| Step | Hyp | Ref
| Expression |
| 1 | | rrndistlt.i |
. . . . 5
⊢ (𝜑 → 𝐼 ∈ Fin) |
| 2 | | rrndistlt.z |
. . . . 5
⊢ (𝜑 → 𝐼 ≠ ∅) |
| 3 | | rrndistlt.x |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑋 ∈ (ℝ ↑m 𝐼)) |
| 4 | | elmapi 8889 |
. . . . . . . . . . 11
⊢ (𝑋 ∈ (ℝ
↑m 𝐼)
→ 𝑋:𝐼⟶ℝ) |
| 5 | 3, 4 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑋:𝐼⟶ℝ) |
| 6 | | ax-resscn 11212 |
. . . . . . . . . . 11
⊢ ℝ
⊆ ℂ |
| 7 | 6 | a1i 11 |
. . . . . . . . . 10
⊢ (𝜑 → ℝ ⊆
ℂ) |
| 8 | 5, 7 | fssd 6753 |
. . . . . . . . 9
⊢ (𝜑 → 𝑋:𝐼⟶ℂ) |
| 9 | 8 | ffvelcdmda 7104 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (𝑋‘𝑖) ∈ ℂ) |
| 10 | | rrndistlt.y |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑌 ∈ (ℝ ↑m 𝐼)) |
| 11 | | elmapi 8889 |
. . . . . . . . . . 11
⊢ (𝑌 ∈ (ℝ
↑m 𝐼)
→ 𝑌:𝐼⟶ℝ) |
| 12 | 10, 11 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑌:𝐼⟶ℝ) |
| 13 | 12, 7 | fssd 6753 |
. . . . . . . . 9
⊢ (𝜑 → 𝑌:𝐼⟶ℂ) |
| 14 | 13 | ffvelcdmda 7104 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (𝑌‘𝑖) ∈ ℂ) |
| 15 | 9, 14 | subcld 11620 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝑋‘𝑖) − (𝑌‘𝑖)) ∈ ℂ) |
| 16 | 15 | abscld 15475 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (abs‘((𝑋‘𝑖) − (𝑌‘𝑖))) ∈ ℝ) |
| 17 | 16 | resqcld 14165 |
. . . . 5
⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((abs‘((𝑋‘𝑖) − (𝑌‘𝑖)))↑2) ∈ ℝ) |
| 18 | | rrndistlt.e |
. . . . . . . 8
⊢ (𝜑 → 𝐸 ∈
ℝ+) |
| 19 | 18 | rpred 13077 |
. . . . . . 7
⊢ (𝜑 → 𝐸 ∈ ℝ) |
| 20 | 19 | resqcld 14165 |
. . . . . 6
⊢ (𝜑 → (𝐸↑2) ∈ ℝ) |
| 21 | 20 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (𝐸↑2) ∈ ℝ) |
| 22 | | rrndistlt.l |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (abs‘((𝑋‘𝑖) − (𝑌‘𝑖))) < 𝐸) |
| 23 | 15 | absge0d 15483 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 0 ≤ (abs‘((𝑋‘𝑖) − (𝑌‘𝑖)))) |
| 24 | 19 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝐸 ∈ ℝ) |
| 25 | 18 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝐸 ∈
ℝ+) |
| 26 | 25 | rpge0d 13081 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 0 ≤ 𝐸) |
| 27 | | lt2sq 14173 |
. . . . . . 7
⊢
((((abs‘((𝑋‘𝑖) − (𝑌‘𝑖))) ∈ ℝ ∧ 0 ≤
(abs‘((𝑋‘𝑖) − (𝑌‘𝑖)))) ∧ (𝐸 ∈ ℝ ∧ 0 ≤ 𝐸)) → ((abs‘((𝑋‘𝑖) − (𝑌‘𝑖))) < 𝐸 ↔ ((abs‘((𝑋‘𝑖) − (𝑌‘𝑖)))↑2) < (𝐸↑2))) |
| 28 | 16, 23, 24, 26, 27 | syl22anc 839 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((abs‘((𝑋‘𝑖) − (𝑌‘𝑖))) < 𝐸 ↔ ((abs‘((𝑋‘𝑖) − (𝑌‘𝑖)))↑2) < (𝐸↑2))) |
| 29 | 22, 28 | mpbid 232 |
. . . . 5
⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((abs‘((𝑋‘𝑖) − (𝑌‘𝑖)))↑2) < (𝐸↑2)) |
| 30 | 1, 2, 17, 21, 29 | fsumlt 15836 |
. . . 4
⊢ (𝜑 → Σ𝑖 ∈ 𝐼 ((abs‘((𝑋‘𝑖) − (𝑌‘𝑖)))↑2) < Σ𝑖 ∈ 𝐼 (𝐸↑2)) |
| 31 | 5 | ffvelcdmda 7104 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (𝑋‘𝑖) ∈ ℝ) |
| 32 | 12 | ffvelcdmda 7104 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (𝑌‘𝑖) ∈ ℝ) |
| 33 | 31, 32 | resubcld 11691 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝑋‘𝑖) − (𝑌‘𝑖)) ∈ ℝ) |
| 34 | | absresq 15341 |
. . . . . . . 8
⊢ (((𝑋‘𝑖) − (𝑌‘𝑖)) ∈ ℝ → ((abs‘((𝑋‘𝑖) − (𝑌‘𝑖)))↑2) = (((𝑋‘𝑖) − (𝑌‘𝑖))↑2)) |
| 35 | 33, 34 | syl 17 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((abs‘((𝑋‘𝑖) − (𝑌‘𝑖)))↑2) = (((𝑋‘𝑖) − (𝑌‘𝑖))↑2)) |
| 36 | 35 | eqcomd 2743 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (((𝑋‘𝑖) − (𝑌‘𝑖))↑2) = ((abs‘((𝑋‘𝑖) − (𝑌‘𝑖)))↑2)) |
| 37 | 36 | sumeq2dv 15738 |
. . . . 5
⊢ (𝜑 → Σ𝑖 ∈ 𝐼 (((𝑋‘𝑖) − (𝑌‘𝑖))↑2) = Σ𝑖 ∈ 𝐼 ((abs‘((𝑋‘𝑖) − (𝑌‘𝑖)))↑2)) |
| 38 | 6, 20 | sselid 3981 |
. . . . . . 7
⊢ (𝜑 → (𝐸↑2) ∈ ℂ) |
| 39 | | fsumconst 15826 |
. . . . . . 7
⊢ ((𝐼 ∈ Fin ∧ (𝐸↑2) ∈ ℂ) →
Σ𝑖 ∈ 𝐼 (𝐸↑2) = ((♯‘𝐼) · (𝐸↑2))) |
| 40 | 1, 38, 39 | syl2anc 584 |
. . . . . 6
⊢ (𝜑 → Σ𝑖 ∈ 𝐼 (𝐸↑2) = ((♯‘𝐼) · (𝐸↑2))) |
| 41 | | rrndistlt.n |
. . . . . . . . 9
⊢ 𝑁 = (♯‘𝐼) |
| 42 | | eqcom 2744 |
. . . . . . . . 9
⊢ (𝑁 = (♯‘𝐼) ↔ (♯‘𝐼) = 𝑁) |
| 43 | 41, 42 | mpbi 230 |
. . . . . . . 8
⊢
(♯‘𝐼) =
𝑁 |
| 44 | 43 | oveq1i 7441 |
. . . . . . 7
⊢
((♯‘𝐼)
· (𝐸↑2)) =
(𝑁 · (𝐸↑2)) |
| 45 | 44 | a1i 11 |
. . . . . 6
⊢ (𝜑 → ((♯‘𝐼) · (𝐸↑2)) = (𝑁 · (𝐸↑2))) |
| 46 | 40, 45 | eqtr2d 2778 |
. . . . 5
⊢ (𝜑 → (𝑁 · (𝐸↑2)) = Σ𝑖 ∈ 𝐼 (𝐸↑2)) |
| 47 | 37, 46 | breq12d 5156 |
. . . 4
⊢ (𝜑 → (Σ𝑖 ∈ 𝐼 (((𝑋‘𝑖) − (𝑌‘𝑖))↑2) < (𝑁 · (𝐸↑2)) ↔ Σ𝑖 ∈ 𝐼 ((abs‘((𝑋‘𝑖) − (𝑌‘𝑖)))↑2) < Σ𝑖 ∈ 𝐼 (𝐸↑2))) |
| 48 | 30, 47 | mpbird 257 |
. . 3
⊢ (𝜑 → Σ𝑖 ∈ 𝐼 (((𝑋‘𝑖) − (𝑌‘𝑖))↑2) < (𝑁 · (𝐸↑2))) |
| 49 | | nfv 1914 |
. . . . 5
⊢
Ⅎ𝑖𝜑 |
| 50 | 33 | resqcld 14165 |
. . . . 5
⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (((𝑋‘𝑖) − (𝑌‘𝑖))↑2) ∈ ℝ) |
| 51 | 49, 1, 50 | fsumreclf 45591 |
. . . 4
⊢ (𝜑 → Σ𝑖 ∈ 𝐼 (((𝑋‘𝑖) − (𝑌‘𝑖))↑2) ∈ ℝ) |
| 52 | 33 | sqge0d 14177 |
. . . . 5
⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 0 ≤ (((𝑋‘𝑖) − (𝑌‘𝑖))↑2)) |
| 53 | 1, 50, 52 | fsumge0 15831 |
. . . 4
⊢ (𝜑 → 0 ≤ Σ𝑖 ∈ 𝐼 (((𝑋‘𝑖) − (𝑌‘𝑖))↑2)) |
| 54 | | hashcl 14395 |
. . . . . . . 8
⊢ (𝐼 ∈ Fin →
(♯‘𝐼) ∈
ℕ0) |
| 55 | 1, 54 | syl 17 |
. . . . . . 7
⊢ (𝜑 → (♯‘𝐼) ∈
ℕ0) |
| 56 | 41, 55 | eqeltrid 2845 |
. . . . . 6
⊢ (𝜑 → 𝑁 ∈
ℕ0) |
| 57 | 56 | nn0red 12588 |
. . . . 5
⊢ (𝜑 → 𝑁 ∈ ℝ) |
| 58 | 57, 20 | remulcld 11291 |
. . . 4
⊢ (𝜑 → (𝑁 · (𝐸↑2)) ∈ ℝ) |
| 59 | 56 | nn0ge0d 12590 |
. . . . 5
⊢ (𝜑 → 0 ≤ 𝑁) |
| 60 | 19 | sqge0d 14177 |
. . . . 5
⊢ (𝜑 → 0 ≤ (𝐸↑2)) |
| 61 | 57, 20, 59, 60 | mulge0d 11840 |
. . . 4
⊢ (𝜑 → 0 ≤ (𝑁 · (𝐸↑2))) |
| 62 | 51, 53, 58, 61 | sqrtltd 15466 |
. . 3
⊢ (𝜑 → (Σ𝑖 ∈ 𝐼 (((𝑋‘𝑖) − (𝑌‘𝑖))↑2) < (𝑁 · (𝐸↑2)) ↔ (√‘Σ𝑖 ∈ 𝐼 (((𝑋‘𝑖) − (𝑌‘𝑖))↑2)) < (√‘(𝑁 · (𝐸↑2))))) |
| 63 | 48, 62 | mpbid 232 |
. 2
⊢ (𝜑 →
(√‘Σ𝑖
∈ 𝐼 (((𝑋‘𝑖) − (𝑌‘𝑖))↑2)) < (√‘(𝑁 · (𝐸↑2)))) |
| 64 | | rrndistlt.d |
. . . . . 6
⊢ 𝐷 =
(dist‘(ℝ^‘𝐼)) |
| 65 | 64 | a1i 11 |
. . . . 5
⊢ (𝜑 → 𝐷 = (dist‘(ℝ^‘𝐼))) |
| 66 | | eqid 2737 |
. . . . . . 7
⊢
(ℝ^‘𝐼) =
(ℝ^‘𝐼) |
| 67 | | eqid 2737 |
. . . . . . 7
⊢ (ℝ
↑m 𝐼) =
(ℝ ↑m 𝐼) |
| 68 | 66, 67 | rrxdsfi 25445 |
. . . . . 6
⊢ (𝐼 ∈ Fin →
(dist‘(ℝ^‘𝐼)) = (𝑓 ∈ (ℝ ↑m 𝐼), 𝑔 ∈ (ℝ ↑m 𝐼) ↦
(√‘Σ𝑖
∈ 𝐼 (((𝑓‘𝑖) − (𝑔‘𝑖))↑2)))) |
| 69 | 1, 68 | syl 17 |
. . . . 5
⊢ (𝜑 →
(dist‘(ℝ^‘𝐼)) = (𝑓 ∈ (ℝ ↑m 𝐼), 𝑔 ∈ (ℝ ↑m 𝐼) ↦
(√‘Σ𝑖
∈ 𝐼 (((𝑓‘𝑖) − (𝑔‘𝑖))↑2)))) |
| 70 | 65, 69 | eqtrd 2777 |
. . . 4
⊢ (𝜑 → 𝐷 = (𝑓 ∈ (ℝ ↑m 𝐼), 𝑔 ∈ (ℝ ↑m 𝐼) ↦
(√‘Σ𝑖
∈ 𝐼 (((𝑓‘𝑖) − (𝑔‘𝑖))↑2)))) |
| 71 | | fveq1 6905 |
. . . . . . . . . 10
⊢ (𝑓 = 𝑋 → (𝑓‘𝑖) = (𝑋‘𝑖)) |
| 72 | 71 | adantr 480 |
. . . . . . . . 9
⊢ ((𝑓 = 𝑋 ∧ 𝑔 = 𝑌) → (𝑓‘𝑖) = (𝑋‘𝑖)) |
| 73 | | fveq1 6905 |
. . . . . . . . . 10
⊢ (𝑔 = 𝑌 → (𝑔‘𝑖) = (𝑌‘𝑖)) |
| 74 | 73 | adantl 481 |
. . . . . . . . 9
⊢ ((𝑓 = 𝑋 ∧ 𝑔 = 𝑌) → (𝑔‘𝑖) = (𝑌‘𝑖)) |
| 75 | 72, 74 | oveq12d 7449 |
. . . . . . . 8
⊢ ((𝑓 = 𝑋 ∧ 𝑔 = 𝑌) → ((𝑓‘𝑖) − (𝑔‘𝑖)) = ((𝑋‘𝑖) − (𝑌‘𝑖))) |
| 76 | 75 | oveq1d 7446 |
. . . . . . 7
⊢ ((𝑓 = 𝑋 ∧ 𝑔 = 𝑌) → (((𝑓‘𝑖) − (𝑔‘𝑖))↑2) = (((𝑋‘𝑖) − (𝑌‘𝑖))↑2)) |
| 77 | 76 | sumeq2sdv 15739 |
. . . . . 6
⊢ ((𝑓 = 𝑋 ∧ 𝑔 = 𝑌) → Σ𝑖 ∈ 𝐼 (((𝑓‘𝑖) − (𝑔‘𝑖))↑2) = Σ𝑖 ∈ 𝐼 (((𝑋‘𝑖) − (𝑌‘𝑖))↑2)) |
| 78 | 77 | fveq2d 6910 |
. . . . 5
⊢ ((𝑓 = 𝑋 ∧ 𝑔 = 𝑌) → (√‘Σ𝑖 ∈ 𝐼 (((𝑓‘𝑖) − (𝑔‘𝑖))↑2)) = (√‘Σ𝑖 ∈ 𝐼 (((𝑋‘𝑖) − (𝑌‘𝑖))↑2))) |
| 79 | 78 | adantl 481 |
. . . 4
⊢ ((𝜑 ∧ (𝑓 = 𝑋 ∧ 𝑔 = 𝑌)) → (√‘Σ𝑖 ∈ 𝐼 (((𝑓‘𝑖) − (𝑔‘𝑖))↑2)) = (√‘Σ𝑖 ∈ 𝐼 (((𝑋‘𝑖) − (𝑌‘𝑖))↑2))) |
| 80 | 51, 53 | resqrtcld 15456 |
. . . 4
⊢ (𝜑 →
(√‘Σ𝑖
∈ 𝐼 (((𝑋‘𝑖) − (𝑌‘𝑖))↑2)) ∈ ℝ) |
| 81 | 70, 79, 3, 10, 80 | ovmpod 7585 |
. . 3
⊢ (𝜑 → (𝑋𝐷𝑌) = (√‘Σ𝑖 ∈ 𝐼 (((𝑋‘𝑖) − (𝑌‘𝑖))↑2))) |
| 82 | | sqrtmul 15298 |
. . . . 5
⊢ (((𝑁 ∈ ℝ ∧ 0 ≤
𝑁) ∧ ((𝐸↑2) ∈ ℝ ∧ 0
≤ (𝐸↑2))) →
(√‘(𝑁 ·
(𝐸↑2))) =
((√‘𝑁) ·
(√‘(𝐸↑2)))) |
| 83 | 57, 59, 20, 60, 82 | syl22anc 839 |
. . . 4
⊢ (𝜑 → (√‘(𝑁 · (𝐸↑2))) = ((√‘𝑁) · (√‘(𝐸↑2)))) |
| 84 | 18 | rpge0d 13081 |
. . . . . 6
⊢ (𝜑 → 0 ≤ 𝐸) |
| 85 | 19, 84 | sqrtsqd 15458 |
. . . . 5
⊢ (𝜑 → (√‘(𝐸↑2)) = 𝐸) |
| 86 | 85 | oveq2d 7447 |
. . . 4
⊢ (𝜑 → ((√‘𝑁) · (√‘(𝐸↑2))) =
((√‘𝑁) ·
𝐸)) |
| 87 | 83, 86 | eqtr2d 2778 |
. . 3
⊢ (𝜑 → ((√‘𝑁) · 𝐸) = (√‘(𝑁 · (𝐸↑2)))) |
| 88 | 81, 87 | breq12d 5156 |
. 2
⊢ (𝜑 → ((𝑋𝐷𝑌) < ((√‘𝑁) · 𝐸) ↔ (√‘Σ𝑖 ∈ 𝐼 (((𝑋‘𝑖) − (𝑌‘𝑖))↑2)) < (√‘(𝑁 · (𝐸↑2))))) |
| 89 | 63, 88 | mpbird 257 |
1
⊢ (𝜑 → (𝑋𝐷𝑌) < ((√‘𝑁) · 𝐸)) |