| Step | Hyp | Ref
| Expression |
| 1 | | stoweidlem17.2 |
. . 3
⊢ (𝜑 → 𝑁 ∈ ℕ) |
| 2 | 1 | nnnn0d 12587 |
. 2
⊢ (𝜑 → 𝑁 ∈
ℕ0) |
| 3 | | nn0uz 12920 |
. . . . 5
⊢
ℕ0 = (ℤ≥‘0) |
| 4 | 2, 3 | eleqtrdi 2851 |
. . . 4
⊢ (𝜑 → 𝑁 ∈
(ℤ≥‘0)) |
| 5 | | eluzfz2 13572 |
. . . 4
⊢ (𝑁 ∈
(ℤ≥‘0) → 𝑁 ∈ (0...𝑁)) |
| 6 | 4, 5 | syl 17 |
. . 3
⊢ (𝜑 → 𝑁 ∈ (0...𝑁)) |
| 7 | 6 | ancli 548 |
. 2
⊢ (𝜑 → (𝜑 ∧ 𝑁 ∈ (0...𝑁))) |
| 8 | | eleq1 2829 |
. . . . 5
⊢ (𝑛 = 0 → (𝑛 ∈ (0...𝑁) ↔ 0 ∈ (0...𝑁))) |
| 9 | 8 | anbi2d 630 |
. . . 4
⊢ (𝑛 = 0 → ((𝜑 ∧ 𝑛 ∈ (0...𝑁)) ↔ (𝜑 ∧ 0 ∈ (0...𝑁)))) |
| 10 | | oveq2 7439 |
. . . . . . 7
⊢ (𝑛 = 0 → (0...𝑛) = (0...0)) |
| 11 | 10 | sumeq1d 15736 |
. . . . . 6
⊢ (𝑛 = 0 → Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋‘𝑖)‘𝑡)) = Σ𝑖 ∈ (0...0)(𝐸 · ((𝑋‘𝑖)‘𝑡))) |
| 12 | 11 | mpteq2dv 5244 |
. . . . 5
⊢ (𝑛 = 0 → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋‘𝑖)‘𝑡))) = (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...0)(𝐸 · ((𝑋‘𝑖)‘𝑡)))) |
| 13 | 12 | eleq1d 2826 |
. . . 4
⊢ (𝑛 = 0 → ((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴 ↔ (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...0)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴)) |
| 14 | 9, 13 | imbi12d 344 |
. . 3
⊢ (𝑛 = 0 → (((𝜑 ∧ 𝑛 ∈ (0...𝑁)) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴) ↔ ((𝜑 ∧ 0 ∈ (0...𝑁)) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...0)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴))) |
| 15 | | eleq1 2829 |
. . . . 5
⊢ (𝑛 = 𝑚 → (𝑛 ∈ (0...𝑁) ↔ 𝑚 ∈ (0...𝑁))) |
| 16 | 15 | anbi2d 630 |
. . . 4
⊢ (𝑛 = 𝑚 → ((𝜑 ∧ 𝑛 ∈ (0...𝑁)) ↔ (𝜑 ∧ 𝑚 ∈ (0...𝑁)))) |
| 17 | | oveq2 7439 |
. . . . . . 7
⊢ (𝑛 = 𝑚 → (0...𝑛) = (0...𝑚)) |
| 18 | 17 | sumeq1d 15736 |
. . . . . 6
⊢ (𝑛 = 𝑚 → Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋‘𝑖)‘𝑡)) = Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡))) |
| 19 | 18 | mpteq2dv 5244 |
. . . . 5
⊢ (𝑛 = 𝑚 → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋‘𝑖)‘𝑡))) = (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)))) |
| 20 | 19 | eleq1d 2826 |
. . . 4
⊢ (𝑛 = 𝑚 → ((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴 ↔ (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴)) |
| 21 | 16, 20 | imbi12d 344 |
. . 3
⊢ (𝑛 = 𝑚 → (((𝜑 ∧ 𝑛 ∈ (0...𝑁)) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴) ↔ ((𝜑 ∧ 𝑚 ∈ (0...𝑁)) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴))) |
| 22 | | eleq1 2829 |
. . . . 5
⊢ (𝑛 = (𝑚 + 1) → (𝑛 ∈ (0...𝑁) ↔ (𝑚 + 1) ∈ (0...𝑁))) |
| 23 | 22 | anbi2d 630 |
. . . 4
⊢ (𝑛 = (𝑚 + 1) → ((𝜑 ∧ 𝑛 ∈ (0...𝑁)) ↔ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)))) |
| 24 | | oveq2 7439 |
. . . . . . 7
⊢ (𝑛 = (𝑚 + 1) → (0...𝑛) = (0...(𝑚 + 1))) |
| 25 | 24 | sumeq1d 15736 |
. . . . . 6
⊢ (𝑛 = (𝑚 + 1) → Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋‘𝑖)‘𝑡)) = Σ𝑖 ∈ (0...(𝑚 + 1))(𝐸 · ((𝑋‘𝑖)‘𝑡))) |
| 26 | 25 | mpteq2dv 5244 |
. . . . 5
⊢ (𝑛 = (𝑚 + 1) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋‘𝑖)‘𝑡))) = (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...(𝑚 + 1))(𝐸 · ((𝑋‘𝑖)‘𝑡)))) |
| 27 | 26 | eleq1d 2826 |
. . . 4
⊢ (𝑛 = (𝑚 + 1) → ((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴 ↔ (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...(𝑚 + 1))(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴)) |
| 28 | 23, 27 | imbi12d 344 |
. . 3
⊢ (𝑛 = (𝑚 + 1) → (((𝜑 ∧ 𝑛 ∈ (0...𝑁)) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴) ↔ ((𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...(𝑚 + 1))(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴))) |
| 29 | | eleq1 2829 |
. . . . 5
⊢ (𝑛 = 𝑁 → (𝑛 ∈ (0...𝑁) ↔ 𝑁 ∈ (0...𝑁))) |
| 30 | 29 | anbi2d 630 |
. . . 4
⊢ (𝑛 = 𝑁 → ((𝜑 ∧ 𝑛 ∈ (0...𝑁)) ↔ (𝜑 ∧ 𝑁 ∈ (0...𝑁)))) |
| 31 | | oveq2 7439 |
. . . . . . 7
⊢ (𝑛 = 𝑁 → (0...𝑛) = (0...𝑁)) |
| 32 | 31 | sumeq1d 15736 |
. . . . . 6
⊢ (𝑛 = 𝑁 → Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋‘𝑖)‘𝑡)) = Σ𝑖 ∈ (0...𝑁)(𝐸 · ((𝑋‘𝑖)‘𝑡))) |
| 33 | 32 | mpteq2dv 5244 |
. . . . 5
⊢ (𝑛 = 𝑁 → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋‘𝑖)‘𝑡))) = (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑁)(𝐸 · ((𝑋‘𝑖)‘𝑡)))) |
| 34 | 33 | eleq1d 2826 |
. . . 4
⊢ (𝑛 = 𝑁 → ((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴 ↔ (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑁)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴)) |
| 35 | 30, 34 | imbi12d 344 |
. . 3
⊢ (𝑛 = 𝑁 → (((𝜑 ∧ 𝑛 ∈ (0...𝑁)) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴) ↔ ((𝜑 ∧ 𝑁 ∈ (0...𝑁)) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑁)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴))) |
| 36 | | 0z 12624 |
. . . . . . . . 9
⊢ 0 ∈
ℤ |
| 37 | | fzsn 13606 |
. . . . . . . . 9
⊢ (0 ∈
ℤ → (0...0) = {0}) |
| 38 | 36, 37 | ax-mp 5 |
. . . . . . . 8
⊢ (0...0) =
{0} |
| 39 | 38 | sumeq1i 15733 |
. . . . . . 7
⊢
Σ𝑖 ∈
(0...0)(𝐸 · ((𝑋‘𝑖)‘𝑡)) = Σ𝑖 ∈ {0} (𝐸 · ((𝑋‘𝑖)‘𝑡)) |
| 40 | 39 | mpteq2i 5247 |
. . . . . 6
⊢ (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...0)(𝐸 · ((𝑋‘𝑖)‘𝑡))) = (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ {0} (𝐸 · ((𝑋‘𝑖)‘𝑡))) |
| 41 | | stoweidlem17.1 |
. . . . . . 7
⊢
Ⅎ𝑡𝜑 |
| 42 | | stoweidlem17.7 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐸 ∈ ℝ) |
| 43 | 42 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝐸 ∈ ℝ) |
| 44 | 43 | recnd 11289 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝐸 ∈ ℂ) |
| 45 | | stoweidlem17.3 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑋:(0...𝑁)⟶𝐴) |
| 46 | | nnz 12634 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑁 ∈ ℕ → 𝑁 ∈
ℤ) |
| 47 | | nngt0 12297 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑁 ∈ ℕ → 0 <
𝑁) |
| 48 | | 0re 11263 |
. . . . . . . . . . . . . . . . . . 19
⊢ 0 ∈
ℝ |
| 49 | | nnre 12273 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑁 ∈ ℕ → 𝑁 ∈
ℝ) |
| 50 | | ltle 11349 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((0
∈ ℝ ∧ 𝑁
∈ ℝ) → (0 < 𝑁 → 0 ≤ 𝑁)) |
| 51 | 48, 49, 50 | sylancr 587 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑁 ∈ ℕ → (0 <
𝑁 → 0 ≤ 𝑁)) |
| 52 | 47, 51 | mpd 15 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑁 ∈ ℕ → 0 ≤
𝑁) |
| 53 | 46, 52 | jca 511 |
. . . . . . . . . . . . . . . 16
⊢ (𝑁 ∈ ℕ → (𝑁 ∈ ℤ ∧ 0 ≤
𝑁)) |
| 54 | 1, 53 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝑁 ∈ ℤ ∧ 0 ≤ 𝑁)) |
| 55 | 36 | eluz1i 12886 |
. . . . . . . . . . . . . . 15
⊢ (𝑁 ∈
(ℤ≥‘0) ↔ (𝑁 ∈ ℤ ∧ 0 ≤ 𝑁)) |
| 56 | 54, 55 | sylibr 234 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑁 ∈
(ℤ≥‘0)) |
| 57 | | eluzfz1 13571 |
. . . . . . . . . . . . . 14
⊢ (𝑁 ∈
(ℤ≥‘0) → 0 ∈ (0...𝑁)) |
| 58 | 56, 57 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 0 ∈ (0...𝑁)) |
| 59 | 45, 58 | ffvelcdmd 7105 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑋‘0) ∈ 𝐴) |
| 60 | | feq1 6716 |
. . . . . . . . . . . . . 14
⊢ (𝑓 = (𝑋‘0) → (𝑓:𝑇⟶ℝ ↔ (𝑋‘0):𝑇⟶ℝ)) |
| 61 | 60 | imbi2d 340 |
. . . . . . . . . . . . 13
⊢ (𝑓 = (𝑋‘0) → ((𝜑 → 𝑓:𝑇⟶ℝ) ↔ (𝜑 → (𝑋‘0):𝑇⟶ℝ))) |
| 62 | | stoweidlem17.8 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ) |
| 63 | 62 | expcom 413 |
. . . . . . . . . . . . 13
⊢ (𝑓 ∈ 𝐴 → (𝜑 → 𝑓:𝑇⟶ℝ)) |
| 64 | 61, 63 | vtoclga 3577 |
. . . . . . . . . . . 12
⊢ ((𝑋‘0) ∈ 𝐴 → (𝜑 → (𝑋‘0):𝑇⟶ℝ)) |
| 65 | 59, 64 | mpcom 38 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑋‘0):𝑇⟶ℝ) |
| 66 | 65 | ffvelcdmda 7104 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝑋‘0)‘𝑡) ∈ ℝ) |
| 67 | 66 | recnd 11289 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝑋‘0)‘𝑡) ∈ ℂ) |
| 68 | 44, 67 | mulcld 11281 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐸 · ((𝑋‘0)‘𝑡)) ∈ ℂ) |
| 69 | | fveq2 6906 |
. . . . . . . . . . 11
⊢ (𝑖 = 0 → (𝑋‘𝑖) = (𝑋‘0)) |
| 70 | 69 | fveq1d 6908 |
. . . . . . . . . 10
⊢ (𝑖 = 0 → ((𝑋‘𝑖)‘𝑡) = ((𝑋‘0)‘𝑡)) |
| 71 | 70 | oveq2d 7447 |
. . . . . . . . 9
⊢ (𝑖 = 0 → (𝐸 · ((𝑋‘𝑖)‘𝑡)) = (𝐸 · ((𝑋‘0)‘𝑡))) |
| 72 | 71 | sumsn 15782 |
. . . . . . . 8
⊢ ((0
∈ ℤ ∧ (𝐸
· ((𝑋‘0)‘𝑡)) ∈ ℂ) → Σ𝑖 ∈ {0} (𝐸 · ((𝑋‘𝑖)‘𝑡)) = (𝐸 · ((𝑋‘0)‘𝑡))) |
| 73 | 36, 68, 72 | sylancr 587 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → Σ𝑖 ∈ {0} (𝐸 · ((𝑋‘𝑖)‘𝑡)) = (𝐸 · ((𝑋‘0)‘𝑡))) |
| 74 | 41, 73 | mpteq2da 5240 |
. . . . . 6
⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ {0} (𝐸 · ((𝑋‘𝑖)‘𝑡))) = (𝑡 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘0)‘𝑡)))) |
| 75 | 40, 74 | eqtrid 2789 |
. . . . 5
⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...0)(𝐸 · ((𝑋‘𝑖)‘𝑡))) = (𝑡 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘0)‘𝑡)))) |
| 76 | | stoweidlem17.5 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴) |
| 77 | | stoweidlem17.6 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴) |
| 78 | 41, 76, 77, 62, 42, 59 | stoweidlem2 46017 |
. . . . 5
⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘0)‘𝑡))) ∈ 𝐴) |
| 79 | 75, 78 | eqeltrd 2841 |
. . . 4
⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...0)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴) |
| 80 | 79 | adantr 480 |
. . 3
⊢ ((𝜑 ∧ 0 ∈ (0...𝑁)) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...0)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴) |
| 81 | | eqidd 2738 |
. . . . . . . . . . . . . . 15
⊢ (𝑟 = 𝑡 → 𝐸 = 𝐸) |
| 82 | 81 | cbvmptv 5255 |
. . . . . . . . . . . . . 14
⊢ (𝑟 ∈ 𝑇 ↦ 𝐸) = (𝑡 ∈ 𝑇 ↦ 𝐸) |
| 83 | 82 | eqcomi 2746 |
. . . . . . . . . . . . 13
⊢ (𝑡 ∈ 𝑇 ↦ 𝐸) = (𝑟 ∈ 𝑇 ↦ 𝐸) |
| 84 | | simpr 484 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑡 ∈ 𝑇) |
| 85 | 83, 81, 84, 43 | fvmptd3 7039 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡) = 𝐸) |
| 86 | 85 | oveq1d 7446 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡) · ((𝑋‘(𝑚 + 1))‘𝑡)) = (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡))) |
| 87 | 41, 86 | mpteq2da 5240 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡) · ((𝑋‘(𝑚 + 1))‘𝑡))) = (𝑡 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))) |
| 88 | 87 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡) · ((𝑋‘(𝑚 + 1))‘𝑡))) = (𝑡 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))) |
| 89 | 45 | ffvelcdmda 7104 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → (𝑋‘(𝑚 + 1)) ∈ 𝐴) |
| 90 | | simpl 482 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → 𝜑) |
| 91 | | id 22 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 = 𝐸 → 𝑥 = 𝐸) |
| 92 | 91 | mpteq2dv 5244 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 = 𝐸 → (𝑡 ∈ 𝑇 ↦ 𝑥) = (𝑡 ∈ 𝑇 ↦ 𝐸)) |
| 93 | 92 | eleq1d 2826 |
. . . . . . . . . . . . . 14
⊢ (𝑥 = 𝐸 → ((𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴 ↔ (𝑡 ∈ 𝑇 ↦ 𝐸) ∈ 𝐴)) |
| 94 | 93 | imbi2d 340 |
. . . . . . . . . . . . 13
⊢ (𝑥 = 𝐸 → ((𝜑 → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴) ↔ (𝜑 → (𝑡 ∈ 𝑇 ↦ 𝐸) ∈ 𝐴))) |
| 95 | 77 | expcom 413 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ ℝ → (𝜑 → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴)) |
| 96 | 94, 95 | vtoclga 3577 |
. . . . . . . . . . . 12
⊢ (𝐸 ∈ ℝ → (𝜑 → (𝑡 ∈ 𝑇 ↦ 𝐸) ∈ 𝐴)) |
| 97 | 42, 96 | mpcom 38 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ 𝐸) ∈ 𝐴) |
| 98 | 97 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → (𝑡 ∈ 𝑇 ↦ 𝐸) ∈ 𝐴) |
| 99 | | fveq1 6905 |
. . . . . . . . . . . . . . . 16
⊢ (𝑔 = (𝑋‘(𝑚 + 1)) → (𝑔‘𝑡) = ((𝑋‘(𝑚 + 1))‘𝑡)) |
| 100 | 99 | oveq2d 7447 |
. . . . . . . . . . . . . . 15
⊢ (𝑔 = (𝑋‘(𝑚 + 1)) → (((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡) · (𝑔‘𝑡)) = (((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡) · ((𝑋‘(𝑚 + 1))‘𝑡))) |
| 101 | 100 | mpteq2dv 5244 |
. . . . . . . . . . . . . 14
⊢ (𝑔 = (𝑋‘(𝑚 + 1)) → (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡) · (𝑔‘𝑡))) = (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡) · ((𝑋‘(𝑚 + 1))‘𝑡)))) |
| 102 | 101 | eleq1d 2826 |
. . . . . . . . . . . . 13
⊢ (𝑔 = (𝑋‘(𝑚 + 1)) → ((𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴 ↔ (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡) · ((𝑋‘(𝑚 + 1))‘𝑡))) ∈ 𝐴)) |
| 103 | 102 | imbi2d 340 |
. . . . . . . . . . . 12
⊢ (𝑔 = (𝑋‘(𝑚 + 1)) → (((𝜑 ∧ (𝑡 ∈ 𝑇 ↦ 𝐸) ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴) ↔ ((𝜑 ∧ (𝑡 ∈ 𝑇 ↦ 𝐸) ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡) · ((𝑋‘(𝑚 + 1))‘𝑡))) ∈ 𝐴))) |
| 104 | 82 | eleq1i 2832 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑟 ∈ 𝑇 ↦ 𝐸) ∈ 𝐴 ↔ (𝑡 ∈ 𝑇 ↦ 𝐸) ∈ 𝐴) |
| 105 | | fveq1 6905 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑓 = (𝑟 ∈ 𝑇 ↦ 𝐸) → (𝑓‘𝑡) = ((𝑟 ∈ 𝑇 ↦ 𝐸)‘𝑡)) |
| 106 | 82 | fveq1i 6907 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑟 ∈ 𝑇 ↦ 𝐸)‘𝑡) = ((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡) |
| 107 | 105, 106 | eqtrdi 2793 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑓 = (𝑟 ∈ 𝑇 ↦ 𝐸) → (𝑓‘𝑡) = ((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡)) |
| 108 | 107 | oveq1d 7446 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑓 = (𝑟 ∈ 𝑇 ↦ 𝐸) → ((𝑓‘𝑡) · (𝑔‘𝑡)) = (((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡) · (𝑔‘𝑡))) |
| 109 | 108 | mpteq2dv 5244 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑓 = (𝑟 ∈ 𝑇 ↦ 𝐸) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) = (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡) · (𝑔‘𝑡)))) |
| 110 | 109 | eleq1d 2826 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑓 = (𝑟 ∈ 𝑇 ↦ 𝐸) → ((𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴 ↔ (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴)) |
| 111 | 110 | imbi2d 340 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑓 = (𝑟 ∈ 𝑇 ↦ 𝐸) → (((𝜑 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴) ↔ ((𝜑 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴))) |
| 112 | 76 | 3com12 1124 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑓 ∈ 𝐴 ∧ 𝜑 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴) |
| 113 | 112 | 3expib 1123 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑓 ∈ 𝐴 → ((𝜑 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴)) |
| 114 | 111, 113 | vtoclga 3577 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑟 ∈ 𝑇 ↦ 𝐸) ∈ 𝐴 → ((𝜑 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴)) |
| 115 | 104, 114 | sylbir 235 |
. . . . . . . . . . . . . . 15
⊢ ((𝑡 ∈ 𝑇 ↦ 𝐸) ∈ 𝐴 → ((𝜑 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴)) |
| 116 | 115 | 3impib 1117 |
. . . . . . . . . . . . . 14
⊢ (((𝑡 ∈ 𝑇 ↦ 𝐸) ∈ 𝐴 ∧ 𝜑 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴) |
| 117 | 116 | 3com13 1125 |
. . . . . . . . . . . . 13
⊢ ((𝑔 ∈ 𝐴 ∧ 𝜑 ∧ (𝑡 ∈ 𝑇 ↦ 𝐸) ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴) |
| 118 | 117 | 3expib 1123 |
. . . . . . . . . . . 12
⊢ (𝑔 ∈ 𝐴 → ((𝜑 ∧ (𝑡 ∈ 𝑇 ↦ 𝐸) ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴)) |
| 119 | 103, 118 | vtoclga 3577 |
. . . . . . . . . . 11
⊢ ((𝑋‘(𝑚 + 1)) ∈ 𝐴 → ((𝜑 ∧ (𝑡 ∈ 𝑇 ↦ 𝐸) ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡) · ((𝑋‘(𝑚 + 1))‘𝑡))) ∈ 𝐴)) |
| 120 | 119 | 3impib 1117 |
. . . . . . . . . 10
⊢ (((𝑋‘(𝑚 + 1)) ∈ 𝐴 ∧ 𝜑 ∧ (𝑡 ∈ 𝑇 ↦ 𝐸) ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡) · ((𝑋‘(𝑚 + 1))‘𝑡))) ∈ 𝐴) |
| 121 | 89, 90, 98, 120 | syl3anc 1373 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ 𝐸)‘𝑡) · ((𝑋‘(𝑚 + 1))‘𝑡))) ∈ 𝐴) |
| 122 | 88, 121 | eqeltrrd 2842 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → (𝑡 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡))) ∈ 𝐴) |
| 123 | 122 | ad2antll 729 |
. . . . . . 7
⊢ (((𝑚 ∈ ℕ0
→ ((𝜑 ∧ 𝑚 ∈ (0...𝑁)) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴)) ∧ (𝑚 ∈ ℕ0 ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)))) → (𝑡 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡))) ∈ 𝐴) |
| 124 | | simprrl 781 |
. . . . . . 7
⊢ (((𝑚 ∈ ℕ0
→ ((𝜑 ∧ 𝑚 ∈ (0...𝑁)) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴)) ∧ (𝑚 ∈ ℕ0 ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)))) → 𝜑) |
| 125 | | simpl 482 |
. . . . . . . . . 10
⊢ ((𝑚 ∈ ℕ0
∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → 𝑚 ∈ ℕ0) |
| 126 | | simprl 771 |
. . . . . . . . . 10
⊢ ((𝑚 ∈ ℕ0
∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → 𝜑) |
| 127 | 1 | ad2antrl 728 |
. . . . . . . . . . . 12
⊢ ((𝑚 ∈ ℕ0
∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → 𝑁 ∈ ℕ) |
| 128 | 127 | nnnn0d 12587 |
. . . . . . . . . . 11
⊢ ((𝑚 ∈ ℕ0
∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → 𝑁 ∈
ℕ0) |
| 129 | | nn0re 12535 |
. . . . . . . . . . . . 13
⊢ (𝑚 ∈ ℕ0
→ 𝑚 ∈
ℝ) |
| 130 | 129 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝑚 ∈ ℕ0
∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → 𝑚 ∈ ℝ) |
| 131 | | peano2nn0 12566 |
. . . . . . . . . . . . . 14
⊢ (𝑚 ∈ ℕ0
→ (𝑚 + 1) ∈
ℕ0) |
| 132 | 131 | nn0red 12588 |
. . . . . . . . . . . . 13
⊢ (𝑚 ∈ ℕ0
→ (𝑚 + 1) ∈
ℝ) |
| 133 | 132 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝑚 ∈ ℕ0
∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → (𝑚 + 1) ∈ ℝ) |
| 134 | 1 | nnred 12281 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑁 ∈ ℝ) |
| 135 | 134 | ad2antrl 728 |
. . . . . . . . . . . 12
⊢ ((𝑚 ∈ ℕ0
∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → 𝑁 ∈ ℝ) |
| 136 | | lep1 12108 |
. . . . . . . . . . . . 13
⊢ (𝑚 ∈ ℝ → 𝑚 ≤ (𝑚 + 1)) |
| 137 | 125, 129,
136 | 3syl 18 |
. . . . . . . . . . . 12
⊢ ((𝑚 ∈ ℕ0
∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → 𝑚 ≤ (𝑚 + 1)) |
| 138 | | elfzle2 13568 |
. . . . . . . . . . . . 13
⊢ ((𝑚 + 1) ∈ (0...𝑁) → (𝑚 + 1) ≤ 𝑁) |
| 139 | 138 | ad2antll 729 |
. . . . . . . . . . . 12
⊢ ((𝑚 ∈ ℕ0
∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → (𝑚 + 1) ≤ 𝑁) |
| 140 | 130, 133,
135, 137, 139 | letrd 11418 |
. . . . . . . . . . 11
⊢ ((𝑚 ∈ ℕ0
∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → 𝑚 ≤ 𝑁) |
| 141 | | elfz2nn0 13658 |
. . . . . . . . . . 11
⊢ (𝑚 ∈ (0...𝑁) ↔ (𝑚 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0
∧ 𝑚 ≤ 𝑁)) |
| 142 | 125, 128,
140, 141 | syl3anbrc 1344 |
. . . . . . . . . 10
⊢ ((𝑚 ∈ ℕ0
∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → 𝑚 ∈ (0...𝑁)) |
| 143 | 125, 126,
142 | jca32 515 |
. . . . . . . . 9
⊢ ((𝑚 ∈ ℕ0
∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → (𝑚 ∈ ℕ0 ∧ (𝜑 ∧ 𝑚 ∈ (0...𝑁)))) |
| 144 | 143 | adantl 481 |
. . . . . . . 8
⊢ (((𝑚 ∈ ℕ0
→ ((𝜑 ∧ 𝑚 ∈ (0...𝑁)) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴)) ∧ (𝑚 ∈ ℕ0 ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)))) → (𝑚 ∈ ℕ0 ∧ (𝜑 ∧ 𝑚 ∈ (0...𝑁)))) |
| 145 | | pm3.31 449 |
. . . . . . . . 9
⊢ ((𝑚 ∈ ℕ0
→ ((𝜑 ∧ 𝑚 ∈ (0...𝑁)) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴)) → ((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ 𝑚 ∈ (0...𝑁))) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴)) |
| 146 | 145 | adantr 480 |
. . . . . . . 8
⊢ (((𝑚 ∈ ℕ0
→ ((𝜑 ∧ 𝑚 ∈ (0...𝑁)) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴)) ∧ (𝑚 ∈ ℕ0 ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)))) → ((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ 𝑚 ∈ (0...𝑁))) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴)) |
| 147 | 144, 146 | mpd 15 |
. . . . . . 7
⊢ (((𝑚 ∈ ℕ0
→ ((𝜑 ∧ 𝑚 ∈ (0...𝑁)) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴)) ∧ (𝑚 ∈ ℕ0 ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)))) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴) |
| 148 | | fveq2 6906 |
. . . . . . . . . . . 12
⊢ (𝑟 = 𝑡 → ((𝑋‘(𝑚 + 1))‘𝑟) = ((𝑋‘(𝑚 + 1))‘𝑡)) |
| 149 | 148 | oveq2d 7447 |
. . . . . . . . . . 11
⊢ (𝑟 = 𝑡 → (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑟)) = (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡))) |
| 150 | 149 | cbvmptv 5255 |
. . . . . . . . . 10
⊢ (𝑟 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑟))) = (𝑡 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡))) |
| 151 | 150 | eleq1i 2832 |
. . . . . . . . 9
⊢ ((𝑟 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑟))) ∈ 𝐴 ↔ (𝑡 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡))) ∈ 𝐴) |
| 152 | | fveq1 6905 |
. . . . . . . . . . . . . . 15
⊢ (𝑔 = (𝑟 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑟))) → (𝑔‘𝑡) = ((𝑟 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑟)))‘𝑡)) |
| 153 | 150 | fveq1i 6907 |
. . . . . . . . . . . . . . 15
⊢ ((𝑟 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑟)))‘𝑡) = ((𝑡 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))‘𝑡) |
| 154 | 152, 153 | eqtrdi 2793 |
. . . . . . . . . . . . . 14
⊢ (𝑔 = (𝑟 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑟))) → (𝑔‘𝑡) = ((𝑡 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))‘𝑡)) |
| 155 | 154 | oveq2d 7447 |
. . . . . . . . . . . . 13
⊢ (𝑔 = (𝑟 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑟))) → (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)))‘𝑡) + (𝑔‘𝑡)) = (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)))‘𝑡) + ((𝑡 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))‘𝑡))) |
| 156 | 155 | mpteq2dv 5244 |
. . . . . . . . . . . 12
⊢ (𝑔 = (𝑟 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑟))) → (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)))‘𝑡) + (𝑔‘𝑡))) = (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)))‘𝑡) + ((𝑡 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))‘𝑡)))) |
| 157 | 156 | eleq1d 2826 |
. . . . . . . . . . 11
⊢ (𝑔 = (𝑟 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑟))) → ((𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)))‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴 ↔ (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)))‘𝑡) + ((𝑡 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))‘𝑡))) ∈ 𝐴)) |
| 158 | 157 | imbi2d 340 |
. . . . . . . . . 10
⊢ (𝑔 = (𝑟 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑟))) → (((𝜑 ∧ (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)))‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴) ↔ ((𝜑 ∧ (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)))‘𝑡) + ((𝑡 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))‘𝑡))) ∈ 𝐴))) |
| 159 | | fveq2 6906 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑟 = 𝑡 → ((𝑋‘𝑖)‘𝑟) = ((𝑋‘𝑖)‘𝑡)) |
| 160 | 159 | oveq2d 7447 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑟 = 𝑡 → (𝐸 · ((𝑋‘𝑖)‘𝑟)) = (𝐸 · ((𝑋‘𝑖)‘𝑡))) |
| 161 | 160 | sumeq2sdv 15739 |
. . . . . . . . . . . . . . . 16
⊢ (𝑟 = 𝑡 → Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑟)) = Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡))) |
| 162 | 161 | cbvmptv 5255 |
. . . . . . . . . . . . . . 15
⊢ (𝑟 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑟))) = (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡))) |
| 163 | 162 | eleq1i 2832 |
. . . . . . . . . . . . . 14
⊢ ((𝑟 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑟))) ∈ 𝐴 ↔ (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴) |
| 164 | | fveq1 6905 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑓 = (𝑟 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑟))) → (𝑓‘𝑡) = ((𝑟 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑟)))‘𝑡)) |
| 165 | 162 | fveq1i 6907 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑟 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑟)))‘𝑡) = ((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)))‘𝑡) |
| 166 | 164, 165 | eqtrdi 2793 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑓 = (𝑟 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑟))) → (𝑓‘𝑡) = ((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)))‘𝑡)) |
| 167 | 166 | oveq1d 7446 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑓 = (𝑟 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑟))) → ((𝑓‘𝑡) + (𝑔‘𝑡)) = (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)))‘𝑡) + (𝑔‘𝑡))) |
| 168 | 167 | mpteq2dv 5244 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑓 = (𝑟 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑟))) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) = (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)))‘𝑡) + (𝑔‘𝑡)))) |
| 169 | 168 | eleq1d 2826 |
. . . . . . . . . . . . . . . 16
⊢ (𝑓 = (𝑟 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑟))) → ((𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴 ↔ (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)))‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴)) |
| 170 | 169 | imbi2d 340 |
. . . . . . . . . . . . . . 15
⊢ (𝑓 = (𝑟 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑟))) → (((𝜑 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴) ↔ ((𝜑 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)))‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴))) |
| 171 | | stoweidlem17.4 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴) |
| 172 | 171 | 3com12 1124 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑓 ∈ 𝐴 ∧ 𝜑 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴) |
| 173 | 172 | 3expib 1123 |
. . . . . . . . . . . . . . 15
⊢ (𝑓 ∈ 𝐴 → ((𝜑 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴)) |
| 174 | 170, 173 | vtoclga 3577 |
. . . . . . . . . . . . . 14
⊢ ((𝑟 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑟))) ∈ 𝐴 → ((𝜑 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)))‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴)) |
| 175 | 163, 174 | sylbir 235 |
. . . . . . . . . . . . 13
⊢ ((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴 → ((𝜑 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)))‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴)) |
| 176 | 175 | 3impib 1117 |
. . . . . . . . . . . 12
⊢ (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴 ∧ 𝜑 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)))‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴) |
| 177 | 176 | 3com13 1125 |
. . . . . . . . . . 11
⊢ ((𝑔 ∈ 𝐴 ∧ 𝜑 ∧ (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)))‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴) |
| 178 | 177 | 3expib 1123 |
. . . . . . . . . 10
⊢ (𝑔 ∈ 𝐴 → ((𝜑 ∧ (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)))‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴)) |
| 179 | 158, 178 | vtoclga 3577 |
. . . . . . . . 9
⊢ ((𝑟 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑟))) ∈ 𝐴 → ((𝜑 ∧ (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)))‘𝑡) + ((𝑡 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))‘𝑡))) ∈ 𝐴)) |
| 180 | 151, 179 | sylbir 235 |
. . . . . . . 8
⊢ ((𝑡 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡))) ∈ 𝐴 → ((𝜑 ∧ (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)))‘𝑡) + ((𝑡 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))‘𝑡))) ∈ 𝐴)) |
| 181 | 180 | 3impib 1117 |
. . . . . . 7
⊢ (((𝑡 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡))) ∈ 𝐴 ∧ 𝜑 ∧ (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)))‘𝑡) + ((𝑡 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))‘𝑡))) ∈ 𝐴) |
| 182 | 123, 124,
147, 181 | syl3anc 1373 |
. . . . . 6
⊢ (((𝑚 ∈ ℕ0
→ ((𝜑 ∧ 𝑚 ∈ (0...𝑁)) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴)) ∧ (𝑚 ∈ ℕ0 ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)))) → (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)))‘𝑡) + ((𝑡 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))‘𝑡))) ∈ 𝐴) |
| 183 | | 3anass 1095 |
. . . . . . . . 9
⊢ ((𝑚 ∈ ℕ0
∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ↔ (𝑚 ∈ ℕ0 ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)))) |
| 184 | 183 | biimpri 228 |
. . . . . . . 8
⊢ ((𝑚 ∈ ℕ0
∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) → (𝑚 ∈ ℕ0 ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) |
| 185 | 184 | adantl 481 |
. . . . . . 7
⊢ (((𝑚 ∈ ℕ0
→ ((𝜑 ∧ 𝑚 ∈ (0...𝑁)) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴)) ∧ (𝑚 ∈ ℕ0 ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)))) → (𝑚 ∈ ℕ0 ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) |
| 186 | | nfv 1914 |
. . . . . . . . . 10
⊢
Ⅎ𝑡 𝑚 ∈
ℕ0 |
| 187 | | nfv 1914 |
. . . . . . . . . 10
⊢
Ⅎ𝑡(𝑚 + 1) ∈ (0...𝑁) |
| 188 | 186, 41, 187 | nf3an 1901 |
. . . . . . . . 9
⊢
Ⅎ𝑡(𝑚 ∈ ℕ0
∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) |
| 189 | | simpr 484 |
. . . . . . . . . . . 12
⊢ (((𝑚 ∈ ℕ0
∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) → 𝑡 ∈ 𝑇) |
| 190 | | fzfid 14014 |
. . . . . . . . . . . . 13
⊢ (((𝑚 ∈ ℕ0
∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) → (0...𝑚) ∈ Fin) |
| 191 | 42 | 3ad2ant2 1135 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑚 ∈ ℕ0
∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → 𝐸 ∈ ℝ) |
| 192 | 191 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ (((𝑚 ∈ ℕ0
∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) → 𝐸 ∈ ℝ) |
| 193 | 192 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((((𝑚 ∈ ℕ0
∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (0...𝑚)) → 𝐸 ∈ ℝ) |
| 194 | | fzelp1 13616 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑖 ∈ (0...𝑚) → 𝑖 ∈ (0...(𝑚 + 1))) |
| 195 | 194 | anim2i 617 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝑚 ∈ ℕ0
∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (0...𝑚)) → (((𝑚 ∈ ℕ0 ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (0...(𝑚 + 1)))) |
| 196 | | an32 646 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝑚 ∈ ℕ0
∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (0...(𝑚 + 1))) ↔ (((𝑚 ∈ ℕ0 ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑖 ∈ (0...(𝑚 + 1))) ∧ 𝑡 ∈ 𝑇)) |
| 197 | 195, 196 | sylib 218 |
. . . . . . . . . . . . . . 15
⊢ ((((𝑚 ∈ ℕ0
∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (0...𝑚)) → (((𝑚 ∈ ℕ0 ∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑖 ∈ (0...(𝑚 + 1))) ∧ 𝑡 ∈ 𝑇)) |
| 198 | 45 | 3ad2ant2 1135 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑚 ∈ ℕ0
∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → 𝑋:(0...𝑁)⟶𝐴) |
| 199 | 198 | adantr 480 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑚 ∈ ℕ0
∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑖 ∈ (0...(𝑚 + 1))) → 𝑋:(0...𝑁)⟶𝐴) |
| 200 | | elfzuz3 13561 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑚 + 1) ∈ (0...𝑁) → 𝑁 ∈ (ℤ≥‘(𝑚 + 1))) |
| 201 | | fzss2 13604 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑁 ∈
(ℤ≥‘(𝑚 + 1)) → (0...(𝑚 + 1)) ⊆ (0...𝑁)) |
| 202 | 200, 201 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑚 + 1) ∈ (0...𝑁) → (0...(𝑚 + 1)) ⊆ (0...𝑁)) |
| 203 | 202 | sselda 3983 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑚 + 1) ∈ (0...𝑁) ∧ 𝑖 ∈ (0...(𝑚 + 1))) → 𝑖 ∈ (0...𝑁)) |
| 204 | 203 | 3ad2antl3 1188 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑚 ∈ ℕ0
∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑖 ∈ (0...(𝑚 + 1))) → 𝑖 ∈ (0...𝑁)) |
| 205 | 199, 204 | ffvelcdmd 7105 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑚 ∈ ℕ0
∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑖 ∈ (0...(𝑚 + 1))) → (𝑋‘𝑖) ∈ 𝐴) |
| 206 | | simpl2 1193 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑚 ∈ ℕ0
∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑖 ∈ (0...(𝑚 + 1))) → 𝜑) |
| 207 | | feq1 6716 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑓 = (𝑋‘𝑖) → (𝑓:𝑇⟶ℝ ↔ (𝑋‘𝑖):𝑇⟶ℝ)) |
| 208 | 207 | imbi2d 340 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑓 = (𝑋‘𝑖) → ((𝜑 → 𝑓:𝑇⟶ℝ) ↔ (𝜑 → (𝑋‘𝑖):𝑇⟶ℝ))) |
| 209 | 208, 63 | vtoclga 3577 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑋‘𝑖) ∈ 𝐴 → (𝜑 → (𝑋‘𝑖):𝑇⟶ℝ)) |
| 210 | 205, 206,
209 | sylc 65 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑚 ∈ ℕ0
∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑖 ∈ (0...(𝑚 + 1))) → (𝑋‘𝑖):𝑇⟶ℝ) |
| 211 | 210 | ffvelcdmda 7104 |
. . . . . . . . . . . . . . 15
⊢ ((((𝑚 ∈ ℕ0
∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑖 ∈ (0...(𝑚 + 1))) ∧ 𝑡 ∈ 𝑇) → ((𝑋‘𝑖)‘𝑡) ∈ ℝ) |
| 212 | 197, 211 | syl 17 |
. . . . . . . . . . . . . 14
⊢ ((((𝑚 ∈ ℕ0
∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (0...𝑚)) → ((𝑋‘𝑖)‘𝑡) ∈ ℝ) |
| 213 | 193, 212 | remulcld 11291 |
. . . . . . . . . . . . 13
⊢ ((((𝑚 ∈ ℕ0
∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (0...𝑚)) → (𝐸 · ((𝑋‘𝑖)‘𝑡)) ∈ ℝ) |
| 214 | 190, 213 | fsumrecl 15770 |
. . . . . . . . . . . 12
⊢ (((𝑚 ∈ ℕ0
∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) → Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)) ∈ ℝ) |
| 215 | | eqid 2737 |
. . . . . . . . . . . . 13
⊢ (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡))) = (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡))) |
| 216 | 215 | fvmpt2 7027 |
. . . . . . . . . . . 12
⊢ ((𝑡 ∈ 𝑇 ∧ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)) ∈ ℝ) → ((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)))‘𝑡) = Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡))) |
| 217 | 189, 214,
216 | syl2anc 584 |
. . . . . . . . . . 11
⊢ (((𝑚 ∈ ℕ0
∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) → ((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)))‘𝑡) = Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡))) |
| 218 | 217 | oveq1d 7446 |
. . . . . . . . . 10
⊢ (((𝑚 ∈ ℕ0
∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) → (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)))‘𝑡) + (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡))) = (Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)) + (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))) |
| 219 | | 3simpc 1151 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑚 ∈ ℕ0
∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) |
| 220 | 219 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ (((𝑚 ∈ ℕ0
∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) → (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁))) |
| 221 | | feq1 6716 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑓 = (𝑋‘(𝑚 + 1)) → (𝑓:𝑇⟶ℝ ↔ (𝑋‘(𝑚 + 1)):𝑇⟶ℝ)) |
| 222 | 221 | imbi2d 340 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑓 = (𝑋‘(𝑚 + 1)) → ((𝜑 → 𝑓:𝑇⟶ℝ) ↔ (𝜑 → (𝑋‘(𝑚 + 1)):𝑇⟶ℝ))) |
| 223 | 222, 63 | vtoclga 3577 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑋‘(𝑚 + 1)) ∈ 𝐴 → (𝜑 → (𝑋‘(𝑚 + 1)):𝑇⟶ℝ)) |
| 224 | 89, 90, 223 | sylc 65 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → (𝑋‘(𝑚 + 1)):𝑇⟶ℝ) |
| 225 | 220, 224 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (((𝑚 ∈ ℕ0
∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) → (𝑋‘(𝑚 + 1)):𝑇⟶ℝ) |
| 226 | 225, 189 | ffvelcdmd 7105 |
. . . . . . . . . . . . 13
⊢ (((𝑚 ∈ ℕ0
∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) → ((𝑋‘(𝑚 + 1))‘𝑡) ∈ ℝ) |
| 227 | 192, 226 | remulcld 11291 |
. . . . . . . . . . . 12
⊢ (((𝑚 ∈ ℕ0
∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) → (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)) ∈ ℝ) |
| 228 | | eqid 2737 |
. . . . . . . . . . . . 13
⊢ (𝑡 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡))) = (𝑡 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡))) |
| 229 | 228 | fvmpt2 7027 |
. . . . . . . . . . . 12
⊢ ((𝑡 ∈ 𝑇 ∧ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)) ∈ ℝ) → ((𝑡 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))‘𝑡) = (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡))) |
| 230 | 189, 227,
229 | syl2anc 584 |
. . . . . . . . . . 11
⊢ (((𝑚 ∈ ℕ0
∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) → ((𝑡 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))‘𝑡) = (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡))) |
| 231 | 230 | oveq2d 7447 |
. . . . . . . . . 10
⊢ (((𝑚 ∈ ℕ0
∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) → (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)))‘𝑡) + ((𝑡 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))‘𝑡)) = (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)))‘𝑡) + (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))) |
| 232 | | elfzuz 13560 |
. . . . . . . . . . . . . 14
⊢ ((𝑚 + 1) ∈ (0...𝑁) → (𝑚 + 1) ∈
(ℤ≥‘0)) |
| 233 | 232 | 3ad2ant3 1136 |
. . . . . . . . . . . . 13
⊢ ((𝑚 ∈ ℕ0
∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → (𝑚 + 1) ∈
(ℤ≥‘0)) |
| 234 | 233 | adantr 480 |
. . . . . . . . . . . 12
⊢ (((𝑚 ∈ ℕ0
∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) → (𝑚 + 1) ∈
(ℤ≥‘0)) |
| 235 | 192 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((((𝑚 ∈ ℕ0
∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (0...(𝑚 + 1))) → 𝐸 ∈ ℝ) |
| 236 | 211 | an32s 652 |
. . . . . . . . . . . . 13
⊢ ((((𝑚 ∈ ℕ0
∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (0...(𝑚 + 1))) → ((𝑋‘𝑖)‘𝑡) ∈ ℝ) |
| 237 | | remulcl 11240 |
. . . . . . . . . . . . . 14
⊢ ((𝐸 ∈ ℝ ∧ ((𝑋‘𝑖)‘𝑡) ∈ ℝ) → (𝐸 · ((𝑋‘𝑖)‘𝑡)) ∈ ℝ) |
| 238 | 237 | recnd 11289 |
. . . . . . . . . . . . 13
⊢ ((𝐸 ∈ ℝ ∧ ((𝑋‘𝑖)‘𝑡) ∈ ℝ) → (𝐸 · ((𝑋‘𝑖)‘𝑡)) ∈ ℂ) |
| 239 | 235, 236,
238 | syl2anc 584 |
. . . . . . . . . . . 12
⊢ ((((𝑚 ∈ ℕ0
∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (0...(𝑚 + 1))) → (𝐸 · ((𝑋‘𝑖)‘𝑡)) ∈ ℂ) |
| 240 | | fveq2 6906 |
. . . . . . . . . . . . . 14
⊢ (𝑖 = (𝑚 + 1) → (𝑋‘𝑖) = (𝑋‘(𝑚 + 1))) |
| 241 | 240 | fveq1d 6908 |
. . . . . . . . . . . . 13
⊢ (𝑖 = (𝑚 + 1) → ((𝑋‘𝑖)‘𝑡) = ((𝑋‘(𝑚 + 1))‘𝑡)) |
| 242 | 241 | oveq2d 7447 |
. . . . . . . . . . . 12
⊢ (𝑖 = (𝑚 + 1) → (𝐸 · ((𝑋‘𝑖)‘𝑡)) = (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡))) |
| 243 | 234, 239,
242 | fsumm1 15787 |
. . . . . . . . . . 11
⊢ (((𝑚 ∈ ℕ0
∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) → Σ𝑖 ∈ (0...(𝑚 + 1))(𝐸 · ((𝑋‘𝑖)‘𝑡)) = (Σ𝑖 ∈ (0...((𝑚 + 1) − 1))(𝐸 · ((𝑋‘𝑖)‘𝑡)) + (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))) |
| 244 | | nn0cn 12536 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑚 ∈ ℕ0
→ 𝑚 ∈
ℂ) |
| 245 | 244 | 3ad2ant1 1134 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑚 ∈ ℕ0
∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → 𝑚 ∈ ℂ) |
| 246 | 245 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ (((𝑚 ∈ ℕ0
∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) → 𝑚 ∈ ℂ) |
| 247 | | 1cnd 11256 |
. . . . . . . . . . . . . . 15
⊢ (((𝑚 ∈ ℕ0
∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) → 1 ∈ ℂ) |
| 248 | 246, 247 | pncand 11621 |
. . . . . . . . . . . . . 14
⊢ (((𝑚 ∈ ℕ0
∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) → ((𝑚 + 1) − 1) = 𝑚) |
| 249 | 248 | oveq2d 7447 |
. . . . . . . . . . . . 13
⊢ (((𝑚 ∈ ℕ0
∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) → (0...((𝑚 + 1) − 1)) = (0...𝑚)) |
| 250 | 249 | sumeq1d 15736 |
. . . . . . . . . . . 12
⊢ (((𝑚 ∈ ℕ0
∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) → Σ𝑖 ∈ (0...((𝑚 + 1) − 1))(𝐸 · ((𝑋‘𝑖)‘𝑡)) = Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡))) |
| 251 | 250 | oveq1d 7446 |
. . . . . . . . . . 11
⊢ (((𝑚 ∈ ℕ0
∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) → (Σ𝑖 ∈ (0...((𝑚 + 1) − 1))(𝐸 · ((𝑋‘𝑖)‘𝑡)) + (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡))) = (Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)) + (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))) |
| 252 | 243, 251 | eqtrd 2777 |
. . . . . . . . . 10
⊢ (((𝑚 ∈ ℕ0
∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) → Σ𝑖 ∈ (0...(𝑚 + 1))(𝐸 · ((𝑋‘𝑖)‘𝑡)) = (Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)) + (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))) |
| 253 | 218, 231,
252 | 3eqtr4rd 2788 |
. . . . . . . . 9
⊢ (((𝑚 ∈ ℕ0
∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) → Σ𝑖 ∈ (0...(𝑚 + 1))(𝐸 · ((𝑋‘𝑖)‘𝑡)) = (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)))‘𝑡) + ((𝑡 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))‘𝑡))) |
| 254 | 188, 253 | mpteq2da 5240 |
. . . . . . . 8
⊢ ((𝑚 ∈ ℕ0
∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...(𝑚 + 1))(𝐸 · ((𝑋‘𝑖)‘𝑡))) = (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)))‘𝑡) + ((𝑡 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))‘𝑡)))) |
| 255 | 254 | eleq1d 2826 |
. . . . . . 7
⊢ ((𝑚 ∈ ℕ0
∧ 𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → ((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...(𝑚 + 1))(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴 ↔ (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)))‘𝑡) + ((𝑡 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))‘𝑡))) ∈ 𝐴)) |
| 256 | 185, 255 | syl 17 |
. . . . . 6
⊢ (((𝑚 ∈ ℕ0
→ ((𝜑 ∧ 𝑚 ∈ (0...𝑁)) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴)) ∧ (𝑚 ∈ ℕ0 ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)))) → ((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...(𝑚 + 1))(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴 ↔ (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡)))‘𝑡) + ((𝑡 ∈ 𝑇 ↦ (𝐸 · ((𝑋‘(𝑚 + 1))‘𝑡)))‘𝑡))) ∈ 𝐴)) |
| 257 | 182, 256 | mpbird 257 |
. . . . 5
⊢ (((𝑚 ∈ ℕ0
→ ((𝜑 ∧ 𝑚 ∈ (0...𝑁)) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴)) ∧ (𝑚 ∈ ℕ0 ∧ (𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)))) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...(𝑚 + 1))(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴) |
| 258 | 257 | exp32 420 |
. . . 4
⊢ ((𝑚 ∈ ℕ0
→ ((𝜑 ∧ 𝑚 ∈ (0...𝑁)) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴)) → (𝑚 ∈ ℕ0 → ((𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...(𝑚 + 1))(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴))) |
| 259 | 258 | pm2.86i 110 |
. . 3
⊢ (𝑚 ∈ ℕ0
→ (((𝜑 ∧ 𝑚 ∈ (0...𝑁)) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑚)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴) → ((𝜑 ∧ (𝑚 + 1) ∈ (0...𝑁)) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...(𝑚 + 1))(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴))) |
| 260 | 14, 21, 28, 35, 80, 259 | nn0ind 12713 |
. 2
⊢ (𝑁 ∈ ℕ0
→ ((𝜑 ∧ 𝑁 ∈ (0...𝑁)) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑁)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴)) |
| 261 | 2, 7, 260 | sylc 65 |
1
⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑁)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴) |