Step | Hyp | Ref
| Expression |
1 | | stoweidlem32.2 |
. . 3
⊢ 𝑃 = (𝑡 ∈ 𝑇 ↦ (𝑌 · Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑡))) |
2 | | stoweidlem32.1 |
. . . 4
⊢
Ⅎ𝑡𝜑 |
3 | | stoweidlem32.3 |
. . . . . . . . . 10
⊢ 𝐹 = (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑡)) |
4 | | fveq2 6717 |
. . . . . . . . . . . 12
⊢ (𝑡 = 𝑠 → ((𝐺‘𝑖)‘𝑡) = ((𝐺‘𝑖)‘𝑠)) |
5 | 4 | sumeq2sdv 15268 |
. . . . . . . . . . 11
⊢ (𝑡 = 𝑠 → Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑡) = Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑠)) |
6 | 5 | cbvmptv 5158 |
. . . . . . . . . 10
⊢ (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑡)) = (𝑠 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑠)) |
7 | 3, 6 | eqtri 2765 |
. . . . . . . . 9
⊢ 𝐹 = (𝑠 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑠)) |
8 | | fveq2 6717 |
. . . . . . . . . 10
⊢ (𝑠 = 𝑡 → ((𝐺‘𝑖)‘𝑠) = ((𝐺‘𝑖)‘𝑡)) |
9 | 8 | sumeq2sdv 15268 |
. . . . . . . . 9
⊢ (𝑠 = 𝑡 → Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑠) = Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑡)) |
10 | | simpr 488 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑡 ∈ 𝑇) |
11 | | fzfid 13546 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (1...𝑀) ∈ Fin) |
12 | | simpl 486 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → 𝜑) |
13 | | stoweidlem32.7 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐺:(1...𝑀)⟶𝐴) |
14 | 13 | ffvelrnda 6904 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐺‘𝑖) ∈ 𝐴) |
15 | | eleq1 2825 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑓 = (𝐺‘𝑖) → (𝑓 ∈ 𝐴 ↔ (𝐺‘𝑖) ∈ 𝐴)) |
16 | 15 | anbi2d 632 |
. . . . . . . . . . . . . . . 16
⊢ (𝑓 = (𝐺‘𝑖) → ((𝜑 ∧ 𝑓 ∈ 𝐴) ↔ (𝜑 ∧ (𝐺‘𝑖) ∈ 𝐴))) |
17 | | feq1 6526 |
. . . . . . . . . . . . . . . 16
⊢ (𝑓 = (𝐺‘𝑖) → (𝑓:𝑇⟶ℝ ↔ (𝐺‘𝑖):𝑇⟶ℝ)) |
18 | 16, 17 | imbi12d 348 |
. . . . . . . . . . . . . . 15
⊢ (𝑓 = (𝐺‘𝑖) → (((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ) ↔ ((𝜑 ∧ (𝐺‘𝑖) ∈ 𝐴) → (𝐺‘𝑖):𝑇⟶ℝ))) |
19 | | stoweidlem32.11 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ) |
20 | 18, 19 | vtoclg 3481 |
. . . . . . . . . . . . . 14
⊢ ((𝐺‘𝑖) ∈ 𝐴 → ((𝜑 ∧ (𝐺‘𝑖) ∈ 𝐴) → (𝐺‘𝑖):𝑇⟶ℝ)) |
21 | 14, 20 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ((𝜑 ∧ (𝐺‘𝑖) ∈ 𝐴) → (𝐺‘𝑖):𝑇⟶ℝ)) |
22 | 12, 14, 21 | mp2and 699 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐺‘𝑖):𝑇⟶ℝ) |
23 | 22 | adantlr 715 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (1...𝑀)) → (𝐺‘𝑖):𝑇⟶ℝ) |
24 | | simplr 769 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (1...𝑀)) → 𝑡 ∈ 𝑇) |
25 | 23, 24 | ffvelrnd 6905 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (1...𝑀)) → ((𝐺‘𝑖)‘𝑡) ∈ ℝ) |
26 | 11, 25 | fsumrecl 15298 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑡) ∈ ℝ) |
27 | 7, 9, 10, 26 | fvmptd3 6841 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) = Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑡)) |
28 | 27, 26 | eqeltrd 2838 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) ∈ ℝ) |
29 | 28 | recnd 10861 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) ∈ ℂ) |
30 | | stoweidlem32.4 |
. . . . . . . . . 10
⊢ 𝐻 = (𝑡 ∈ 𝑇 ↦ 𝑌) |
31 | | eqidd 2738 |
. . . . . . . . . . 11
⊢ (𝑠 = 𝑡 → 𝑌 = 𝑌) |
32 | 31 | cbvmptv 5158 |
. . . . . . . . . 10
⊢ (𝑠 ∈ 𝑇 ↦ 𝑌) = (𝑡 ∈ 𝑇 ↦ 𝑌) |
33 | 30, 32 | eqtr4i 2768 |
. . . . . . . . 9
⊢ 𝐻 = (𝑠 ∈ 𝑇 ↦ 𝑌) |
34 | | stoweidlem32.6 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑌 ∈ ℝ) |
35 | 34 | adantr 484 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑌 ∈ ℝ) |
36 | 33, 31, 10, 35 | fvmptd3 6841 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐻‘𝑡) = 𝑌) |
37 | 36, 35 | eqeltrd 2838 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐻‘𝑡) ∈ ℝ) |
38 | 37 | recnd 10861 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐻‘𝑡) ∈ ℂ) |
39 | 29, 38 | mulcomd 10854 |
. . . . 5
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐹‘𝑡) · (𝐻‘𝑡)) = ((𝐻‘𝑡) · (𝐹‘𝑡))) |
40 | 36, 27 | oveq12d 7231 |
. . . . 5
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐻‘𝑡) · (𝐹‘𝑡)) = (𝑌 · Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑡))) |
41 | 39, 40 | eqtr2d 2778 |
. . . 4
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝑌 · Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑡)) = ((𝐹‘𝑡) · (𝐻‘𝑡))) |
42 | 2, 41 | mpteq2da 5149 |
. . 3
⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ (𝑌 · Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑡))) = (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) · (𝐻‘𝑡)))) |
43 | 1, 42 | syl5eq 2790 |
. 2
⊢ (𝜑 → 𝑃 = (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) · (𝐻‘𝑡)))) |
44 | | stoweidlem32.5 |
. . . 4
⊢ (𝜑 → 𝑀 ∈ ℕ) |
45 | | stoweidlem32.8 |
. . . 4
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴) |
46 | 2, 3, 44, 13, 45, 19 | stoweidlem20 43236 |
. . 3
⊢ (𝜑 → 𝐹 ∈ 𝐴) |
47 | | stoweidlem32.10 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴) |
48 | 47 | stoweidlem4 43220 |
. . . . 5
⊢ ((𝜑 ∧ 𝑌 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑌) ∈ 𝐴) |
49 | 34, 48 | mpdan 687 |
. . . 4
⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ 𝑌) ∈ 𝐴) |
50 | 30, 49 | eqeltrid 2842 |
. . 3
⊢ (𝜑 → 𝐻 ∈ 𝐴) |
51 | | nfmpt1 5153 |
. . . . . 6
⊢
Ⅎ𝑡(𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑡)) |
52 | 3, 51 | nfcxfr 2902 |
. . . . 5
⊢
Ⅎ𝑡𝐹 |
53 | 52 | nfeq2 2921 |
. . . 4
⊢
Ⅎ𝑡 𝑓 = 𝐹 |
54 | | nfmpt1 5153 |
. . . . . 6
⊢
Ⅎ𝑡(𝑡 ∈ 𝑇 ↦ 𝑌) |
55 | 30, 54 | nfcxfr 2902 |
. . . . 5
⊢
Ⅎ𝑡𝐻 |
56 | 55 | nfeq2 2921 |
. . . 4
⊢
Ⅎ𝑡 𝑔 = 𝐻 |
57 | | stoweidlem32.9 |
. . . 4
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴) |
58 | 53, 56, 57 | stoweidlem6 43222 |
. . 3
⊢ ((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐻 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) · (𝐻‘𝑡))) ∈ 𝐴) |
59 | 46, 50, 58 | mpd3an23 1465 |
. 2
⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) · (𝐻‘𝑡))) ∈ 𝐴) |
60 | 43, 59 | eqeltrd 2838 |
1
⊢ (𝜑 → 𝑃 ∈ 𝐴) |