| 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 25445 |
. . . . . . . . 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 480 |
. . . . . . 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 6905 |
. . . . . . . . . . . . 13
⊢ (𝑔 = 𝑌 → (𝑔‘𝑖) = (𝑌‘𝑖)) |
| 8 | 7 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝑔 = 𝑌 ∧ ℎ = 𝑓) → (𝑔‘𝑖) = (𝑌‘𝑖)) |
| 9 | | fveq1 6905 |
. . . . . . . . . . . . 13
⊢ (ℎ = 𝑓 → (ℎ‘𝑖) = (𝑓‘𝑖)) |
| 10 | 9 | adantl 481 |
. . . . . . . . . . . 12
⊢ ((𝑔 = 𝑌 ∧ ℎ = 𝑓) → (ℎ‘𝑖) = (𝑓‘𝑖)) |
| 11 | 8, 10 | oveq12d 7449 |
. . . . . . . . . . 11
⊢ ((𝑔 = 𝑌 ∧ ℎ = 𝑓) → ((𝑔‘𝑖) − (ℎ‘𝑖)) = ((𝑌‘𝑖) − (𝑓‘𝑖))) |
| 12 | 11 | oveq1d 7446 |
. . . . . . . . . 10
⊢ ((𝑔 = 𝑌 ∧ ℎ = 𝑓) → (((𝑔‘𝑖) − (ℎ‘𝑖))↑2) = (((𝑌‘𝑖) − (𝑓‘𝑖))↑2)) |
| 13 | 12 | sumeq2sdv 15739 |
. . . . . . . . 9
⊢ ((𝑔 = 𝑌 ∧ ℎ = 𝑓) → Σ𝑖 ∈ 𝑋 (((𝑔‘𝑖) − (ℎ‘𝑖))↑2) = Σ𝑖 ∈ 𝑋 (((𝑌‘𝑖) − (𝑓‘𝑖))↑2)) |
| 14 | 13 | fveq2d 6910 |
. . . . . . . 8
⊢ ((𝑔 = 𝑌 ∧ ℎ = 𝑓) → (√‘Σ𝑖 ∈ 𝑋 (((𝑔‘𝑖) − (ℎ‘𝑖))↑2)) = (√‘Σ𝑖 ∈ 𝑋 (((𝑌‘𝑖) − (𝑓‘𝑖))↑2))) |
| 15 | 14 | adantl 481 |
. . . . . . 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 480 |
. . . . . . 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 | ffvelcdmda 7104 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐶‘𝑖) ∈ ℝ) |
| 21 | | hoiqssbllem2.d |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐷:𝑋⟶ℝ) |
| 22 | 21 | ffvelcdmda 7104 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐷‘𝑖) ∈ ℝ) |
| 23 | 22 | rexrd 11311 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐷‘𝑖) ∈
ℝ*) |
| 24 | 18, 20, 23 | hoissrrn2 46593 |
. . . . . . . . 9
⊢ (𝜑 → X𝑖 ∈
𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖)) ⊆ (ℝ ↑m 𝑋)) |
| 25 | 24 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖)) ⊆ (ℝ ↑m 𝑋)) |
| 26 | | simpr 484 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) |
| 27 | 25, 26 | sseldd 3984 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → 𝑓 ∈ (ℝ ↑m 𝑋)) |
| 28 | | fvexd 6921 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → (√‘Σ𝑖 ∈ 𝑋 (((𝑌‘𝑖) − (𝑓‘𝑖))↑2)) ∈ V) |
| 29 | 6, 15, 17, 27, 28 | ovmpod 7585 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → (𝑌(dist‘(ℝ^‘𝑋))𝑓) = (√‘Σ𝑖 ∈ 𝑋 (((𝑌‘𝑖) − (𝑓‘𝑖))↑2))) |
| 30 | | nfcv 2905 |
. . . . . . . . . 10
⊢
Ⅎ𝑖𝑓 |
| 31 | | nfixp1 8958 |
. . . . . . . . . 10
⊢
Ⅎ𝑖X𝑖 ∈
𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖)) |
| 32 | 30, 31 | nfel 2920 |
. . . . . . . . 9
⊢
Ⅎ𝑖 𝑓 ∈ X𝑖 ∈
𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖)) |
| 33 | 18, 32 | nfan 1899 |
. . . . . . . 8
⊢
Ⅎ𝑖(𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) |
| 34 | | simpl 482 |
. . . . . . . . 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 8889 |
. . . . . . . . . . . . 13
⊢ (𝑌 ∈ (ℝ
↑m 𝑋)
→ 𝑌:𝑋⟶ℝ) |
| 37 | 16, 36 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑌:𝑋⟶ℝ) |
| 38 | 37 | ffvelcdmda 7104 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝑌‘𝑖) ∈ ℝ) |
| 39 | 34, 38 | sylan 580 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) ∧ 𝑖 ∈ 𝑋) → (𝑌‘𝑖) ∈ ℝ) |
| 40 | | icossre 13468 |
. . . . . . . . . . . . 13
⊢ (((𝐶‘𝑖) ∈ ℝ ∧ (𝐷‘𝑖) ∈ ℝ*) → ((𝐶‘𝑖)[,)(𝐷‘𝑖)) ⊆ ℝ) |
| 41 | 20, 23, 40 | syl2anc 584 |
. . . . . . . . . . . 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 45204 |
. . . . . . . . . . . 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 3984 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) ∧ 𝑖 ∈ 𝑋) → (𝑓‘𝑖) ∈ ℝ) |
| 46 | 39, 45 | resubcld 11691 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) ∧ 𝑖 ∈ 𝑋) → ((𝑌‘𝑖) − (𝑓‘𝑖)) ∈ ℝ) |
| 47 | | 2nn0 12543 |
. . . . . . . . . 10
⊢ 2 ∈
ℕ0 |
| 48 | 47 | a1i 11 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) ∧ 𝑖 ∈ 𝑋) → 2 ∈
ℕ0) |
| 49 | 46, 48 | reexpcld 14203 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) ∧ 𝑖 ∈ 𝑋) → (((𝑌‘𝑖) − (𝑓‘𝑖))↑2) ∈ ℝ) |
| 50 | 33, 35, 49 | fsumreclf 45591 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → Σ𝑖 ∈ 𝑋 (((𝑌‘𝑖) − (𝑓‘𝑖))↑2) ∈ ℝ) |
| 51 | | fveq2 6906 |
. . . . . . . . . . . . 13
⊢ (𝑖 = 𝑗 → (𝐶‘𝑖) = (𝐶‘𝑗)) |
| 52 | | fveq2 6906 |
. . . . . . . . . . . . 13
⊢ (𝑖 = 𝑗 → (𝐷‘𝑖) = (𝐷‘𝑗)) |
| 53 | 51, 52 | oveq12d 7449 |
. . . . . . . . . . . 12
⊢ (𝑖 = 𝑗 → ((𝐶‘𝑖)[,)(𝐷‘𝑖)) = ((𝐶‘𝑗)[,)(𝐷‘𝑗))) |
| 54 | 53 | cbvixpv 8955 |
. . . . . . . . . . 11
⊢ X𝑖 ∈
𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖)) = X𝑗 ∈ 𝑋 ((𝐶‘𝑗)[,)(𝐷‘𝑗)) |
| 55 | 54 | eleq2i 2833 |
. . . . . . . . . 10
⊢ (𝑓 ∈ X𝑖 ∈
𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖)) ↔ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐶‘𝑗)[,)(𝐷‘𝑗))) |
| 56 | 55 | biimpi 216 |
. . . . . . . . 9
⊢ (𝑓 ∈ X𝑖 ∈
𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖)) → 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐶‘𝑗)[,)(𝐷‘𝑗))) |
| 57 | 56 | adantl 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐶‘𝑗)[,)(𝐷‘𝑗))) |
| 58 | 1 | adantr 480 |
. . . . . . . . 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 228 |
. . . . . . . . . . 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 484 |
. . . . . . . . . 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 14177 |
. . . . . . . . . 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 15831 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐶‘𝑗)[,)(𝐷‘𝑗))) → 0 ≤ Σ𝑖 ∈ 𝑋 (((𝑌‘𝑖) − (𝑓‘𝑖))↑2)) |
| 67 | 34, 57, 66 | syl2anc 584 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → 0 ≤ Σ𝑖 ∈ 𝑋 (((𝑌‘𝑖) − (𝑓‘𝑖))↑2)) |
| 68 | 50, 67 | resqrtcld 15456 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → (√‘Σ𝑖 ∈ 𝑋 (((𝑌‘𝑖) − (𝑓‘𝑖))↑2)) ∈ ℝ) |
| 69 | 29, 68 | eqeltrd 2841 |
. . . . 5
⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → (𝑌(dist‘(ℝ^‘𝑋))𝑓) ∈ ℝ) |
| 70 | 22, 20 | resubcld 11691 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐷‘𝑖) − (𝐶‘𝑖)) ∈ ℝ) |
| 71 | 70 | resqcld 14165 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (((𝐷‘𝑖) − (𝐶‘𝑖))↑2) ∈ ℝ) |
| 72 | 1, 71 | fsumrecl 15770 |
. . . . . . 7
⊢ (𝜑 → Σ𝑖 ∈ 𝑋 (((𝐷‘𝑖) − (𝐶‘𝑖))↑2) ∈ ℝ) |
| 73 | 70 | sqge0d 14177 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 0 ≤ (((𝐷‘𝑖) − (𝐶‘𝑖))↑2)) |
| 74 | 1, 71, 73 | fsumge0 15831 |
. . . . . . 7
⊢ (𝜑 → 0 ≤ Σ𝑖 ∈ 𝑋 (((𝐷‘𝑖) − (𝐶‘𝑖))↑2)) |
| 75 | 72, 74 | resqrtcld 15456 |
. . . . . 6
⊢ (𝜑 →
(√‘Σ𝑖
∈ 𝑋 (((𝐷‘𝑖) − (𝐶‘𝑖))↑2)) ∈ ℝ) |
| 76 | 75 | adantr 480 |
. . . . 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 13077 |
. . . . . 6
⊢ (𝜑 → 𝐸 ∈ ℝ) |
| 79 | 78 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → 𝐸 ∈ ℝ) |
| 80 | | hoiqssbllem2.n |
. . . . . . . . . 10
⊢ (𝜑 → 𝑋 ≠ ∅) |
| 81 | 80 | adantr 480 |
. . . . . . . . 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 580 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) ∧ 𝑖 ∈ 𝑋) → (𝐷‘𝑖) ∈ ℝ) |
| 84 | 34, 20 | sylan 580 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) ∧ 𝑖 ∈ 𝑋) → (𝐶‘𝑖) ∈ ℝ) |
| 85 | 83, 84 | resubcld 11691 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) ∧ 𝑖 ∈ 𝑋) → ((𝐷‘𝑖) − (𝐶‘𝑖)) ∈ ℝ) |
| 86 | 20 | rexrd 11311 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐶‘𝑖) ∈
ℝ*) |
| 87 | 38 | rexrd 11311 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝑌‘𝑖) ∈
ℝ*) |
| 88 | | 2rp 13039 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ 2 ∈
ℝ+ |
| 89 | 88 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → 2 ∈
ℝ+) |
| 90 | | hashnncl 14405 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑋 ∈ Fin →
((♯‘𝑋) ∈
ℕ ↔ 𝑋 ≠
∅)) |
| 91 | 1, 90 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝜑 → ((♯‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅)) |
| 92 | 80, 91 | mpbird 257 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝜑 → (♯‘𝑋) ∈
ℕ) |
| 93 | 92 | nnred 12281 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝜑 → (♯‘𝑋) ∈
ℝ) |
| 94 | 92 | nngt0d 12315 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝜑 → 0 <
(♯‘𝑋)) |
| 95 | 93, 94 | elrpd 13074 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → (♯‘𝑋) ∈
ℝ+) |
| 96 | 95 | rpsqrtcld 15450 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 →
(√‘(♯‘𝑋)) ∈
ℝ+) |
| 97 | 89, 96 | rpmulcld 13093 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → (2 ·
(√‘(♯‘𝑋))) ∈
ℝ+) |
| 98 | 77, 97 | rpdivcld 13094 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (𝐸 / (2 ·
(√‘(♯‘𝑋)))) ∈
ℝ+) |
| 99 | 98 | rpred 13077 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (𝐸 / (2 ·
(√‘(♯‘𝑋)))) ∈ ℝ) |
| 100 | 99 | adantr 480 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐸 / (2 ·
(√‘(♯‘𝑋)))) ∈ ℝ) |
| 101 | 38, 100 | resubcld 11691 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝑌‘𝑖) − (𝐸 / (2 ·
(√‘(♯‘𝑋))))) ∈ ℝ) |
| 102 | 101 | rexrd 11311 |
. . . . . . . . . . . . . . . . 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 45523 |
. . . . . . . . . . . . . . . . 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 11409 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐶‘𝑖) ≤ (𝑌‘𝑖)) |
| 107 | 38, 100 | readdcld 11290 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝑌‘𝑖) + (𝐸 / (2 ·
(√‘(♯‘𝑋))))) ∈ ℝ) |
| 108 | 107 | rexrd 11311 |
. . . . . . . . . . . . . . . 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 45508 |
. . . . . . . . . . . . . . . 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 13437 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝑌‘𝑖) ∈ ((𝐶‘𝑖)[,)(𝐷‘𝑖))) |
| 113 | 34, 112 | sylan 580 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) ∧ 𝑖 ∈ 𝑋) → (𝑌‘𝑖) ∈ ((𝐶‘𝑖)[,)(𝐷‘𝑖))) |
| 114 | | icodiamlt 15474 |
. . . . . . . . . . . . 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 11264 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 0 ∈ ℝ) |
| 117 | 20, 38, 22, 106, 111 | lelttrd 11419 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐶‘𝑖) < (𝐷‘𝑖)) |
| 118 | 20, 22 | posdifd 11850 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐶‘𝑖) < (𝐷‘𝑖) ↔ 0 < ((𝐷‘𝑖) − (𝐶‘𝑖)))) |
| 119 | 117, 118 | mpbid 232 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 0 < ((𝐷‘𝑖) − (𝐶‘𝑖))) |
| 120 | 116, 70, 119 | ltled 11409 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 0 ≤ ((𝐷‘𝑖) − (𝐶‘𝑖))) |
| 121 | 70, 120 | absidd 15461 |
. . . . . . . . . . . . . 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 5169 |
. . . . . . . . . . 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 45371 |
. . . . . . . . . 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 15836 |
. . . . . . . 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 584 |
. . . . . . 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 15466 |
. . . . . . 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 232 |
. . . . . 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 5165 |
. . . . 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 13108 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐸 / (√‘(♯‘𝑋))) ∈
ℝ) |
| 135 | 134 | resqcld 14165 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝐸 / (√‘(♯‘𝑋)))↑2) ∈
ℝ) |
| 136 | 135 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐸 / (√‘(♯‘𝑋)))↑2) ∈
ℝ) |
| 137 | 22, 20 | jca 511 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐷‘𝑖) ∈ ℝ ∧ (𝐶‘𝑖) ∈ ℝ)) |
| 138 | 107, 101 | jca 511 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (((𝑌‘𝑖) + (𝐸 / (2 ·
(√‘(♯‘𝑋))))) ∈ ℝ ∧ ((𝑌‘𝑖) − (𝐸 / (2 ·
(√‘(♯‘𝑋))))) ∈ ℝ)) |
| 139 | 137, 138 | jca 511 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (((𝐷‘𝑖) ∈ ℝ ∧ (𝐶‘𝑖) ∈ ℝ) ∧ (((𝑌‘𝑖) + (𝐸 / (2 ·
(√‘(♯‘𝑋))))) ∈ ℝ ∧ ((𝑌‘𝑖) − (𝐸 / (2 ·
(√‘(♯‘𝑋))))) ∈ ℝ))) |
| 140 | | iooltub 45523 |
. . . . . . . . . . . . . 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 45508 |
. . . . . . . . . . . . . 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 511 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐷‘𝑖) < ((𝑌‘𝑖) + (𝐸 / (2 ·
(√‘(♯‘𝑋))))) ∧ ((𝑌‘𝑖) − (𝐸 / (2 ·
(√‘(♯‘𝑋))))) < (𝐶‘𝑖))) |
| 145 | | lt2sub 11761 |
. . . . . . . . . . . 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 11289 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝑌‘𝑖) ∈ ℂ) |
| 148 | 100 | recnd 11289 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐸 / (2 ·
(√‘(♯‘𝑋)))) ∈ ℂ) |
| 149 | 147, 148,
148 | pnncand 11659 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (((𝑌‘𝑖) + (𝐸 / (2 ·
(√‘(♯‘𝑋))))) − ((𝑌‘𝑖) − (𝐸 / (2 ·
(√‘(♯‘𝑋)))))) = ((𝐸 / (2 ·
(√‘(♯‘𝑋)))) + (𝐸 / (2 ·
(√‘(♯‘𝑋)))))) |
| 150 | 78 | recnd 11289 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝐸 ∈ ℂ) |
| 151 | 96 | rpcnd 13079 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 →
(√‘(♯‘𝑋)) ∈ ℂ) |
| 152 | | 2cnd 12344 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 2 ∈
ℂ) |
| 153 | 96 | rpne0d 13082 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 →
(√‘(♯‘𝑋)) ≠ 0) |
| 154 | 89 | rpne0d 13082 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 2 ≠ 0) |
| 155 | 150, 151,
152, 153, 154 | divdiv3d 45370 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((𝐸 / (√‘(♯‘𝑋))) / 2) = (𝐸 / (2 ·
(√‘(♯‘𝑋))))) |
| 156 | 155 | eqcomd 2743 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝐸 / (2 ·
(√‘(♯‘𝑋)))) = ((𝐸 / (√‘(♯‘𝑋))) / 2)) |
| 157 | 156, 156 | oveq12d 7449 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((𝐸 / (2 ·
(√‘(♯‘𝑋)))) + (𝐸 / (2 ·
(√‘(♯‘𝑋))))) = (((𝐸 / (√‘(♯‘𝑋))) / 2) + ((𝐸 / (√‘(♯‘𝑋))) / 2))) |
| 158 | 150, 151,
153 | divcld 12043 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝐸 / (√‘(♯‘𝑋))) ∈
ℂ) |
| 159 | 158 | 2halvesd 12512 |
. . . . . . . . . . . . . 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 480 |
. . . . . . . . . . . 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 5169 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐷‘𝑖) − (𝐶‘𝑖)) < (𝐸 / (√‘(♯‘𝑋)))) |
| 164 | 134 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐸 / (√‘(♯‘𝑋))) ∈
ℝ) |
| 165 | | 0red 11264 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 0 ∈
ℝ) |
| 166 | 96 | rpred 13077 |
. . . . . . . . . . . . . 14
⊢ (𝜑 →
(√‘(♯‘𝑋)) ∈ ℝ) |
| 167 | 77 | rpgt0d 13080 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 0 < 𝐸) |
| 168 | 96 | rpgt0d 13080 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 0 <
(√‘(♯‘𝑋))) |
| 169 | 78, 166, 167, 168 | divgt0d 12203 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 0 < (𝐸 / (√‘(♯‘𝑋)))) |
| 170 | 165, 134,
169 | ltled 11409 |
. . . . . . . . . . . 12
⊢ (𝜑 → 0 ≤ (𝐸 / (√‘(♯‘𝑋)))) |
| 171 | 170 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 0 ≤ (𝐸 / (√‘(♯‘𝑋)))) |
| 172 | | lt2sq 14173 |
. . . . . . . . . . 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 232 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (((𝐷‘𝑖) − (𝐶‘𝑖))↑2) < ((𝐸 / (√‘(♯‘𝑋)))↑2)) |
| 175 | 1, 80, 71, 136, 174 | fsumlt 15836 |
. . . . . . . 8
⊢ (𝜑 → Σ𝑖 ∈ 𝑋 (((𝐷‘𝑖) − (𝐶‘𝑖))↑2) < Σ𝑖 ∈ 𝑋 ((𝐸 / (√‘(♯‘𝑋)))↑2)) |
| 176 | 1, 136 | fsumrecl 15770 |
. . . . . . . . 9
⊢ (𝜑 → Σ𝑖 ∈ 𝑋 ((𝐸 / (√‘(♯‘𝑋)))↑2) ∈
ℝ) |
| 177 | 164 | sqge0d 14177 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 0 ≤ ((𝐸 / (√‘(♯‘𝑋)))↑2)) |
| 178 | 1, 136, 177 | fsumge0 15831 |
. . . . . . . . 9
⊢ (𝜑 → 0 ≤ Σ𝑖 ∈ 𝑋 ((𝐸 / (√‘(♯‘𝑋)))↑2)) |
| 179 | 72, 74, 176, 178 | sqrtltd 15466 |
. . . . . . . 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 232 |
. . . . . . 7
⊢ (𝜑 →
(√‘Σ𝑖
∈ 𝑋 (((𝐷‘𝑖) − (𝐶‘𝑖))↑2)) < (√‘Σ𝑖 ∈ 𝑋 ((𝐸 / (√‘(♯‘𝑋)))↑2))) |
| 181 | 135 | recnd 11289 |
. . . . . . . . . . 11
⊢ (𝜑 → ((𝐸 / (√‘(♯‘𝑋)))↑2) ∈
ℂ) |
| 182 | | fsumconst 15826 |
. . . . . . . . . . 11
⊢ ((𝑋 ∈ Fin ∧ ((𝐸 /
(√‘(♯‘𝑋)))↑2) ∈ ℂ) →
Σ𝑖 ∈ 𝑋 ((𝐸 / (√‘(♯‘𝑋)))↑2) =
((♯‘𝑋) ·
((𝐸 /
(√‘(♯‘𝑋)))↑2))) |
| 183 | 1, 181, 182 | syl2anc 584 |
. . . . . . . . . 10
⊢ (𝜑 → Σ𝑖 ∈ 𝑋 ((𝐸 / (√‘(♯‘𝑋)))↑2) =
((♯‘𝑋) ·
((𝐸 /
(√‘(♯‘𝑋)))↑2))) |
| 184 | | sqdiv 14161 |
. . . . . . . . . . . . 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 11289 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (♯‘𝑋) ∈
ℂ) |
| 187 | | sqrtth 15403 |
. . . . . . . . . . . . . 14
⊢
((♯‘𝑋)
∈ ℂ → ((√‘(♯‘𝑋))↑2) = (♯‘𝑋)) |
| 188 | 186, 187 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 →
((√‘(♯‘𝑋))↑2) = (♯‘𝑋)) |
| 189 | 188 | oveq2d 7447 |
. . . . . . . . . . . 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 7447 |
. . . . . . . . . 10
⊢ (𝜑 → ((♯‘𝑋) · ((𝐸 / (√‘(♯‘𝑋)))↑2)) =
((♯‘𝑋) ·
((𝐸↑2) /
(♯‘𝑋)))) |
| 192 | 150 | sqcld 14184 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐸↑2) ∈ ℂ) |
| 193 | 165, 94 | gtned 11396 |
. . . . . . . . . . 11
⊢ (𝜑 → (♯‘𝑋) ≠ 0) |
| 194 | 192, 186,
193 | divcan2d 12045 |
. . . . . . . . . 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 6910 |
. . . . . . . 8
⊢ (𝜑 →
(√‘Σ𝑖
∈ 𝑋 ((𝐸 /
(√‘(♯‘𝑋)))↑2)) = (√‘(𝐸↑2))) |
| 197 | 165, 78, 167 | ltled 11409 |
. . . . . . . . 9
⊢ (𝜑 → 0 ≤ 𝐸) |
| 198 | | sqrtsq 15308 |
. . . . . . . . 9
⊢ ((𝐸 ∈ ℝ ∧ 0 ≤
𝐸) →
(√‘(𝐸↑2))
= 𝐸) |
| 199 | 78, 197, 198 | syl2anc 584 |
. . . . . . . 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 5169 |
. . . . . 6
⊢ (𝜑 →
(√‘Σ𝑖
∈ 𝑋 (((𝐷‘𝑖) − (𝐶‘𝑖))↑2)) < 𝐸) |
| 203 | 202 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → (√‘Σ𝑖 ∈ 𝑋 (((𝐷‘𝑖) − (𝐶‘𝑖))↑2)) < 𝐸) |
| 204 | 69, 76, 79, 133, 203 | lttrd 11422 |
. . . 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 25446 |
. . . . . . 7
⊢ (𝑋 ∈ Fin →
(dist‘(ℝ^‘𝑋)) ∈ (Met‘(ℝ
↑m 𝑋))) |
| 207 | | metxmet 24344 |
. . . . . . 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 480 |
. . . . 5
⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → (dist‘(ℝ^‘𝑋)) ∈
(∞Met‘(ℝ ↑m 𝑋))) |
| 210 | 79 | rexrd 11311 |
. . . . 5
⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → 𝐸 ∈
ℝ*) |
| 211 | 27, 3 | eleqtrdi 2851 |
. . . . 5
⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → 𝑓 ∈ (ℝ ↑m 𝑋)) |
| 212 | | elbl2 24400 |
. . . . 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 257 |
. . 3
⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → 𝑓 ∈ (𝑌(ball‘(dist‘(ℝ^‘𝑋)))𝐸)) |
| 215 | 214 | ralrimiva 3146 |
. 2
⊢ (𝜑 → ∀𝑓 ∈ X 𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))𝑓 ∈ (𝑌(ball‘(dist‘(ℝ^‘𝑋)))𝐸)) |
| 216 | | dfss3 3972 |
. 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 234 |
1
⊢ (𝜑 → X𝑖 ∈
𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖)) ⊆ (𝑌(ball‘(dist‘(ℝ^‘𝑋)))𝐸)) |