Step | Hyp | Ref
| Expression |
1 | | prdsbl.y |
. . . . . . . . 9
⊢ 𝑌 = (𝑆Xs(𝑥 ∈ 𝐼 ↦ 𝑅)) |
2 | | prdsbl.b |
. . . . . . . . 9
⊢ 𝐵 = (Base‘𝑌) |
3 | | prdsbl.s |
. . . . . . . . 9
⊢ (𝜑 → 𝑆 ∈ 𝑊) |
4 | | prdsbl.i |
. . . . . . . . 9
⊢ (𝜑 → 𝐼 ∈ Fin) |
5 | | prdsbl.r |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑅 ∈ 𝑍) |
6 | 5 | ralrimiva 3103 |
. . . . . . . . 9
⊢ (𝜑 → ∀𝑥 ∈ 𝐼 𝑅 ∈ 𝑍) |
7 | | prdsbl.v |
. . . . . . . . 9
⊢ 𝑉 = (Base‘𝑅) |
8 | 1, 2, 3, 4, 6, 7 | prdsbas3 17192 |
. . . . . . . 8
⊢ (𝜑 → 𝐵 = X𝑥 ∈ 𝐼 𝑉) |
9 | 8 | eleq2d 2824 |
. . . . . . 7
⊢ (𝜑 → (𝑓 ∈ 𝐵 ↔ 𝑓 ∈ X𝑥 ∈ 𝐼 𝑉)) |
10 | 9 | biimpa 477 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → 𝑓 ∈ X𝑥 ∈ 𝐼 𝑉) |
11 | | ixpfn 8691 |
. . . . . 6
⊢ (𝑓 ∈ X𝑥 ∈
𝐼 𝑉 → 𝑓 Fn 𝐼) |
12 | | vex 3436 |
. . . . . . . 8
⊢ 𝑓 ∈ V |
13 | 12 | elixp 8692 |
. . . . . . 7
⊢ (𝑓 ∈ X𝑥 ∈
𝐼 ((𝑃‘𝑥)(ball‘𝐸)𝐴) ↔ (𝑓 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑓‘𝑥) ∈ ((𝑃‘𝑥)(ball‘𝐸)𝐴))) |
14 | 13 | baib 536 |
. . . . . 6
⊢ (𝑓 Fn 𝐼 → (𝑓 ∈ X𝑥 ∈ 𝐼 ((𝑃‘𝑥)(ball‘𝐸)𝐴) ↔ ∀𝑥 ∈ 𝐼 (𝑓‘𝑥) ∈ ((𝑃‘𝑥)(ball‘𝐸)𝐴))) |
15 | 10, 11, 14 | 3syl 18 |
. . . . 5
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → (𝑓 ∈ X𝑥 ∈ 𝐼 ((𝑃‘𝑥)(ball‘𝐸)𝐴) ↔ ∀𝑥 ∈ 𝐼 (𝑓‘𝑥) ∈ ((𝑃‘𝑥)(ball‘𝐸)𝐴))) |
16 | | prdsbl.m |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐸 ∈ (∞Met‘𝑉)) |
17 | 16 | adantlr 712 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → 𝐸 ∈ (∞Met‘𝑉)) |
18 | | prdsbl.a |
. . . . . . . 8
⊢ (𝜑 → 𝐴 ∈
ℝ*) |
19 | 18 | ad2antrr 723 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → 𝐴 ∈
ℝ*) |
20 | | prdsbl.p |
. . . . . . . . . 10
⊢ (𝜑 → 𝑃 ∈ 𝐵) |
21 | 1, 2, 3, 4, 6, 7, 20 | prdsbascl 17194 |
. . . . . . . . 9
⊢ (𝜑 → ∀𝑥 ∈ 𝐼 (𝑃‘𝑥) ∈ 𝑉) |
22 | 21 | adantr 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → ∀𝑥 ∈ 𝐼 (𝑃‘𝑥) ∈ 𝑉) |
23 | 22 | r19.21bi 3134 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → (𝑃‘𝑥) ∈ 𝑉) |
24 | 3 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → 𝑆 ∈ 𝑊) |
25 | 4 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → 𝐼 ∈ Fin) |
26 | 6 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → ∀𝑥 ∈ 𝐼 𝑅 ∈ 𝑍) |
27 | | simpr 485 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → 𝑓 ∈ 𝐵) |
28 | 1, 2, 24, 25, 26, 7, 27 | prdsbascl 17194 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → ∀𝑥 ∈ 𝐼 (𝑓‘𝑥) ∈ 𝑉) |
29 | 28 | r19.21bi 3134 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → (𝑓‘𝑥) ∈ 𝑉) |
30 | | elbl2 23543 |
. . . . . . 7
⊢ (((𝐸 ∈ (∞Met‘𝑉) ∧ 𝐴 ∈ ℝ*) ∧ ((𝑃‘𝑥) ∈ 𝑉 ∧ (𝑓‘𝑥) ∈ 𝑉)) → ((𝑓‘𝑥) ∈ ((𝑃‘𝑥)(ball‘𝐸)𝐴) ↔ ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) < 𝐴)) |
31 | 17, 19, 23, 29, 30 | syl22anc 836 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → ((𝑓‘𝑥) ∈ ((𝑃‘𝑥)(ball‘𝐸)𝐴) ↔ ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) < 𝐴)) |
32 | 31 | ralbidva 3111 |
. . . . 5
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → (∀𝑥 ∈ 𝐼 (𝑓‘𝑥) ∈ ((𝑃‘𝑥)(ball‘𝐸)𝐴) ↔ ∀𝑥 ∈ 𝐼 ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) < 𝐴)) |
33 | | xmetcl 23484 |
. . . . . . . . . 10
⊢ ((𝐸 ∈ (∞Met‘𝑉) ∧ (𝑃‘𝑥) ∈ 𝑉 ∧ (𝑓‘𝑥) ∈ 𝑉) → ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) ∈
ℝ*) |
34 | 17, 23, 29, 33 | syl3anc 1370 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) ∈
ℝ*) |
35 | 34 | ralrimiva 3103 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → ∀𝑥 ∈ 𝐼 ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) ∈
ℝ*) |
36 | | eqid 2738 |
. . . . . . . . 9
⊢ (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) = (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) |
37 | | breq1 5077 |
. . . . . . . . 9
⊢ (𝑧 = ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) → (𝑧 < 𝐴 ↔ ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) < 𝐴)) |
38 | 36, 37 | ralrnmptw 6970 |
. . . . . . . 8
⊢
(∀𝑥 ∈
𝐼 ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) ∈ ℝ* →
(∀𝑧 ∈ ran
(𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥)))𝑧 < 𝐴 ↔ ∀𝑥 ∈ 𝐼 ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) < 𝐴)) |
39 | 35, 38 | syl 17 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → (∀𝑧 ∈ ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥)))𝑧 < 𝐴 ↔ ∀𝑥 ∈ 𝐼 ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) < 𝐴)) |
40 | | prdsbl.g |
. . . . . . . . . 10
⊢ (𝜑 → 0 < 𝐴) |
41 | 40 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → 0 < 𝐴) |
42 | | c0ex 10969 |
. . . . . . . . . 10
⊢ 0 ∈
V |
43 | | breq1 5077 |
. . . . . . . . . 10
⊢ (𝑧 = 0 → (𝑧 < 𝐴 ↔ 0 < 𝐴)) |
44 | 42, 43 | ralsn 4617 |
. . . . . . . . 9
⊢
(∀𝑧 ∈
{0}𝑧 < 𝐴 ↔ 0 < 𝐴) |
45 | 41, 44 | sylibr 233 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → ∀𝑧 ∈ {0}𝑧 < 𝐴) |
46 | | ralunb 4125 |
. . . . . . . . 9
⊢
(∀𝑧 ∈
(ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0})𝑧 < 𝐴 ↔ (∀𝑧 ∈ ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥)))𝑧 < 𝐴 ∧ ∀𝑧 ∈ {0}𝑧 < 𝐴)) |
47 | 20 | adantr 481 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → 𝑃 ∈ 𝐵) |
48 | | prdsbl.e |
. . . . . . . . . . . 12
⊢ 𝐸 = ((dist‘𝑅) ↾ (𝑉 × 𝑉)) |
49 | | prdsbl.d |
. . . . . . . . . . . 12
⊢ 𝐷 = (dist‘𝑌) |
50 | 1, 2, 24, 25, 26, 47, 27, 7, 48, 49 | prdsdsval3 17196 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → (𝑃𝐷𝑓) = sup((ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0}), ℝ*, <
)) |
51 | | xrltso 12875 |
. . . . . . . . . . . . 13
⊢ < Or
ℝ* |
52 | 51 | a1i 11 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → < Or
ℝ*) |
53 | 36 | rnmpt 5864 |
. . . . . . . . . . . . . . 15
⊢ ran
(𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) = {𝑦 ∣ ∃𝑥 ∈ 𝐼 𝑦 = ((𝑃‘𝑥)𝐸(𝑓‘𝑥))} |
54 | | abrexfi 9119 |
. . . . . . . . . . . . . . 15
⊢ (𝐼 ∈ Fin → {𝑦 ∣ ∃𝑥 ∈ 𝐼 𝑦 = ((𝑃‘𝑥)𝐸(𝑓‘𝑥))} ∈ Fin) |
55 | 53, 54 | eqeltrid 2843 |
. . . . . . . . . . . . . 14
⊢ (𝐼 ∈ Fin → ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∈ Fin) |
56 | 25, 55 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∈ Fin) |
57 | | snfi 8834 |
. . . . . . . . . . . . 13
⊢ {0}
∈ Fin |
58 | | unfi 8955 |
. . . . . . . . . . . . 13
⊢ ((ran
(𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∈ Fin ∧ {0} ∈ Fin) →
(ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0}) ∈ Fin) |
59 | 56, 57, 58 | sylancl 586 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → (ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0}) ∈ Fin) |
60 | | ssun2 4107 |
. . . . . . . . . . . . . 14
⊢ {0}
⊆ (ran (𝑥 ∈
𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0}) |
61 | 42 | snss 4719 |
. . . . . . . . . . . . . 14
⊢ (0 ∈
(ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0}) ↔ {0} ⊆ (ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0})) |
62 | 60, 61 | mpbir 230 |
. . . . . . . . . . . . 13
⊢ 0 ∈
(ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0}) |
63 | | ne0i 4268 |
. . . . . . . . . . . . 13
⊢ (0 ∈
(ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0}) → (ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0}) ≠ ∅) |
64 | 62, 63 | mp1i 13 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → (ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0}) ≠ ∅) |
65 | 34 | fmpttd 6989 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))):𝐼⟶ℝ*) |
66 | 65 | frnd 6608 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ⊆
ℝ*) |
67 | | 0xr 11022 |
. . . . . . . . . . . . . . 15
⊢ 0 ∈
ℝ* |
68 | 67 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → 0 ∈
ℝ*) |
69 | 68 | snssd 4742 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → {0} ⊆
ℝ*) |
70 | 66, 69 | unssd 4120 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → (ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0}) ⊆
ℝ*) |
71 | | fisupcl 9228 |
. . . . . . . . . . . 12
⊢ (( <
Or ℝ* ∧ ((ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0}) ∈ Fin ∧ (ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0}) ≠ ∅ ∧ (ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0}) ⊆ ℝ*))
→ sup((ran (𝑥 ∈
𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0}), ℝ*, < )
∈ (ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0})) |
72 | 52, 59, 64, 70, 71 | syl13anc 1371 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → sup((ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0}), ℝ*, < )
∈ (ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0})) |
73 | 50, 72 | eqeltrd 2839 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → (𝑃𝐷𝑓) ∈ (ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0})) |
74 | | breq1 5077 |
. . . . . . . . . . 11
⊢ (𝑧 = (𝑃𝐷𝑓) → (𝑧 < 𝐴 ↔ (𝑃𝐷𝑓) < 𝐴)) |
75 | 74 | rspcv 3557 |
. . . . . . . . . 10
⊢ ((𝑃𝐷𝑓) ∈ (ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0}) → (∀𝑧 ∈ (ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0})𝑧 < 𝐴 → (𝑃𝐷𝑓) < 𝐴)) |
76 | 73, 75 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → (∀𝑧 ∈ (ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0})𝑧 < 𝐴 → (𝑃𝐷𝑓) < 𝐴)) |
77 | 46, 76 | syl5bir 242 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → ((∀𝑧 ∈ ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥)))𝑧 < 𝐴 ∧ ∀𝑧 ∈ {0}𝑧 < 𝐴) → (𝑃𝐷𝑓) < 𝐴)) |
78 | 45, 77 | mpan2d 691 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → (∀𝑧 ∈ ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥)))𝑧 < 𝐴 → (𝑃𝐷𝑓) < 𝐴)) |
79 | 39, 78 | sylbird 259 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → (∀𝑥 ∈ 𝐼 ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) < 𝐴 → (𝑃𝐷𝑓) < 𝐴)) |
80 | | ssun1 4106 |
. . . . . . . . . . 11
⊢ ran
(𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ⊆ (ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0}) |
81 | | ovex 7308 |
. . . . . . . . . . . . . 14
⊢ ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) ∈ V |
82 | 81 | elabrex 7116 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ 𝐼 → ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐼 𝑦 = ((𝑃‘𝑥)𝐸(𝑓‘𝑥))}) |
83 | 82 | adantl 482 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐼 𝑦 = ((𝑃‘𝑥)𝐸(𝑓‘𝑥))}) |
84 | 83, 53 | eleqtrrdi 2850 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) ∈ ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥)))) |
85 | 80, 84 | sselid 3919 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) ∈ (ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0})) |
86 | | supxrub 13058 |
. . . . . . . . . 10
⊢ (((ran
(𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0}) ⊆ ℝ*
∧ ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) ∈ (ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0})) → ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) ≤ sup((ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0}), ℝ*, <
)) |
87 | 70, 85, 86 | syl2an2r 682 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) ≤ sup((ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0}), ℝ*, <
)) |
88 | 50 | adantr 481 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → (𝑃𝐷𝑓) = sup((ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0}), ℝ*, <
)) |
89 | 87, 88 | breqtrrd 5102 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) ≤ (𝑃𝐷𝑓)) |
90 | 1, 2, 7, 48, 49, 3, 4, 5, 16 | prdsxmet 23522 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝐵)) |
91 | 90 | ad2antrr 723 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → 𝐷 ∈ (∞Met‘𝐵)) |
92 | 20 | ad2antrr 723 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → 𝑃 ∈ 𝐵) |
93 | 27 | adantr 481 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → 𝑓 ∈ 𝐵) |
94 | | xmetcl 23484 |
. . . . . . . . . 10
⊢ ((𝐷 ∈ (∞Met‘𝐵) ∧ 𝑃 ∈ 𝐵 ∧ 𝑓 ∈ 𝐵) → (𝑃𝐷𝑓) ∈
ℝ*) |
95 | 91, 92, 93, 94 | syl3anc 1370 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → (𝑃𝐷𝑓) ∈
ℝ*) |
96 | | xrlelttr 12890 |
. . . . . . . . 9
⊢ ((((𝑃‘𝑥)𝐸(𝑓‘𝑥)) ∈ ℝ* ∧ (𝑃𝐷𝑓) ∈ ℝ* ∧ 𝐴 ∈ ℝ*)
→ ((((𝑃‘𝑥)𝐸(𝑓‘𝑥)) ≤ (𝑃𝐷𝑓) ∧ (𝑃𝐷𝑓) < 𝐴) → ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) < 𝐴)) |
97 | 34, 95, 19, 96 | syl3anc 1370 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → ((((𝑃‘𝑥)𝐸(𝑓‘𝑥)) ≤ (𝑃𝐷𝑓) ∧ (𝑃𝐷𝑓) < 𝐴) → ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) < 𝐴)) |
98 | 89, 97 | mpand 692 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → ((𝑃𝐷𝑓) < 𝐴 → ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) < 𝐴)) |
99 | 98 | ralrimdva 3106 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → ((𝑃𝐷𝑓) < 𝐴 → ∀𝑥 ∈ 𝐼 ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) < 𝐴)) |
100 | 79, 99 | impbid 211 |
. . . . 5
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → (∀𝑥 ∈ 𝐼 ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) < 𝐴 ↔ (𝑃𝐷𝑓) < 𝐴)) |
101 | 15, 32, 100 | 3bitrrd 306 |
. . . 4
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → ((𝑃𝐷𝑓) < 𝐴 ↔ 𝑓 ∈ X𝑥 ∈ 𝐼 ((𝑃‘𝑥)(ball‘𝐸)𝐴))) |
102 | 101 | pm5.32da 579 |
. . 3
⊢ (𝜑 → ((𝑓 ∈ 𝐵 ∧ (𝑃𝐷𝑓) < 𝐴) ↔ (𝑓 ∈ 𝐵 ∧ 𝑓 ∈ X𝑥 ∈ 𝐼 ((𝑃‘𝑥)(ball‘𝐸)𝐴)))) |
103 | | elbl 23541 |
. . . 4
⊢ ((𝐷 ∈ (∞Met‘𝐵) ∧ 𝑃 ∈ 𝐵 ∧ 𝐴 ∈ ℝ*) → (𝑓 ∈ (𝑃(ball‘𝐷)𝐴) ↔ (𝑓 ∈ 𝐵 ∧ (𝑃𝐷𝑓) < 𝐴))) |
104 | 90, 20, 18, 103 | syl3anc 1370 |
. . 3
⊢ (𝜑 → (𝑓 ∈ (𝑃(ball‘𝐷)𝐴) ↔ (𝑓 ∈ 𝐵 ∧ (𝑃𝐷𝑓) < 𝐴))) |
105 | 21 | r19.21bi 3134 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑃‘𝑥) ∈ 𝑉) |
106 | 18 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐴 ∈
ℝ*) |
107 | | blssm 23571 |
. . . . . . . . 9
⊢ ((𝐸 ∈ (∞Met‘𝑉) ∧ (𝑃‘𝑥) ∈ 𝑉 ∧ 𝐴 ∈ ℝ*) → ((𝑃‘𝑥)(ball‘𝐸)𝐴) ⊆ 𝑉) |
108 | 16, 105, 106, 107 | syl3anc 1370 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ((𝑃‘𝑥)(ball‘𝐸)𝐴) ⊆ 𝑉) |
109 | 108 | ralrimiva 3103 |
. . . . . . 7
⊢ (𝜑 → ∀𝑥 ∈ 𝐼 ((𝑃‘𝑥)(ball‘𝐸)𝐴) ⊆ 𝑉) |
110 | | ss2ixp 8698 |
. . . . . . 7
⊢
(∀𝑥 ∈
𝐼 ((𝑃‘𝑥)(ball‘𝐸)𝐴) ⊆ 𝑉 → X𝑥 ∈ 𝐼 ((𝑃‘𝑥)(ball‘𝐸)𝐴) ⊆ X𝑥 ∈ 𝐼 𝑉) |
111 | 109, 110 | syl 17 |
. . . . . 6
⊢ (𝜑 → X𝑥 ∈
𝐼 ((𝑃‘𝑥)(ball‘𝐸)𝐴) ⊆ X𝑥 ∈ 𝐼 𝑉) |
112 | 111, 8 | sseqtrrd 3962 |
. . . . 5
⊢ (𝜑 → X𝑥 ∈
𝐼 ((𝑃‘𝑥)(ball‘𝐸)𝐴) ⊆ 𝐵) |
113 | 112 | sseld 3920 |
. . . 4
⊢ (𝜑 → (𝑓 ∈ X𝑥 ∈ 𝐼 ((𝑃‘𝑥)(ball‘𝐸)𝐴) → 𝑓 ∈ 𝐵)) |
114 | 113 | pm4.71rd 563 |
. . 3
⊢ (𝜑 → (𝑓 ∈ X𝑥 ∈ 𝐼 ((𝑃‘𝑥)(ball‘𝐸)𝐴) ↔ (𝑓 ∈ 𝐵 ∧ 𝑓 ∈ X𝑥 ∈ 𝐼 ((𝑃‘𝑥)(ball‘𝐸)𝐴)))) |
115 | 102, 104,
114 | 3bitr4d 311 |
. 2
⊢ (𝜑 → (𝑓 ∈ (𝑃(ball‘𝐷)𝐴) ↔ 𝑓 ∈ X𝑥 ∈ 𝐼 ((𝑃‘𝑥)(ball‘𝐸)𝐴))) |
116 | 115 | eqrdv 2736 |
1
⊢ (𝜑 → (𝑃(ball‘𝐷)𝐴) = X𝑥 ∈ 𝐼 ((𝑃‘𝑥)(ball‘𝐸)𝐴)) |