| 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 3146 |
. . . . . . . . 9
⊢ (𝜑 → ∀𝑥 ∈ 𝐼 𝑅 ∈ 𝑍) |
| 7 | | prdsbl.v |
. . . . . . . . 9
⊢ 𝑉 = (Base‘𝑅) |
| 8 | 1, 2, 3, 4, 6, 7 | prdsbas3 17526 |
. . . . . . . 8
⊢ (𝜑 → 𝐵 = X𝑥 ∈ 𝐼 𝑉) |
| 9 | 8 | eleq2d 2827 |
. . . . . . 7
⊢ (𝜑 → (𝑓 ∈ 𝐵 ↔ 𝑓 ∈ X𝑥 ∈ 𝐼 𝑉)) |
| 10 | 9 | biimpa 476 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → 𝑓 ∈ X𝑥 ∈ 𝐼 𝑉) |
| 11 | | ixpfn 8943 |
. . . . . 6
⊢ (𝑓 ∈ X𝑥 ∈
𝐼 𝑉 → 𝑓 Fn 𝐼) |
| 12 | | vex 3484 |
. . . . . . . 8
⊢ 𝑓 ∈ V |
| 13 | 12 | elixp 8944 |
. . . . . . 7
⊢ (𝑓 ∈ X𝑥 ∈
𝐼 ((𝑃‘𝑥)(ball‘𝐸)𝐴) ↔ (𝑓 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑓‘𝑥) ∈ ((𝑃‘𝑥)(ball‘𝐸)𝐴))) |
| 14 | 13 | baib 535 |
. . . . . 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 715 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → 𝐸 ∈ (∞Met‘𝑉)) |
| 18 | | prdsbl.a |
. . . . . . . 8
⊢ (𝜑 → 𝐴 ∈
ℝ*) |
| 19 | 18 | ad2antrr 726 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → 𝐴 ∈
ℝ*) |
| 20 | | prdsbl.p |
. . . . . . . . . 10
⊢ (𝜑 → 𝑃 ∈ 𝐵) |
| 21 | 1, 2, 3, 4, 6, 7, 20 | prdsbascl 17528 |
. . . . . . . . 9
⊢ (𝜑 → ∀𝑥 ∈ 𝐼 (𝑃‘𝑥) ∈ 𝑉) |
| 22 | 21 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → ∀𝑥 ∈ 𝐼 (𝑃‘𝑥) ∈ 𝑉) |
| 23 | 22 | r19.21bi 3251 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → (𝑃‘𝑥) ∈ 𝑉) |
| 24 | 3 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → 𝑆 ∈ 𝑊) |
| 25 | 4 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → 𝐼 ∈ Fin) |
| 26 | 6 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → ∀𝑥 ∈ 𝐼 𝑅 ∈ 𝑍) |
| 27 | | simpr 484 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → 𝑓 ∈ 𝐵) |
| 28 | 1, 2, 24, 25, 26, 7, 27 | prdsbascl 17528 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → ∀𝑥 ∈ 𝐼 (𝑓‘𝑥) ∈ 𝑉) |
| 29 | 28 | r19.21bi 3251 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → (𝑓‘𝑥) ∈ 𝑉) |
| 30 | | elbl2 24400 |
. . . . . . 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 839 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → ((𝑓‘𝑥) ∈ ((𝑃‘𝑥)(ball‘𝐸)𝐴) ↔ ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) < 𝐴)) |
| 32 | 31 | ralbidva 3176 |
. . . . 5
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → (∀𝑥 ∈ 𝐼 (𝑓‘𝑥) ∈ ((𝑃‘𝑥)(ball‘𝐸)𝐴) ↔ ∀𝑥 ∈ 𝐼 ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) < 𝐴)) |
| 33 | | xmetcl 24341 |
. . . . . . . . . 10
⊢ ((𝐸 ∈ (∞Met‘𝑉) ∧ (𝑃‘𝑥) ∈ 𝑉 ∧ (𝑓‘𝑥) ∈ 𝑉) → ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) ∈
ℝ*) |
| 34 | 17, 23, 29, 33 | syl3anc 1373 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) ∈
ℝ*) |
| 35 | 34 | ralrimiva 3146 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → ∀𝑥 ∈ 𝐼 ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) ∈
ℝ*) |
| 36 | | eqid 2737 |
. . . . . . . . 9
⊢ (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) = (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) |
| 37 | | breq1 5146 |
. . . . . . . . 9
⊢ (𝑧 = ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) → (𝑧 < 𝐴 ↔ ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) < 𝐴)) |
| 38 | 36, 37 | ralrnmptw 7114 |
. . . . . . . 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 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → 0 < 𝐴) |
| 42 | | c0ex 11255 |
. . . . . . . . . 10
⊢ 0 ∈
V |
| 43 | | breq1 5146 |
. . . . . . . . . 10
⊢ (𝑧 = 0 → (𝑧 < 𝐴 ↔ 0 < 𝐴)) |
| 44 | 42, 43 | ralsn 4681 |
. . . . . . . . 9
⊢
(∀𝑧 ∈
{0}𝑧 < 𝐴 ↔ 0 < 𝐴) |
| 45 | 41, 44 | sylibr 234 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → ∀𝑧 ∈ {0}𝑧 < 𝐴) |
| 46 | | ralunb 4197 |
. . . . . . . . 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 480 |
. . . . . . . . . . . 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 17530 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → (𝑃𝐷𝑓) = sup((ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0}), ℝ*, <
)) |
| 51 | | xrltso 13183 |
. . . . . . . . . . . . 13
⊢ < Or
ℝ* |
| 52 | 51 | a1i 11 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → < Or
ℝ*) |
| 53 | 36 | rnmpt 5968 |
. . . . . . . . . . . . . . 15
⊢ ran
(𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) = {𝑦 ∣ ∃𝑥 ∈ 𝐼 𝑦 = ((𝑃‘𝑥)𝐸(𝑓‘𝑥))} |
| 54 | | abrexfi 9392 |
. . . . . . . . . . . . . . 15
⊢ (𝐼 ∈ Fin → {𝑦 ∣ ∃𝑥 ∈ 𝐼 𝑦 = ((𝑃‘𝑥)𝐸(𝑓‘𝑥))} ∈ Fin) |
| 55 | 53, 54 | eqeltrid 2845 |
. . . . . . . . . . . . . 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 9083 |
. . . . . . . . . . . . 13
⊢ {0}
∈ Fin |
| 58 | | unfi 9211 |
. . . . . . . . . . . . 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 4179 |
. . . . . . . . . . . . . 14
⊢ {0}
⊆ (ran (𝑥 ∈
𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0}) |
| 61 | 42 | snss 4785 |
. . . . . . . . . . . . . 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 231 |
. . . . . . . . . . . . 13
⊢ 0 ∈
(ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0}) |
| 63 | | ne0i 4341 |
. . . . . . . . . . . . 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 7135 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))):𝐼⟶ℝ*) |
| 66 | 65 | frnd 6744 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ⊆
ℝ*) |
| 67 | | 0xr 11308 |
. . . . . . . . . . . . . . 15
⊢ 0 ∈
ℝ* |
| 68 | 67 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → 0 ∈
ℝ*) |
| 69 | 68 | snssd 4809 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → {0} ⊆
ℝ*) |
| 70 | 66, 69 | unssd 4192 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → (ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0}) ⊆
ℝ*) |
| 71 | | fisupcl 9509 |
. . . . . . . . . . . 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 1374 |
. . . . . . . . . . 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 2841 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → (𝑃𝐷𝑓) ∈ (ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0})) |
| 74 | | breq1 5146 |
. . . . . . . . . . 11
⊢ (𝑧 = (𝑃𝐷𝑓) → (𝑧 < 𝐴 ↔ (𝑃𝐷𝑓) < 𝐴)) |
| 75 | 74 | rspcv 3618 |
. . . . . . . . . 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 | biimtrrid 243 |
. . . . . . . 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 694 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → (∀𝑧 ∈ ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥)))𝑧 < 𝐴 → (𝑃𝐷𝑓) < 𝐴)) |
| 79 | 39, 78 | sylbird 260 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → (∀𝑥 ∈ 𝐼 ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) < 𝐴 → (𝑃𝐷𝑓) < 𝐴)) |
| 80 | | ssun1 4178 |
. . . . . . . . . . 11
⊢ ran
(𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ⊆ (ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0}) |
| 81 | | ovex 7464 |
. . . . . . . . . . . . . 14
⊢ ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) ∈ V |
| 82 | 81 | elabrex 7262 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ 𝐼 → ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐼 𝑦 = ((𝑃‘𝑥)𝐸(𝑓‘𝑥))}) |
| 83 | 82 | adantl 481 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐼 𝑦 = ((𝑃‘𝑥)𝐸(𝑓‘𝑥))}) |
| 84 | 83, 53 | eleqtrrdi 2852 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) ∈ ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥)))) |
| 85 | 80, 84 | sselid 3981 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) ∈ (ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0})) |
| 86 | | supxrub 13366 |
. . . . . . . . . 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 685 |
. . . . . . . . 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 480 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → (𝑃𝐷𝑓) = sup((ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0}), ℝ*, <
)) |
| 89 | 87, 88 | breqtrrd 5171 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) ≤ (𝑃𝐷𝑓)) |
| 90 | 1, 2, 7, 48, 49, 3, 4, 5, 16 | prdsxmet 24379 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝐵)) |
| 91 | 90 | ad2antrr 726 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → 𝐷 ∈ (∞Met‘𝐵)) |
| 92 | 20 | ad2antrr 726 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → 𝑃 ∈ 𝐵) |
| 93 | 27 | adantr 480 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → 𝑓 ∈ 𝐵) |
| 94 | | xmetcl 24341 |
. . . . . . . . . 10
⊢ ((𝐷 ∈ (∞Met‘𝐵) ∧ 𝑃 ∈ 𝐵 ∧ 𝑓 ∈ 𝐵) → (𝑃𝐷𝑓) ∈
ℝ*) |
| 95 | 91, 92, 93, 94 | syl3anc 1373 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → (𝑃𝐷𝑓) ∈
ℝ*) |
| 96 | | xrlelttr 13198 |
. . . . . . . . 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 1373 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → ((((𝑃‘𝑥)𝐸(𝑓‘𝑥)) ≤ (𝑃𝐷𝑓) ∧ (𝑃𝐷𝑓) < 𝐴) → ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) < 𝐴)) |
| 98 | 89, 97 | mpand 695 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → ((𝑃𝐷𝑓) < 𝐴 → ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) < 𝐴)) |
| 99 | 98 | ralrimdva 3154 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → ((𝑃𝐷𝑓) < 𝐴 → ∀𝑥 ∈ 𝐼 ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) < 𝐴)) |
| 100 | 79, 99 | impbid 212 |
. . . . 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 24398 |
. . . 4
⊢ ((𝐷 ∈ (∞Met‘𝐵) ∧ 𝑃 ∈ 𝐵 ∧ 𝐴 ∈ ℝ*) → (𝑓 ∈ (𝑃(ball‘𝐷)𝐴) ↔ (𝑓 ∈ 𝐵 ∧ (𝑃𝐷𝑓) < 𝐴))) |
| 104 | 90, 20, 18, 103 | syl3anc 1373 |
. . 3
⊢ (𝜑 → (𝑓 ∈ (𝑃(ball‘𝐷)𝐴) ↔ (𝑓 ∈ 𝐵 ∧ (𝑃𝐷𝑓) < 𝐴))) |
| 105 | 21 | r19.21bi 3251 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑃‘𝑥) ∈ 𝑉) |
| 106 | 18 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐴 ∈
ℝ*) |
| 107 | | blssm 24428 |
. . . . . . . . 9
⊢ ((𝐸 ∈ (∞Met‘𝑉) ∧ (𝑃‘𝑥) ∈ 𝑉 ∧ 𝐴 ∈ ℝ*) → ((𝑃‘𝑥)(ball‘𝐸)𝐴) ⊆ 𝑉) |
| 108 | 16, 105, 106, 107 | syl3anc 1373 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ((𝑃‘𝑥)(ball‘𝐸)𝐴) ⊆ 𝑉) |
| 109 | 108 | ralrimiva 3146 |
. . . . . . 7
⊢ (𝜑 → ∀𝑥 ∈ 𝐼 ((𝑃‘𝑥)(ball‘𝐸)𝐴) ⊆ 𝑉) |
| 110 | | ss2ixp 8950 |
. . . . . . 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 4021 |
. . . . 5
⊢ (𝜑 → X𝑥 ∈
𝐼 ((𝑃‘𝑥)(ball‘𝐸)𝐴) ⊆ 𝐵) |
| 113 | 112 | sseld 3982 |
. . . 4
⊢ (𝜑 → (𝑓 ∈ X𝑥 ∈ 𝐼 ((𝑃‘𝑥)(ball‘𝐸)𝐴) → 𝑓 ∈ 𝐵)) |
| 114 | 113 | pm4.71rd 562 |
. . 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 2735 |
1
⊢ (𝜑 → (𝑃(ball‘𝐷)𝐴) = X𝑥 ∈ 𝐼 ((𝑃‘𝑥)(ball‘𝐸)𝐴)) |