Proof of Theorem bcthlem2
Step | Hyp | Ref
| Expression |
1 | | bcthlem.11 |
. . . . 5
⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔‘𝑘))) |
2 | | oveq1 6657 |
. . . . . . . 8
⊢ (𝑘 = 𝑛 → (𝑘 + 1) = (𝑛 + 1)) |
3 | 2 | fveq2d 6195 |
. . . . . . 7
⊢ (𝑘 = 𝑛 → (𝑔‘(𝑘 + 1)) = (𝑔‘(𝑛 + 1))) |
4 | | id 22 |
. . . . . . . 8
⊢ (𝑘 = 𝑛 → 𝑘 = 𝑛) |
5 | | fveq2 6191 |
. . . . . . . 8
⊢ (𝑘 = 𝑛 → (𝑔‘𝑘) = (𝑔‘𝑛)) |
6 | 4, 5 | oveq12d 6668 |
. . . . . . 7
⊢ (𝑘 = 𝑛 → (𝑘𝐹(𝑔‘𝑘)) = (𝑛𝐹(𝑔‘𝑛))) |
7 | 3, 6 | eleq12d 2695 |
. . . . . 6
⊢ (𝑘 = 𝑛 → ((𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔‘𝑘)) ↔ (𝑔‘(𝑛 + 1)) ∈ (𝑛𝐹(𝑔‘𝑛)))) |
8 | 7 | rspccva 3308 |
. . . . 5
⊢
((∀𝑘 ∈
ℕ (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔‘𝑘)) ∧ 𝑛 ∈ ℕ) → (𝑔‘(𝑛 + 1)) ∈ (𝑛𝐹(𝑔‘𝑛))) |
9 | 1, 8 | sylan 488 |
. . . 4
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑔‘(𝑛 + 1)) ∈ (𝑛𝐹(𝑔‘𝑛))) |
10 | | bcthlem.9 |
. . . . . 6
⊢ (𝜑 → 𝑔:ℕ⟶(𝑋 ×
ℝ+)) |
11 | 10 | ffvelrnda 6359 |
. . . . 5
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑔‘𝑛) ∈ (𝑋 ×
ℝ+)) |
12 | | bcth.2 |
. . . . . . 7
⊢ 𝐽 = (MetOpen‘𝐷) |
13 | | bcthlem.4 |
. . . . . . 7
⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋)) |
14 | | 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))))}) |
15 | 12, 13, 14 | bcthlem1 23121 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ (𝑔‘𝑛) ∈ (𝑋 × ℝ+))) →
((𝑔‘(𝑛 + 1)) ∈ (𝑛𝐹(𝑔‘𝑛)) ↔ ((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ+) ∧
(2nd ‘(𝑔‘(𝑛 + 1))) < (1 / 𝑛) ∧ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔‘𝑛)) ∖ (𝑀‘𝑛))))) |
16 | 15 | expr 643 |
. . . . 5
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑔‘𝑛) ∈ (𝑋 × ℝ+) → ((𝑔‘(𝑛 + 1)) ∈ (𝑛𝐹(𝑔‘𝑛)) ↔ ((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ+) ∧
(2nd ‘(𝑔‘(𝑛 + 1))) < (1 / 𝑛) ∧ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔‘𝑛)) ∖ (𝑀‘𝑛)))))) |
17 | 11, 16 | mpd 15 |
. . . 4
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑔‘(𝑛 + 1)) ∈ (𝑛𝐹(𝑔‘𝑛)) ↔ ((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ+) ∧
(2nd ‘(𝑔‘(𝑛 + 1))) < (1 / 𝑛) ∧ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔‘𝑛)) ∖ (𝑀‘𝑛))))) |
18 | 9, 17 | mpbid 222 |
. . 3
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ+) ∧
(2nd ‘(𝑔‘(𝑛 + 1))) < (1 / 𝑛) ∧ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔‘𝑛)) ∖ (𝑀‘𝑛)))) |
19 | | cmetmet 23084 |
. . . . . . . . . . . . 13
⊢ (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋)) |
20 | 13, 19 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋)) |
21 | | metxmet 22139 |
. . . . . . . . . . . 12
⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋)) |
22 | 20, 21 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋)) |
23 | 12 | mopntop 22245 |
. . . . . . . . . . 11
⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top) |
24 | 22, 23 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐽 ∈ Top) |
25 | 24 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ+)) → 𝐽 ∈ Top) |
26 | | xp1st 7198 |
. . . . . . . . . . . 12
⊢ ((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ+) →
(1st ‘(𝑔‘(𝑛 + 1))) ∈ 𝑋) |
27 | | xp2nd 7199 |
. . . . . . . . . . . . 13
⊢ ((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ+) →
(2nd ‘(𝑔‘(𝑛 + 1))) ∈
ℝ+) |
28 | 27 | rpxrd 11873 |
. . . . . . . . . . . 12
⊢ ((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ+) →
(2nd ‘(𝑔‘(𝑛 + 1))) ∈
ℝ*) |
29 | 26, 28 | jca 554 |
. . . . . . . . . . 11
⊢ ((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ+) →
((1st ‘(𝑔‘(𝑛 + 1))) ∈ 𝑋 ∧ (2nd ‘(𝑔‘(𝑛 + 1))) ∈
ℝ*)) |
30 | | blssm 22223 |
. . . . . . . . . . . 12
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (1st
‘(𝑔‘(𝑛 + 1))) ∈ 𝑋 ∧ (2nd ‘(𝑔‘(𝑛 + 1))) ∈ ℝ*) →
((1st ‘(𝑔‘(𝑛 + 1)))(ball‘𝐷)(2nd ‘(𝑔‘(𝑛 + 1)))) ⊆ 𝑋) |
31 | 30 | 3expb 1266 |
. . . . . . . . . . 11
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ ((1st
‘(𝑔‘(𝑛 + 1))) ∈ 𝑋 ∧ (2nd ‘(𝑔‘(𝑛 + 1))) ∈ ℝ*)) →
((1st ‘(𝑔‘(𝑛 + 1)))(ball‘𝐷)(2nd ‘(𝑔‘(𝑛 + 1)))) ⊆ 𝑋) |
32 | 22, 29, 31 | syl2an 494 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ+)) →
((1st ‘(𝑔‘(𝑛 + 1)))(ball‘𝐷)(2nd ‘(𝑔‘(𝑛 + 1)))) ⊆ 𝑋) |
33 | | 1st2nd2 7205 |
. . . . . . . . . . . . 13
⊢ ((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ+) → (𝑔‘(𝑛 + 1)) = 〈(1st ‘(𝑔‘(𝑛 + 1))), (2nd ‘(𝑔‘(𝑛 + 1)))〉) |
34 | 33 | fveq2d 6195 |
. . . . . . . . . . . 12
⊢ ((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ+) →
((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) = ((ball‘𝐷)‘〈(1st ‘(𝑔‘(𝑛 + 1))), (2nd ‘(𝑔‘(𝑛 + 1)))〉)) |
35 | | df-ov 6653 |
. . . . . . . . . . . 12
⊢
((1st ‘(𝑔‘(𝑛 + 1)))(ball‘𝐷)(2nd ‘(𝑔‘(𝑛 + 1)))) = ((ball‘𝐷)‘〈(1st ‘(𝑔‘(𝑛 + 1))), (2nd ‘(𝑔‘(𝑛 + 1)))〉) |
36 | 34, 35 | syl6reqr 2675 |
. . . . . . . . . . 11
⊢ ((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ+) →
((1st ‘(𝑔‘(𝑛 + 1)))(ball‘𝐷)(2nd ‘(𝑔‘(𝑛 + 1)))) = ((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))) |
37 | 36 | adantl 482 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ+)) →
((1st ‘(𝑔‘(𝑛 + 1)))(ball‘𝐷)(2nd ‘(𝑔‘(𝑛 + 1)))) = ((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))) |
38 | 12 | mopnuni 22246 |
. . . . . . . . . . . 12
⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = ∪ 𝐽) |
39 | 22, 38 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑋 = ∪ 𝐽) |
40 | 39 | adantr 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ+)) → 𝑋 = ∪
𝐽) |
41 | 32, 37, 40 | 3sstr3d 3647 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ+)) →
((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ∪
𝐽) |
42 | | eqid 2622 |
. . . . . . . . . 10
⊢ ∪ 𝐽 =
∪ 𝐽 |
43 | 42 | sscls 20860 |
. . . . . . . . 9
⊢ ((𝐽 ∈ Top ∧
((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ∪
𝐽) →
((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1))))) |
44 | 25, 41, 43 | syl2anc 693 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ+)) →
((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1))))) |
45 | | difss2 3739 |
. . . . . . . 8
⊢
(((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔‘𝑛)) ∖ (𝑀‘𝑛)) → ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))) ⊆ ((ball‘𝐷)‘(𝑔‘𝑛))) |
46 | | sstr2 3610 |
. . . . . . . 8
⊢
(((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))) → (((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))) ⊆ ((ball‘𝐷)‘(𝑔‘𝑛)) → ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝑔‘𝑛)))) |
47 | 44, 45, 46 | syl2im 40 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ+)) →
(((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔‘𝑛)) ∖ (𝑀‘𝑛)) → ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝑔‘𝑛)))) |
48 | 47 | a1d 25 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ+)) →
((2nd ‘(𝑔‘(𝑛 + 1))) < (1 / 𝑛) → (((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔‘𝑛)) ∖ (𝑀‘𝑛)) → ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝑔‘𝑛))))) |
49 | 48 | ex 450 |
. . . . 5
⊢ (𝜑 → ((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ+) →
((2nd ‘(𝑔‘(𝑛 + 1))) < (1 / 𝑛) → (((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔‘𝑛)) ∖ (𝑀‘𝑛)) → ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝑔‘𝑛)))))) |
50 | 49 | 3impd 1281 |
. . . 4
⊢ (𝜑 → (((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ+) ∧
(2nd ‘(𝑔‘(𝑛 + 1))) < (1 / 𝑛) ∧ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔‘𝑛)) ∖ (𝑀‘𝑛))) → ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝑔‘𝑛)))) |
51 | 50 | adantr 481 |
. . 3
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ+) ∧
(2nd ‘(𝑔‘(𝑛 + 1))) < (1 / 𝑛) ∧ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔‘𝑛)) ∖ (𝑀‘𝑛))) → ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝑔‘𝑛)))) |
52 | 18, 51 | mpd 15 |
. 2
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝑔‘𝑛))) |
53 | 52 | ralrimiva 2966 |
1
⊢ (𝜑 → ∀𝑛 ∈ ℕ ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝑔‘𝑛))) |