Step | Hyp | Ref
| Expression |
1 | | hoiqssbllem2.x |
. . . . . . . . 9
⊢ (𝜑 → 𝑋 ∈ Fin) |
2 | | eqid 2737 |
. . . . . . . . . 10
⊢
(ℝ^‘𝑋) =
(ℝ^‘𝑋) |
3 | | eqid 2737 |
. . . . . . . . . 10
⊢ (ℝ
↑m 𝑋) =
(ℝ ↑m 𝑋) |
4 | 2, 3 | rrxdsfi 24308 |
. . . . . . . . 9
⊢ (𝑋 ∈ Fin →
(dist‘(ℝ^‘𝑋)) = (𝑔 ∈ (ℝ ↑m 𝑋), ℎ ∈ (ℝ ↑m 𝑋) ↦
(√‘Σ𝑖
∈ 𝑋 (((𝑔‘𝑖) − (ℎ‘𝑖))↑2)))) |
5 | 1, 4 | syl 17 |
. . . . . . . 8
⊢ (𝜑 →
(dist‘(ℝ^‘𝑋)) = (𝑔 ∈ (ℝ ↑m 𝑋), ℎ ∈ (ℝ ↑m 𝑋) ↦
(√‘Σ𝑖
∈ 𝑋 (((𝑔‘𝑖) − (ℎ‘𝑖))↑2)))) |
6 | 5 | adantr 484 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → (dist‘(ℝ^‘𝑋)) = (𝑔 ∈ (ℝ ↑m 𝑋), ℎ ∈ (ℝ ↑m 𝑋) ↦
(√‘Σ𝑖
∈ 𝑋 (((𝑔‘𝑖) − (ℎ‘𝑖))↑2)))) |
7 | | fveq1 6716 |
. . . . . . . . . . . . 13
⊢ (𝑔 = 𝑌 → (𝑔‘𝑖) = (𝑌‘𝑖)) |
8 | 7 | adantr 484 |
. . . . . . . . . . . 12
⊢ ((𝑔 = 𝑌 ∧ ℎ = 𝑓) → (𝑔‘𝑖) = (𝑌‘𝑖)) |
9 | | fveq1 6716 |
. . . . . . . . . . . . 13
⊢ (ℎ = 𝑓 → (ℎ‘𝑖) = (𝑓‘𝑖)) |
10 | 9 | adantl 485 |
. . . . . . . . . . . 12
⊢ ((𝑔 = 𝑌 ∧ ℎ = 𝑓) → (ℎ‘𝑖) = (𝑓‘𝑖)) |
11 | 8, 10 | oveq12d 7231 |
. . . . . . . . . . 11
⊢ ((𝑔 = 𝑌 ∧ ℎ = 𝑓) → ((𝑔‘𝑖) − (ℎ‘𝑖)) = ((𝑌‘𝑖) − (𝑓‘𝑖))) |
12 | 11 | oveq1d 7228 |
. . . . . . . . . 10
⊢ ((𝑔 = 𝑌 ∧ ℎ = 𝑓) → (((𝑔‘𝑖) − (ℎ‘𝑖))↑2) = (((𝑌‘𝑖) − (𝑓‘𝑖))↑2)) |
13 | 12 | sumeq2sdv 15268 |
. . . . . . . . 9
⊢ ((𝑔 = 𝑌 ∧ ℎ = 𝑓) → Σ𝑖 ∈ 𝑋 (((𝑔‘𝑖) − (ℎ‘𝑖))↑2) = Σ𝑖 ∈ 𝑋 (((𝑌‘𝑖) − (𝑓‘𝑖))↑2)) |
14 | 13 | fveq2d 6721 |
. . . . . . . 8
⊢ ((𝑔 = 𝑌 ∧ ℎ = 𝑓) → (√‘Σ𝑖 ∈ 𝑋 (((𝑔‘𝑖) − (ℎ‘𝑖))↑2)) = (√‘Σ𝑖 ∈ 𝑋 (((𝑌‘𝑖) − (𝑓‘𝑖))↑2))) |
15 | 14 | adantl 485 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) ∧ (𝑔 = 𝑌 ∧ ℎ = 𝑓)) → (√‘Σ𝑖 ∈ 𝑋 (((𝑔‘𝑖) − (ℎ‘𝑖))↑2)) = (√‘Σ𝑖 ∈ 𝑋 (((𝑌‘𝑖) − (𝑓‘𝑖))↑2))) |
16 | | hoiqssbllem2.y |
. . . . . . . 8
⊢ (𝜑 → 𝑌 ∈ (ℝ ↑m 𝑋)) |
17 | 16 | adantr 484 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → 𝑌 ∈ (ℝ ↑m 𝑋)) |
18 | | hoiqssbllem2.i |
. . . . . . . . . 10
⊢
Ⅎ𝑖𝜑 |
19 | | hoiqssbllem2.c |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐶:𝑋⟶ℝ) |
20 | 19 | ffvelrnda 6904 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐶‘𝑖) ∈ ℝ) |
21 | | hoiqssbllem2.d |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐷:𝑋⟶ℝ) |
22 | 21 | ffvelrnda 6904 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐷‘𝑖) ∈ ℝ) |
23 | 22 | rexrd 10883 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐷‘𝑖) ∈
ℝ*) |
24 | 18, 20, 23 | hoissrrn2 43791 |
. . . . . . . . 9
⊢ (𝜑 → X𝑖 ∈
𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖)) ⊆ (ℝ ↑m 𝑋)) |
25 | 24 | adantr 484 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖)) ⊆ (ℝ ↑m 𝑋)) |
26 | | simpr 488 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) |
27 | 25, 26 | sseldd 3902 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → 𝑓 ∈ (ℝ ↑m 𝑋)) |
28 | | fvexd 6732 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → (√‘Σ𝑖 ∈ 𝑋 (((𝑌‘𝑖) − (𝑓‘𝑖))↑2)) ∈ V) |
29 | 6, 15, 17, 27, 28 | ovmpod 7361 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → (𝑌(dist‘(ℝ^‘𝑋))𝑓) = (√‘Σ𝑖 ∈ 𝑋 (((𝑌‘𝑖) − (𝑓‘𝑖))↑2))) |
30 | | nfcv 2904 |
. . . . . . . . . 10
⊢
Ⅎ𝑖𝑓 |
31 | | nfixp1 8599 |
. . . . . . . . . 10
⊢
Ⅎ𝑖X𝑖 ∈
𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖)) |
32 | 30, 31 | nfel 2918 |
. . . . . . . . 9
⊢
Ⅎ𝑖 𝑓 ∈ X𝑖 ∈
𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖)) |
33 | 18, 32 | nfan 1907 |
. . . . . . . 8
⊢
Ⅎ𝑖(𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) |
34 | | simpl 486 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → 𝜑) |
35 | 34, 1 | syl 17 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → 𝑋 ∈ Fin) |
36 | | elmapi 8530 |
. . . . . . . . . . . . 13
⊢ (𝑌 ∈ (ℝ
↑m 𝑋)
→ 𝑌:𝑋⟶ℝ) |
37 | 16, 36 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑌:𝑋⟶ℝ) |
38 | 37 | ffvelrnda 6904 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝑌‘𝑖) ∈ ℝ) |
39 | 34, 38 | sylan 583 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) ∧ 𝑖 ∈ 𝑋) → (𝑌‘𝑖) ∈ ℝ) |
40 | | icossre 13016 |
. . . . . . . . . . . . 13
⊢ (((𝐶‘𝑖) ∈ ℝ ∧ (𝐷‘𝑖) ∈ ℝ*) → ((𝐶‘𝑖)[,)(𝐷‘𝑖)) ⊆ ℝ) |
41 | 20, 23, 40 | syl2anc 587 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐶‘𝑖)[,)(𝐷‘𝑖)) ⊆ ℝ) |
42 | 41 | adantlr 715 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) ∧ 𝑖 ∈ 𝑋) → ((𝐶‘𝑖)[,)(𝐷‘𝑖)) ⊆ ℝ) |
43 | | fvixp2 42411 |
. . . . . . . . . . . 12
⊢ ((𝑓 ∈ X𝑖 ∈
𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖)) ∧ 𝑖 ∈ 𝑋) → (𝑓‘𝑖) ∈ ((𝐶‘𝑖)[,)(𝐷‘𝑖))) |
44 | 43 | adantll 714 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) ∧ 𝑖 ∈ 𝑋) → (𝑓‘𝑖) ∈ ((𝐶‘𝑖)[,)(𝐷‘𝑖))) |
45 | 42, 44 | sseldd 3902 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) ∧ 𝑖 ∈ 𝑋) → (𝑓‘𝑖) ∈ ℝ) |
46 | 39, 45 | resubcld 11260 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) ∧ 𝑖 ∈ 𝑋) → ((𝑌‘𝑖) − (𝑓‘𝑖)) ∈ ℝ) |
47 | | 2nn0 12107 |
. . . . . . . . . 10
⊢ 2 ∈
ℕ0 |
48 | 47 | a1i 11 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) ∧ 𝑖 ∈ 𝑋) → 2 ∈
ℕ0) |
49 | 46, 48 | reexpcld 13733 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) ∧ 𝑖 ∈ 𝑋) → (((𝑌‘𝑖) − (𝑓‘𝑖))↑2) ∈ ℝ) |
50 | 33, 35, 49 | fsumreclf 42792 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → Σ𝑖 ∈ 𝑋 (((𝑌‘𝑖) − (𝑓‘𝑖))↑2) ∈ ℝ) |
51 | | fveq2 6717 |
. . . . . . . . . . . . 13
⊢ (𝑖 = 𝑗 → (𝐶‘𝑖) = (𝐶‘𝑗)) |
52 | | fveq2 6717 |
. . . . . . . . . . . . 13
⊢ (𝑖 = 𝑗 → (𝐷‘𝑖) = (𝐷‘𝑗)) |
53 | 51, 52 | oveq12d 7231 |
. . . . . . . . . . . 12
⊢ (𝑖 = 𝑗 → ((𝐶‘𝑖)[,)(𝐷‘𝑖)) = ((𝐶‘𝑗)[,)(𝐷‘𝑗))) |
54 | 53 | cbvixpv 8596 |
. . . . . . . . . . 11
⊢ X𝑖 ∈
𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖)) = X𝑗 ∈ 𝑋 ((𝐶‘𝑗)[,)(𝐷‘𝑗)) |
55 | 54 | eleq2i 2829 |
. . . . . . . . . 10
⊢ (𝑓 ∈ X𝑖 ∈
𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖)) ↔ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐶‘𝑗)[,)(𝐷‘𝑗))) |
56 | 55 | biimpi 219 |
. . . . . . . . 9
⊢ (𝑓 ∈ X𝑖 ∈
𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖)) → 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐶‘𝑗)[,)(𝐷‘𝑗))) |
57 | 56 | adantl 485 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐶‘𝑗)[,)(𝐷‘𝑗))) |
58 | 1 | adantr 484 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐶‘𝑗)[,)(𝐷‘𝑗))) → 𝑋 ∈ Fin) |
59 | | simpll 767 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐶‘𝑗)[,)(𝐷‘𝑗))) ∧ 𝑖 ∈ 𝑋) → 𝜑) |
60 | 55 | biimpri 231 |
. . . . . . . . . . 11
⊢ (𝑓 ∈ X𝑗 ∈
𝑋 ((𝐶‘𝑗)[,)(𝐷‘𝑗)) → 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) |
61 | 60 | ad2antlr 727 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐶‘𝑗)[,)(𝐷‘𝑗))) ∧ 𝑖 ∈ 𝑋) → 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) |
62 | | simpr 488 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐶‘𝑗)[,)(𝐷‘𝑗))) ∧ 𝑖 ∈ 𝑋) → 𝑖 ∈ 𝑋) |
63 | 59, 61, 62, 49 | syl21anc 838 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐶‘𝑗)[,)(𝐷‘𝑗))) ∧ 𝑖 ∈ 𝑋) → (((𝑌‘𝑖) − (𝑓‘𝑖))↑2) ∈ ℝ) |
64 | 46 | sqge0d 13818 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) ∧ 𝑖 ∈ 𝑋) → 0 ≤ (((𝑌‘𝑖) − (𝑓‘𝑖))↑2)) |
65 | 59, 61, 62, 64 | syl21anc 838 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐶‘𝑗)[,)(𝐷‘𝑗))) ∧ 𝑖 ∈ 𝑋) → 0 ≤ (((𝑌‘𝑖) − (𝑓‘𝑖))↑2)) |
66 | 58, 63, 65 | fsumge0 15359 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐶‘𝑗)[,)(𝐷‘𝑗))) → 0 ≤ Σ𝑖 ∈ 𝑋 (((𝑌‘𝑖) − (𝑓‘𝑖))↑2)) |
67 | 34, 57, 66 | syl2anc 587 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → 0 ≤ Σ𝑖 ∈ 𝑋 (((𝑌‘𝑖) − (𝑓‘𝑖))↑2)) |
68 | 50, 67 | resqrtcld 14981 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → (√‘Σ𝑖 ∈ 𝑋 (((𝑌‘𝑖) − (𝑓‘𝑖))↑2)) ∈ ℝ) |
69 | 29, 68 | eqeltrd 2838 |
. . . . 5
⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → (𝑌(dist‘(ℝ^‘𝑋))𝑓) ∈ ℝ) |
70 | 22, 20 | resubcld 11260 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐷‘𝑖) − (𝐶‘𝑖)) ∈ ℝ) |
71 | 70 | resqcld 13817 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (((𝐷‘𝑖) − (𝐶‘𝑖))↑2) ∈ ℝ) |
72 | 1, 71 | fsumrecl 15298 |
. . . . . . 7
⊢ (𝜑 → Σ𝑖 ∈ 𝑋 (((𝐷‘𝑖) − (𝐶‘𝑖))↑2) ∈ ℝ) |
73 | 70 | sqge0d 13818 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 0 ≤ (((𝐷‘𝑖) − (𝐶‘𝑖))↑2)) |
74 | 1, 71, 73 | fsumge0 15359 |
. . . . . . 7
⊢ (𝜑 → 0 ≤ Σ𝑖 ∈ 𝑋 (((𝐷‘𝑖) − (𝐶‘𝑖))↑2)) |
75 | 72, 74 | resqrtcld 14981 |
. . . . . 6
⊢ (𝜑 →
(√‘Σ𝑖
∈ 𝑋 (((𝐷‘𝑖) − (𝐶‘𝑖))↑2)) ∈ ℝ) |
76 | 75 | adantr 484 |
. . . . 5
⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → (√‘Σ𝑖 ∈ 𝑋 (((𝐷‘𝑖) − (𝐶‘𝑖))↑2)) ∈ ℝ) |
77 | | hoiqssbllem2.e |
. . . . . . 7
⊢ (𝜑 → 𝐸 ∈
ℝ+) |
78 | 77 | rpred 12628 |
. . . . . 6
⊢ (𝜑 → 𝐸 ∈ ℝ) |
79 | 78 | adantr 484 |
. . . . 5
⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → 𝐸 ∈ ℝ) |
80 | | hoiqssbllem2.n |
. . . . . . . . . 10
⊢ (𝜑 → 𝑋 ≠ ∅) |
81 | 80 | adantr 484 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐶‘𝑗)[,)(𝐷‘𝑗))) → 𝑋 ≠ ∅) |
82 | 71 | adantlr 715 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐶‘𝑗)[,)(𝐷‘𝑗))) ∧ 𝑖 ∈ 𝑋) → (((𝐷‘𝑖) − (𝐶‘𝑖))↑2) ∈ ℝ) |
83 | 34, 22 | sylan 583 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) ∧ 𝑖 ∈ 𝑋) → (𝐷‘𝑖) ∈ ℝ) |
84 | 34, 20 | sylan 583 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) ∧ 𝑖 ∈ 𝑋) → (𝐶‘𝑖) ∈ ℝ) |
85 | 83, 84 | resubcld 11260 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) ∧ 𝑖 ∈ 𝑋) → ((𝐷‘𝑖) − (𝐶‘𝑖)) ∈ ℝ) |
86 | 20 | rexrd 10883 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐶‘𝑖) ∈
ℝ*) |
87 | 38 | rexrd 10883 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝑌‘𝑖) ∈
ℝ*) |
88 | | 2rp 12591 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ 2 ∈
ℝ+ |
89 | 88 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → 2 ∈
ℝ+) |
90 | | hashnncl 13933 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑋 ∈ Fin →
((♯‘𝑋) ∈
ℕ ↔ 𝑋 ≠
∅)) |
91 | 1, 90 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝜑 → ((♯‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅)) |
92 | 80, 91 | mpbird 260 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝜑 → (♯‘𝑋) ∈
ℕ) |
93 | 92 | nnred 11845 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝜑 → (♯‘𝑋) ∈
ℝ) |
94 | 92 | nngt0d 11879 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝜑 → 0 <
(♯‘𝑋)) |
95 | 93, 94 | elrpd 12625 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → (♯‘𝑋) ∈
ℝ+) |
96 | 95 | rpsqrtcld 14975 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 →
(√‘(♯‘𝑋)) ∈
ℝ+) |
97 | 89, 96 | rpmulcld 12644 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → (2 ·
(√‘(♯‘𝑋))) ∈
ℝ+) |
98 | 77, 97 | rpdivcld 12645 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (𝐸 / (2 ·
(√‘(♯‘𝑋)))) ∈
ℝ+) |
99 | 98 | rpred 12628 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (𝐸 / (2 ·
(√‘(♯‘𝑋)))) ∈ ℝ) |
100 | 99 | adantr 484 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐸 / (2 ·
(√‘(♯‘𝑋)))) ∈ ℝ) |
101 | 38, 100 | resubcld 11260 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝑌‘𝑖) − (𝐸 / (2 ·
(√‘(♯‘𝑋))))) ∈ ℝ) |
102 | 101 | rexrd 10883 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝑌‘𝑖) − (𝐸 / (2 ·
(√‘(♯‘𝑋))))) ∈
ℝ*) |
103 | | hoiqssbllem2.l |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐶‘𝑖) ∈ (((𝑌‘𝑖) − (𝐸 / (2 ·
(√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))) |
104 | | iooltub 42723 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝑌‘𝑖) − (𝐸 / (2 ·
(√‘(♯‘𝑋))))) ∈ ℝ* ∧
(𝑌‘𝑖) ∈ ℝ* ∧ (𝐶‘𝑖) ∈ (((𝑌‘𝑖) − (𝐸 / (2 ·
(√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))) → (𝐶‘𝑖) < (𝑌‘𝑖)) |
105 | 102, 87, 103, 104 | syl3anc 1373 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐶‘𝑖) < (𝑌‘𝑖)) |
106 | 20, 38, 105 | ltled 10980 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐶‘𝑖) ≤ (𝑌‘𝑖)) |
107 | 38, 100 | readdcld 10862 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝑌‘𝑖) + (𝐸 / (2 ·
(√‘(♯‘𝑋))))) ∈ ℝ) |
108 | 107 | rexrd 10883 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝑌‘𝑖) + (𝐸 / (2 ·
(√‘(♯‘𝑋))))) ∈
ℝ*) |
109 | | hoiqssbllem2.r |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐷‘𝑖) ∈ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 ·
(√‘(♯‘𝑋))))))) |
110 | | ioogtlb 42708 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑌‘𝑖) ∈ ℝ* ∧ ((𝑌‘𝑖) + (𝐸 / (2 ·
(√‘(♯‘𝑋))))) ∈ ℝ* ∧
(𝐷‘𝑖) ∈ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 ·
(√‘(♯‘𝑋))))))) → (𝑌‘𝑖) < (𝐷‘𝑖)) |
111 | 87, 108, 109, 110 | syl3anc 1373 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝑌‘𝑖) < (𝐷‘𝑖)) |
112 | 86, 23, 87, 106, 111 | elicod 12985 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝑌‘𝑖) ∈ ((𝐶‘𝑖)[,)(𝐷‘𝑖))) |
113 | 34, 112 | sylan 583 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) ∧ 𝑖 ∈ 𝑋) → (𝑌‘𝑖) ∈ ((𝐶‘𝑖)[,)(𝐷‘𝑖))) |
114 | | icodiamlt 14999 |
. . . . . . . . . . . . 13
⊢ ((((𝐶‘𝑖) ∈ ℝ ∧ (𝐷‘𝑖) ∈ ℝ) ∧ ((𝑌‘𝑖) ∈ ((𝐶‘𝑖)[,)(𝐷‘𝑖)) ∧ (𝑓‘𝑖) ∈ ((𝐶‘𝑖)[,)(𝐷‘𝑖)))) → (abs‘((𝑌‘𝑖) − (𝑓‘𝑖))) < ((𝐷‘𝑖) − (𝐶‘𝑖))) |
115 | 84, 83, 113, 44, 114 | syl22anc 839 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) ∧ 𝑖 ∈ 𝑋) → (abs‘((𝑌‘𝑖) − (𝑓‘𝑖))) < ((𝐷‘𝑖) − (𝐶‘𝑖))) |
116 | | 0red 10836 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 0 ∈ ℝ) |
117 | 20, 38, 22, 106, 111 | lelttrd 10990 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐶‘𝑖) < (𝐷‘𝑖)) |
118 | 20, 22 | posdifd 11419 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐶‘𝑖) < (𝐷‘𝑖) ↔ 0 < ((𝐷‘𝑖) − (𝐶‘𝑖)))) |
119 | 117, 118 | mpbid 235 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 0 < ((𝐷‘𝑖) − (𝐶‘𝑖))) |
120 | 116, 70, 119 | ltled 10980 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 0 ≤ ((𝐷‘𝑖) − (𝐶‘𝑖))) |
121 | 70, 120 | absidd 14986 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (abs‘((𝐷‘𝑖) − (𝐶‘𝑖))) = ((𝐷‘𝑖) − (𝐶‘𝑖))) |
122 | 121 | eqcomd 2743 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐷‘𝑖) − (𝐶‘𝑖)) = (abs‘((𝐷‘𝑖) − (𝐶‘𝑖)))) |
123 | 122 | adantlr 715 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) ∧ 𝑖 ∈ 𝑋) → ((𝐷‘𝑖) − (𝐶‘𝑖)) = (abs‘((𝐷‘𝑖) − (𝐶‘𝑖)))) |
124 | 115, 123 | breqtrd 5079 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) ∧ 𝑖 ∈ 𝑋) → (abs‘((𝑌‘𝑖) − (𝑓‘𝑖))) < (abs‘((𝐷‘𝑖) − (𝐶‘𝑖)))) |
125 | 46, 85, 124 | abslt2sqd 42572 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) ∧ 𝑖 ∈ 𝑋) → (((𝑌‘𝑖) − (𝑓‘𝑖))↑2) < (((𝐷‘𝑖) − (𝐶‘𝑖))↑2)) |
126 | 59, 61, 62, 125 | syl21anc 838 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐶‘𝑗)[,)(𝐷‘𝑗))) ∧ 𝑖 ∈ 𝑋) → (((𝑌‘𝑖) − (𝑓‘𝑖))↑2) < (((𝐷‘𝑖) − (𝐶‘𝑖))↑2)) |
127 | 58, 81, 63, 82, 126 | fsumlt 15364 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐶‘𝑗)[,)(𝐷‘𝑗))) → Σ𝑖 ∈ 𝑋 (((𝑌‘𝑖) − (𝑓‘𝑖))↑2) < Σ𝑖 ∈ 𝑋 (((𝐷‘𝑖) − (𝐶‘𝑖))↑2)) |
128 | 34, 57, 127 | syl2anc 587 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → Σ𝑖 ∈ 𝑋 (((𝑌‘𝑖) − (𝑓‘𝑖))↑2) < Σ𝑖 ∈ 𝑋 (((𝐷‘𝑖) − (𝐶‘𝑖))↑2)) |
129 | 34, 72 | syl 17 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → Σ𝑖 ∈ 𝑋 (((𝐷‘𝑖) − (𝐶‘𝑖))↑2) ∈ ℝ) |
130 | 34, 74 | syl 17 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → 0 ≤ Σ𝑖 ∈ 𝑋 (((𝐷‘𝑖) − (𝐶‘𝑖))↑2)) |
131 | 50, 67, 129, 130 | sqrtltd 14991 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → (Σ𝑖 ∈ 𝑋 (((𝑌‘𝑖) − (𝑓‘𝑖))↑2) < Σ𝑖 ∈ 𝑋 (((𝐷‘𝑖) − (𝐶‘𝑖))↑2) ↔ (√‘Σ𝑖 ∈ 𝑋 (((𝑌‘𝑖) − (𝑓‘𝑖))↑2)) < (√‘Σ𝑖 ∈ 𝑋 (((𝐷‘𝑖) − (𝐶‘𝑖))↑2)))) |
132 | 128, 131 | mpbid 235 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → (√‘Σ𝑖 ∈ 𝑋 (((𝑌‘𝑖) − (𝑓‘𝑖))↑2)) < (√‘Σ𝑖 ∈ 𝑋 (((𝐷‘𝑖) − (𝐶‘𝑖))↑2))) |
133 | 29, 132 | eqbrtrd 5075 |
. . . . 5
⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → (𝑌(dist‘(ℝ^‘𝑋))𝑓) < (√‘Σ𝑖 ∈ 𝑋 (((𝐷‘𝑖) − (𝐶‘𝑖))↑2))) |
134 | 78, 96 | rerpdivcld 12659 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐸 / (√‘(♯‘𝑋))) ∈
ℝ) |
135 | 134 | resqcld 13817 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝐸 / (√‘(♯‘𝑋)))↑2) ∈
ℝ) |
136 | 135 | adantr 484 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐸 / (√‘(♯‘𝑋)))↑2) ∈
ℝ) |
137 | 22, 20 | jca 515 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐷‘𝑖) ∈ ℝ ∧ (𝐶‘𝑖) ∈ ℝ)) |
138 | 107, 101 | jca 515 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (((𝑌‘𝑖) + (𝐸 / (2 ·
(√‘(♯‘𝑋))))) ∈ ℝ ∧ ((𝑌‘𝑖) − (𝐸 / (2 ·
(√‘(♯‘𝑋))))) ∈ ℝ)) |
139 | 137, 138 | jca 515 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (((𝐷‘𝑖) ∈ ℝ ∧ (𝐶‘𝑖) ∈ ℝ) ∧ (((𝑌‘𝑖) + (𝐸 / (2 ·
(√‘(♯‘𝑋))))) ∈ ℝ ∧ ((𝑌‘𝑖) − (𝐸 / (2 ·
(√‘(♯‘𝑋))))) ∈ ℝ))) |
140 | | iooltub 42723 |
. . . . . . . . . . . . . 14
⊢ (((𝑌‘𝑖) ∈ ℝ* ∧ ((𝑌‘𝑖) + (𝐸 / (2 ·
(√‘(♯‘𝑋))))) ∈ ℝ* ∧
(𝐷‘𝑖) ∈ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 ·
(√‘(♯‘𝑋))))))) → (𝐷‘𝑖) < ((𝑌‘𝑖) + (𝐸 / (2 ·
(√‘(♯‘𝑋)))))) |
141 | 87, 108, 109, 140 | syl3anc 1373 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐷‘𝑖) < ((𝑌‘𝑖) + (𝐸 / (2 ·
(√‘(♯‘𝑋)))))) |
142 | | ioogtlb 42708 |
. . . . . . . . . . . . . 14
⊢ ((((𝑌‘𝑖) − (𝐸 / (2 ·
(√‘(♯‘𝑋))))) ∈ ℝ* ∧
(𝑌‘𝑖) ∈ ℝ* ∧ (𝐶‘𝑖) ∈ (((𝑌‘𝑖) − (𝐸 / (2 ·
(√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))) → ((𝑌‘𝑖) − (𝐸 / (2 ·
(√‘(♯‘𝑋))))) < (𝐶‘𝑖)) |
143 | 102, 87, 103, 142 | syl3anc 1373 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝑌‘𝑖) − (𝐸 / (2 ·
(√‘(♯‘𝑋))))) < (𝐶‘𝑖)) |
144 | 141, 143 | jca 515 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐷‘𝑖) < ((𝑌‘𝑖) + (𝐸 / (2 ·
(√‘(♯‘𝑋))))) ∧ ((𝑌‘𝑖) − (𝐸 / (2 ·
(√‘(♯‘𝑋))))) < (𝐶‘𝑖))) |
145 | | lt2sub 11330 |
. . . . . . . . . . . 12
⊢ ((((𝐷‘𝑖) ∈ ℝ ∧ (𝐶‘𝑖) ∈ ℝ) ∧ (((𝑌‘𝑖) + (𝐸 / (2 ·
(√‘(♯‘𝑋))))) ∈ ℝ ∧ ((𝑌‘𝑖) − (𝐸 / (2 ·
(√‘(♯‘𝑋))))) ∈ ℝ)) → (((𝐷‘𝑖) < ((𝑌‘𝑖) + (𝐸 / (2 ·
(√‘(♯‘𝑋))))) ∧ ((𝑌‘𝑖) − (𝐸 / (2 ·
(√‘(♯‘𝑋))))) < (𝐶‘𝑖)) → ((𝐷‘𝑖) − (𝐶‘𝑖)) < (((𝑌‘𝑖) + (𝐸 / (2 ·
(√‘(♯‘𝑋))))) − ((𝑌‘𝑖) − (𝐸 / (2 ·
(√‘(♯‘𝑋)))))))) |
146 | 139, 144,
145 | sylc 65 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐷‘𝑖) − (𝐶‘𝑖)) < (((𝑌‘𝑖) + (𝐸 / (2 ·
(√‘(♯‘𝑋))))) − ((𝑌‘𝑖) − (𝐸 / (2 ·
(√‘(♯‘𝑋))))))) |
147 | 38 | recnd 10861 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝑌‘𝑖) ∈ ℂ) |
148 | 100 | recnd 10861 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐸 / (2 ·
(√‘(♯‘𝑋)))) ∈ ℂ) |
149 | 147, 148,
148 | pnncand 11228 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (((𝑌‘𝑖) + (𝐸 / (2 ·
(√‘(♯‘𝑋))))) − ((𝑌‘𝑖) − (𝐸 / (2 ·
(√‘(♯‘𝑋)))))) = ((𝐸 / (2 ·
(√‘(♯‘𝑋)))) + (𝐸 / (2 ·
(√‘(♯‘𝑋)))))) |
150 | 78 | recnd 10861 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝐸 ∈ ℂ) |
151 | 96 | rpcnd 12630 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 →
(√‘(♯‘𝑋)) ∈ ℂ) |
152 | | 2cnd 11908 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 2 ∈
ℂ) |
153 | 96 | rpne0d 12633 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 →
(√‘(♯‘𝑋)) ≠ 0) |
154 | 89 | rpne0d 12633 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 2 ≠ 0) |
155 | 150, 151,
152, 153, 154 | divdiv3d 42571 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((𝐸 / (√‘(♯‘𝑋))) / 2) = (𝐸 / (2 ·
(√‘(♯‘𝑋))))) |
156 | 155 | eqcomd 2743 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝐸 / (2 ·
(√‘(♯‘𝑋)))) = ((𝐸 / (√‘(♯‘𝑋))) / 2)) |
157 | 156, 156 | oveq12d 7231 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((𝐸 / (2 ·
(√‘(♯‘𝑋)))) + (𝐸 / (2 ·
(√‘(♯‘𝑋))))) = (((𝐸 / (√‘(♯‘𝑋))) / 2) + ((𝐸 / (√‘(♯‘𝑋))) / 2))) |
158 | 150, 151,
153 | divcld 11608 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝐸 / (√‘(♯‘𝑋))) ∈
ℂ) |
159 | 158 | 2halvesd 12076 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (((𝐸 / (√‘(♯‘𝑋))) / 2) + ((𝐸 / (√‘(♯‘𝑋))) / 2)) = (𝐸 / (√‘(♯‘𝑋)))) |
160 | 157, 159 | eqtrd 2777 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((𝐸 / (2 ·
(√‘(♯‘𝑋)))) + (𝐸 / (2 ·
(√‘(♯‘𝑋))))) = (𝐸 / (√‘(♯‘𝑋)))) |
161 | 160 | adantr 484 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐸 / (2 ·
(√‘(♯‘𝑋)))) + (𝐸 / (2 ·
(√‘(♯‘𝑋))))) = (𝐸 / (√‘(♯‘𝑋)))) |
162 | 149, 161 | eqtrd 2777 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (((𝑌‘𝑖) + (𝐸 / (2 ·
(√‘(♯‘𝑋))))) − ((𝑌‘𝑖) − (𝐸 / (2 ·
(√‘(♯‘𝑋)))))) = (𝐸 / (√‘(♯‘𝑋)))) |
163 | 146, 162 | breqtrd 5079 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐷‘𝑖) − (𝐶‘𝑖)) < (𝐸 / (√‘(♯‘𝑋)))) |
164 | 134 | adantr 484 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐸 / (√‘(♯‘𝑋))) ∈
ℝ) |
165 | | 0red 10836 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 0 ∈
ℝ) |
166 | 96 | rpred 12628 |
. . . . . . . . . . . . . 14
⊢ (𝜑 →
(√‘(♯‘𝑋)) ∈ ℝ) |
167 | 77 | rpgt0d 12631 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 0 < 𝐸) |
168 | 96 | rpgt0d 12631 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 0 <
(√‘(♯‘𝑋))) |
169 | 78, 166, 167, 168 | divgt0d 11767 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 0 < (𝐸 / (√‘(♯‘𝑋)))) |
170 | 165, 134,
169 | ltled 10980 |
. . . . . . . . . . . 12
⊢ (𝜑 → 0 ≤ (𝐸 / (√‘(♯‘𝑋)))) |
171 | 170 | adantr 484 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 0 ≤ (𝐸 / (√‘(♯‘𝑋)))) |
172 | | lt2sq 13704 |
. . . . . . . . . . 11
⊢
(((((𝐷‘𝑖) − (𝐶‘𝑖)) ∈ ℝ ∧ 0 ≤ ((𝐷‘𝑖) − (𝐶‘𝑖))) ∧ ((𝐸 / (√‘(♯‘𝑋))) ∈ ℝ ∧ 0 ≤
(𝐸 /
(√‘(♯‘𝑋))))) → (((𝐷‘𝑖) − (𝐶‘𝑖)) < (𝐸 / (√‘(♯‘𝑋))) ↔ (((𝐷‘𝑖) − (𝐶‘𝑖))↑2) < ((𝐸 / (√‘(♯‘𝑋)))↑2))) |
173 | 70, 120, 164, 171, 172 | syl22anc 839 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (((𝐷‘𝑖) − (𝐶‘𝑖)) < (𝐸 / (√‘(♯‘𝑋))) ↔ (((𝐷‘𝑖) − (𝐶‘𝑖))↑2) < ((𝐸 / (√‘(♯‘𝑋)))↑2))) |
174 | 163, 173 | mpbid 235 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (((𝐷‘𝑖) − (𝐶‘𝑖))↑2) < ((𝐸 / (√‘(♯‘𝑋)))↑2)) |
175 | 1, 80, 71, 136, 174 | fsumlt 15364 |
. . . . . . . 8
⊢ (𝜑 → Σ𝑖 ∈ 𝑋 (((𝐷‘𝑖) − (𝐶‘𝑖))↑2) < Σ𝑖 ∈ 𝑋 ((𝐸 / (√‘(♯‘𝑋)))↑2)) |
176 | 1, 136 | fsumrecl 15298 |
. . . . . . . . 9
⊢ (𝜑 → Σ𝑖 ∈ 𝑋 ((𝐸 / (√‘(♯‘𝑋)))↑2) ∈
ℝ) |
177 | 164 | sqge0d 13818 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 0 ≤ ((𝐸 / (√‘(♯‘𝑋)))↑2)) |
178 | 1, 136, 177 | fsumge0 15359 |
. . . . . . . . 9
⊢ (𝜑 → 0 ≤ Σ𝑖 ∈ 𝑋 ((𝐸 / (√‘(♯‘𝑋)))↑2)) |
179 | 72, 74, 176, 178 | sqrtltd 14991 |
. . . . . . . 8
⊢ (𝜑 → (Σ𝑖 ∈ 𝑋 (((𝐷‘𝑖) − (𝐶‘𝑖))↑2) < Σ𝑖 ∈ 𝑋 ((𝐸 / (√‘(♯‘𝑋)))↑2) ↔
(√‘Σ𝑖
∈ 𝑋 (((𝐷‘𝑖) − (𝐶‘𝑖))↑2)) < (√‘Σ𝑖 ∈ 𝑋 ((𝐸 / (√‘(♯‘𝑋)))↑2)))) |
180 | 175, 179 | mpbid 235 |
. . . . . . 7
⊢ (𝜑 →
(√‘Σ𝑖
∈ 𝑋 (((𝐷‘𝑖) − (𝐶‘𝑖))↑2)) < (√‘Σ𝑖 ∈ 𝑋 ((𝐸 / (√‘(♯‘𝑋)))↑2))) |
181 | 135 | recnd 10861 |
. . . . . . . . . . 11
⊢ (𝜑 → ((𝐸 / (√‘(♯‘𝑋)))↑2) ∈
ℂ) |
182 | | fsumconst 15354 |
. . . . . . . . . . 11
⊢ ((𝑋 ∈ Fin ∧ ((𝐸 /
(√‘(♯‘𝑋)))↑2) ∈ ℂ) →
Σ𝑖 ∈ 𝑋 ((𝐸 / (√‘(♯‘𝑋)))↑2) =
((♯‘𝑋) ·
((𝐸 /
(√‘(♯‘𝑋)))↑2))) |
183 | 1, 181, 182 | syl2anc 587 |
. . . . . . . . . 10
⊢ (𝜑 → Σ𝑖 ∈ 𝑋 ((𝐸 / (√‘(♯‘𝑋)))↑2) =
((♯‘𝑋) ·
((𝐸 /
(√‘(♯‘𝑋)))↑2))) |
184 | | sqdiv 13693 |
. . . . . . . . . . . . 13
⊢ ((𝐸 ∈ ℂ ∧
(√‘(♯‘𝑋)) ∈ ℂ ∧
(√‘(♯‘𝑋)) ≠ 0) → ((𝐸 / (√‘(♯‘𝑋)))↑2) = ((𝐸↑2) /
((√‘(♯‘𝑋))↑2))) |
185 | 150, 151,
153, 184 | syl3anc 1373 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((𝐸 / (√‘(♯‘𝑋)))↑2) = ((𝐸↑2) /
((√‘(♯‘𝑋))↑2))) |
186 | 93 | recnd 10861 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (♯‘𝑋) ∈
ℂ) |
187 | | sqrtth 14928 |
. . . . . . . . . . . . . 14
⊢
((♯‘𝑋)
∈ ℂ → ((√‘(♯‘𝑋))↑2) = (♯‘𝑋)) |
188 | 186, 187 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 →
((√‘(♯‘𝑋))↑2) = (♯‘𝑋)) |
189 | 188 | oveq2d 7229 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((𝐸↑2) /
((√‘(♯‘𝑋))↑2)) = ((𝐸↑2) / (♯‘𝑋))) |
190 | 185, 189 | eqtrd 2777 |
. . . . . . . . . . 11
⊢ (𝜑 → ((𝐸 / (√‘(♯‘𝑋)))↑2) = ((𝐸↑2) / (♯‘𝑋))) |
191 | 190 | oveq2d 7229 |
. . . . . . . . . 10
⊢ (𝜑 → ((♯‘𝑋) · ((𝐸 / (√‘(♯‘𝑋)))↑2)) =
((♯‘𝑋) ·
((𝐸↑2) /
(♯‘𝑋)))) |
192 | 150 | sqcld 13714 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐸↑2) ∈ ℂ) |
193 | 165, 94 | gtned 10967 |
. . . . . . . . . . 11
⊢ (𝜑 → (♯‘𝑋) ≠ 0) |
194 | 192, 186,
193 | divcan2d 11610 |
. . . . . . . . . 10
⊢ (𝜑 → ((♯‘𝑋) · ((𝐸↑2) / (♯‘𝑋))) = (𝐸↑2)) |
195 | 183, 191,
194 | 3eqtrd 2781 |
. . . . . . . . 9
⊢ (𝜑 → Σ𝑖 ∈ 𝑋 ((𝐸 / (√‘(♯‘𝑋)))↑2) = (𝐸↑2)) |
196 | 195 | fveq2d 6721 |
. . . . . . . 8
⊢ (𝜑 →
(√‘Σ𝑖
∈ 𝑋 ((𝐸 /
(√‘(♯‘𝑋)))↑2)) = (√‘(𝐸↑2))) |
197 | 165, 78, 167 | ltled 10980 |
. . . . . . . . 9
⊢ (𝜑 → 0 ≤ 𝐸) |
198 | | sqrtsq 14833 |
. . . . . . . . 9
⊢ ((𝐸 ∈ ℝ ∧ 0 ≤
𝐸) →
(√‘(𝐸↑2))
= 𝐸) |
199 | 78, 197, 198 | syl2anc 587 |
. . . . . . . 8
⊢ (𝜑 → (√‘(𝐸↑2)) = 𝐸) |
200 | | eqidd 2738 |
. . . . . . . 8
⊢ (𝜑 → 𝐸 = 𝐸) |
201 | 196, 199,
200 | 3eqtrd 2781 |
. . . . . . 7
⊢ (𝜑 →
(√‘Σ𝑖
∈ 𝑋 ((𝐸 /
(√‘(♯‘𝑋)))↑2)) = 𝐸) |
202 | 180, 201 | breqtrd 5079 |
. . . . . 6
⊢ (𝜑 →
(√‘Σ𝑖
∈ 𝑋 (((𝐷‘𝑖) − (𝐶‘𝑖))↑2)) < 𝐸) |
203 | 202 | adantr 484 |
. . . . 5
⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → (√‘Σ𝑖 ∈ 𝑋 (((𝐷‘𝑖) − (𝐶‘𝑖))↑2)) < 𝐸) |
204 | 69, 76, 79, 133, 203 | lttrd 10993 |
. . . 4
⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → (𝑌(dist‘(ℝ^‘𝑋))𝑓) < 𝐸) |
205 | | eqid 2737 |
. . . . . . . 8
⊢
(dist‘(ℝ^‘𝑋)) = (dist‘(ℝ^‘𝑋)) |
206 | 205 | rrxmetfi 24309 |
. . . . . . 7
⊢ (𝑋 ∈ Fin →
(dist‘(ℝ^‘𝑋)) ∈ (Met‘(ℝ
↑m 𝑋))) |
207 | | metxmet 23232 |
. . . . . . 7
⊢
((dist‘(ℝ^‘𝑋)) ∈ (Met‘(ℝ
↑m 𝑋))
→ (dist‘(ℝ^‘𝑋)) ∈ (∞Met‘(ℝ
↑m 𝑋))) |
208 | 1, 206, 207 | 3syl 18 |
. . . . . 6
⊢ (𝜑 →
(dist‘(ℝ^‘𝑋)) ∈ (∞Met‘(ℝ
↑m 𝑋))) |
209 | 208 | adantr 484 |
. . . . 5
⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → (dist‘(ℝ^‘𝑋)) ∈
(∞Met‘(ℝ ↑m 𝑋))) |
210 | 79 | rexrd 10883 |
. . . . 5
⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → 𝐸 ∈
ℝ*) |
211 | 27, 3 | eleqtrdi 2848 |
. . . . 5
⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → 𝑓 ∈ (ℝ ↑m 𝑋)) |
212 | | elbl2 23288 |
. . . . 5
⊢
((((dist‘(ℝ^‘𝑋)) ∈ (∞Met‘(ℝ
↑m 𝑋))
∧ 𝐸 ∈
ℝ*) ∧ (𝑌 ∈ (ℝ ↑m 𝑋) ∧ 𝑓 ∈ (ℝ ↑m 𝑋))) → (𝑓 ∈ (𝑌(ball‘(dist‘(ℝ^‘𝑋)))𝐸) ↔ (𝑌(dist‘(ℝ^‘𝑋))𝑓) < 𝐸)) |
213 | 209, 210,
17, 211, 212 | syl22anc 839 |
. . . 4
⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → (𝑓 ∈ (𝑌(ball‘(dist‘(ℝ^‘𝑋)))𝐸) ↔ (𝑌(dist‘(ℝ^‘𝑋))𝑓) < 𝐸)) |
214 | 204, 213 | mpbird 260 |
. . 3
⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → 𝑓 ∈ (𝑌(ball‘(dist‘(ℝ^‘𝑋)))𝐸)) |
215 | 214 | ralrimiva 3105 |
. 2
⊢ (𝜑 → ∀𝑓 ∈ X 𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))𝑓 ∈ (𝑌(ball‘(dist‘(ℝ^‘𝑋)))𝐸)) |
216 | | dfss3 3888 |
. 2
⊢ (X𝑖 ∈
𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖)) ⊆ (𝑌(ball‘(dist‘(ℝ^‘𝑋)))𝐸) ↔ ∀𝑓 ∈ X 𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))𝑓 ∈ (𝑌(ball‘(dist‘(ℝ^‘𝑋)))𝐸)) |
217 | 215, 216 | sylibr 237 |
1
⊢ (𝜑 → X𝑖 ∈
𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖)) ⊆ (𝑌(ball‘(dist‘(ℝ^‘𝑋)))𝐸)) |