Proof of Theorem ipasslem2
Step | Hyp | Ref
| Expression |
1 | | nn0cn 11929 |
. . . . 5
⊢ (𝑁 ∈ ℕ0
→ 𝑁 ∈
ℂ) |
2 | 1 | negcld 11007 |
. . . 4
⊢ (𝑁 ∈ ℕ0
→ -𝑁 ∈
ℂ) |
3 | | ip1i.9 |
. . . . . 6
⊢ 𝑈 ∈
CPreHilOLD |
4 | 3 | phnvi 28683 |
. . . . 5
⊢ 𝑈 ∈ NrmCVec |
5 | | ipasslem1.b |
. . . . 5
⊢ 𝐵 ∈ 𝑋 |
6 | | ip1i.1 |
. . . . . 6
⊢ 𝑋 = (BaseSet‘𝑈) |
7 | | ip1i.7 |
. . . . . 6
⊢ 𝑃 =
(·𝑖OLD‘𝑈) |
8 | 6, 7 | dipcl 28579 |
. . . . 5
⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑃𝐵) ∈ ℂ) |
9 | 4, 5, 8 | mp3an13 1450 |
. . . 4
⊢ (𝐴 ∈ 𝑋 → (𝐴𝑃𝐵) ∈ ℂ) |
10 | | mulcl 10644 |
. . . 4
⊢ ((-𝑁 ∈ ℂ ∧ (𝐴𝑃𝐵) ∈ ℂ) → (-𝑁 · (𝐴𝑃𝐵)) ∈ ℂ) |
11 | 2, 9, 10 | syl2an 599 |
. . 3
⊢ ((𝑁 ∈ ℕ0
∧ 𝐴 ∈ 𝑋) → (-𝑁 · (𝐴𝑃𝐵)) ∈ ℂ) |
12 | | ip1i.4 |
. . . . . . 7
⊢ 𝑆 = (
·𝑠OLD ‘𝑈) |
13 | 6, 12 | nvscl 28493 |
. . . . . 6
⊢ ((𝑈 ∈ NrmCVec ∧ -𝑁 ∈ ℂ ∧ 𝐴 ∈ 𝑋) → (-𝑁𝑆𝐴) ∈ 𝑋) |
14 | 4, 13 | mp3an1 1446 |
. . . . 5
⊢ ((-𝑁 ∈ ℂ ∧ 𝐴 ∈ 𝑋) → (-𝑁𝑆𝐴) ∈ 𝑋) |
15 | 2, 14 | sylan 584 |
. . . 4
⊢ ((𝑁 ∈ ℕ0
∧ 𝐴 ∈ 𝑋) → (-𝑁𝑆𝐴) ∈ 𝑋) |
16 | 6, 7 | dipcl 28579 |
. . . . 5
⊢ ((𝑈 ∈ NrmCVec ∧ (-𝑁𝑆𝐴) ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((-𝑁𝑆𝐴)𝑃𝐵) ∈ ℂ) |
17 | 4, 5, 16 | mp3an13 1450 |
. . . 4
⊢ ((-𝑁𝑆𝐴) ∈ 𝑋 → ((-𝑁𝑆𝐴)𝑃𝐵) ∈ ℂ) |
18 | 15, 17 | syl 17 |
. . 3
⊢ ((𝑁 ∈ ℕ0
∧ 𝐴 ∈ 𝑋) → ((-𝑁𝑆𝐴)𝑃𝐵) ∈ ℂ) |
19 | | ax-1cn 10618 |
. . . . . . . . . . . . 13
⊢ 1 ∈
ℂ |
20 | | mulneg2 11100 |
. . . . . . . . . . . . 13
⊢ ((𝑁 ∈ ℂ ∧ 1 ∈
ℂ) → (𝑁 ·
-1) = -(𝑁 ·
1)) |
21 | 19, 20 | mpan2 691 |
. . . . . . . . . . . 12
⊢ (𝑁 ∈ ℂ → (𝑁 · -1) = -(𝑁 · 1)) |
22 | | mulid1 10662 |
. . . . . . . . . . . . 13
⊢ (𝑁 ∈ ℂ → (𝑁 · 1) = 𝑁) |
23 | 22 | negeqd 10903 |
. . . . . . . . . . . 12
⊢ (𝑁 ∈ ℂ → -(𝑁 · 1) = -𝑁) |
24 | 21, 23 | eqtr2d 2795 |
. . . . . . . . . . 11
⊢ (𝑁 ∈ ℂ → -𝑁 = (𝑁 · -1)) |
25 | 24 | adantr 485 |
. . . . . . . . . 10
⊢ ((𝑁 ∈ ℂ ∧ 𝐴 ∈ 𝑋) → -𝑁 = (𝑁 · -1)) |
26 | 25 | oveq1d 7158 |
. . . . . . . . 9
⊢ ((𝑁 ∈ ℂ ∧ 𝐴 ∈ 𝑋) → (-𝑁𝑆𝐴) = ((𝑁 · -1)𝑆𝐴)) |
27 | | neg1cn 11773 |
. . . . . . . . . 10
⊢ -1 ∈
ℂ |
28 | 6, 12 | nvsass 28495 |
. . . . . . . . . . 11
⊢ ((𝑈 ∈ NrmCVec ∧ (𝑁 ∈ ℂ ∧ -1 ∈
ℂ ∧ 𝐴 ∈
𝑋)) → ((𝑁 · -1)𝑆𝐴) = (𝑁𝑆(-1𝑆𝐴))) |
29 | 4, 28 | mpan 690 |
. . . . . . . . . 10
⊢ ((𝑁 ∈ ℂ ∧ -1 ∈
ℂ ∧ 𝐴 ∈
𝑋) → ((𝑁 · -1)𝑆𝐴) = (𝑁𝑆(-1𝑆𝐴))) |
30 | 27, 29 | mp3an2 1447 |
. . . . . . . . 9
⊢ ((𝑁 ∈ ℂ ∧ 𝐴 ∈ 𝑋) → ((𝑁 · -1)𝑆𝐴) = (𝑁𝑆(-1𝑆𝐴))) |
31 | 26, 30 | eqtrd 2794 |
. . . . . . . 8
⊢ ((𝑁 ∈ ℂ ∧ 𝐴 ∈ 𝑋) → (-𝑁𝑆𝐴) = (𝑁𝑆(-1𝑆𝐴))) |
32 | 1, 31 | sylan 584 |
. . . . . . 7
⊢ ((𝑁 ∈ ℕ0
∧ 𝐴 ∈ 𝑋) → (-𝑁𝑆𝐴) = (𝑁𝑆(-1𝑆𝐴))) |
33 | 32 | oveq1d 7158 |
. . . . . 6
⊢ ((𝑁 ∈ ℕ0
∧ 𝐴 ∈ 𝑋) → ((-𝑁𝑆𝐴)𝑃𝐵) = ((𝑁𝑆(-1𝑆𝐴))𝑃𝐵)) |
34 | 6, 12 | nvscl 28493 |
. . . . . . . 8
⊢ ((𝑈 ∈ NrmCVec ∧ -1 ∈
ℂ ∧ 𝐴 ∈
𝑋) → (-1𝑆𝐴) ∈ 𝑋) |
35 | 4, 27, 34 | mp3an12 1449 |
. . . . . . 7
⊢ (𝐴 ∈ 𝑋 → (-1𝑆𝐴) ∈ 𝑋) |
36 | | ip1i.2 |
. . . . . . . 8
⊢ 𝐺 = ( +𝑣
‘𝑈) |
37 | 6, 36, 12, 7, 3, 5 | ipasslem1 28698 |
. . . . . . 7
⊢ ((𝑁 ∈ ℕ0
∧ (-1𝑆𝐴) ∈ 𝑋) → ((𝑁𝑆(-1𝑆𝐴))𝑃𝐵) = (𝑁 · ((-1𝑆𝐴)𝑃𝐵))) |
38 | 35, 37 | sylan2 596 |
. . . . . 6
⊢ ((𝑁 ∈ ℕ0
∧ 𝐴 ∈ 𝑋) → ((𝑁𝑆(-1𝑆𝐴))𝑃𝐵) = (𝑁 · ((-1𝑆𝐴)𝑃𝐵))) |
39 | 33, 38 | eqtrd 2794 |
. . . . 5
⊢ ((𝑁 ∈ ℕ0
∧ 𝐴 ∈ 𝑋) → ((-𝑁𝑆𝐴)𝑃𝐵) = (𝑁 · ((-1𝑆𝐴)𝑃𝐵))) |
40 | 39 | oveq2d 7159 |
. . . 4
⊢ ((𝑁 ∈ ℕ0
∧ 𝐴 ∈ 𝑋) → ((-𝑁 · (𝐴𝑃𝐵)) − ((-𝑁𝑆𝐴)𝑃𝐵)) = ((-𝑁 · (𝐴𝑃𝐵)) − (𝑁 · ((-1𝑆𝐴)𝑃𝐵)))) |
41 | 6, 7 | dipcl 28579 |
. . . . . . . 8
⊢ ((𝑈 ∈ NrmCVec ∧ (-1𝑆𝐴) ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((-1𝑆𝐴)𝑃𝐵) ∈ ℂ) |
42 | 4, 5, 41 | mp3an13 1450 |
. . . . . . 7
⊢ ((-1𝑆𝐴) ∈ 𝑋 → ((-1𝑆𝐴)𝑃𝐵) ∈ ℂ) |
43 | 35, 42 | syl 17 |
. . . . . 6
⊢ (𝐴 ∈ 𝑋 → ((-1𝑆𝐴)𝑃𝐵) ∈ ℂ) |
44 | | mulcl 10644 |
. . . . . 6
⊢ ((𝑁 ∈ ℂ ∧ ((-1𝑆𝐴)𝑃𝐵) ∈ ℂ) → (𝑁 · ((-1𝑆𝐴)𝑃𝐵)) ∈ ℂ) |
45 | 1, 43, 44 | syl2an 599 |
. . . . 5
⊢ ((𝑁 ∈ ℕ0
∧ 𝐴 ∈ 𝑋) → (𝑁 · ((-1𝑆𝐴)𝑃𝐵)) ∈ ℂ) |
46 | 11, 45 | negsubd 11026 |
. . . 4
⊢ ((𝑁 ∈ ℕ0
∧ 𝐴 ∈ 𝑋) → ((-𝑁 · (𝐴𝑃𝐵)) + -(𝑁 · ((-1𝑆𝐴)𝑃𝐵))) = ((-𝑁 · (𝐴𝑃𝐵)) − (𝑁 · ((-1𝑆𝐴)𝑃𝐵)))) |
47 | | mulneg1 11099 |
. . . . . . 7
⊢ ((𝑁 ∈ ℂ ∧ ((-1𝑆𝐴)𝑃𝐵) ∈ ℂ) → (-𝑁 · ((-1𝑆𝐴)𝑃𝐵)) = -(𝑁 · ((-1𝑆𝐴)𝑃𝐵))) |
48 | 1, 43, 47 | syl2an 599 |
. . . . . 6
⊢ ((𝑁 ∈ ℕ0
∧ 𝐴 ∈ 𝑋) → (-𝑁 · ((-1𝑆𝐴)𝑃𝐵)) = -(𝑁 · ((-1𝑆𝐴)𝑃𝐵))) |
49 | 48 | oveq2d 7159 |
. . . . 5
⊢ ((𝑁 ∈ ℕ0
∧ 𝐴 ∈ 𝑋) → ((-𝑁 · (𝐴𝑃𝐵)) + (-𝑁 · ((-1𝑆𝐴)𝑃𝐵))) = ((-𝑁 · (𝐴𝑃𝐵)) + -(𝑁 · ((-1𝑆𝐴)𝑃𝐵)))) |
50 | 2 | adantr 485 |
. . . . . . 7
⊢ ((𝑁 ∈ ℕ0
∧ 𝐴 ∈ 𝑋) → -𝑁 ∈ ℂ) |
51 | 9 | adantl 486 |
. . . . . . 7
⊢ ((𝑁 ∈ ℕ0
∧ 𝐴 ∈ 𝑋) → (𝐴𝑃𝐵) ∈ ℂ) |
52 | 43 | adantl 486 |
. . . . . . 7
⊢ ((𝑁 ∈ ℕ0
∧ 𝐴 ∈ 𝑋) → ((-1𝑆𝐴)𝑃𝐵) ∈ ℂ) |
53 | 50, 51, 52 | adddid 10688 |
. . . . . 6
⊢ ((𝑁 ∈ ℕ0
∧ 𝐴 ∈ 𝑋) → (-𝑁 · ((𝐴𝑃𝐵) + ((-1𝑆𝐴)𝑃𝐵))) = ((-𝑁 · (𝐴𝑃𝐵)) + (-𝑁 · ((-1𝑆𝐴)𝑃𝐵)))) |
54 | 6, 36, 12, 7, 3 | ipdiri 28697 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ 𝑋 ∧ (-1𝑆𝐴) ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝐺(-1𝑆𝐴))𝑃𝐵) = ((𝐴𝑃𝐵) + ((-1𝑆𝐴)𝑃𝐵))) |
55 | 5, 54 | mp3an3 1448 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ 𝑋 ∧ (-1𝑆𝐴) ∈ 𝑋) → ((𝐴𝐺(-1𝑆𝐴))𝑃𝐵) = ((𝐴𝑃𝐵) + ((-1𝑆𝐴)𝑃𝐵))) |
56 | 35, 55 | mpdan 687 |
. . . . . . . . 9
⊢ (𝐴 ∈ 𝑋 → ((𝐴𝐺(-1𝑆𝐴))𝑃𝐵) = ((𝐴𝑃𝐵) + ((-1𝑆𝐴)𝑃𝐵))) |
57 | | eqid 2759 |
. . . . . . . . . . . . 13
⊢
(0vec‘𝑈) = (0vec‘𝑈) |
58 | 6, 36, 12, 57 | nvrinv 28518 |
. . . . . . . . . . . 12
⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺(-1𝑆𝐴)) = (0vec‘𝑈)) |
59 | 4, 58 | mpan 690 |
. . . . . . . . . . 11
⊢ (𝐴 ∈ 𝑋 → (𝐴𝐺(-1𝑆𝐴)) = (0vec‘𝑈)) |
60 | 59 | oveq1d 7158 |
. . . . . . . . . 10
⊢ (𝐴 ∈ 𝑋 → ((𝐴𝐺(-1𝑆𝐴))𝑃𝐵) = ((0vec‘𝑈)𝑃𝐵)) |
61 | 6, 57, 7 | dip0l 28585 |
. . . . . . . . . . 11
⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋) → ((0vec‘𝑈)𝑃𝐵) = 0) |
62 | 4, 5, 61 | mp2an 692 |
. . . . . . . . . 10
⊢
((0vec‘𝑈)𝑃𝐵) = 0 |
63 | 60, 62 | eqtrdi 2810 |
. . . . . . . . 9
⊢ (𝐴 ∈ 𝑋 → ((𝐴𝐺(-1𝑆𝐴))𝑃𝐵) = 0) |
64 | 56, 63 | eqtr3d 2796 |
. . . . . . . 8
⊢ (𝐴 ∈ 𝑋 → ((𝐴𝑃𝐵) + ((-1𝑆𝐴)𝑃𝐵)) = 0) |
65 | 64 | oveq2d 7159 |
. . . . . . 7
⊢ (𝐴 ∈ 𝑋 → (-𝑁 · ((𝐴𝑃𝐵) + ((-1𝑆𝐴)𝑃𝐵))) = (-𝑁 · 0)) |
66 | 2 | mul01d 10862 |
. . . . . . 7
⊢ (𝑁 ∈ ℕ0
→ (-𝑁 · 0) =
0) |
67 | 65, 66 | sylan9eqr 2816 |
. . . . . 6
⊢ ((𝑁 ∈ ℕ0
∧ 𝐴 ∈ 𝑋) → (-𝑁 · ((𝐴𝑃𝐵) + ((-1𝑆𝐴)𝑃𝐵))) = 0) |
68 | 53, 67 | eqtr3d 2796 |
. . . . 5
⊢ ((𝑁 ∈ ℕ0
∧ 𝐴 ∈ 𝑋) → ((-𝑁 · (𝐴𝑃𝐵)) + (-𝑁 · ((-1𝑆𝐴)𝑃𝐵))) = 0) |
69 | 49, 68 | eqtr3d 2796 |
. . . 4
⊢ ((𝑁 ∈ ℕ0
∧ 𝐴 ∈ 𝑋) → ((-𝑁 · (𝐴𝑃𝐵)) + -(𝑁 · ((-1𝑆𝐴)𝑃𝐵))) = 0) |
70 | 40, 46, 69 | 3eqtr2d 2800 |
. . 3
⊢ ((𝑁 ∈ ℕ0
∧ 𝐴 ∈ 𝑋) → ((-𝑁 · (𝐴𝑃𝐵)) − ((-𝑁𝑆𝐴)𝑃𝐵)) = 0) |
71 | 11, 18, 70 | subeq0d 11028 |
. 2
⊢ ((𝑁 ∈ ℕ0
∧ 𝐴 ∈ 𝑋) → (-𝑁 · (𝐴𝑃𝐵)) = ((-𝑁𝑆𝐴)𝑃𝐵)) |
72 | 71 | eqcomd 2765 |
1
⊢ ((𝑁 ∈ ℕ0
∧ 𝐴 ∈ 𝑋) → ((-𝑁𝑆𝐴)𝑃𝐵) = (-𝑁 · (𝐴𝑃𝐵))) |