∧ (1st ∘
𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) → (2nd
‘(𝐹‘𝐴)) ∈
ℝ+) | 
| 69 | 68 | rpxrd 13078 | . . . 4
⊢ ((𝜑 ∧ (1st ∘
𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) → (2nd
‘(𝐹‘𝐴)) ∈
ℝ*) | 
| 70 |  | blssm 24428 | . . . 4
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (1st
‘(𝐹‘𝐴)) ∈ 𝑋 ∧ (2nd ‘(𝐹‘𝐴)) ∈ ℝ*) →
((1st ‘(𝐹‘𝐴))(ball‘𝐷)(2nd ‘(𝐹‘𝐴))) ⊆ 𝑋) | 
| 71 | 3, 66, 69, 70 | syl3anc 1373 | . . 3
⊢ ((𝜑 ∧ (1st ∘
𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) → ((1st
‘(𝐹‘𝐴))(ball‘𝐷)(2nd ‘(𝐹‘𝐴))) ⊆ 𝑋) | 
| 72 | 64, 71 | eqsstrd 4018 | . 2
⊢ ((𝜑 ∧ (1st ∘
𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘𝐴)) ⊆ 𝑋) | 
| 73 | 1, 6, 8, 9, 57, 72 | lmcls 23310 | 1
⊢ ((𝜑 ∧ (1st ∘
𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) → 𝑃 ∈ ((cls‘𝐽)‘((ball‘𝐷)‘(𝐹‘𝐴)))) |