Step | Hyp | Ref
| Expression |
1 | | heibor1.4 |
. . 3
⊢ (𝜑 → 𝐽 ∈ Comp) |
2 | | heibor1.3 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋)) |
3 | | metxmet 21949 |
. . . . . . . . . 10
⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋)) |
4 | 2, 3 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋)) |
5 | | heibor.1 |
. . . . . . . . . 10
⊢ 𝐽 = (MetOpen‘𝐷) |
6 | 5 | mopntop 22055 |
. . . . . . . . 9
⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top) |
7 | 4, 6 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 𝐽 ∈ Top) |
8 | | imassrn 5396 |
. . . . . . . . 9
⊢ (𝐹 “ 𝑢) ⊆ ran 𝐹 |
9 | | heibor1.6 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐹:ℕ⟶𝑋) |
10 | | frn 5966 |
. . . . . . . . . . 11
⊢ (𝐹:ℕ⟶𝑋 → ran 𝐹 ⊆ 𝑋) |
11 | 9, 10 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → ran 𝐹 ⊆ 𝑋) |
12 | 5 | mopnuni 22056 |
. . . . . . . . . . 11
⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = ∪ 𝐽) |
13 | 4, 12 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑋 = ∪ 𝐽) |
14 | 11, 13 | sseqtrd 3604 |
. . . . . . . . 9
⊢ (𝜑 → ran 𝐹 ⊆ ∪ 𝐽) |
15 | 8, 14 | syl5ss 3579 |
. . . . . . . 8
⊢ (𝜑 → (𝐹 “ 𝑢) ⊆ ∪ 𝐽) |
16 | | eqid 2610 |
. . . . . . . . 9
⊢ ∪ 𝐽 =
∪ 𝐽 |
17 | 16 | clscld 20661 |
. . . . . . . 8
⊢ ((𝐽 ∈ Top ∧ (𝐹 “ 𝑢) ⊆ ∪ 𝐽) → ((cls‘𝐽)‘(𝐹 “ 𝑢)) ∈ (Clsd‘𝐽)) |
18 | 7, 15, 17 | syl2anc 691 |
. . . . . . 7
⊢ (𝜑 → ((cls‘𝐽)‘(𝐹 “ 𝑢)) ∈ (Clsd‘𝐽)) |
19 | | eleq1a 2683 |
. . . . . . 7
⊢
(((cls‘𝐽)‘(𝐹 “ 𝑢)) ∈ (Clsd‘𝐽) → (𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢)) → 𝑘 ∈ (Clsd‘𝐽))) |
20 | 18, 19 | syl 17 |
. . . . . 6
⊢ (𝜑 → (𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢)) → 𝑘 ∈ (Clsd‘𝐽))) |
21 | 20 | rexlimdvw 3016 |
. . . . 5
⊢ (𝜑 → (∃𝑢 ∈ ran ℤ≥𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢)) → 𝑘 ∈ (Clsd‘𝐽))) |
22 | 21 | abssdv 3639 |
. . . 4
⊢ (𝜑 → {𝑘 ∣ ∃𝑢 ∈ ran ℤ≥𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢))} ⊆ (Clsd‘𝐽)) |
23 | | fvex 6113 |
. . . . 5
⊢
(Clsd‘𝐽)
∈ V |
24 | 23 | elpw2 4755 |
. . . 4
⊢ ({𝑘 ∣ ∃𝑢 ∈ ran
ℤ≥𝑘 =
((cls‘𝐽)‘(𝐹 “ 𝑢))} ∈ 𝒫 (Clsd‘𝐽) ↔ {𝑘 ∣ ∃𝑢 ∈ ran ℤ≥𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢))} ⊆ (Clsd‘𝐽)) |
25 | 22, 24 | sylibr 223 |
. . 3
⊢ (𝜑 → {𝑘 ∣ ∃𝑢 ∈ ran ℤ≥𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢))} ∈ 𝒫 (Clsd‘𝐽)) |
26 | | elin 3758 |
. . . . . . 7
⊢ (𝑟 ∈ (𝒫 {𝑘 ∣ ∃𝑢 ∈ ran
ℤ≥𝑘 =
((cls‘𝐽)‘(𝐹 “ 𝑢))} ∩ Fin) ↔ (𝑟 ∈ 𝒫 {𝑘 ∣ ∃𝑢 ∈ ran ℤ≥𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢))} ∧ 𝑟 ∈ Fin)) |
27 | | selpw 4115 |
. . . . . . . . 9
⊢ (𝑟 ∈ 𝒫 {𝑘 ∣ ∃𝑢 ∈ ran
ℤ≥𝑘 =
((cls‘𝐽)‘(𝐹 “ 𝑢))} ↔ 𝑟 ⊆ {𝑘 ∣ ∃𝑢 ∈ ran ℤ≥𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢))}) |
28 | | ssabral 3636 |
. . . . . . . . 9
⊢ (𝑟 ⊆ {𝑘 ∣ ∃𝑢 ∈ ran ℤ≥𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢))} ↔ ∀𝑘 ∈ 𝑟 ∃𝑢 ∈ ran ℤ≥𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢))) |
29 | 27, 28 | bitri 263 |
. . . . . . . 8
⊢ (𝑟 ∈ 𝒫 {𝑘 ∣ ∃𝑢 ∈ ran
ℤ≥𝑘 =
((cls‘𝐽)‘(𝐹 “ 𝑢))} ↔ ∀𝑘 ∈ 𝑟 ∃𝑢 ∈ ran ℤ≥𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢))) |
30 | 29 | anbi1i 727 |
. . . . . . 7
⊢ ((𝑟 ∈ 𝒫 {𝑘 ∣ ∃𝑢 ∈ ran
ℤ≥𝑘 =
((cls‘𝐽)‘(𝐹 “ 𝑢))} ∧ 𝑟 ∈ Fin) ↔ (∀𝑘 ∈ 𝑟 ∃𝑢 ∈ ran ℤ≥𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢)) ∧ 𝑟 ∈ Fin)) |
31 | 26, 30 | bitri 263 |
. . . . . 6
⊢ (𝑟 ∈ (𝒫 {𝑘 ∣ ∃𝑢 ∈ ran
ℤ≥𝑘 =
((cls‘𝐽)‘(𝐹 “ 𝑢))} ∩ Fin) ↔ (∀𝑘 ∈ 𝑟 ∃𝑢 ∈ ran ℤ≥𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢)) ∧ 𝑟 ∈ Fin)) |
32 | | raleq 3115 |
. . . . . . . . . . . . . 14
⊢ (𝑚 = ∅ → (∀𝑘 ∈ 𝑚 ∃𝑢 ∈ ran ℤ≥𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢)) ↔ ∀𝑘 ∈ ∅ ∃𝑢 ∈ ran ℤ≥𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢)))) |
33 | 32 | anbi2d 736 |
. . . . . . . . . . . . 13
⊢ (𝑚 = ∅ → ((𝜑 ∧ ∀𝑘 ∈ 𝑚 ∃𝑢 ∈ ran ℤ≥𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢))) ↔ (𝜑 ∧ ∀𝑘 ∈ ∅ ∃𝑢 ∈ ran ℤ≥𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢))))) |
34 | | inteq 4413 |
. . . . . . . . . . . . . . 15
⊢ (𝑚 = ∅ → ∩ 𝑚 =
∩ ∅) |
35 | 34 | sseq2d 3596 |
. . . . . . . . . . . . . 14
⊢ (𝑚 = ∅ → ((𝐹 “ 𝑘) ⊆ ∩ 𝑚 ↔ (𝐹 “ 𝑘) ⊆ ∩
∅)) |
36 | 35 | rexbidv 3034 |
. . . . . . . . . . . . 13
⊢ (𝑚 = ∅ → (∃𝑘 ∈ ran
ℤ≥(𝐹
“ 𝑘) ⊆ ∩ 𝑚
↔ ∃𝑘 ∈ ran
ℤ≥(𝐹
“ 𝑘) ⊆ ∩ ∅)) |
37 | 33, 36 | imbi12d 333 |
. . . . . . . . . . . 12
⊢ (𝑚 = ∅ → (((𝜑 ∧ ∀𝑘 ∈ 𝑚 ∃𝑢 ∈ ran ℤ≥𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢))) → ∃𝑘 ∈ ran ℤ≥(𝐹 “ 𝑘) ⊆ ∩ 𝑚) ↔ ((𝜑 ∧ ∀𝑘 ∈ ∅ ∃𝑢 ∈ ran ℤ≥𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢))) → ∃𝑘 ∈ ran ℤ≥(𝐹 “ 𝑘) ⊆ ∩
∅))) |
38 | | raleq 3115 |
. . . . . . . . . . . . . 14
⊢ (𝑚 = 𝑦 → (∀𝑘 ∈ 𝑚 ∃𝑢 ∈ ran ℤ≥𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢)) ↔ ∀𝑘 ∈ 𝑦 ∃𝑢 ∈ ran ℤ≥𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢)))) |
39 | 38 | anbi2d 736 |
. . . . . . . . . . . . 13
⊢ (𝑚 = 𝑦 → ((𝜑 ∧ ∀𝑘 ∈ 𝑚 ∃𝑢 ∈ ran ℤ≥𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢))) ↔ (𝜑 ∧ ∀𝑘 ∈ 𝑦 ∃𝑢 ∈ ran ℤ≥𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢))))) |
40 | | inteq 4413 |
. . . . . . . . . . . . . . 15
⊢ (𝑚 = 𝑦 → ∩ 𝑚 = ∩
𝑦) |
41 | 40 | sseq2d 3596 |
. . . . . . . . . . . . . 14
⊢ (𝑚 = 𝑦 → ((𝐹 “ 𝑘) ⊆ ∩ 𝑚 ↔ (𝐹 “ 𝑘) ⊆ ∩ 𝑦)) |
42 | 41 | rexbidv 3034 |
. . . . . . . . . . . . 13
⊢ (𝑚 = 𝑦 → (∃𝑘 ∈ ran ℤ≥(𝐹 “ 𝑘) ⊆ ∩ 𝑚 ↔ ∃𝑘 ∈ ran
ℤ≥(𝐹
“ 𝑘) ⊆ ∩ 𝑦)) |
43 | 39, 42 | imbi12d 333 |
. . . . . . . . . . . 12
⊢ (𝑚 = 𝑦 → (((𝜑 ∧ ∀𝑘 ∈ 𝑚 ∃𝑢 ∈ ran ℤ≥𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢))) → ∃𝑘 ∈ ran ℤ≥(𝐹 “ 𝑘) ⊆ ∩ 𝑚) ↔ ((𝜑 ∧ ∀𝑘 ∈ 𝑦 ∃𝑢 ∈ ran ℤ≥𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢))) → ∃𝑘 ∈ ran ℤ≥(𝐹 “ 𝑘) ⊆ ∩ 𝑦))) |
44 | | raleq 3115 |
. . . . . . . . . . . . . 14
⊢ (𝑚 = (𝑦 ∪ {𝑛}) → (∀𝑘 ∈ 𝑚 ∃𝑢 ∈ ran ℤ≥𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢)) ↔ ∀𝑘 ∈ (𝑦 ∪ {𝑛})∃𝑢 ∈ ran ℤ≥𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢)))) |
45 | 44 | anbi2d 736 |
. . . . . . . . . . . . 13
⊢ (𝑚 = (𝑦 ∪ {𝑛}) → ((𝜑 ∧ ∀𝑘 ∈ 𝑚 ∃𝑢 ∈ ran ℤ≥𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢))) ↔ (𝜑 ∧ ∀𝑘 ∈ (𝑦 ∪ {𝑛})∃𝑢 ∈ ran ℤ≥𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢))))) |
46 | | inteq 4413 |
. . . . . . . . . . . . . . 15
⊢ (𝑚 = (𝑦 ∪ {𝑛}) → ∩ 𝑚 = ∩
(𝑦 ∪ {𝑛})) |
47 | 46 | sseq2d 3596 |
. . . . . . . . . . . . . 14
⊢ (𝑚 = (𝑦 ∪ {𝑛}) → ((𝐹 “ 𝑘) ⊆ ∩ 𝑚 ↔ (𝐹 “ 𝑘) ⊆ ∩ (𝑦 ∪ {𝑛}))) |
48 | 47 | rexbidv 3034 |
. . . . . . . . . . . . 13
⊢ (𝑚 = (𝑦 ∪ {𝑛}) → (∃𝑘 ∈ ran ℤ≥(𝐹 “ 𝑘) ⊆ ∩ 𝑚 ↔ ∃𝑘 ∈ ran
ℤ≥(𝐹
“ 𝑘) ⊆ ∩ (𝑦
∪ {𝑛}))) |
49 | 45, 48 | imbi12d 333 |
. . . . . . . . . . . 12
⊢ (𝑚 = (𝑦 ∪ {𝑛}) → (((𝜑 ∧ ∀𝑘 ∈ 𝑚 ∃𝑢 ∈ ran ℤ≥𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢))) → ∃𝑘 ∈ ran ℤ≥(𝐹 “ 𝑘) ⊆ ∩ 𝑚) ↔ ((𝜑 ∧ ∀𝑘 ∈ (𝑦 ∪ {𝑛})∃𝑢 ∈ ran ℤ≥𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢))) → ∃𝑘 ∈ ran ℤ≥(𝐹 “ 𝑘) ⊆ ∩ (𝑦 ∪ {𝑛})))) |
50 | | raleq 3115 |
. . . . . . . . . . . . . 14
⊢ (𝑚 = 𝑟 → (∀𝑘 ∈ 𝑚 ∃𝑢 ∈ ran ℤ≥𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢)) ↔ ∀𝑘 ∈ 𝑟 ∃𝑢 ∈ ran ℤ≥𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢)))) |
51 | 50 | anbi2d 736 |
. . . . . . . . . . . . 13
⊢ (𝑚 = 𝑟 → ((𝜑 ∧ ∀𝑘 ∈ 𝑚 ∃𝑢 ∈ ran ℤ≥𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢))) ↔ (𝜑 ∧ ∀𝑘 ∈ 𝑟 ∃𝑢 ∈ ran ℤ≥𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢))))) |
52 | | inteq 4413 |
. . . . . . . . . . . . . . 15
⊢ (𝑚 = 𝑟 → ∩ 𝑚 = ∩
𝑟) |
53 | 52 | sseq2d 3596 |
. . . . . . . . . . . . . 14
⊢ (𝑚 = 𝑟 → ((𝐹 “ 𝑘) ⊆ ∩ 𝑚 ↔ (𝐹 “ 𝑘) ⊆ ∩ 𝑟)) |
54 | 53 | rexbidv 3034 |
. . . . . . . . . . . . 13
⊢ (𝑚 = 𝑟 → (∃𝑘 ∈ ran ℤ≥(𝐹 “ 𝑘) ⊆ ∩ 𝑚 ↔ ∃𝑘 ∈ ran
ℤ≥(𝐹
“ 𝑘) ⊆ ∩ 𝑟)) |
55 | 51, 54 | imbi12d 333 |
. . . . . . . . . . . 12
⊢ (𝑚 = 𝑟 → (((𝜑 ∧ ∀𝑘 ∈ 𝑚 ∃𝑢 ∈ ran ℤ≥𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢))) → ∃𝑘 ∈ ran ℤ≥(𝐹 “ 𝑘) ⊆ ∩ 𝑚) ↔ ((𝜑 ∧ ∀𝑘 ∈ 𝑟 ∃𝑢 ∈ ran ℤ≥𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢))) → ∃𝑘 ∈ ran ℤ≥(𝐹 “ 𝑘) ⊆ ∩ 𝑟))) |
56 | | uzf 11566 |
. . . . . . . . . . . . . . . 16
⊢
ℤ≥:ℤ⟶𝒫 ℤ |
57 | | ffn 5958 |
. . . . . . . . . . . . . . . 16
⊢
(ℤ≥:ℤ⟶𝒫 ℤ →
ℤ≥ Fn ℤ) |
58 | 56, 57 | ax-mp 5 |
. . . . . . . . . . . . . . 15
⊢
ℤ≥ Fn ℤ |
59 | | 0z 11265 |
. . . . . . . . . . . . . . 15
⊢ 0 ∈
ℤ |
60 | | fnfvelrn 6264 |
. . . . . . . . . . . . . . 15
⊢
((ℤ≥ Fn ℤ ∧ 0 ∈ ℤ) →
(ℤ≥‘0) ∈ ran
ℤ≥) |
61 | 58, 59, 60 | mp2an 704 |
. . . . . . . . . . . . . 14
⊢
(ℤ≥‘0) ∈ ran
ℤ≥ |
62 | | ssv 3588 |
. . . . . . . . . . . . . . 15
⊢ (𝐹 “
(ℤ≥‘0)) ⊆ V |
63 | | int0 4425 |
. . . . . . . . . . . . . . 15
⊢ ∩ ∅ = V |
64 | 62, 63 | sseqtr4i 3601 |
. . . . . . . . . . . . . 14
⊢ (𝐹 “
(ℤ≥‘0)) ⊆ ∩
∅ |
65 | | imaeq2 5381 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 =
(ℤ≥‘0) → (𝐹 “ 𝑘) = (𝐹 “
(ℤ≥‘0))) |
66 | 65 | sseq1d 3595 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 =
(ℤ≥‘0) → ((𝐹 “ 𝑘) ⊆ ∩
∅ ↔ (𝐹 “
(ℤ≥‘0)) ⊆ ∩
∅)) |
67 | 66 | rspcev 3282 |
. . . . . . . . . . . . . 14
⊢
(((ℤ≥‘0) ∈ ran ℤ≥
∧ (𝐹 “
(ℤ≥‘0)) ⊆ ∩ ∅)
→ ∃𝑘 ∈ ran
ℤ≥(𝐹
“ 𝑘) ⊆ ∩ ∅) |
68 | 61, 64, 67 | mp2an 704 |
. . . . . . . . . . . . 13
⊢
∃𝑘 ∈ ran
ℤ≥(𝐹
“ 𝑘) ⊆ ∩ ∅ |
69 | 68 | a1i 11 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ∀𝑘 ∈ ∅ ∃𝑢 ∈ ran ℤ≥𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢))) → ∃𝑘 ∈ ran ℤ≥(𝐹 “ 𝑘) ⊆ ∩
∅) |
70 | | ssun1 3738 |
. . . . . . . . . . . . . . . . 17
⊢ 𝑦 ⊆ (𝑦 ∪ {𝑛}) |
71 | | ssralv 3629 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑦 ⊆ (𝑦 ∪ {𝑛}) → (∀𝑘 ∈ (𝑦 ∪ {𝑛})∃𝑢 ∈ ran ℤ≥𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢)) → ∀𝑘 ∈ 𝑦 ∃𝑢 ∈ ran ℤ≥𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢)))) |
72 | 70, 71 | ax-mp 5 |
. . . . . . . . . . . . . . . 16
⊢
(∀𝑘 ∈
(𝑦 ∪ {𝑛})∃𝑢 ∈ ran ℤ≥𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢)) → ∀𝑘 ∈ 𝑦 ∃𝑢 ∈ ran ℤ≥𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢))) |
73 | 72 | anim2i 591 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ ∀𝑘 ∈ (𝑦 ∪ {𝑛})∃𝑢 ∈ ran ℤ≥𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢))) → (𝜑 ∧ ∀𝑘 ∈ 𝑦 ∃𝑢 ∈ ran ℤ≥𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢)))) |
74 | 73 | imim1i 61 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ ∀𝑘 ∈ 𝑦 ∃𝑢 ∈ ran ℤ≥𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢))) → ∃𝑘 ∈ ran ℤ≥(𝐹 “ 𝑘) ⊆ ∩ 𝑦) → ((𝜑 ∧ ∀𝑘 ∈ (𝑦 ∪ {𝑛})∃𝑢 ∈ ran ℤ≥𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢))) → ∃𝑘 ∈ ran ℤ≥(𝐹 “ 𝑘) ⊆ ∩ 𝑦)) |
75 | | ssun2 3739 |
. . . . . . . . . . . . . . . . . 18
⊢ {𝑛} ⊆ (𝑦 ∪ {𝑛}) |
76 | | ssralv 3629 |
. . . . . . . . . . . . . . . . . 18
⊢ ({𝑛} ⊆ (𝑦 ∪ {𝑛}) → (∀𝑘 ∈ (𝑦 ∪ {𝑛})∃𝑢 ∈ ran ℤ≥𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢)) → ∀𝑘 ∈ {𝑛}∃𝑢 ∈ ran ℤ≥𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢)))) |
77 | 75, 76 | ax-mp 5 |
. . . . . . . . . . . . . . . . 17
⊢
(∀𝑘 ∈
(𝑦 ∪ {𝑛})∃𝑢 ∈ ran ℤ≥𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢)) → ∀𝑘 ∈ {𝑛}∃𝑢 ∈ ran ℤ≥𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢))) |
78 | | vex 3176 |
. . . . . . . . . . . . . . . . . 18
⊢ 𝑛 ∈ V |
79 | | eqeq1 2614 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 = 𝑛 → (𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢)) ↔ 𝑛 = ((cls‘𝐽)‘(𝐹 “ 𝑢)))) |
80 | 79 | rexbidv 3034 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 = 𝑛 → (∃𝑢 ∈ ran ℤ≥𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢)) ↔ ∃𝑢 ∈ ran ℤ≥𝑛 = ((cls‘𝐽)‘(𝐹 “ 𝑢)))) |
81 | 78, 80 | ralsn 4169 |
. . . . . . . . . . . . . . . . 17
⊢
(∀𝑘 ∈
{𝑛}∃𝑢 ∈ ran
ℤ≥𝑘 =
((cls‘𝐽)‘(𝐹 “ 𝑢)) ↔ ∃𝑢 ∈ ran ℤ≥𝑛 = ((cls‘𝐽)‘(𝐹 “ 𝑢))) |
82 | 77, 81 | sylib 207 |
. . . . . . . . . . . . . . . 16
⊢
(∀𝑘 ∈
(𝑦 ∪ {𝑛})∃𝑢 ∈ ran ℤ≥𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢)) → ∃𝑢 ∈ ran ℤ≥𝑛 = ((cls‘𝐽)‘(𝐹 “ 𝑢))) |
83 | | uzin2 13932 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑢 ∈ ran
ℤ≥ ∧ 𝑘 ∈ ran ℤ≥) →
(𝑢 ∩ 𝑘) ∈ ran
ℤ≥) |
84 | 8, 11 | syl5ss 3579 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝜑 → (𝐹 “ 𝑢) ⊆ 𝑋) |
85 | 84, 13 | sseqtrd 3604 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝜑 → (𝐹 “ 𝑢) ⊆ ∪ 𝐽) |
86 | 16 | sscls 20670 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝐽 ∈ Top ∧ (𝐹 “ 𝑢) ⊆ ∪ 𝐽) → (𝐹 “ 𝑢) ⊆ ((cls‘𝐽)‘(𝐹 “ 𝑢))) |
87 | 7, 85, 86 | syl2anc 691 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝜑 → (𝐹 “ 𝑢) ⊆ ((cls‘𝐽)‘(𝐹 “ 𝑢))) |
88 | | sseq2 3590 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑛 = ((cls‘𝐽)‘(𝐹 “ 𝑢)) → ((𝐹 “ 𝑢) ⊆ 𝑛 ↔ (𝐹 “ 𝑢) ⊆ ((cls‘𝐽)‘(𝐹 “ 𝑢)))) |
89 | 87, 88 | syl5ibrcom 236 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → (𝑛 = ((cls‘𝐽)‘(𝐹 “ 𝑢)) → (𝐹 “ 𝑢) ⊆ 𝑛)) |
90 | | inss2 3796 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑢 ∩ 𝑘) ⊆ 𝑘 |
91 | | inss1 3795 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑢 ∩ 𝑘) ⊆ 𝑢 |
92 | | imass2 5420 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ ((𝑢 ∩ 𝑘) ⊆ 𝑘 → (𝐹 “ (𝑢 ∩ 𝑘)) ⊆ (𝐹 “ 𝑘)) |
93 | | imass2 5420 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ ((𝑢 ∩ 𝑘) ⊆ 𝑢 → (𝐹 “ (𝑢 ∩ 𝑘)) ⊆ (𝐹 “ 𝑢)) |
94 | 92, 93 | anim12i 588 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (((𝑢 ∩ 𝑘) ⊆ 𝑘 ∧ (𝑢 ∩ 𝑘) ⊆ 𝑢) → ((𝐹 “ (𝑢 ∩ 𝑘)) ⊆ (𝐹 “ 𝑘) ∧ (𝐹 “ (𝑢 ∩ 𝑘)) ⊆ (𝐹 “ 𝑢))) |
95 | | ssin 3797 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (((𝐹 “ (𝑢 ∩ 𝑘)) ⊆ (𝐹 “ 𝑘) ∧ (𝐹 “ (𝑢 ∩ 𝑘)) ⊆ (𝐹 “ 𝑢)) ↔ (𝐹 “ (𝑢 ∩ 𝑘)) ⊆ ((𝐹 “ 𝑘) ∩ (𝐹 “ 𝑢))) |
96 | 94, 95 | sylib 207 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (((𝑢 ∩ 𝑘) ⊆ 𝑘 ∧ (𝑢 ∩ 𝑘) ⊆ 𝑢) → (𝐹 “ (𝑢 ∩ 𝑘)) ⊆ ((𝐹 “ 𝑘) ∩ (𝐹 “ 𝑢))) |
97 | 90, 91, 96 | mp2an 704 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝐹 “ (𝑢 ∩ 𝑘)) ⊆ ((𝐹 “ 𝑘) ∩ (𝐹 “ 𝑢)) |
98 | | ss2in 3802 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (((𝐹 “ 𝑘) ⊆ ∩ 𝑦 ∧ (𝐹 “ 𝑢) ⊆ 𝑛) → ((𝐹 “ 𝑘) ∩ (𝐹 “ 𝑢)) ⊆ (∩
𝑦 ∩ 𝑛)) |
99 | 97, 98 | syl5ss 3579 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝐹 “ 𝑘) ⊆ ∩ 𝑦 ∧ (𝐹 “ 𝑢) ⊆ 𝑛) → (𝐹 “ (𝑢 ∩ 𝑘)) ⊆ (∩
𝑦 ∩ 𝑛)) |
100 | 78 | intunsn 4451 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ∩ (𝑦
∪ {𝑛}) = (∩ 𝑦
∩ 𝑛) |
101 | 99, 100 | syl6sseqr 3615 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝐹 “ 𝑘) ⊆ ∩ 𝑦 ∧ (𝐹 “ 𝑢) ⊆ 𝑛) → (𝐹 “ (𝑢 ∩ 𝑘)) ⊆ ∩
(𝑦 ∪ {𝑛})) |
102 | 101 | expcom 450 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝐹 “ 𝑢) ⊆ 𝑛 → ((𝐹 “ 𝑘) ⊆ ∩ 𝑦 → (𝐹 “ (𝑢 ∩ 𝑘)) ⊆ ∩
(𝑦 ∪ {𝑛}))) |
103 | 89, 102 | syl6 34 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → (𝑛 = ((cls‘𝐽)‘(𝐹 “ 𝑢)) → ((𝐹 “ 𝑘) ⊆ ∩ 𝑦 → (𝐹 “ (𝑢 ∩ 𝑘)) ⊆ ∩
(𝑦 ∪ {𝑛})))) |
104 | 103 | impd 446 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → ((𝑛 = ((cls‘𝐽)‘(𝐹 “ 𝑢)) ∧ (𝐹 “ 𝑘) ⊆ ∩ 𝑦) → (𝐹 “ (𝑢 ∩ 𝑘)) ⊆ ∩
(𝑦 ∪ {𝑛}))) |
105 | | imaeq2 5381 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑚 = (𝑢 ∩ 𝑘) → (𝐹 “ 𝑚) = (𝐹 “ (𝑢 ∩ 𝑘))) |
106 | 105 | sseq1d 3595 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑚 = (𝑢 ∩ 𝑘) → ((𝐹 “ 𝑚) ⊆ ∩ (𝑦 ∪ {𝑛}) ↔ (𝐹 “ (𝑢 ∩ 𝑘)) ⊆ ∩
(𝑦 ∪ {𝑛}))) |
107 | 106 | rspcev 3282 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝑢 ∩ 𝑘) ∈ ran ℤ≥ ∧
(𝐹 “ (𝑢 ∩ 𝑘)) ⊆ ∩
(𝑦 ∪ {𝑛})) → ∃𝑚 ∈ ran
ℤ≥(𝐹
“ 𝑚) ⊆ ∩ (𝑦
∪ {𝑛})) |
108 | 107 | expcom 450 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝐹 “ (𝑢 ∩ 𝑘)) ⊆ ∩
(𝑦 ∪ {𝑛}) → ((𝑢 ∩ 𝑘) ∈ ran ℤ≥ →
∃𝑚 ∈ ran
ℤ≥(𝐹
“ 𝑚) ⊆ ∩ (𝑦
∪ {𝑛}))) |
109 | 104, 108 | syl6 34 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → ((𝑛 = ((cls‘𝐽)‘(𝐹 “ 𝑢)) ∧ (𝐹 “ 𝑘) ⊆ ∩ 𝑦) → ((𝑢 ∩ 𝑘) ∈ ran ℤ≥ →
∃𝑚 ∈ ran
ℤ≥(𝐹
“ 𝑚) ⊆ ∩ (𝑦
∪ {𝑛})))) |
110 | 109 | com23 84 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → ((𝑢 ∩ 𝑘) ∈ ran ℤ≥ →
((𝑛 = ((cls‘𝐽)‘(𝐹 “ 𝑢)) ∧ (𝐹 “ 𝑘) ⊆ ∩ 𝑦) → ∃𝑚 ∈ ran
ℤ≥(𝐹
“ 𝑚) ⊆ ∩ (𝑦
∪ {𝑛})))) |
111 | 83, 110 | syl5 33 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → ((𝑢 ∈ ran ℤ≥ ∧
𝑘 ∈ ran
ℤ≥) → ((𝑛 = ((cls‘𝐽)‘(𝐹 “ 𝑢)) ∧ (𝐹 “ 𝑘) ⊆ ∩ 𝑦) → ∃𝑚 ∈ ran
ℤ≥(𝐹
“ 𝑚) ⊆ ∩ (𝑦
∪ {𝑛})))) |
112 | 111 | rexlimdvv 3019 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (∃𝑢 ∈ ran
ℤ≥∃𝑘 ∈ ran ℤ≥(𝑛 = ((cls‘𝐽)‘(𝐹 “ 𝑢)) ∧ (𝐹 “ 𝑘) ⊆ ∩ 𝑦) → ∃𝑚 ∈ ran
ℤ≥(𝐹
“ 𝑚) ⊆ ∩ (𝑦
∪ {𝑛}))) |
113 | | reeanv 3086 |
. . . . . . . . . . . . . . . . . 18
⊢
(∃𝑢 ∈ ran
ℤ≥∃𝑘 ∈ ran ℤ≥(𝑛 = ((cls‘𝐽)‘(𝐹 “ 𝑢)) ∧ (𝐹 “ 𝑘) ⊆ ∩ 𝑦) ↔ (∃𝑢 ∈ ran
ℤ≥𝑛 =
((cls‘𝐽)‘(𝐹 “ 𝑢)) ∧ ∃𝑘 ∈ ran ℤ≥(𝐹 “ 𝑘) ⊆ ∩ 𝑦)) |
114 | | imaeq2 5381 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑚 = 𝑘 → (𝐹 “ 𝑚) = (𝐹 “ 𝑘)) |
115 | 114 | sseq1d 3595 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑚 = 𝑘 → ((𝐹 “ 𝑚) ⊆ ∩ (𝑦 ∪ {𝑛}) ↔ (𝐹 “ 𝑘) ⊆ ∩ (𝑦 ∪ {𝑛}))) |
116 | 115 | cbvrexv 3148 |
. . . . . . . . . . . . . . . . . 18
⊢
(∃𝑚 ∈ ran
ℤ≥(𝐹
“ 𝑚) ⊆ ∩ (𝑦
∪ {𝑛}) ↔
∃𝑘 ∈ ran
ℤ≥(𝐹
“ 𝑘) ⊆ ∩ (𝑦
∪ {𝑛})) |
117 | 112, 113,
116 | 3imtr3g 283 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → ((∃𝑢 ∈ ran
ℤ≥𝑛 =
((cls‘𝐽)‘(𝐹 “ 𝑢)) ∧ ∃𝑘 ∈ ran ℤ≥(𝐹 “ 𝑘) ⊆ ∩ 𝑦) → ∃𝑘 ∈ ran
ℤ≥(𝐹
“ 𝑘) ⊆ ∩ (𝑦
∪ {𝑛}))) |
118 | 117 | expd 451 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (∃𝑢 ∈ ran ℤ≥𝑛 = ((cls‘𝐽)‘(𝐹 “ 𝑢)) → (∃𝑘 ∈ ran ℤ≥(𝐹 “ 𝑘) ⊆ ∩ 𝑦 → ∃𝑘 ∈ ran
ℤ≥(𝐹
“ 𝑘) ⊆ ∩ (𝑦
∪ {𝑛})))) |
119 | 82, 118 | syl5 33 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (∀𝑘 ∈ (𝑦 ∪ {𝑛})∃𝑢 ∈ ran ℤ≥𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢)) → (∃𝑘 ∈ ran ℤ≥(𝐹 “ 𝑘) ⊆ ∩ 𝑦 → ∃𝑘 ∈ ran
ℤ≥(𝐹
“ 𝑘) ⊆ ∩ (𝑦
∪ {𝑛})))) |
120 | 119 | imp 444 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ ∀𝑘 ∈ (𝑦 ∪ {𝑛})∃𝑢 ∈ ran ℤ≥𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢))) → (∃𝑘 ∈ ran ℤ≥(𝐹 “ 𝑘) ⊆ ∩ 𝑦 → ∃𝑘 ∈ ran
ℤ≥(𝐹
“ 𝑘) ⊆ ∩ (𝑦
∪ {𝑛}))) |
121 | 74, 120 | sylcom 30 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ ∀𝑘 ∈ 𝑦 ∃𝑢 ∈ ran ℤ≥𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢))) → ∃𝑘 ∈ ran ℤ≥(𝐹 “ 𝑘) ⊆ ∩ 𝑦) → ((𝜑 ∧ ∀𝑘 ∈ (𝑦 ∪ {𝑛})∃𝑢 ∈ ran ℤ≥𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢))) → ∃𝑘 ∈ ran ℤ≥(𝐹 “ 𝑘) ⊆ ∩ (𝑦 ∪ {𝑛}))) |
122 | 121 | a1i 11 |
. . . . . . . . . . . 12
⊢ (𝑦 ∈ Fin → (((𝜑 ∧ ∀𝑘 ∈ 𝑦 ∃𝑢 ∈ ran ℤ≥𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢))) → ∃𝑘 ∈ ran ℤ≥(𝐹 “ 𝑘) ⊆ ∩ 𝑦) → ((𝜑 ∧ ∀𝑘 ∈ (𝑦 ∪ {𝑛})∃𝑢 ∈ ran ℤ≥𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢))) → ∃𝑘 ∈ ran ℤ≥(𝐹 “ 𝑘) ⊆ ∩ (𝑦 ∪ {𝑛})))) |
123 | 37, 43, 49, 55, 69, 122 | findcard2 8085 |
. . . . . . . . . . 11
⊢ (𝑟 ∈ Fin → ((𝜑 ∧ ∀𝑘 ∈ 𝑟 ∃𝑢 ∈ ran ℤ≥𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢))) → ∃𝑘 ∈ ran ℤ≥(𝐹 “ 𝑘) ⊆ ∩ 𝑟)) |
124 | 123 | com12 32 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ∀𝑘 ∈ 𝑟 ∃𝑢 ∈ ran ℤ≥𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢))) → (𝑟 ∈ Fin → ∃𝑘 ∈ ran ℤ≥(𝐹 “ 𝑘) ⊆ ∩ 𝑟)) |
125 | 124 | impr 647 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (∀𝑘 ∈ 𝑟 ∃𝑢 ∈ ran ℤ≥𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢)) ∧ 𝑟 ∈ Fin)) → ∃𝑘 ∈ ran
ℤ≥(𝐹
“ 𝑘) ⊆ ∩ 𝑟) |
126 | | ffn 5958 |
. . . . . . . . . . . 12
⊢ (𝐹:ℕ⟶𝑋 → 𝐹 Fn ℕ) |
127 | 9, 126 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐹 Fn ℕ) |
128 | | inss1 3795 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 ∩ ℕ) ⊆ 𝑘 |
129 | | imass2 5420 |
. . . . . . . . . . . . . . 15
⊢ ((𝑘 ∩ ℕ) ⊆ 𝑘 → (𝐹 “ (𝑘 ∩ ℕ)) ⊆ (𝐹 “ 𝑘)) |
130 | 128, 129 | ax-mp 5 |
. . . . . . . . . . . . . 14
⊢ (𝐹 “ (𝑘 ∩ ℕ)) ⊆ (𝐹 “ 𝑘) |
131 | | nnuz 11599 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ℕ =
(ℤ≥‘1) |
132 | | 1z 11284 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ 1 ∈
ℤ |
133 | | fnfvelrn 6264 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
((ℤ≥ Fn ℤ ∧ 1 ∈ ℤ) →
(ℤ≥‘1) ∈ ran
ℤ≥) |
134 | 58, 132, 133 | mp2an 704 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(ℤ≥‘1) ∈ ran
ℤ≥ |
135 | 131, 134 | eqeltri 2684 |
. . . . . . . . . . . . . . . . . . 19
⊢ ℕ
∈ ran ℤ≥ |
136 | | uzin2 13932 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑘 ∈ ran
ℤ≥ ∧ ℕ ∈ ran ℤ≥) →
(𝑘 ∩ ℕ) ∈
ran ℤ≥) |
137 | 135, 136 | mpan2 703 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 ∈ ran
ℤ≥ → (𝑘 ∩ ℕ) ∈ ran
ℤ≥) |
138 | | uzn0 11579 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑘 ∩ ℕ) ∈ ran
ℤ≥ → (𝑘 ∩ ℕ) ≠
∅) |
139 | 137, 138 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 ∈ ran
ℤ≥ → (𝑘 ∩ ℕ) ≠
∅) |
140 | | n0 3890 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑘 ∩ ℕ) ≠ ∅
↔ ∃𝑦 𝑦 ∈ (𝑘 ∩ ℕ)) |
141 | 139, 140 | sylib 207 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 ∈ ran
ℤ≥ → ∃𝑦 𝑦 ∈ (𝑘 ∩ ℕ)) |
142 | | fnfun 5902 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝐹 Fn ℕ → Fun 𝐹) |
143 | | inss2 3796 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 ∩ ℕ) ⊆
ℕ |
144 | | fndm 5904 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝐹 Fn ℕ → dom 𝐹 = ℕ) |
145 | 143, 144 | syl5sseqr 3617 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝐹 Fn ℕ → (𝑘 ∩ ℕ) ⊆ dom
𝐹) |
146 | | funfvima2 6397 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((Fun
𝐹 ∧ (𝑘 ∩ ℕ) ⊆ dom 𝐹) → (𝑦 ∈ (𝑘 ∩ ℕ) → (𝐹‘𝑦) ∈ (𝐹 “ (𝑘 ∩ ℕ)))) |
147 | 142, 145,
146 | syl2anc 691 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝐹 Fn ℕ → (𝑦 ∈ (𝑘 ∩ ℕ) → (𝐹‘𝑦) ∈ (𝐹 “ (𝑘 ∩ ℕ)))) |
148 | | ne0i 3880 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝐹‘𝑦) ∈ (𝐹 “ (𝑘 ∩ ℕ)) → (𝐹 “ (𝑘 ∩ ℕ)) ≠
∅) |
149 | 147, 148 | syl6 34 |
. . . . . . . . . . . . . . . . 17
⊢ (𝐹 Fn ℕ → (𝑦 ∈ (𝑘 ∩ ℕ) → (𝐹 “ (𝑘 ∩ ℕ)) ≠
∅)) |
150 | 149 | exlimdv 1848 |
. . . . . . . . . . . . . . . 16
⊢ (𝐹 Fn ℕ → (∃𝑦 𝑦 ∈ (𝑘 ∩ ℕ) → (𝐹 “ (𝑘 ∩ ℕ)) ≠
∅)) |
151 | 141, 150 | syl5 33 |
. . . . . . . . . . . . . . 15
⊢ (𝐹 Fn ℕ → (𝑘 ∈ ran
ℤ≥ → (𝐹 “ (𝑘 ∩ ℕ)) ≠
∅)) |
152 | 151 | imp 444 |
. . . . . . . . . . . . . 14
⊢ ((𝐹 Fn ℕ ∧ 𝑘 ∈ ran
ℤ≥) → (𝐹 “ (𝑘 ∩ ℕ)) ≠
∅) |
153 | | ssn0 3928 |
. . . . . . . . . . . . . 14
⊢ (((𝐹 “ (𝑘 ∩ ℕ)) ⊆ (𝐹 “ 𝑘) ∧ (𝐹 “ (𝑘 ∩ ℕ)) ≠ ∅) → (𝐹 “ 𝑘) ≠ ∅) |
154 | 130, 152,
153 | sylancr 694 |
. . . . . . . . . . . . 13
⊢ ((𝐹 Fn ℕ ∧ 𝑘 ∈ ran
ℤ≥) → (𝐹 “ 𝑘) ≠ ∅) |
155 | | ssn0 3928 |
. . . . . . . . . . . . . 14
⊢ (((𝐹 “ 𝑘) ⊆ ∩ 𝑟 ∧ (𝐹 “ 𝑘) ≠ ∅) → ∩ 𝑟
≠ ∅) |
156 | 155 | expcom 450 |
. . . . . . . . . . . . 13
⊢ ((𝐹 “ 𝑘) ≠ ∅ → ((𝐹 “ 𝑘) ⊆ ∩ 𝑟 → ∩ 𝑟
≠ ∅)) |
157 | 154, 156 | syl 17 |
. . . . . . . . . . . 12
⊢ ((𝐹 Fn ℕ ∧ 𝑘 ∈ ran
ℤ≥) → ((𝐹 “ 𝑘) ⊆ ∩ 𝑟 → ∩ 𝑟
≠ ∅)) |
158 | 157 | rexlimdva 3013 |
. . . . . . . . . . 11
⊢ (𝐹 Fn ℕ → (∃𝑘 ∈ ran
ℤ≥(𝐹
“ 𝑘) ⊆ ∩ 𝑟
→ ∩ 𝑟 ≠ ∅)) |
159 | 127, 158 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → (∃𝑘 ∈ ran ℤ≥(𝐹 “ 𝑘) ⊆ ∩ 𝑟 → ∩ 𝑟
≠ ∅)) |
160 | 159 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (∀𝑘 ∈ 𝑟 ∃𝑢 ∈ ran ℤ≥𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢)) ∧ 𝑟 ∈ Fin)) → (∃𝑘 ∈ ran
ℤ≥(𝐹
“ 𝑘) ⊆ ∩ 𝑟
→ ∩ 𝑟 ≠ ∅)) |
161 | 125, 160 | mpd 15 |
. . . . . . . 8
⊢ ((𝜑 ∧ (∀𝑘 ∈ 𝑟 ∃𝑢 ∈ ran ℤ≥𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢)) ∧ 𝑟 ∈ Fin)) → ∩ 𝑟
≠ ∅) |
162 | 161 | necomd 2837 |
. . . . . . 7
⊢ ((𝜑 ∧ (∀𝑘 ∈ 𝑟 ∃𝑢 ∈ ran ℤ≥𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢)) ∧ 𝑟 ∈ Fin)) → ∅ ≠ ∩ 𝑟) |
163 | 162 | neneqd 2787 |
. . . . . 6
⊢ ((𝜑 ∧ (∀𝑘 ∈ 𝑟 ∃𝑢 ∈ ran ℤ≥𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢)) ∧ 𝑟 ∈ Fin)) → ¬ ∅ = ∩ 𝑟) |
164 | 31, 163 | sylan2b 491 |
. . . . 5
⊢ ((𝜑 ∧ 𝑟 ∈ (𝒫 {𝑘 ∣ ∃𝑢 ∈ ran ℤ≥𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢))} ∩ Fin)) → ¬ ∅ = ∩ 𝑟) |
165 | 164 | nrexdv 2984 |
. . . 4
⊢ (𝜑 → ¬ ∃𝑟 ∈ (𝒫 {𝑘 ∣ ∃𝑢 ∈ ran
ℤ≥𝑘 =
((cls‘𝐽)‘(𝐹 “ 𝑢))} ∩ Fin)∅ = ∩ 𝑟) |
166 | | 0ex 4718 |
. . . . 5
⊢ ∅
∈ V |
167 | | zex 11263 |
. . . . . . . 8
⊢ ℤ
∈ V |
168 | 167 | pwex 4774 |
. . . . . . 7
⊢ 𝒫
ℤ ∈ V |
169 | | frn 5966 |
. . . . . . . 8
⊢
(ℤ≥:ℤ⟶𝒫 ℤ → ran
ℤ≥ ⊆ 𝒫 ℤ) |
170 | 56, 169 | ax-mp 5 |
. . . . . . 7
⊢ ran
ℤ≥ ⊆ 𝒫 ℤ |
171 | 168, 170 | ssexi 4731 |
. . . . . 6
⊢ ran
ℤ≥ ∈ V |
172 | 171 | abrexex 7033 |
. . . . 5
⊢ {𝑘 ∣ ∃𝑢 ∈ ran
ℤ≥𝑘 =
((cls‘𝐽)‘(𝐹 “ 𝑢))} ∈ V |
173 | | elfi 8202 |
. . . . 5
⊢ ((∅
∈ V ∧ {𝑘 ∣
∃𝑢 ∈ ran
ℤ≥𝑘 =
((cls‘𝐽)‘(𝐹 “ 𝑢))} ∈ V) → (∅ ∈
(fi‘{𝑘 ∣
∃𝑢 ∈ ran
ℤ≥𝑘 =
((cls‘𝐽)‘(𝐹 “ 𝑢))}) ↔ ∃𝑟 ∈ (𝒫 {𝑘 ∣ ∃𝑢 ∈ ran ℤ≥𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢))} ∩ Fin)∅ = ∩ 𝑟)) |
174 | 166, 172,
173 | mp2an 704 |
. . . 4
⊢ (∅
∈ (fi‘{𝑘 ∣
∃𝑢 ∈ ran
ℤ≥𝑘 =
((cls‘𝐽)‘(𝐹 “ 𝑢))}) ↔ ∃𝑟 ∈ (𝒫 {𝑘 ∣ ∃𝑢 ∈ ran ℤ≥𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢))} ∩ Fin)∅ = ∩ 𝑟) |
175 | 165, 174 | sylnibr 318 |
. . 3
⊢ (𝜑 → ¬ ∅ ∈
(fi‘{𝑘 ∣
∃𝑢 ∈ ran
ℤ≥𝑘 =
((cls‘𝐽)‘(𝐹 “ 𝑢))})) |
176 | | cmptop 21008 |
. . . . . 6
⊢ (𝐽 ∈ Comp → 𝐽 ∈ Top) |
177 | | cmpfi 21021 |
. . . . . 6
⊢ (𝐽 ∈ Top → (𝐽 ∈ Comp ↔
∀𝑚 ∈ 𝒫
(Clsd‘𝐽)(¬
∅ ∈ (fi‘𝑚)
→ ∩ 𝑚 ≠ ∅))) |
178 | 176, 177 | syl 17 |
. . . . 5
⊢ (𝐽 ∈ Comp → (𝐽 ∈ Comp ↔
∀𝑚 ∈ 𝒫
(Clsd‘𝐽)(¬
∅ ∈ (fi‘𝑚)
→ ∩ 𝑚 ≠ ∅))) |
179 | 178 | ibi 255 |
. . . 4
⊢ (𝐽 ∈ Comp →
∀𝑚 ∈ 𝒫
(Clsd‘𝐽)(¬
∅ ∈ (fi‘𝑚)
→ ∩ 𝑚 ≠ ∅)) |
180 | | fveq2 6103 |
. . . . . . . 8
⊢ (𝑚 = {𝑘 ∣ ∃𝑢 ∈ ran ℤ≥𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢))} → (fi‘𝑚) = (fi‘{𝑘 ∣ ∃𝑢 ∈ ran ℤ≥𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢))})) |
181 | 180 | eleq2d 2673 |
. . . . . . 7
⊢ (𝑚 = {𝑘 ∣ ∃𝑢 ∈ ran ℤ≥𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢))} → (∅ ∈ (fi‘𝑚) ↔ ∅ ∈
(fi‘{𝑘 ∣
∃𝑢 ∈ ran
ℤ≥𝑘 =
((cls‘𝐽)‘(𝐹 “ 𝑢))}))) |
182 | 181 | notbid 307 |
. . . . . 6
⊢ (𝑚 = {𝑘 ∣ ∃𝑢 ∈ ran ℤ≥𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢))} → (¬ ∅ ∈
(fi‘𝑚) ↔ ¬
∅ ∈ (fi‘{𝑘
∣ ∃𝑢 ∈ ran
ℤ≥𝑘 =
((cls‘𝐽)‘(𝐹 “ 𝑢))}))) |
183 | | inteq 4413 |
. . . . . . . 8
⊢ (𝑚 = {𝑘 ∣ ∃𝑢 ∈ ran ℤ≥𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢))} → ∩ 𝑚 = ∩
{𝑘 ∣ ∃𝑢 ∈ ran
ℤ≥𝑘 =
((cls‘𝐽)‘(𝐹 “ 𝑢))}) |
184 | 183 | neeq1d 2841 |
. . . . . . 7
⊢ (𝑚 = {𝑘 ∣ ∃𝑢 ∈ ran ℤ≥𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢))} → (∩
𝑚 ≠ ∅ ↔ ∩ {𝑘
∣ ∃𝑢 ∈ ran
ℤ≥𝑘 =
((cls‘𝐽)‘(𝐹 “ 𝑢))} ≠ ∅)) |
185 | | n0 3890 |
. . . . . . 7
⊢ (∩ {𝑘
∣ ∃𝑢 ∈ ran
ℤ≥𝑘 =
((cls‘𝐽)‘(𝐹 “ 𝑢))} ≠ ∅ ↔ ∃𝑦 𝑦 ∈ ∩ {𝑘 ∣ ∃𝑢 ∈ ran
ℤ≥𝑘 =
((cls‘𝐽)‘(𝐹 “ 𝑢))}) |
186 | 184, 185 | syl6bb 275 |
. . . . . 6
⊢ (𝑚 = {𝑘 ∣ ∃𝑢 ∈ ran ℤ≥𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢))} → (∩
𝑚 ≠ ∅ ↔
∃𝑦 𝑦 ∈ ∩ {𝑘 ∣ ∃𝑢 ∈ ran
ℤ≥𝑘 =
((cls‘𝐽)‘(𝐹 “ 𝑢))})) |
187 | 182, 186 | imbi12d 333 |
. . . . 5
⊢ (𝑚 = {𝑘 ∣ ∃𝑢 ∈ ran ℤ≥𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢))} → ((¬ ∅ ∈
(fi‘𝑚) → ∩ 𝑚
≠ ∅) ↔ (¬ ∅ ∈ (fi‘{𝑘 ∣ ∃𝑢 ∈ ran ℤ≥𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢))}) → ∃𝑦 𝑦 ∈ ∩ {𝑘 ∣ ∃𝑢 ∈ ran
ℤ≥𝑘 =
((cls‘𝐽)‘(𝐹 “ 𝑢))}))) |
188 | 187 | rspccv 3279 |
. . . 4
⊢
(∀𝑚 ∈
𝒫 (Clsd‘𝐽)(¬ ∅ ∈ (fi‘𝑚) → ∩ 𝑚
≠ ∅) → ({𝑘
∣ ∃𝑢 ∈ ran
ℤ≥𝑘 =
((cls‘𝐽)‘(𝐹 “ 𝑢))} ∈ 𝒫 (Clsd‘𝐽) → (¬ ∅ ∈
(fi‘{𝑘 ∣
∃𝑢 ∈ ran
ℤ≥𝑘 =
((cls‘𝐽)‘(𝐹 “ 𝑢))}) → ∃𝑦 𝑦 ∈ ∩ {𝑘 ∣ ∃𝑢 ∈ ran
ℤ≥𝑘 =
((cls‘𝐽)‘(𝐹 “ 𝑢))}))) |
189 | 179, 188 | syl 17 |
. . 3
⊢ (𝐽 ∈ Comp → ({𝑘 ∣ ∃𝑢 ∈ ran
ℤ≥𝑘 =
((cls‘𝐽)‘(𝐹 “ 𝑢))} ∈ 𝒫 (Clsd‘𝐽) → (¬ ∅ ∈
(fi‘{𝑘 ∣
∃𝑢 ∈ ran
ℤ≥𝑘 =
((cls‘𝐽)‘(𝐹 “ 𝑢))}) → ∃𝑦 𝑦 ∈ ∩ {𝑘 ∣ ∃𝑢 ∈ ran
ℤ≥𝑘 =
((cls‘𝐽)‘(𝐹 “ 𝑢))}))) |
190 | 1, 25, 175, 189 | syl3c 64 |
. 2
⊢ (𝜑 → ∃𝑦 𝑦 ∈ ∩ {𝑘 ∣ ∃𝑢 ∈ ran
ℤ≥𝑘 =
((cls‘𝐽)‘(𝐹 “ 𝑢))}) |
191 | | lmrel 20844 |
. . 3
⊢ Rel
(⇝𝑡‘𝐽) |
192 | | r19.23v 3005 |
. . . . . 6
⊢
(∀𝑢 ∈
ran ℤ≥(𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢)) → 𝑦 ∈ 𝑘) ↔ (∃𝑢 ∈ ran ℤ≥𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢)) → 𝑦 ∈ 𝑘)) |
193 | 192 | albii 1737 |
. . . . 5
⊢
(∀𝑘∀𝑢 ∈ ran ℤ≥(𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢)) → 𝑦 ∈ 𝑘) ↔ ∀𝑘(∃𝑢 ∈ ran ℤ≥𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢)) → 𝑦 ∈ 𝑘)) |
194 | | fvex 6113 |
. . . . . . . 8
⊢
((cls‘𝐽)‘(𝐹 “ 𝑢)) ∈ V |
195 | | eleq2 2677 |
. . . . . . . 8
⊢ (𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢)) → (𝑦 ∈ 𝑘 ↔ 𝑦 ∈ ((cls‘𝐽)‘(𝐹 “ 𝑢)))) |
196 | 194, 195 | ceqsalv 3206 |
. . . . . . 7
⊢
(∀𝑘(𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢)) → 𝑦 ∈ 𝑘) ↔ 𝑦 ∈ ((cls‘𝐽)‘(𝐹 “ 𝑢))) |
197 | 196 | ralbii 2963 |
. . . . . 6
⊢
(∀𝑢 ∈
ran ℤ≥∀𝑘(𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢)) → 𝑦 ∈ 𝑘) ↔ ∀𝑢 ∈ ran ℤ≥𝑦 ∈ ((cls‘𝐽)‘(𝐹 “ 𝑢))) |
198 | | ralcom4 3197 |
. . . . . 6
⊢
(∀𝑢 ∈
ran ℤ≥∀𝑘(𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢)) → 𝑦 ∈ 𝑘) ↔ ∀𝑘∀𝑢 ∈ ran ℤ≥(𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢)) → 𝑦 ∈ 𝑘)) |
199 | 197, 198 | bitr3i 265 |
. . . . 5
⊢
(∀𝑢 ∈
ran ℤ≥𝑦
∈ ((cls‘𝐽)‘(𝐹 “ 𝑢)) ↔ ∀𝑘∀𝑢 ∈ ran ℤ≥(𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢)) → 𝑦 ∈ 𝑘)) |
200 | | vex 3176 |
. . . . . 6
⊢ 𝑦 ∈ V |
201 | 200 | elintab 4422 |
. . . . 5
⊢ (𝑦 ∈ ∩ {𝑘
∣ ∃𝑢 ∈ ran
ℤ≥𝑘 =
((cls‘𝐽)‘(𝐹 “ 𝑢))} ↔ ∀𝑘(∃𝑢 ∈ ran ℤ≥𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢)) → 𝑦 ∈ 𝑘)) |
202 | 193, 199,
201 | 3bitr4i 291 |
. . . 4
⊢
(∀𝑢 ∈
ran ℤ≥𝑦
∈ ((cls‘𝐽)‘(𝐹 “ 𝑢)) ↔ 𝑦 ∈ ∩ {𝑘 ∣ ∃𝑢 ∈ ran
ℤ≥𝑘 =
((cls‘𝐽)‘(𝐹 “ 𝑢))}) |
203 | | eqid 2610 |
. . . . . . . . . . 11
⊢
((cls‘𝐽)‘(𝐹 “ ℕ)) = ((cls‘𝐽)‘(𝐹 “ ℕ)) |
204 | | imaeq2 5381 |
. . . . . . . . . . . . . 14
⊢ (𝑢 = ℕ → (𝐹 “ 𝑢) = (𝐹 “ ℕ)) |
205 | 204 | fveq2d 6107 |
. . . . . . . . . . . . 13
⊢ (𝑢 = ℕ →
((cls‘𝐽)‘(𝐹 “ 𝑢)) = ((cls‘𝐽)‘(𝐹 “ ℕ))) |
206 | 205 | eqeq2d 2620 |
. . . . . . . . . . . 12
⊢ (𝑢 = ℕ →
(((cls‘𝐽)‘(𝐹 “ ℕ)) = ((cls‘𝐽)‘(𝐹 “ 𝑢)) ↔ ((cls‘𝐽)‘(𝐹 “ ℕ)) = ((cls‘𝐽)‘(𝐹 “ ℕ)))) |
207 | 206 | rspcev 3282 |
. . . . . . . . . . 11
⊢ ((ℕ
∈ ran ℤ≥ ∧ ((cls‘𝐽)‘(𝐹 “ ℕ)) = ((cls‘𝐽)‘(𝐹 “ ℕ))) → ∃𝑢 ∈ ran
ℤ≥((cls‘𝐽)‘(𝐹 “ ℕ)) = ((cls‘𝐽)‘(𝐹 “ 𝑢))) |
208 | 135, 203,
207 | mp2an 704 |
. . . . . . . . . 10
⊢
∃𝑢 ∈ ran
ℤ≥((cls‘𝐽)‘(𝐹 “ ℕ)) = ((cls‘𝐽)‘(𝐹 “ 𝑢)) |
209 | | fvex 6113 |
. . . . . . . . . . 11
⊢
((cls‘𝐽)‘(𝐹 “ ℕ)) ∈ V |
210 | | eqeq1 2614 |
. . . . . . . . . . . 12
⊢ (𝑘 = ((cls‘𝐽)‘(𝐹 “ ℕ)) → (𝑘 = ((cls‘𝐽)‘(𝐹 “ 𝑢)) ↔ ((cls‘𝐽)‘(𝐹 “ ℕ)) = ((cls‘𝐽)‘(𝐹 “ 𝑢)))) |
211 | 210 | rexbidv 3034 |
. . . . . . . . . . 11
⊢ (𝑘 = ((cls‘𝐽)‘(𝐹 “ ℕ)) → (∃𝑢 ∈ ran
ℤ≥𝑘 =
((cls‘𝐽)‘(𝐹 “ 𝑢)) ↔ ∃𝑢 ∈ ran
ℤ≥((cls‘𝐽)‘(𝐹 “ ℕ)) = ((cls‘𝐽)‘(𝐹 “ 𝑢)))) |
212 | 209, 211 | elab 3319 |
. . . . . . . . . 10
⊢
(((cls‘𝐽)‘(𝐹 “ ℕ)) ∈ {𝑘 ∣ ∃𝑢 ∈ ran
ℤ≥𝑘 =
((cls‘𝐽)‘(𝐹 “ 𝑢))} ↔ ∃𝑢 ∈ ran
ℤ≥((cls‘𝐽)‘(𝐹 “ ℕ)) = ((cls‘𝐽)‘(𝐹 “ 𝑢))) |
213 | 208, 212 | mpbir 220 |
. . . . . . . . 9
⊢
((cls‘𝐽)‘(𝐹 “ ℕ)) ∈ {𝑘 ∣ ∃𝑢 ∈ ran
ℤ≥𝑘 =
((cls‘𝐽)‘(𝐹 “ 𝑢))} |
214 | | intss1 4427 |
. . . . . . . . 9
⊢
(((cls‘𝐽)‘(𝐹 “ ℕ)) ∈ {𝑘 ∣ ∃𝑢 ∈ ran
ℤ≥𝑘 =
((cls‘𝐽)‘(𝐹 “ 𝑢))} → ∩
{𝑘 ∣ ∃𝑢 ∈ ran
ℤ≥𝑘 =
((cls‘𝐽)‘(𝐹 “ 𝑢))} ⊆ ((cls‘𝐽)‘(𝐹 “ ℕ))) |
215 | 213, 214 | ax-mp 5 |
. . . . . . . 8
⊢ ∩ {𝑘
∣ ∃𝑢 ∈ ran
ℤ≥𝑘 =
((cls‘𝐽)‘(𝐹 “ 𝑢))} ⊆ ((cls‘𝐽)‘(𝐹 “ ℕ)) |
216 | | imassrn 5396 |
. . . . . . . . . . 11
⊢ (𝐹 “ ℕ) ⊆ ran
𝐹 |
217 | 216, 14 | syl5ss 3579 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐹 “ ℕ) ⊆ ∪ 𝐽) |
218 | 16 | clsss3 20673 |
. . . . . . . . . 10
⊢ ((𝐽 ∈ Top ∧ (𝐹 “ ℕ) ⊆ ∪ 𝐽)
→ ((cls‘𝐽)‘(𝐹 “ ℕ)) ⊆ ∪ 𝐽) |
219 | 7, 217, 218 | syl2anc 691 |
. . . . . . . . 9
⊢ (𝜑 → ((cls‘𝐽)‘(𝐹 “ ℕ)) ⊆ ∪ 𝐽) |
220 | 219, 13 | sseqtr4d 3605 |
. . . . . . . 8
⊢ (𝜑 → ((cls‘𝐽)‘(𝐹 “ ℕ)) ⊆ 𝑋) |
221 | 215, 220 | syl5ss 3579 |
. . . . . . 7
⊢ (𝜑 → ∩ {𝑘
∣ ∃𝑢 ∈ ran
ℤ≥𝑘 =
((cls‘𝐽)‘(𝐹 “ 𝑢))} ⊆ 𝑋) |
222 | 221 | sselda 3568 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ ∩ {𝑘 ∣ ∃𝑢 ∈ ran
ℤ≥𝑘 =
((cls‘𝐽)‘(𝐹 “ 𝑢))}) → 𝑦 ∈ 𝑋) |
223 | 202, 222 | sylan2b 491 |
. . . . 5
⊢ ((𝜑 ∧ ∀𝑢 ∈ ran ℤ≥𝑦 ∈ ((cls‘𝐽)‘(𝐹 “ 𝑢))) → 𝑦 ∈ 𝑋) |
224 | | heibor1.5 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐹 ∈ (Cau‘𝐷)) |
225 | | 1zzd 11285 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 1 ∈
ℤ) |
226 | 131, 4, 225 | iscau3 22884 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝐹 ∈ (Cau‘𝐷) ↔ (𝐹 ∈ (𝑋 ↑pm ℂ) ∧
∀𝑦 ∈
ℝ+ ∃𝑚 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑚)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ∀𝑛 ∈ (ℤ≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑛)) < 𝑦)))) |
227 | 224, 226 | mpbid 221 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐹 ∈ (𝑋 ↑pm ℂ) ∧
∀𝑦 ∈
ℝ+ ∃𝑚 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑚)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ∀𝑛 ∈ (ℤ≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑛)) < 𝑦))) |
228 | 227 | simprd 478 |
. . . . . . . . . 10
⊢ (𝜑 → ∀𝑦 ∈ ℝ+ ∃𝑚 ∈ ℕ ∀𝑘 ∈
(ℤ≥‘𝑚)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ∀𝑛 ∈ (ℤ≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑛)) < 𝑦)) |
229 | | simp3 1056 |
. . . . . . . . . . . . 13
⊢ ((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ∀𝑛 ∈ (ℤ≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑛)) < 𝑦) → ∀𝑛 ∈ (ℤ≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑛)) < 𝑦) |
230 | 229 | ralimi 2936 |
. . . . . . . . . . . 12
⊢
(∀𝑘 ∈
(ℤ≥‘𝑚)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ∀𝑛 ∈ (ℤ≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑛)) < 𝑦) → ∀𝑘 ∈ (ℤ≥‘𝑚)∀𝑛 ∈ (ℤ≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑛)) < 𝑦) |
231 | 230 | reximi 2994 |
. . . . . . . . . . 11
⊢
(∃𝑚 ∈
ℕ ∀𝑘 ∈
(ℤ≥‘𝑚)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ∀𝑛 ∈ (ℤ≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑛)) < 𝑦) → ∃𝑚 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑚)∀𝑛 ∈ (ℤ≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑛)) < 𝑦) |
232 | 231 | ralimi 2936 |
. . . . . . . . . 10
⊢
(∀𝑦 ∈
ℝ+ ∃𝑚 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑚)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ∀𝑛 ∈ (ℤ≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑛)) < 𝑦) → ∀𝑦 ∈ ℝ+ ∃𝑚 ∈ ℕ ∀𝑘 ∈
(ℤ≥‘𝑚)∀𝑛 ∈ (ℤ≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑛)) < 𝑦) |
233 | 228, 232 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → ∀𝑦 ∈ ℝ+ ∃𝑚 ∈ ℕ ∀𝑘 ∈
(ℤ≥‘𝑚)∀𝑛 ∈ (ℤ≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑛)) < 𝑦) |
234 | 233 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ ∀𝑢 ∈ ran ℤ≥𝑦 ∈ ((cls‘𝐽)‘(𝐹 “ 𝑢))) → ∀𝑦 ∈ ℝ+ ∃𝑚 ∈ ℕ ∀𝑘 ∈
(ℤ≥‘𝑚)∀𝑛 ∈ (ℤ≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑛)) < 𝑦) |
235 | | rphalfcl 11734 |
. . . . . . . 8
⊢ (𝑟 ∈ ℝ+
→ (𝑟 / 2) ∈
ℝ+) |
236 | | breq2 4587 |
. . . . . . . . . . 11
⊢ (𝑦 = (𝑟 / 2) → (((𝐹‘𝑘)𝐷(𝐹‘𝑛)) < 𝑦 ↔ ((𝐹‘𝑘)𝐷(𝐹‘𝑛)) < (𝑟 / 2))) |
237 | 236 | 2ralbidv 2972 |
. . . . . . . . . 10
⊢ (𝑦 = (𝑟 / 2) → (∀𝑘 ∈ (ℤ≥‘𝑚)∀𝑛 ∈ (ℤ≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑛)) < 𝑦 ↔ ∀𝑘 ∈ (ℤ≥‘𝑚)∀𝑛 ∈ (ℤ≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑛)) < (𝑟 / 2))) |
238 | 237 | rexbidv 3034 |
. . . . . . . . 9
⊢ (𝑦 = (𝑟 / 2) → (∃𝑚 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑚)∀𝑛 ∈ (ℤ≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑛)) < 𝑦 ↔ ∃𝑚 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑚)∀𝑛 ∈ (ℤ≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑛)) < (𝑟 / 2))) |
239 | 238 | rspccva 3281 |
. . . . . . . 8
⊢
((∀𝑦 ∈
ℝ+ ∃𝑚 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑚)∀𝑛 ∈ (ℤ≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑛)) < 𝑦 ∧ (𝑟 / 2) ∈ ℝ+) →
∃𝑚 ∈ ℕ
∀𝑘 ∈
(ℤ≥‘𝑚)∀𝑛 ∈ (ℤ≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑛)) < (𝑟 / 2)) |
240 | 234, 235,
239 | syl2an 493 |
. . . . . . 7
⊢ (((𝜑 ∧ ∀𝑢 ∈ ran ℤ≥𝑦 ∈ ((cls‘𝐽)‘(𝐹 “ 𝑢))) ∧ 𝑟 ∈ ℝ+) →
∃𝑚 ∈ ℕ
∀𝑘 ∈
(ℤ≥‘𝑚)∀𝑛 ∈ (ℤ≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑛)) < (𝑟 / 2)) |
241 | | ffun 5961 |
. . . . . . . . . . . . 13
⊢ (𝐹:ℕ⟶𝑋 → Fun 𝐹) |
242 | 9, 241 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → Fun 𝐹) |
243 | 242 | ad2antrr 758 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ ∀𝑢 ∈ ran ℤ≥𝑦 ∈ ((cls‘𝐽)‘(𝐹 “ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ 𝑚 ∈ ℕ)) → Fun
𝐹) |
244 | 7 | ad2antrr 758 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ ∀𝑢 ∈ ran ℤ≥𝑦 ∈ ((cls‘𝐽)‘(𝐹 “ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ 𝑚 ∈ ℕ)) → 𝐽 ∈ Top) |
245 | | imassrn 5396 |
. . . . . . . . . . . . . 14
⊢ (𝐹 “
(ℤ≥‘𝑚)) ⊆ ran 𝐹 |
246 | 245, 14 | syl5ss 3579 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝐹 “ (ℤ≥‘𝑚)) ⊆ ∪ 𝐽) |
247 | 246 | ad2antrr 758 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ ∀𝑢 ∈ ran ℤ≥𝑦 ∈ ((cls‘𝐽)‘(𝐹 “ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ 𝑚 ∈ ℕ)) → (𝐹 “
(ℤ≥‘𝑚)) ⊆ ∪ 𝐽) |
248 | | nnz 11276 |
. . . . . . . . . . . . . . 15
⊢ (𝑚 ∈ ℕ → 𝑚 ∈
ℤ) |
249 | | fnfvelrn 6264 |
. . . . . . . . . . . . . . 15
⊢
((ℤ≥ Fn ℤ ∧ 𝑚 ∈ ℤ) →
(ℤ≥‘𝑚) ∈ ran
ℤ≥) |
250 | 58, 248, 249 | sylancr 694 |
. . . . . . . . . . . . . 14
⊢ (𝑚 ∈ ℕ →
(ℤ≥‘𝑚) ∈ ran
ℤ≥) |
251 | 250 | ad2antll 761 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ ∀𝑢 ∈ ran ℤ≥𝑦 ∈ ((cls‘𝐽)‘(𝐹 “ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ 𝑚 ∈ ℕ)) →
(ℤ≥‘𝑚) ∈ ran
ℤ≥) |
252 | | simplr 788 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ ∀𝑢 ∈ ran ℤ≥𝑦 ∈ ((cls‘𝐽)‘(𝐹 “ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ 𝑚 ∈ ℕ)) →
∀𝑢 ∈ ran
ℤ≥𝑦
∈ ((cls‘𝐽)‘(𝐹 “ 𝑢))) |
253 | | imaeq2 5381 |
. . . . . . . . . . . . . . . 16
⊢ (𝑢 =
(ℤ≥‘𝑚) → (𝐹 “ 𝑢) = (𝐹 “ (ℤ≥‘𝑚))) |
254 | 253 | fveq2d 6107 |
. . . . . . . . . . . . . . 15
⊢ (𝑢 =
(ℤ≥‘𝑚) → ((cls‘𝐽)‘(𝐹 “ 𝑢)) = ((cls‘𝐽)‘(𝐹 “ (ℤ≥‘𝑚)))) |
255 | 254 | eleq2d 2673 |
. . . . . . . . . . . . . 14
⊢ (𝑢 =
(ℤ≥‘𝑚) → (𝑦 ∈ ((cls‘𝐽)‘(𝐹 “ 𝑢)) ↔ 𝑦 ∈ ((cls‘𝐽)‘(𝐹 “ (ℤ≥‘𝑚))))) |
256 | 255 | rspcv 3278 |
. . . . . . . . . . . . 13
⊢
((ℤ≥‘𝑚) ∈ ran ℤ≥ →
(∀𝑢 ∈ ran
ℤ≥𝑦
∈ ((cls‘𝐽)‘(𝐹 “ 𝑢)) → 𝑦 ∈ ((cls‘𝐽)‘(𝐹 “ (ℤ≥‘𝑚))))) |
257 | 251, 252,
256 | sylc 63 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ ∀𝑢 ∈ ran ℤ≥𝑦 ∈ ((cls‘𝐽)‘(𝐹 “ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ 𝑚 ∈ ℕ)) → 𝑦 ∈ ((cls‘𝐽)‘(𝐹 “ (ℤ≥‘𝑚)))) |
258 | 4 | ad2antrr 758 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ ∀𝑢 ∈ ran ℤ≥𝑦 ∈ ((cls‘𝐽)‘(𝐹 “ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ 𝑚 ∈ ℕ)) → 𝐷 ∈ (∞Met‘𝑋)) |
259 | 223 | adantr 480 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ ∀𝑢 ∈ ran ℤ≥𝑦 ∈ ((cls‘𝐽)‘(𝐹 “ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ 𝑚 ∈ ℕ)) → 𝑦 ∈ 𝑋) |
260 | 235 | ad2antrl 760 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ ∀𝑢 ∈ ran ℤ≥𝑦 ∈ ((cls‘𝐽)‘(𝐹 “ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ 𝑚 ∈ ℕ)) → (𝑟 / 2) ∈
ℝ+) |
261 | 260 | rpxrd 11749 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ ∀𝑢 ∈ ran ℤ≥𝑦 ∈ ((cls‘𝐽)‘(𝐹 “ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ 𝑚 ∈ ℕ)) → (𝑟 / 2) ∈
ℝ*) |
262 | 5 | blopn 22115 |
. . . . . . . . . . . . 13
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ (𝑟 / 2) ∈ ℝ*) →
(𝑦(ball‘𝐷)(𝑟 / 2)) ∈ 𝐽) |
263 | 258, 259,
261, 262 | syl3anc 1318 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ ∀𝑢 ∈ ran ℤ≥𝑦 ∈ ((cls‘𝐽)‘(𝐹 “ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ 𝑚 ∈ ℕ)) → (𝑦(ball‘𝐷)(𝑟 / 2)) ∈ 𝐽) |
264 | | blcntr 22028 |
. . . . . . . . . . . . 13
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ (𝑟 / 2) ∈ ℝ+) →
𝑦 ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) |
265 | 258, 259,
260, 264 | syl3anc 1318 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ ∀𝑢 ∈ ran ℤ≥𝑦 ∈ ((cls‘𝐽)‘(𝐹 “ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ 𝑚 ∈ ℕ)) → 𝑦 ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) |
266 | 16 | clsndisj 20689 |
. . . . . . . . . . . 12
⊢ (((𝐽 ∈ Top ∧ (𝐹 “
(ℤ≥‘𝑚)) ⊆ ∪ 𝐽 ∧ 𝑦 ∈ ((cls‘𝐽)‘(𝐹 “ (ℤ≥‘𝑚)))) ∧ ((𝑦(ball‘𝐷)(𝑟 / 2)) ∈ 𝐽 ∧ 𝑦 ∈ (𝑦(ball‘𝐷)(𝑟 / 2)))) → ((𝑦(ball‘𝐷)(𝑟 / 2)) ∩ (𝐹 “ (ℤ≥‘𝑚))) ≠
∅) |
267 | 244, 247,
257, 263, 265, 266 | syl32anc 1326 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ ∀𝑢 ∈ ran ℤ≥𝑦 ∈ ((cls‘𝐽)‘(𝐹 “ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ 𝑚 ∈ ℕ)) → ((𝑦(ball‘𝐷)(𝑟 / 2)) ∩ (𝐹 “ (ℤ≥‘𝑚))) ≠
∅) |
268 | | n0 3890 |
. . . . . . . . . . . 12
⊢ (((𝑦(ball‘𝐷)(𝑟 / 2)) ∩ (𝐹 “ (ℤ≥‘𝑚))) ≠ ∅ ↔
∃𝑛 𝑛 ∈ ((𝑦(ball‘𝐷)(𝑟 / 2)) ∩ (𝐹 “ (ℤ≥‘𝑚)))) |
269 | | inss2 3796 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑦(ball‘𝐷)(𝑟 / 2)) ∩ (𝐹 “ (ℤ≥‘𝑚))) ⊆ (𝐹 “ (ℤ≥‘𝑚)) |
270 | 269 | sseli 3564 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 ∈ ((𝑦(ball‘𝐷)(𝑟 / 2)) ∩ (𝐹 “ (ℤ≥‘𝑚))) → 𝑛 ∈ (𝐹 “ (ℤ≥‘𝑚))) |
271 | | fvelima 6158 |
. . . . . . . . . . . . . . . 16
⊢ ((Fun
𝐹 ∧ 𝑛 ∈ (𝐹 “ (ℤ≥‘𝑚))) → ∃𝑘 ∈
(ℤ≥‘𝑚)(𝐹‘𝑘) = 𝑛) |
272 | 270, 271 | sylan2 490 |
. . . . . . . . . . . . . . 15
⊢ ((Fun
𝐹 ∧ 𝑛 ∈ ((𝑦(ball‘𝐷)(𝑟 / 2)) ∩ (𝐹 “ (ℤ≥‘𝑚)))) → ∃𝑘 ∈
(ℤ≥‘𝑚)(𝐹‘𝑘) = 𝑛) |
273 | | inss1 3795 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑦(ball‘𝐷)(𝑟 / 2)) ∩ (𝐹 “ (ℤ≥‘𝑚))) ⊆ (𝑦(ball‘𝐷)(𝑟 / 2)) |
274 | 273 | sseli 3564 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑛 ∈ ((𝑦(ball‘𝐷)(𝑟 / 2)) ∩ (𝐹 “ (ℤ≥‘𝑚))) → 𝑛 ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) |
275 | 274 | adantl 481 |
. . . . . . . . . . . . . . . . 17
⊢ ((Fun
𝐹 ∧ 𝑛 ∈ ((𝑦(ball‘𝐷)(𝑟 / 2)) ∩ (𝐹 “ (ℤ≥‘𝑚)))) → 𝑛 ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) |
276 | | eleq1a 2683 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑛 ∈ (𝑦(ball‘𝐷)(𝑟 / 2)) → ((𝐹‘𝑘) = 𝑛 → (𝐹‘𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2)))) |
277 | 275, 276 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ ((Fun
𝐹 ∧ 𝑛 ∈ ((𝑦(ball‘𝐷)(𝑟 / 2)) ∩ (𝐹 “ (ℤ≥‘𝑚)))) → ((𝐹‘𝑘) = 𝑛 → (𝐹‘𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2)))) |
278 | 277 | reximdv 2999 |
. . . . . . . . . . . . . . 15
⊢ ((Fun
𝐹 ∧ 𝑛 ∈ ((𝑦(ball‘𝐷)(𝑟 / 2)) ∩ (𝐹 “ (ℤ≥‘𝑚)))) → (∃𝑘 ∈
(ℤ≥‘𝑚)(𝐹‘𝑘) = 𝑛 → ∃𝑘 ∈ (ℤ≥‘𝑚)(𝐹‘𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2)))) |
279 | 272, 278 | mpd 15 |
. . . . . . . . . . . . . 14
⊢ ((Fun
𝐹 ∧ 𝑛 ∈ ((𝑦(ball‘𝐷)(𝑟 / 2)) ∩ (𝐹 “ (ℤ≥‘𝑚)))) → ∃𝑘 ∈
(ℤ≥‘𝑚)(𝐹‘𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) |
280 | 279 | ex 449 |
. . . . . . . . . . . . 13
⊢ (Fun
𝐹 → (𝑛 ∈ ((𝑦(ball‘𝐷)(𝑟 / 2)) ∩ (𝐹 “ (ℤ≥‘𝑚))) → ∃𝑘 ∈
(ℤ≥‘𝑚)(𝐹‘𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2)))) |
281 | 280 | exlimdv 1848 |
. . . . . . . . . . . 12
⊢ (Fun
𝐹 → (∃𝑛 𝑛 ∈ ((𝑦(ball‘𝐷)(𝑟 / 2)) ∩ (𝐹 “ (ℤ≥‘𝑚))) → ∃𝑘 ∈
(ℤ≥‘𝑚)(𝐹‘𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2)))) |
282 | 268, 281 | syl5bi 231 |
. . . . . . . . . . 11
⊢ (Fun
𝐹 → (((𝑦(ball‘𝐷)(𝑟 / 2)) ∩ (𝐹 “ (ℤ≥‘𝑚))) ≠ ∅ →
∃𝑘 ∈
(ℤ≥‘𝑚)(𝐹‘𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2)))) |
283 | 243, 267,
282 | sylc 63 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ ∀𝑢 ∈ ran ℤ≥𝑦 ∈ ((cls‘𝐽)‘(𝐹 “ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ 𝑚 ∈ ℕ)) →
∃𝑘 ∈
(ℤ≥‘𝑚)(𝐹‘𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) |
284 | | r19.29 3054 |
. . . . . . . . . . 11
⊢
((∀𝑘 ∈
(ℤ≥‘𝑚)∀𝑛 ∈ (ℤ≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑛)) < (𝑟 / 2) ∧ ∃𝑘 ∈ (ℤ≥‘𝑚)(𝐹‘𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) → ∃𝑘 ∈ (ℤ≥‘𝑚)(∀𝑛 ∈ (ℤ≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑛)) < (𝑟 / 2) ∧ (𝐹‘𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2)))) |
285 | | uznnssnn 11611 |
. . . . . . . . . . . . . 14
⊢ (𝑚 ∈ ℕ →
(ℤ≥‘𝑚) ⊆ ℕ) |
286 | 285 | ad2antll 761 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑟 ∈ ℝ+ ∧ 𝑚 ∈ ℕ)) →
(ℤ≥‘𝑚) ⊆ ℕ) |
287 | | simprlr 799 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑟 ∈ ℝ+ ∧ 𝑚 ∈ ℕ)) ∧ ((𝑘 ∈
(ℤ≥‘𝑚) ∧ (𝐹‘𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ≥‘𝑘))) → (𝐹‘𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) |
288 | 4 | ad3antrrr 762 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑟 ∈ ℝ+ ∧ 𝑚 ∈ ℕ)) ∧ ((𝑘 ∈
(ℤ≥‘𝑚) ∧ (𝐹‘𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ≥‘𝑘))) → 𝐷 ∈ (∞Met‘𝑋)) |
289 | | simplrl 796 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑟 ∈ ℝ+ ∧ 𝑚 ∈ ℕ)) ∧ ((𝑘 ∈
(ℤ≥‘𝑚) ∧ (𝐹‘𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ≥‘𝑘))) → 𝑟 ∈ ℝ+) |
290 | 289, 235 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑟 ∈ ℝ+ ∧ 𝑚 ∈ ℕ)) ∧ ((𝑘 ∈
(ℤ≥‘𝑚) ∧ (𝐹‘𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ≥‘𝑘))) → (𝑟 / 2) ∈
ℝ+) |
291 | 290 | rpxrd 11749 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑟 ∈ ℝ+ ∧ 𝑚 ∈ ℕ)) ∧ ((𝑘 ∈
(ℤ≥‘𝑚) ∧ (𝐹‘𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ≥‘𝑘))) → (𝑟 / 2) ∈
ℝ*) |
292 | | simpllr 795 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑟 ∈ ℝ+ ∧ 𝑚 ∈ ℕ)) ∧ ((𝑘 ∈
(ℤ≥‘𝑚) ∧ (𝐹‘𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ≥‘𝑘))) → 𝑦 ∈ 𝑋) |
293 | 9 | ad3antrrr 762 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑟 ∈ ℝ+ ∧ 𝑚 ∈ ℕ)) ∧ ((𝑘 ∈
(ℤ≥‘𝑚) ∧ (𝐹‘𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ≥‘𝑘))) → 𝐹:ℕ⟶𝑋) |
294 | | eluznn 11634 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑚 ∈ ℕ ∧ 𝑘 ∈
(ℤ≥‘𝑚)) → 𝑘 ∈ ℕ) |
295 | 294 | ad2ant2lr 780 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝑟 ∈ ℝ+
∧ 𝑚 ∈ ℕ)
∧ (𝑘 ∈
(ℤ≥‘𝑚) ∧ (𝐹‘𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2)))) → 𝑘 ∈ ℕ) |
296 | 295 | ad2ant2lr 780 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑟 ∈ ℝ+ ∧ 𝑚 ∈ ℕ)) ∧ ((𝑘 ∈
(ℤ≥‘𝑚) ∧ (𝐹‘𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ≥‘𝑘))) → 𝑘 ∈ ℕ) |
297 | 293, 296 | ffvelrnd 6268 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑟 ∈ ℝ+ ∧ 𝑚 ∈ ℕ)) ∧ ((𝑘 ∈
(ℤ≥‘𝑚) ∧ (𝐹‘𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ≥‘𝑘))) → (𝐹‘𝑘) ∈ 𝑋) |
298 | | elbl3 22007 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑟 / 2) ∈ ℝ*) ∧
(𝑦 ∈ 𝑋 ∧ (𝐹‘𝑘) ∈ 𝑋)) → ((𝐹‘𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2)) ↔ ((𝐹‘𝑘)𝐷𝑦) < (𝑟 / 2))) |
299 | 288, 291,
292, 297, 298 | syl22anc 1319 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑟 ∈ ℝ+ ∧ 𝑚 ∈ ℕ)) ∧ ((𝑘 ∈
(ℤ≥‘𝑚) ∧ (𝐹‘𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ≥‘𝑘))) → ((𝐹‘𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2)) ↔ ((𝐹‘𝑘)𝐷𝑦) < (𝑟 / 2))) |
300 | 287, 299 | mpbid 221 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑟 ∈ ℝ+ ∧ 𝑚 ∈ ℕ)) ∧ ((𝑘 ∈
(ℤ≥‘𝑚) ∧ (𝐹‘𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ≥‘𝑘))) → ((𝐹‘𝑘)𝐷𝑦) < (𝑟 / 2)) |
301 | 2 | ad3antrrr 762 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑟 ∈ ℝ+ ∧ 𝑚 ∈ ℕ)) ∧ ((𝑘 ∈
(ℤ≥‘𝑚) ∧ (𝐹‘𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ≥‘𝑘))) → 𝐷 ∈ (Met‘𝑋)) |
302 | | simprr 792 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑟 ∈ ℝ+ ∧ 𝑚 ∈ ℕ)) ∧ ((𝑘 ∈
(ℤ≥‘𝑚) ∧ (𝐹‘𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ≥‘𝑘))) → 𝑛 ∈ (ℤ≥‘𝑘)) |
303 | | eluznn 11634 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑘 ∈ ℕ ∧ 𝑛 ∈
(ℤ≥‘𝑘)) → 𝑛 ∈ ℕ) |
304 | 296, 302,
303 | syl2anc 691 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑟 ∈ ℝ+ ∧ 𝑚 ∈ ℕ)) ∧ ((𝑘 ∈
(ℤ≥‘𝑚) ∧ (𝐹‘𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ≥‘𝑘))) → 𝑛 ∈ ℕ) |
305 | 293, 304 | ffvelrnd 6268 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑟 ∈ ℝ+ ∧ 𝑚 ∈ ℕ)) ∧ ((𝑘 ∈
(ℤ≥‘𝑚) ∧ (𝐹‘𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ≥‘𝑘))) → (𝐹‘𝑛) ∈ 𝑋) |
306 | | metcl 21947 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝐷 ∈ (Met‘𝑋) ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ (𝐹‘𝑛) ∈ 𝑋) → ((𝐹‘𝑘)𝐷(𝐹‘𝑛)) ∈ ℝ) |
307 | 301, 297,
305, 306 | syl3anc 1318 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑟 ∈ ℝ+ ∧ 𝑚 ∈ ℕ)) ∧ ((𝑘 ∈
(ℤ≥‘𝑚) ∧ (𝐹‘𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ≥‘𝑘))) → ((𝐹‘𝑘)𝐷(𝐹‘𝑛)) ∈ ℝ) |
308 | | metcl 21947 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝐷 ∈ (Met‘𝑋) ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((𝐹‘𝑘)𝐷𝑦) ∈ ℝ) |
309 | 301, 297,
292, 308 | syl3anc 1318 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑟 ∈ ℝ+ ∧ 𝑚 ∈ ℕ)) ∧ ((𝑘 ∈
(ℤ≥‘𝑚) ∧ (𝐹‘𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ≥‘𝑘))) → ((𝐹‘𝑘)𝐷𝑦) ∈ ℝ) |
310 | 290 | rpred 11748 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑟 ∈ ℝ+ ∧ 𝑚 ∈ ℕ)) ∧ ((𝑘 ∈
(ℤ≥‘𝑚) ∧ (𝐹‘𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ≥‘𝑘))) → (𝑟 / 2) ∈ ℝ) |
311 | | lt2add 10392 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(((((𝐹‘𝑘)𝐷(𝐹‘𝑛)) ∈ ℝ ∧ ((𝐹‘𝑘)𝐷𝑦) ∈ ℝ) ∧ ((𝑟 / 2) ∈ ℝ ∧ (𝑟 / 2) ∈ ℝ)) →
((((𝐹‘𝑘)𝐷(𝐹‘𝑛)) < (𝑟 / 2) ∧ ((𝐹‘𝑘)𝐷𝑦) < (𝑟 / 2)) → (((𝐹‘𝑘)𝐷(𝐹‘𝑛)) + ((𝐹‘𝑘)𝐷𝑦)) < ((𝑟 / 2) + (𝑟 / 2)))) |
312 | 307, 309,
310, 310, 311 | syl22anc 1319 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑟 ∈ ℝ+ ∧ 𝑚 ∈ ℕ)) ∧ ((𝑘 ∈
(ℤ≥‘𝑚) ∧ (𝐹‘𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ≥‘𝑘))) → ((((𝐹‘𝑘)𝐷(𝐹‘𝑛)) < (𝑟 / 2) ∧ ((𝐹‘𝑘)𝐷𝑦) < (𝑟 / 2)) → (((𝐹‘𝑘)𝐷(𝐹‘𝑛)) + ((𝐹‘𝑘)𝐷𝑦)) < ((𝑟 / 2) + (𝑟 / 2)))) |
313 | 300, 312 | mpan2d 706 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑟 ∈ ℝ+ ∧ 𝑚 ∈ ℕ)) ∧ ((𝑘 ∈
(ℤ≥‘𝑚) ∧ (𝐹‘𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ≥‘𝑘))) → (((𝐹‘𝑘)𝐷(𝐹‘𝑛)) < (𝑟 / 2) → (((𝐹‘𝑘)𝐷(𝐹‘𝑛)) + ((𝐹‘𝑘)𝐷𝑦)) < ((𝑟 / 2) + (𝑟 / 2)))) |
314 | 289 | rpcnd 11750 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑟 ∈ ℝ+ ∧ 𝑚 ∈ ℕ)) ∧ ((𝑘 ∈
(ℤ≥‘𝑚) ∧ (𝐹‘𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ≥‘𝑘))) → 𝑟 ∈ ℂ) |
315 | 314 | 2halvesd 11155 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑟 ∈ ℝ+ ∧ 𝑚 ∈ ℕ)) ∧ ((𝑘 ∈
(ℤ≥‘𝑚) ∧ (𝐹‘𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ≥‘𝑘))) → ((𝑟 / 2) + (𝑟 / 2)) = 𝑟) |
316 | 315 | breq2d 4595 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑟 ∈ ℝ+ ∧ 𝑚 ∈ ℕ)) ∧ ((𝑘 ∈
(ℤ≥‘𝑚) ∧ (𝐹‘𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ≥‘𝑘))) → ((((𝐹‘𝑘)𝐷(𝐹‘𝑛)) + ((𝐹‘𝑘)𝐷𝑦)) < ((𝑟 / 2) + (𝑟 / 2)) ↔ (((𝐹‘𝑘)𝐷(𝐹‘𝑛)) + ((𝐹‘𝑘)𝐷𝑦)) < 𝑟)) |
317 | 313, 316 | sylibd 228 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑟 ∈ ℝ+ ∧ 𝑚 ∈ ℕ)) ∧ ((𝑘 ∈
(ℤ≥‘𝑚) ∧ (𝐹‘𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ≥‘𝑘))) → (((𝐹‘𝑘)𝐷(𝐹‘𝑛)) < (𝑟 / 2) → (((𝐹‘𝑘)𝐷(𝐹‘𝑛)) + ((𝐹‘𝑘)𝐷𝑦)) < 𝑟)) |
318 | | mettri2 21956 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝐷 ∈ (Met‘𝑋) ∧ ((𝐹‘𝑘) ∈ 𝑋 ∧ (𝐹‘𝑛) ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝐹‘𝑛)𝐷𝑦) ≤ (((𝐹‘𝑘)𝐷(𝐹‘𝑛)) + ((𝐹‘𝑘)𝐷𝑦))) |
319 | 301, 297,
305, 292, 318 | syl13anc 1320 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑟 ∈ ℝ+ ∧ 𝑚 ∈ ℕ)) ∧ ((𝑘 ∈
(ℤ≥‘𝑚) ∧ (𝐹‘𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ≥‘𝑘))) → ((𝐹‘𝑛)𝐷𝑦) ≤ (((𝐹‘𝑘)𝐷(𝐹‘𝑛)) + ((𝐹‘𝑘)𝐷𝑦))) |
320 | | metcl 21947 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝐷 ∈ (Met‘𝑋) ∧ (𝐹‘𝑛) ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((𝐹‘𝑛)𝐷𝑦) ∈ ℝ) |
321 | 301, 305,
292, 320 | syl3anc 1318 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑟 ∈ ℝ+ ∧ 𝑚 ∈ ℕ)) ∧ ((𝑘 ∈
(ℤ≥‘𝑚) ∧ (𝐹‘𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ≥‘𝑘))) → ((𝐹‘𝑛)𝐷𝑦) ∈ ℝ) |
322 | 307, 309 | readdcld 9948 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑟 ∈ ℝ+ ∧ 𝑚 ∈ ℕ)) ∧ ((𝑘 ∈
(ℤ≥‘𝑚) ∧ (𝐹‘𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ≥‘𝑘))) → (((𝐹‘𝑘)𝐷(𝐹‘𝑛)) + ((𝐹‘𝑘)𝐷𝑦)) ∈ ℝ) |
323 | 289 | rpred 11748 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑟 ∈ ℝ+ ∧ 𝑚 ∈ ℕ)) ∧ ((𝑘 ∈
(ℤ≥‘𝑚) ∧ (𝐹‘𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ≥‘𝑘))) → 𝑟 ∈ ℝ) |
324 | | lelttr 10007 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝐹‘𝑛)𝐷𝑦) ∈ ℝ ∧ (((𝐹‘𝑘)𝐷(𝐹‘𝑛)) + ((𝐹‘𝑘)𝐷𝑦)) ∈ ℝ ∧ 𝑟 ∈ ℝ) → ((((𝐹‘𝑛)𝐷𝑦) ≤ (((𝐹‘𝑘)𝐷(𝐹‘𝑛)) + ((𝐹‘𝑘)𝐷𝑦)) ∧ (((𝐹‘𝑘)𝐷(𝐹‘𝑛)) + ((𝐹‘𝑘)𝐷𝑦)) < 𝑟) → ((𝐹‘𝑛)𝐷𝑦) < 𝑟)) |
325 | 321, 322,
323, 324 | syl3anc 1318 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑟 ∈ ℝ+ ∧ 𝑚 ∈ ℕ)) ∧ ((𝑘 ∈
(ℤ≥‘𝑚) ∧ (𝐹‘𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ≥‘𝑘))) → ((((𝐹‘𝑛)𝐷𝑦) ≤ (((𝐹‘𝑘)𝐷(𝐹‘𝑛)) + ((𝐹‘𝑘)𝐷𝑦)) ∧ (((𝐹‘𝑘)𝐷(𝐹‘𝑛)) + ((𝐹‘𝑘)𝐷𝑦)) < 𝑟) → ((𝐹‘𝑛)𝐷𝑦) < 𝑟)) |
326 | 319, 325 | mpand 707 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑟 ∈ ℝ+ ∧ 𝑚 ∈ ℕ)) ∧ ((𝑘 ∈
(ℤ≥‘𝑚) ∧ (𝐹‘𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ≥‘𝑘))) → ((((𝐹‘𝑘)𝐷(𝐹‘𝑛)) + ((𝐹‘𝑘)𝐷𝑦)) < 𝑟 → ((𝐹‘𝑛)𝐷𝑦) < 𝑟)) |
327 | 317, 326 | syld 46 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑟 ∈ ℝ+ ∧ 𝑚 ∈ ℕ)) ∧ ((𝑘 ∈
(ℤ≥‘𝑚) ∧ (𝐹‘𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) ∧ 𝑛 ∈ (ℤ≥‘𝑘))) → (((𝐹‘𝑘)𝐷(𝐹‘𝑛)) < (𝑟 / 2) → ((𝐹‘𝑛)𝐷𝑦) < 𝑟)) |
328 | 327 | anassrs 678 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑟 ∈ ℝ+ ∧ 𝑚 ∈ ℕ)) ∧ (𝑘 ∈
(ℤ≥‘𝑚) ∧ (𝐹‘𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2)))) ∧ 𝑛 ∈ (ℤ≥‘𝑘)) → (((𝐹‘𝑘)𝐷(𝐹‘𝑛)) < (𝑟 / 2) → ((𝐹‘𝑛)𝐷𝑦) < 𝑟)) |
329 | 328 | ralimdva 2945 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑟 ∈ ℝ+ ∧ 𝑚 ∈ ℕ)) ∧ (𝑘 ∈
(ℤ≥‘𝑚) ∧ (𝐹‘𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2)))) → (∀𝑛 ∈ (ℤ≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑛)) < (𝑟 / 2) → ∀𝑛 ∈ (ℤ≥‘𝑘)((𝐹‘𝑛)𝐷𝑦) < 𝑟)) |
330 | 329 | expr 641 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑟 ∈ ℝ+ ∧ 𝑚 ∈ ℕ)) ∧ 𝑘 ∈
(ℤ≥‘𝑚)) → ((𝐹‘𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2)) → (∀𝑛 ∈ (ℤ≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑛)) < (𝑟 / 2) → ∀𝑛 ∈ (ℤ≥‘𝑘)((𝐹‘𝑛)𝐷𝑦) < 𝑟))) |
331 | 330 | com23 84 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑟 ∈ ℝ+ ∧ 𝑚 ∈ ℕ)) ∧ 𝑘 ∈
(ℤ≥‘𝑚)) → (∀𝑛 ∈ (ℤ≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑛)) < (𝑟 / 2) → ((𝐹‘𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2)) → ∀𝑛 ∈ (ℤ≥‘𝑘)((𝐹‘𝑛)𝐷𝑦) < 𝑟))) |
332 | 331 | impd 446 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑟 ∈ ℝ+ ∧ 𝑚 ∈ ℕ)) ∧ 𝑘 ∈
(ℤ≥‘𝑚)) → ((∀𝑛 ∈ (ℤ≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑛)) < (𝑟 / 2) ∧ (𝐹‘𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) → ∀𝑛 ∈ (ℤ≥‘𝑘)((𝐹‘𝑛)𝐷𝑦) < 𝑟)) |
333 | 332 | reximdva 3000 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑟 ∈ ℝ+ ∧ 𝑚 ∈ ℕ)) →
(∃𝑘 ∈
(ℤ≥‘𝑚)(∀𝑛 ∈ (ℤ≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑛)) < (𝑟 / 2) ∧ (𝐹‘𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) → ∃𝑘 ∈ (ℤ≥‘𝑚)∀𝑛 ∈ (ℤ≥‘𝑘)((𝐹‘𝑛)𝐷𝑦) < 𝑟)) |
334 | | ssrexv 3630 |
. . . . . . . . . . . . 13
⊢
((ℤ≥‘𝑚) ⊆ ℕ → (∃𝑘 ∈
(ℤ≥‘𝑚)∀𝑛 ∈ (ℤ≥‘𝑘)((𝐹‘𝑛)𝐷𝑦) < 𝑟 → ∃𝑘 ∈ ℕ ∀𝑛 ∈ (ℤ≥‘𝑘)((𝐹‘𝑛)𝐷𝑦) < 𝑟)) |
335 | 286, 333,
334 | sylsyld 59 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑟 ∈ ℝ+ ∧ 𝑚 ∈ ℕ)) →
(∃𝑘 ∈
(ℤ≥‘𝑚)(∀𝑛 ∈ (ℤ≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑛)) < (𝑟 / 2) ∧ (𝐹‘𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) → ∃𝑘 ∈ ℕ ∀𝑛 ∈ (ℤ≥‘𝑘)((𝐹‘𝑛)𝐷𝑦) < 𝑟)) |
336 | 223, 335 | syldanl 731 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ ∀𝑢 ∈ ran ℤ≥𝑦 ∈ ((cls‘𝐽)‘(𝐹 “ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ 𝑚 ∈ ℕ)) →
(∃𝑘 ∈
(ℤ≥‘𝑚)(∀𝑛 ∈ (ℤ≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑛)) < (𝑟 / 2) ∧ (𝐹‘𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) → ∃𝑘 ∈ ℕ ∀𝑛 ∈ (ℤ≥‘𝑘)((𝐹‘𝑛)𝐷𝑦) < 𝑟)) |
337 | 284, 336 | syl5 33 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ ∀𝑢 ∈ ran ℤ≥𝑦 ∈ ((cls‘𝐽)‘(𝐹 “ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ 𝑚 ∈ ℕ)) →
((∀𝑘 ∈
(ℤ≥‘𝑚)∀𝑛 ∈ (ℤ≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑛)) < (𝑟 / 2) ∧ ∃𝑘 ∈ (ℤ≥‘𝑚)(𝐹‘𝑘) ∈ (𝑦(ball‘𝐷)(𝑟 / 2))) → ∃𝑘 ∈ ℕ ∀𝑛 ∈ (ℤ≥‘𝑘)((𝐹‘𝑛)𝐷𝑦) < 𝑟)) |
338 | 283, 337 | mpan2d 706 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ∀𝑢 ∈ ran ℤ≥𝑦 ∈ ((cls‘𝐽)‘(𝐹 “ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ 𝑚 ∈ ℕ)) →
(∀𝑘 ∈
(ℤ≥‘𝑚)∀𝑛 ∈ (ℤ≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑛)) < (𝑟 / 2) → ∃𝑘 ∈ ℕ ∀𝑛 ∈ (ℤ≥‘𝑘)((𝐹‘𝑛)𝐷𝑦) < 𝑟)) |
339 | 338 | anassrs 678 |
. . . . . . . 8
⊢ ((((𝜑 ∧ ∀𝑢 ∈ ran ℤ≥𝑦 ∈ ((cls‘𝐽)‘(𝐹 “ 𝑢))) ∧ 𝑟 ∈ ℝ+) ∧ 𝑚 ∈ ℕ) →
(∀𝑘 ∈
(ℤ≥‘𝑚)∀𝑛 ∈ (ℤ≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑛)) < (𝑟 / 2) → ∃𝑘 ∈ ℕ ∀𝑛 ∈ (ℤ≥‘𝑘)((𝐹‘𝑛)𝐷𝑦) < 𝑟)) |
340 | 339 | rexlimdva 3013 |
. . . . . . 7
⊢ (((𝜑 ∧ ∀𝑢 ∈ ran ℤ≥𝑦 ∈ ((cls‘𝐽)‘(𝐹 “ 𝑢))) ∧ 𝑟 ∈ ℝ+) →
(∃𝑚 ∈ ℕ
∀𝑘 ∈
(ℤ≥‘𝑚)∀𝑛 ∈ (ℤ≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑛)) < (𝑟 / 2) → ∃𝑘 ∈ ℕ ∀𝑛 ∈ (ℤ≥‘𝑘)((𝐹‘𝑛)𝐷𝑦) < 𝑟)) |
341 | 240, 340 | mpd 15 |
. . . . . 6
⊢ (((𝜑 ∧ ∀𝑢 ∈ ran ℤ≥𝑦 ∈ ((cls‘𝐽)‘(𝐹 “ 𝑢))) ∧ 𝑟 ∈ ℝ+) →
∃𝑘 ∈ ℕ
∀𝑛 ∈
(ℤ≥‘𝑘)((𝐹‘𝑛)𝐷𝑦) < 𝑟) |
342 | 341 | ralrimiva 2949 |
. . . . 5
⊢ ((𝜑 ∧ ∀𝑢 ∈ ran ℤ≥𝑦 ∈ ((cls‘𝐽)‘(𝐹 “ 𝑢))) → ∀𝑟 ∈ ℝ+ ∃𝑘 ∈ ℕ ∀𝑛 ∈
(ℤ≥‘𝑘)((𝐹‘𝑛)𝐷𝑦) < 𝑟) |
343 | | eqidd 2611 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) = (𝐹‘𝑛)) |
344 | 5, 4, 131, 225, 343, 9 | lmmbrf 22868 |
. . . . . 6
⊢ (𝜑 → (𝐹(⇝𝑡‘𝐽)𝑦 ↔ (𝑦 ∈ 𝑋 ∧ ∀𝑟 ∈ ℝ+ ∃𝑘 ∈ ℕ ∀𝑛 ∈
(ℤ≥‘𝑘)((𝐹‘𝑛)𝐷𝑦) < 𝑟))) |
345 | 344 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ ∀𝑢 ∈ ran ℤ≥𝑦 ∈ ((cls‘𝐽)‘(𝐹 “ 𝑢))) → (𝐹(⇝𝑡‘𝐽)𝑦 ↔ (𝑦 ∈ 𝑋 ∧ ∀𝑟 ∈ ℝ+ ∃𝑘 ∈ ℕ ∀𝑛 ∈
(ℤ≥‘𝑘)((𝐹‘𝑛)𝐷𝑦) < 𝑟))) |
346 | 223, 342,
345 | mpbir2and 959 |
. . . 4
⊢ ((𝜑 ∧ ∀𝑢 ∈ ran ℤ≥𝑦 ∈ ((cls‘𝐽)‘(𝐹 “ 𝑢))) → 𝐹(⇝𝑡‘𝐽)𝑦) |
347 | 202, 346 | sylan2br 492 |
. . 3
⊢ ((𝜑 ∧ 𝑦 ∈ ∩ {𝑘 ∣ ∃𝑢 ∈ ran
ℤ≥𝑘 =
((cls‘𝐽)‘(𝐹 “ 𝑢))}) → 𝐹(⇝𝑡‘𝐽)𝑦) |
348 | | releldm 5279 |
. . 3
⊢ ((Rel
(⇝𝑡‘𝐽) ∧ 𝐹(⇝𝑡‘𝐽)𝑦) → 𝐹 ∈ dom
(⇝𝑡‘𝐽)) |
349 | 191, 347,
348 | sylancr 694 |
. 2
⊢ ((𝜑 ∧ 𝑦 ∈ ∩ {𝑘 ∣ ∃𝑢 ∈ ran
ℤ≥𝑘 =
((cls‘𝐽)‘(𝐹 “ 𝑢))}) → 𝐹 ∈ dom
(⇝𝑡‘𝐽)) |
350 | 190, 349 | exlimddv 1850 |
1
⊢ (𝜑 → 𝐹 ∈ dom
(⇝𝑡‘𝐽)) |