Proof of Theorem ax5seglem2
Step | Hyp | Ref
| Expression |
1 | | simpl2l 1114 |
. . . . 5
⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝑇 ∈ (0[,]1) ∧ ∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))))) ∧ 𝑗 ∈ (1...𝑁)) → 𝐴 ∈ (𝔼‘𝑁)) |
2 | | fveecn 25782 |
. . . . 5
⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝑗 ∈ (1...𝑁)) → (𝐴‘𝑗) ∈ ℂ) |
3 | 1, 2 | sylancom 701 |
. . . 4
⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝑇 ∈ (0[,]1) ∧ ∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))))) ∧ 𝑗 ∈ (1...𝑁)) → (𝐴‘𝑗) ∈ ℂ) |
4 | | simpl2r 1115 |
. . . . 5
⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝑇 ∈ (0[,]1) ∧ ∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))))) ∧ 𝑗 ∈ (1...𝑁)) → 𝐶 ∈ (𝔼‘𝑁)) |
5 | | fveecn 25782 |
. . . . 5
⊢ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝑗 ∈ (1...𝑁)) → (𝐶‘𝑗) ∈ ℂ) |
6 | 4, 5 | sylancom 701 |
. . . 4
⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝑇 ∈ (0[,]1) ∧ ∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))))) ∧ 𝑗 ∈ (1...𝑁)) → (𝐶‘𝑗) ∈ ℂ) |
7 | | 0re 10040 |
. . . . . . . . . 10
⊢ 0 ∈
ℝ |
8 | | 1re 10039 |
. . . . . . . . . 10
⊢ 1 ∈
ℝ |
9 | 7, 8 | elicc2i 12239 |
. . . . . . . . 9
⊢ (𝑇 ∈ (0[,]1) ↔ (𝑇 ∈ ℝ ∧ 0 ≤
𝑇 ∧ 𝑇 ≤ 1)) |
10 | 9 | simp1bi 1076 |
. . . . . . . 8
⊢ (𝑇 ∈ (0[,]1) → 𝑇 ∈
ℝ) |
11 | 10 | recnd 10068 |
. . . . . . 7
⊢ (𝑇 ∈ (0[,]1) → 𝑇 ∈
ℂ) |
12 | 11 | adantr 481 |
. . . . . 6
⊢ ((𝑇 ∈ (0[,]1) ∧
∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖)))) → 𝑇 ∈ ℂ) |
13 | 12 | 3ad2ant3 1084 |
. . . . 5
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝑇 ∈ (0[,]1) ∧ ∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))))) → 𝑇 ∈ ℂ) |
14 | 13 | adantr 481 |
. . . 4
⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝑇 ∈ (0[,]1) ∧ ∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))))) ∧ 𝑗 ∈ (1...𝑁)) → 𝑇 ∈ ℂ) |
15 | | fveq2 6191 |
. . . . . . . 8
⊢ (𝑖 = 𝑗 → (𝐵‘𝑖) = (𝐵‘𝑗)) |
16 | | fveq2 6191 |
. . . . . . . . . 10
⊢ (𝑖 = 𝑗 → (𝐴‘𝑖) = (𝐴‘𝑗)) |
17 | 16 | oveq2d 6666 |
. . . . . . . . 9
⊢ (𝑖 = 𝑗 → ((1 − 𝑇) · (𝐴‘𝑖)) = ((1 − 𝑇) · (𝐴‘𝑗))) |
18 | | fveq2 6191 |
. . . . . . . . . 10
⊢ (𝑖 = 𝑗 → (𝐶‘𝑖) = (𝐶‘𝑗)) |
19 | 18 | oveq2d 6666 |
. . . . . . . . 9
⊢ (𝑖 = 𝑗 → (𝑇 · (𝐶‘𝑖)) = (𝑇 · (𝐶‘𝑗))) |
20 | 17, 19 | oveq12d 6668 |
. . . . . . . 8
⊢ (𝑖 = 𝑗 → (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) = (((1 − 𝑇) · (𝐴‘𝑗)) + (𝑇 · (𝐶‘𝑗)))) |
21 | 15, 20 | eqeq12d 2637 |
. . . . . . 7
⊢ (𝑖 = 𝑗 → ((𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ↔ (𝐵‘𝑗) = (((1 − 𝑇) · (𝐴‘𝑗)) + (𝑇 · (𝐶‘𝑗))))) |
22 | 21 | rspccva 3308 |
. . . . . 6
⊢
((∀𝑖 ∈
(1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ 𝑗 ∈ (1...𝑁)) → (𝐵‘𝑗) = (((1 − 𝑇) · (𝐴‘𝑗)) + (𝑇 · (𝐶‘𝑗)))) |
23 | 22 | adantll 750 |
. . . . 5
⊢ (((𝑇 ∈ (0[,]1) ∧
∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖)))) ∧ 𝑗 ∈ (1...𝑁)) → (𝐵‘𝑗) = (((1 − 𝑇) · (𝐴‘𝑗)) + (𝑇 · (𝐶‘𝑗)))) |
24 | 23 | 3ad2antl3 1225 |
. . . 4
⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝑇 ∈ (0[,]1) ∧ ∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))))) ∧ 𝑗 ∈ (1...𝑁)) → (𝐵‘𝑗) = (((1 − 𝑇) · (𝐴‘𝑗)) + (𝑇 · (𝐶‘𝑗)))) |
25 | | oveq1 6657 |
. . . . . 6
⊢ ((𝐵‘𝑗) = (((1 − 𝑇) · (𝐴‘𝑗)) + (𝑇 · (𝐶‘𝑗))) → ((𝐵‘𝑗) − (𝐶‘𝑗)) = ((((1 − 𝑇) · (𝐴‘𝑗)) + (𝑇 · (𝐶‘𝑗))) − (𝐶‘𝑗))) |
26 | 25 | oveq1d 6665 |
. . . . 5
⊢ ((𝐵‘𝑗) = (((1 − 𝑇) · (𝐴‘𝑗)) + (𝑇 · (𝐶‘𝑗))) → (((𝐵‘𝑗) − (𝐶‘𝑗))↑2) = (((((1 − 𝑇) · (𝐴‘𝑗)) + (𝑇 · (𝐶‘𝑗))) − (𝐶‘𝑗))↑2)) |
27 | | ax-1cn 9994 |
. . . . . . . . . . . 12
⊢ 1 ∈
ℂ |
28 | | subcl 10280 |
. . . . . . . . . . . 12
⊢ ((1
∈ ℂ ∧ 𝑇
∈ ℂ) → (1 − 𝑇) ∈ ℂ) |
29 | 27, 28 | mpan 706 |
. . . . . . . . . . 11
⊢ (𝑇 ∈ ℂ → (1
− 𝑇) ∈
ℂ) |
30 | 29 | 3ad2ant3 1084 |
. . . . . . . . . 10
⊢ (((𝐴‘𝑗) ∈ ℂ ∧ (𝐶‘𝑗) ∈ ℂ ∧ 𝑇 ∈ ℂ) → (1 − 𝑇) ∈
ℂ) |
31 | | simp1 1061 |
. . . . . . . . . 10
⊢ (((𝐴‘𝑗) ∈ ℂ ∧ (𝐶‘𝑗) ∈ ℂ ∧ 𝑇 ∈ ℂ) → (𝐴‘𝑗) ∈ ℂ) |
32 | 30, 31 | mulcld 10060 |
. . . . . . . . 9
⊢ (((𝐴‘𝑗) ∈ ℂ ∧ (𝐶‘𝑗) ∈ ℂ ∧ 𝑇 ∈ ℂ) → ((1 − 𝑇) · (𝐴‘𝑗)) ∈ ℂ) |
33 | | simp3 1063 |
. . . . . . . . . 10
⊢ (((𝐴‘𝑗) ∈ ℂ ∧ (𝐶‘𝑗) ∈ ℂ ∧ 𝑇 ∈ ℂ) → 𝑇 ∈ ℂ) |
34 | | simp2 1062 |
. . . . . . . . . 10
⊢ (((𝐴‘𝑗) ∈ ℂ ∧ (𝐶‘𝑗) ∈ ℂ ∧ 𝑇 ∈ ℂ) → (𝐶‘𝑗) ∈ ℂ) |
35 | 33, 34 | mulcld 10060 |
. . . . . . . . 9
⊢ (((𝐴‘𝑗) ∈ ℂ ∧ (𝐶‘𝑗) ∈ ℂ ∧ 𝑇 ∈ ℂ) → (𝑇 · (𝐶‘𝑗)) ∈ ℂ) |
36 | 32, 35, 34 | addsubassd 10412 |
. . . . . . . 8
⊢ (((𝐴‘𝑗) ∈ ℂ ∧ (𝐶‘𝑗) ∈ ℂ ∧ 𝑇 ∈ ℂ) → ((((1 − 𝑇) · (𝐴‘𝑗)) + (𝑇 · (𝐶‘𝑗))) − (𝐶‘𝑗)) = (((1 − 𝑇) · (𝐴‘𝑗)) + ((𝑇 · (𝐶‘𝑗)) − (𝐶‘𝑗)))) |
37 | | subdi 10463 |
. . . . . . . . . . 11
⊢ (((1
− 𝑇) ∈ ℂ
∧ (𝐴‘𝑗) ∈ ℂ ∧ (𝐶‘𝑗) ∈ ℂ) → ((1 − 𝑇) · ((𝐴‘𝑗) − (𝐶‘𝑗))) = (((1 − 𝑇) · (𝐴‘𝑗)) − ((1 − 𝑇) · (𝐶‘𝑗)))) |
38 | 29, 37 | syl3an1 1359 |
. . . . . . . . . 10
⊢ ((𝑇 ∈ ℂ ∧ (𝐴‘𝑗) ∈ ℂ ∧ (𝐶‘𝑗) ∈ ℂ) → ((1 − 𝑇) · ((𝐴‘𝑗) − (𝐶‘𝑗))) = (((1 − 𝑇) · (𝐴‘𝑗)) − ((1 − 𝑇) · (𝐶‘𝑗)))) |
39 | 38 | 3coml 1272 |
. . . . . . . . 9
⊢ (((𝐴‘𝑗) ∈ ℂ ∧ (𝐶‘𝑗) ∈ ℂ ∧ 𝑇 ∈ ℂ) → ((1 − 𝑇) · ((𝐴‘𝑗) − (𝐶‘𝑗))) = (((1 − 𝑇) · (𝐴‘𝑗)) − ((1 − 𝑇) · (𝐶‘𝑗)))) |
40 | | subdir 10464 |
. . . . . . . . . . . . . 14
⊢ ((1
∈ ℂ ∧ 𝑇
∈ ℂ ∧ (𝐶‘𝑗) ∈ ℂ) → ((1 − 𝑇) · (𝐶‘𝑗)) = ((1 · (𝐶‘𝑗)) − (𝑇 · (𝐶‘𝑗)))) |
41 | 27, 40 | mp3an1 1411 |
. . . . . . . . . . . . 13
⊢ ((𝑇 ∈ ℂ ∧ (𝐶‘𝑗) ∈ ℂ) → ((1 − 𝑇) · (𝐶‘𝑗)) = ((1 · (𝐶‘𝑗)) − (𝑇 · (𝐶‘𝑗)))) |
42 | 41 | ancoms 469 |
. . . . . . . . . . . 12
⊢ (((𝐶‘𝑗) ∈ ℂ ∧ 𝑇 ∈ ℂ) → ((1 − 𝑇) · (𝐶‘𝑗)) = ((1 · (𝐶‘𝑗)) − (𝑇 · (𝐶‘𝑗)))) |
43 | 42 | 3adant1 1079 |
. . . . . . . . . . 11
⊢ (((𝐴‘𝑗) ∈ ℂ ∧ (𝐶‘𝑗) ∈ ℂ ∧ 𝑇 ∈ ℂ) → ((1 − 𝑇) · (𝐶‘𝑗)) = ((1 · (𝐶‘𝑗)) − (𝑇 · (𝐶‘𝑗)))) |
44 | | mulid2 10038 |
. . . . . . . . . . . . 13
⊢ ((𝐶‘𝑗) ∈ ℂ → (1 · (𝐶‘𝑗)) = (𝐶‘𝑗)) |
45 | 44 | oveq1d 6665 |
. . . . . . . . . . . 12
⊢ ((𝐶‘𝑗) ∈ ℂ → ((1 · (𝐶‘𝑗)) − (𝑇 · (𝐶‘𝑗))) = ((𝐶‘𝑗) − (𝑇 · (𝐶‘𝑗)))) |
46 | 45 | 3ad2ant2 1083 |
. . . . . . . . . . 11
⊢ (((𝐴‘𝑗) ∈ ℂ ∧ (𝐶‘𝑗) ∈ ℂ ∧ 𝑇 ∈ ℂ) → ((1 · (𝐶‘𝑗)) − (𝑇 · (𝐶‘𝑗))) = ((𝐶‘𝑗) − (𝑇 · (𝐶‘𝑗)))) |
47 | 43, 46 | eqtrd 2656 |
. . . . . . . . . 10
⊢ (((𝐴‘𝑗) ∈ ℂ ∧ (𝐶‘𝑗) ∈ ℂ ∧ 𝑇 ∈ ℂ) → ((1 − 𝑇) · (𝐶‘𝑗)) = ((𝐶‘𝑗) − (𝑇 · (𝐶‘𝑗)))) |
48 | 47 | oveq2d 6666 |
. . . . . . . . 9
⊢ (((𝐴‘𝑗) ∈ ℂ ∧ (𝐶‘𝑗) ∈ ℂ ∧ 𝑇 ∈ ℂ) → (((1 − 𝑇) · (𝐴‘𝑗)) − ((1 − 𝑇) · (𝐶‘𝑗))) = (((1 − 𝑇) · (𝐴‘𝑗)) − ((𝐶‘𝑗) − (𝑇 · (𝐶‘𝑗))))) |
49 | 32, 34, 35 | subsub2d 10421 |
. . . . . . . . 9
⊢ (((𝐴‘𝑗) ∈ ℂ ∧ (𝐶‘𝑗) ∈ ℂ ∧ 𝑇 ∈ ℂ) → (((1 − 𝑇) · (𝐴‘𝑗)) − ((𝐶‘𝑗) − (𝑇 · (𝐶‘𝑗)))) = (((1 − 𝑇) · (𝐴‘𝑗)) + ((𝑇 · (𝐶‘𝑗)) − (𝐶‘𝑗)))) |
50 | 39, 48, 49 | 3eqtrd 2660 |
. . . . . . . 8
⊢ (((𝐴‘𝑗) ∈ ℂ ∧ (𝐶‘𝑗) ∈ ℂ ∧ 𝑇 ∈ ℂ) → ((1 − 𝑇) · ((𝐴‘𝑗) − (𝐶‘𝑗))) = (((1 − 𝑇) · (𝐴‘𝑗)) + ((𝑇 · (𝐶‘𝑗)) − (𝐶‘𝑗)))) |
51 | 36, 50 | eqtr4d 2659 |
. . . . . . 7
⊢ (((𝐴‘𝑗) ∈ ℂ ∧ (𝐶‘𝑗) ∈ ℂ ∧ 𝑇 ∈ ℂ) → ((((1 − 𝑇) · (𝐴‘𝑗)) + (𝑇 · (𝐶‘𝑗))) − (𝐶‘𝑗)) = ((1 − 𝑇) · ((𝐴‘𝑗) − (𝐶‘𝑗)))) |
52 | 51 | oveq1d 6665 |
. . . . . 6
⊢ (((𝐴‘𝑗) ∈ ℂ ∧ (𝐶‘𝑗) ∈ ℂ ∧ 𝑇 ∈ ℂ) → (((((1 − 𝑇) · (𝐴‘𝑗)) + (𝑇 · (𝐶‘𝑗))) − (𝐶‘𝑗))↑2) = (((1 − 𝑇) · ((𝐴‘𝑗) − (𝐶‘𝑗)))↑2)) |
53 | | subcl 10280 |
. . . . . . . 8
⊢ (((𝐴‘𝑗) ∈ ℂ ∧ (𝐶‘𝑗) ∈ ℂ) → ((𝐴‘𝑗) − (𝐶‘𝑗)) ∈ ℂ) |
54 | 53 | 3adant3 1081 |
. . . . . . 7
⊢ (((𝐴‘𝑗) ∈ ℂ ∧ (𝐶‘𝑗) ∈ ℂ ∧ 𝑇 ∈ ℂ) → ((𝐴‘𝑗) − (𝐶‘𝑗)) ∈ ℂ) |
55 | 30, 54 | sqmuld 13020 |
. . . . . 6
⊢ (((𝐴‘𝑗) ∈ ℂ ∧ (𝐶‘𝑗) ∈ ℂ ∧ 𝑇 ∈ ℂ) → (((1 − 𝑇) · ((𝐴‘𝑗) − (𝐶‘𝑗)))↑2) = (((1 − 𝑇)↑2) · (((𝐴‘𝑗) − (𝐶‘𝑗))↑2))) |
56 | 52, 55 | eqtrd 2656 |
. . . . 5
⊢ (((𝐴‘𝑗) ∈ ℂ ∧ (𝐶‘𝑗) ∈ ℂ ∧ 𝑇 ∈ ℂ) → (((((1 − 𝑇) · (𝐴‘𝑗)) + (𝑇 · (𝐶‘𝑗))) − (𝐶‘𝑗))↑2) = (((1 − 𝑇)↑2) · (((𝐴‘𝑗) − (𝐶‘𝑗))↑2))) |
57 | 26, 56 | sylan9eqr 2678 |
. . . 4
⊢ ((((𝐴‘𝑗) ∈ ℂ ∧ (𝐶‘𝑗) ∈ ℂ ∧ 𝑇 ∈ ℂ) ∧ (𝐵‘𝑗) = (((1 − 𝑇) · (𝐴‘𝑗)) + (𝑇 · (𝐶‘𝑗)))) → (((𝐵‘𝑗) − (𝐶‘𝑗))↑2) = (((1 − 𝑇)↑2) · (((𝐴‘𝑗) − (𝐶‘𝑗))↑2))) |
58 | 3, 6, 14, 24, 57 | syl31anc 1329 |
. . 3
⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝑇 ∈ (0[,]1) ∧ ∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))))) ∧ 𝑗 ∈ (1...𝑁)) → (((𝐵‘𝑗) − (𝐶‘𝑗))↑2) = (((1 − 𝑇)↑2) · (((𝐴‘𝑗) − (𝐶‘𝑗))↑2))) |
59 | 58 | sumeq2dv 14433 |
. 2
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝑇 ∈ (0[,]1) ∧ ∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))))) → Σ𝑗 ∈ (1...𝑁)(((𝐵‘𝑗) − (𝐶‘𝑗))↑2) = Σ𝑗 ∈ (1...𝑁)(((1 − 𝑇)↑2) · (((𝐴‘𝑗) − (𝐶‘𝑗))↑2))) |
60 | | fzfid 12772 |
. . 3
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝑇 ∈ (0[,]1) ∧ ∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))))) → (1...𝑁) ∈ Fin) |
61 | | resubcl 10345 |
. . . . . . . 8
⊢ ((1
∈ ℝ ∧ 𝑇
∈ ℝ) → (1 − 𝑇) ∈ ℝ) |
62 | 8, 10, 61 | sylancr 695 |
. . . . . . 7
⊢ (𝑇 ∈ (0[,]1) → (1
− 𝑇) ∈
ℝ) |
63 | 62 | resqcld 13035 |
. . . . . 6
⊢ (𝑇 ∈ (0[,]1) → ((1
− 𝑇)↑2) ∈
ℝ) |
64 | 63 | recnd 10068 |
. . . . 5
⊢ (𝑇 ∈ (0[,]1) → ((1
− 𝑇)↑2) ∈
ℂ) |
65 | 64 | adantr 481 |
. . . 4
⊢ ((𝑇 ∈ (0[,]1) ∧
∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖)))) → ((1 − 𝑇)↑2) ∈ ℂ) |
66 | 65 | 3ad2ant3 1084 |
. . 3
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝑇 ∈ (0[,]1) ∧ ∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))))) → ((1 − 𝑇)↑2) ∈ ℂ) |
67 | 2 | 3adant1 1079 |
. . . . . . . 8
⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝑗 ∈ (1...𝑁)) → (𝐴‘𝑗) ∈ ℂ) |
68 | 67 | 3adant2r 1321 |
. . . . . . 7
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ 𝑗 ∈ (1...𝑁)) → (𝐴‘𝑗) ∈ ℂ) |
69 | 5 | 3adant1 1079 |
. . . . . . . 8
⊢ ((𝑁 ∈ ℕ ∧ 𝐶 ∈ (𝔼‘𝑁) ∧ 𝑗 ∈ (1...𝑁)) → (𝐶‘𝑗) ∈ ℂ) |
70 | 69 | 3adant2l 1320 |
. . . . . . 7
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ 𝑗 ∈ (1...𝑁)) → (𝐶‘𝑗) ∈ ℂ) |
71 | 68, 70 | subcld 10392 |
. . . . . 6
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ 𝑗 ∈ (1...𝑁)) → ((𝐴‘𝑗) − (𝐶‘𝑗)) ∈ ℂ) |
72 | 71 | sqcld 13006 |
. . . . 5
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ 𝑗 ∈ (1...𝑁)) → (((𝐴‘𝑗) − (𝐶‘𝑗))↑2) ∈ ℂ) |
73 | 72 | 3expa 1265 |
. . . 4
⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) ∧ 𝑗 ∈ (1...𝑁)) → (((𝐴‘𝑗) − (𝐶‘𝑗))↑2) ∈ ℂ) |
74 | 73 | 3adantl3 1219 |
. . 3
⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝑇 ∈ (0[,]1) ∧ ∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))))) ∧ 𝑗 ∈ (1...𝑁)) → (((𝐴‘𝑗) − (𝐶‘𝑗))↑2) ∈ ℂ) |
75 | 60, 66, 74 | fsummulc2 14516 |
. 2
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝑇 ∈ (0[,]1) ∧ ∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))))) → (((1 − 𝑇)↑2) · Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐶‘𝑗))↑2)) = Σ𝑗 ∈ (1...𝑁)(((1 − 𝑇)↑2) · (((𝐴‘𝑗) − (𝐶‘𝑗))↑2))) |
76 | 59, 75 | eqtr4d 2659 |
1
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝑇 ∈ (0[,]1) ∧ ∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))))) → Σ𝑗 ∈ (1...𝑁)(((𝐵‘𝑗) − (𝐶‘𝑗))↑2) = (((1 − 𝑇)↑2) · Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐶‘𝑗))↑2))) |