Step | Hyp | Ref
| Expression |
1 | | caufpm 24179 |
. . . . . . . 8
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (Cau‘𝐷)) → 𝐹 ∈ (𝑋 ↑pm
ℂ)) |
2 | | elfvdm 6749 |
. . . . . . . . . 10
⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 ∈ dom ∞Met) |
3 | | cnex 10810 |
. . . . . . . . . 10
⊢ ℂ
∈ V |
4 | | elpmg 8524 |
. . . . . . . . . 10
⊢ ((𝑋 ∈ dom ∞Met ∧
ℂ ∈ V) → (𝐹
∈ (𝑋
↑pm ℂ) ↔ (Fun 𝐹 ∧ 𝐹 ⊆ (ℂ × 𝑋)))) |
5 | 2, 3, 4 | sylancl 589 |
. . . . . . . . 9
⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐹 ∈ (𝑋 ↑pm ℂ) ↔ (Fun
𝐹 ∧ 𝐹 ⊆ (ℂ × 𝑋)))) |
6 | 5 | biimpa 480 |
. . . . . . . 8
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (𝑋 ↑pm ℂ)) → (Fun
𝐹 ∧ 𝐹 ⊆ (ℂ × 𝑋))) |
7 | 1, 6 | syldan 594 |
. . . . . . 7
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (Cau‘𝐷)) → (Fun 𝐹 ∧ 𝐹 ⊆ (ℂ × 𝑋))) |
8 | | rnss 5808 |
. . . . . . 7
⊢ (𝐹 ⊆ (ℂ × 𝑋) → ran 𝐹 ⊆ ran (ℂ × 𝑋)) |
9 | 7, 8 | simpl2im 507 |
. . . . . 6
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (Cau‘𝐷)) → ran 𝐹 ⊆ ran (ℂ × 𝑋)) |
10 | | rnxpss 6035 |
. . . . . 6
⊢ ran
(ℂ × 𝑋) ⊆
𝑋 |
11 | 9, 10 | sstrdi 3913 |
. . . . 5
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (Cau‘𝐷)) → ran 𝐹 ⊆ 𝑋) |
12 | 11 | adantlr 715 |
. . . 4
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹:ℕ⟶𝑌) ∧ 𝐹 ∈ (Cau‘𝐷)) → ran 𝐹 ⊆ 𝑋) |
13 | | frn 6552 |
. . . . 5
⊢ (𝐹:ℕ⟶𝑌 → ran 𝐹 ⊆ 𝑌) |
14 | 13 | ad2antlr 727 |
. . . 4
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹:ℕ⟶𝑌) ∧ 𝐹 ∈ (Cau‘𝐷)) → ran 𝐹 ⊆ 𝑌) |
15 | 12, 14 | ssind 4147 |
. . 3
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹:ℕ⟶𝑌) ∧ 𝐹 ∈ (Cau‘𝐷)) → ran 𝐹 ⊆ (𝑋 ∩ 𝑌)) |
16 | 15 | ex 416 |
. 2
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹:ℕ⟶𝑌) → (𝐹 ∈ (Cau‘𝐷) → ran 𝐹 ⊆ (𝑋 ∩ 𝑌))) |
17 | | xmetres 23262 |
. . . . . . . 8
⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐷 ↾ (𝑌 × 𝑌)) ∈ (∞Met‘(𝑋 ∩ 𝑌))) |
18 | | caufpm 24179 |
. . . . . . . 8
⊢ (((𝐷 ↾ (𝑌 × 𝑌)) ∈ (∞Met‘(𝑋 ∩ 𝑌)) ∧ 𝐹 ∈ (Cau‘(𝐷 ↾ (𝑌 × 𝑌)))) → 𝐹 ∈ ((𝑋 ∩ 𝑌) ↑pm
ℂ)) |
19 | 17, 18 | sylan 583 |
. . . . . . 7
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (Cau‘(𝐷 ↾ (𝑌 × 𝑌)))) → 𝐹 ∈ ((𝑋 ∩ 𝑌) ↑pm
ℂ)) |
20 | | inex1g 5212 |
. . . . . . . . . 10
⊢ (𝑋 ∈ dom ∞Met →
(𝑋 ∩ 𝑌) ∈ V) |
21 | 2, 20 | syl 17 |
. . . . . . . . 9
⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝑋 ∩ 𝑌) ∈ V) |
22 | | elpmg 8524 |
. . . . . . . . 9
⊢ (((𝑋 ∩ 𝑌) ∈ V ∧ ℂ ∈ V) →
(𝐹 ∈ ((𝑋 ∩ 𝑌) ↑pm ℂ) ↔ (Fun
𝐹 ∧ 𝐹 ⊆ (ℂ × (𝑋 ∩ 𝑌))))) |
23 | 21, 3, 22 | sylancl 589 |
. . . . . . . 8
⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐹 ∈ ((𝑋 ∩ 𝑌) ↑pm ℂ) ↔ (Fun
𝐹 ∧ 𝐹 ⊆ (ℂ × (𝑋 ∩ 𝑌))))) |
24 | 23 | biimpa 480 |
. . . . . . 7
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ ((𝑋 ∩ 𝑌) ↑pm ℂ)) → (Fun
𝐹 ∧ 𝐹 ⊆ (ℂ × (𝑋 ∩ 𝑌)))) |
25 | 19, 24 | syldan 594 |
. . . . . 6
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (Cau‘(𝐷 ↾ (𝑌 × 𝑌)))) → (Fun 𝐹 ∧ 𝐹 ⊆ (ℂ × (𝑋 ∩ 𝑌)))) |
26 | | rnss 5808 |
. . . . . 6
⊢ (𝐹 ⊆ (ℂ × (𝑋 ∩ 𝑌)) → ran 𝐹 ⊆ ran (ℂ × (𝑋 ∩ 𝑌))) |
27 | 25, 26 | simpl2im 507 |
. . . . 5
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (Cau‘(𝐷 ↾ (𝑌 × 𝑌)))) → ran 𝐹 ⊆ ran (ℂ × (𝑋 ∩ 𝑌))) |
28 | | rnxpss 6035 |
. . . . 5
⊢ ran
(ℂ × (𝑋 ∩
𝑌)) ⊆ (𝑋 ∩ 𝑌) |
29 | 27, 28 | sstrdi 3913 |
. . . 4
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (Cau‘(𝐷 ↾ (𝑌 × 𝑌)))) → ran 𝐹 ⊆ (𝑋 ∩ 𝑌)) |
30 | 29 | ex 416 |
. . 3
⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐹 ∈ (Cau‘(𝐷 ↾ (𝑌 × 𝑌))) → ran 𝐹 ⊆ (𝑋 ∩ 𝑌))) |
31 | 30 | adantr 484 |
. 2
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹:ℕ⟶𝑌) → (𝐹 ∈ (Cau‘(𝐷 ↾ (𝑌 × 𝑌))) → ran 𝐹 ⊆ (𝑋 ∩ 𝑌))) |
32 | | ffn 6545 |
. . . 4
⊢ (𝐹:ℕ⟶𝑌 → 𝐹 Fn ℕ) |
33 | | df-f 6384 |
. . . . 5
⊢ (𝐹:ℕ⟶(𝑋 ∩ 𝑌) ↔ (𝐹 Fn ℕ ∧ ran 𝐹 ⊆ (𝑋 ∩ 𝑌))) |
34 | 33 | simplbi2 504 |
. . . 4
⊢ (𝐹 Fn ℕ → (ran 𝐹 ⊆ (𝑋 ∩ 𝑌) → 𝐹:ℕ⟶(𝑋 ∩ 𝑌))) |
35 | 32, 34 | syl 17 |
. . 3
⊢ (𝐹:ℕ⟶𝑌 → (ran 𝐹 ⊆ (𝑋 ∩ 𝑌) → 𝐹:ℕ⟶(𝑋 ∩ 𝑌))) |
36 | | inss2 4144 |
. . . . . . . . 9
⊢ (𝑋 ∩ 𝑌) ⊆ 𝑌 |
37 | 36 | a1i 11 |
. . . . . . . 8
⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝑋 ∩ 𝑌) ⊆ 𝑌) |
38 | | fss 6562 |
. . . . . . . 8
⊢ ((𝐹:ℕ⟶(𝑋 ∩ 𝑌) ∧ (𝑋 ∩ 𝑌) ⊆ 𝑌) → 𝐹:ℕ⟶𝑌) |
39 | 37, 38 | sylan2 596 |
. . . . . . 7
⊢ ((𝐹:ℕ⟶(𝑋 ∩ 𝑌) ∧ 𝐷 ∈ (∞Met‘𝑋)) → 𝐹:ℕ⟶𝑌) |
40 | 39 | ancoms 462 |
. . . . . 6
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹:ℕ⟶(𝑋 ∩ 𝑌)) → 𝐹:ℕ⟶𝑌) |
41 | | ffvelrn 6902 |
. . . . . . . . . . . 12
⊢ ((𝐹:ℕ⟶𝑌 ∧ 𝑦 ∈ ℕ) → (𝐹‘𝑦) ∈ 𝑌) |
42 | 41 | adantr 484 |
. . . . . . . . . . 11
⊢ (((𝐹:ℕ⟶𝑌 ∧ 𝑦 ∈ ℕ) ∧ 𝑧 ∈ (ℤ≥‘𝑦)) → (𝐹‘𝑦) ∈ 𝑌) |
43 | | eluznn 12514 |
. . . . . . . . . . . . 13
⊢ ((𝑦 ∈ ℕ ∧ 𝑧 ∈
(ℤ≥‘𝑦)) → 𝑧 ∈ ℕ) |
44 | | ffvelrn 6902 |
. . . . . . . . . . . . 13
⊢ ((𝐹:ℕ⟶𝑌 ∧ 𝑧 ∈ ℕ) → (𝐹‘𝑧) ∈ 𝑌) |
45 | 43, 44 | sylan2 596 |
. . . . . . . . . . . 12
⊢ ((𝐹:ℕ⟶𝑌 ∧ (𝑦 ∈ ℕ ∧ 𝑧 ∈ (ℤ≥‘𝑦))) → (𝐹‘𝑧) ∈ 𝑌) |
46 | 45 | anassrs 471 |
. . . . . . . . . . 11
⊢ (((𝐹:ℕ⟶𝑌 ∧ 𝑦 ∈ ℕ) ∧ 𝑧 ∈ (ℤ≥‘𝑦)) → (𝐹‘𝑧) ∈ 𝑌) |
47 | 42, 46 | ovresd 7375 |
. . . . . . . . . 10
⊢ (((𝐹:ℕ⟶𝑌 ∧ 𝑦 ∈ ℕ) ∧ 𝑧 ∈ (ℤ≥‘𝑦)) → ((𝐹‘𝑦)(𝐷 ↾ (𝑌 × 𝑌))(𝐹‘𝑧)) = ((𝐹‘𝑦)𝐷(𝐹‘𝑧))) |
48 | 47 | breq1d 5063 |
. . . . . . . . 9
⊢ (((𝐹:ℕ⟶𝑌 ∧ 𝑦 ∈ ℕ) ∧ 𝑧 ∈ (ℤ≥‘𝑦)) → (((𝐹‘𝑦)(𝐷 ↾ (𝑌 × 𝑌))(𝐹‘𝑧)) < 𝑥 ↔ ((𝐹‘𝑦)𝐷(𝐹‘𝑧)) < 𝑥)) |
49 | 48 | ralbidva 3117 |
. . . . . . . 8
⊢ ((𝐹:ℕ⟶𝑌 ∧ 𝑦 ∈ ℕ) → (∀𝑧 ∈
(ℤ≥‘𝑦)((𝐹‘𝑦)(𝐷 ↾ (𝑌 × 𝑌))(𝐹‘𝑧)) < 𝑥 ↔ ∀𝑧 ∈ (ℤ≥‘𝑦)((𝐹‘𝑦)𝐷(𝐹‘𝑧)) < 𝑥)) |
50 | 49 | rexbidva 3215 |
. . . . . . 7
⊢ (𝐹:ℕ⟶𝑌 → (∃𝑦 ∈ ℕ ∀𝑧 ∈
(ℤ≥‘𝑦)((𝐹‘𝑦)(𝐷 ↾ (𝑌 × 𝑌))(𝐹‘𝑧)) < 𝑥 ↔ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)((𝐹‘𝑦)𝐷(𝐹‘𝑧)) < 𝑥)) |
51 | 50 | ralbidv 3118 |
. . . . . 6
⊢ (𝐹:ℕ⟶𝑌 → (∀𝑥 ∈ ℝ+
∃𝑦 ∈ ℕ
∀𝑧 ∈
(ℤ≥‘𝑦)((𝐹‘𝑦)(𝐷 ↾ (𝑌 × 𝑌))(𝐹‘𝑧)) < 𝑥 ↔ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℕ ∀𝑧 ∈
(ℤ≥‘𝑦)((𝐹‘𝑦)𝐷(𝐹‘𝑧)) < 𝑥)) |
52 | 40, 51 | syl 17 |
. . . . 5
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹:ℕ⟶(𝑋 ∩ 𝑌)) → (∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℕ ∀𝑧 ∈
(ℤ≥‘𝑦)((𝐹‘𝑦)(𝐷 ↾ (𝑌 × 𝑌))(𝐹‘𝑧)) < 𝑥 ↔ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℕ ∀𝑧 ∈
(ℤ≥‘𝑦)((𝐹‘𝑦)𝐷(𝐹‘𝑧)) < 𝑥)) |
53 | | nnuz 12477 |
. . . . . 6
⊢ ℕ =
(ℤ≥‘1) |
54 | 17 | adantr 484 |
. . . . . 6
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹:ℕ⟶(𝑋 ∩ 𝑌)) → (𝐷 ↾ (𝑌 × 𝑌)) ∈ (∞Met‘(𝑋 ∩ 𝑌))) |
55 | | 1zzd 12208 |
. . . . . 6
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹:ℕ⟶(𝑋 ∩ 𝑌)) → 1 ∈ ℤ) |
56 | | eqidd 2738 |
. . . . . 6
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹:ℕ⟶(𝑋 ∩ 𝑌)) ∧ 𝑧 ∈ ℕ) → (𝐹‘𝑧) = (𝐹‘𝑧)) |
57 | | eqidd 2738 |
. . . . . 6
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹:ℕ⟶(𝑋 ∩ 𝑌)) ∧ 𝑦 ∈ ℕ) → (𝐹‘𝑦) = (𝐹‘𝑦)) |
58 | | simpr 488 |
. . . . . 6
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹:ℕ⟶(𝑋 ∩ 𝑌)) → 𝐹:ℕ⟶(𝑋 ∩ 𝑌)) |
59 | 53, 54, 55, 56, 57, 58 | iscauf 24177 |
. . . . 5
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹:ℕ⟶(𝑋 ∩ 𝑌)) → (𝐹 ∈ (Cau‘(𝐷 ↾ (𝑌 × 𝑌))) ↔ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℕ ∀𝑧 ∈
(ℤ≥‘𝑦)((𝐹‘𝑦)(𝐷 ↾ (𝑌 × 𝑌))(𝐹‘𝑧)) < 𝑥)) |
60 | | simpl 486 |
. . . . . 6
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹:ℕ⟶(𝑋 ∩ 𝑌)) → 𝐷 ∈ (∞Met‘𝑋)) |
61 | | id 22 |
. . . . . . 7
⊢ (𝐹:ℕ⟶(𝑋 ∩ 𝑌) → 𝐹:ℕ⟶(𝑋 ∩ 𝑌)) |
62 | | inss1 4143 |
. . . . . . . 8
⊢ (𝑋 ∩ 𝑌) ⊆ 𝑋 |
63 | 62 | a1i 11 |
. . . . . . 7
⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝑋 ∩ 𝑌) ⊆ 𝑋) |
64 | | fss 6562 |
. . . . . . 7
⊢ ((𝐹:ℕ⟶(𝑋 ∩ 𝑌) ∧ (𝑋 ∩ 𝑌) ⊆ 𝑋) → 𝐹:ℕ⟶𝑋) |
65 | 61, 63, 64 | syl2anr 600 |
. . . . . 6
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹:ℕ⟶(𝑋 ∩ 𝑌)) → 𝐹:ℕ⟶𝑋) |
66 | 53, 60, 55, 56, 57, 65 | iscauf 24177 |
. . . . 5
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹:ℕ⟶(𝑋 ∩ 𝑌)) → (𝐹 ∈ (Cau‘𝐷) ↔ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℕ ∀𝑧 ∈
(ℤ≥‘𝑦)((𝐹‘𝑦)𝐷(𝐹‘𝑧)) < 𝑥)) |
67 | 52, 59, 66 | 3bitr4rd 315 |
. . . 4
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹:ℕ⟶(𝑋 ∩ 𝑌)) → (𝐹 ∈ (Cau‘𝐷) ↔ 𝐹 ∈ (Cau‘(𝐷 ↾ (𝑌 × 𝑌))))) |
68 | 67 | ex 416 |
. . 3
⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐹:ℕ⟶(𝑋 ∩ 𝑌) → (𝐹 ∈ (Cau‘𝐷) ↔ 𝐹 ∈ (Cau‘(𝐷 ↾ (𝑌 × 𝑌)))))) |
69 | 35, 68 | sylan9r 512 |
. 2
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹:ℕ⟶𝑌) → (ran 𝐹 ⊆ (𝑋 ∩ 𝑌) → (𝐹 ∈ (Cau‘𝐷) ↔ 𝐹 ∈ (Cau‘(𝐷 ↾ (𝑌 × 𝑌)))))) |
70 | 16, 31, 69 | pm5.21ndd 384 |
1
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹:ℕ⟶𝑌) → (𝐹 ∈ (Cau‘𝐷) ↔ 𝐹 ∈ (Cau‘(𝐷 ↾ (𝑌 × 𝑌))))) |