Proof of Theorem pjthlem1
Step | Hyp | Ref
| Expression |
1 | | pjthlem.1 |
. . . 4
⊢ (𝜑 → 𝑊 ∈ ℂHil) |
2 | | hlcph 24266 |
. . . 4
⊢ (𝑊 ∈ ℂHil → 𝑊 ∈
ℂPreHil) |
3 | 1, 2 | syl 17 |
. . 3
⊢ (𝜑 → 𝑊 ∈ ℂPreHil) |
4 | | pjthlem.4 |
. . 3
⊢ (𝜑 → 𝐴 ∈ 𝑉) |
5 | | pjthlem.2 |
. . . . 5
⊢ (𝜑 → 𝑈 ∈ 𝐿) |
6 | | pjthlem.v |
. . . . . 6
⊢ 𝑉 = (Base‘𝑊) |
7 | | pjthlem.l |
. . . . . 6
⊢ 𝐿 = (LSubSp‘𝑊) |
8 | 6, 7 | lssss 19978 |
. . . . 5
⊢ (𝑈 ∈ 𝐿 → 𝑈 ⊆ 𝑉) |
9 | 5, 8 | syl 17 |
. . . 4
⊢ (𝜑 → 𝑈 ⊆ 𝑉) |
10 | | pjthlem.5 |
. . . 4
⊢ (𝜑 → 𝐵 ∈ 𝑈) |
11 | 9, 10 | sseldd 3907 |
. . 3
⊢ (𝜑 → 𝐵 ∈ 𝑉) |
12 | | pjthlem.h |
. . . 4
⊢ , =
(·𝑖‘𝑊) |
13 | 6, 12 | cphipcl 24093 |
. . 3
⊢ ((𝑊 ∈ ℂPreHil ∧
𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴 , 𝐵) ∈ ℂ) |
14 | 3, 4, 11, 13 | syl3anc 1373 |
. 2
⊢ (𝜑 → (𝐴 , 𝐵) ∈ ℂ) |
15 | 14 | abscld 15005 |
. . . 4
⊢ (𝜑 → (abs‘(𝐴 , 𝐵)) ∈ ℝ) |
16 | 15 | recnd 10866 |
. . 3
⊢ (𝜑 → (abs‘(𝐴 , 𝐵)) ∈ ℂ) |
17 | 15 | resqcld 13822 |
. . . . . . 7
⊢ (𝜑 → ((abs‘(𝐴 , 𝐵))↑2) ∈ ℝ) |
18 | 17 | renegcld 11264 |
. . . . . 6
⊢ (𝜑 → -((abs‘(𝐴 , 𝐵))↑2) ∈ ℝ) |
19 | 6, 12 | reipcl 24099 |
. . . . . . . . 9
⊢ ((𝑊 ∈ ℂPreHil ∧
𝐵 ∈ 𝑉) → (𝐵 , 𝐵) ∈ ℝ) |
20 | 3, 11, 19 | syl2anc 587 |
. . . . . . . 8
⊢ (𝜑 → (𝐵 , 𝐵) ∈ ℝ) |
21 | | 2re 11909 |
. . . . . . . 8
⊢ 2 ∈
ℝ |
22 | | readdcl 10817 |
. . . . . . . 8
⊢ (((𝐵 , 𝐵) ∈ ℝ ∧ 2 ∈ ℝ)
→ ((𝐵 , 𝐵) + 2) ∈
ℝ) |
23 | 20, 21, 22 | sylancl 589 |
. . . . . . 7
⊢ (𝜑 → ((𝐵 , 𝐵) + 2) ∈ ℝ) |
24 | | 0red 10841 |
. . . . . . . 8
⊢ (𝜑 → 0 ∈
ℝ) |
25 | | peano2re 11010 |
. . . . . . . . 9
⊢ ((𝐵 , 𝐵) ∈ ℝ → ((𝐵 , 𝐵) + 1) ∈ ℝ) |
26 | 20, 25 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → ((𝐵 , 𝐵) + 1) ∈ ℝ) |
27 | 6, 12 | ipge0 24100 |
. . . . . . . . . 10
⊢ ((𝑊 ∈ ℂPreHil ∧
𝐵 ∈ 𝑉) → 0 ≤ (𝐵 , 𝐵)) |
28 | 3, 11, 27 | syl2anc 587 |
. . . . . . . . 9
⊢ (𝜑 → 0 ≤ (𝐵 , 𝐵)) |
29 | 20 | ltp1d 11767 |
. . . . . . . . 9
⊢ (𝜑 → (𝐵 , 𝐵) < ((𝐵 , 𝐵) + 1)) |
30 | 24, 20, 26, 28, 29 | lelttrd 10995 |
. . . . . . . 8
⊢ (𝜑 → 0 < ((𝐵 , 𝐵) + 1)) |
31 | 26 | ltp1d 11767 |
. . . . . . . . 9
⊢ (𝜑 → ((𝐵 , 𝐵) + 1) < (((𝐵 , 𝐵) + 1) + 1)) |
32 | | df-2 11898 |
. . . . . . . . . . 11
⊢ 2 = (1 +
1) |
33 | 32 | oveq2i 7229 |
. . . . . . . . . 10
⊢ ((𝐵 , 𝐵) + 2) = ((𝐵 , 𝐵) + (1 + 1)) |
34 | 20 | recnd 10866 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐵 , 𝐵) ∈ ℂ) |
35 | | ax-1cn 10792 |
. . . . . . . . . . . 12
⊢ 1 ∈
ℂ |
36 | | addass 10821 |
. . . . . . . . . . . 12
⊢ (((𝐵 , 𝐵) ∈ ℂ ∧ 1 ∈ ℂ
∧ 1 ∈ ℂ) → (((𝐵 , 𝐵) + 1) + 1) = ((𝐵 , 𝐵) + (1 + 1))) |
37 | 35, 35, 36 | mp3an23 1455 |
. . . . . . . . . . 11
⊢ ((𝐵 , 𝐵) ∈ ℂ → (((𝐵 , 𝐵) + 1) + 1) = ((𝐵 , 𝐵) + (1 + 1))) |
38 | 34, 37 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → (((𝐵 , 𝐵) + 1) + 1) = ((𝐵 , 𝐵) + (1 + 1))) |
39 | 33, 38 | eqtr4id 2797 |
. . . . . . . . 9
⊢ (𝜑 → ((𝐵 , 𝐵) + 2) = (((𝐵 , 𝐵) + 1) + 1)) |
40 | 31, 39 | breqtrrd 5086 |
. . . . . . . 8
⊢ (𝜑 → ((𝐵 , 𝐵) + 1) < ((𝐵 , 𝐵) + 2)) |
41 | 24, 26, 23, 30, 40 | lttrd 10998 |
. . . . . . 7
⊢ (𝜑 → 0 < ((𝐵 , 𝐵) + 2)) |
42 | 23, 41 | elrpd 12630 |
. . . . . 6
⊢ (𝜑 → ((𝐵 , 𝐵) + 2) ∈
ℝ+) |
43 | | oveq2 7226 |
. . . . . . . . . . . . . 14
⊢ (𝑥 = (𝑇( ·𝑠
‘𝑊)𝐵) → (𝐴 − 𝑥) = (𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵))) |
44 | 43 | fveq2d 6726 |
. . . . . . . . . . . . 13
⊢ (𝑥 = (𝑇( ·𝑠
‘𝑊)𝐵) → (𝑁‘(𝐴 − 𝑥)) = (𝑁‘(𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵)))) |
45 | 44 | breq2d 5070 |
. . . . . . . . . . . 12
⊢ (𝑥 = (𝑇( ·𝑠
‘𝑊)𝐵) → ((𝑁‘𝐴) ≤ (𝑁‘(𝐴 − 𝑥)) ↔ (𝑁‘𝐴) ≤ (𝑁‘(𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵))))) |
46 | | pjthlem.7 |
. . . . . . . . . . . 12
⊢ (𝜑 → ∀𝑥 ∈ 𝑈 (𝑁‘𝐴) ≤ (𝑁‘(𝐴 − 𝑥))) |
47 | | cphlmod 24076 |
. . . . . . . . . . . . . 14
⊢ (𝑊 ∈ ℂPreHil →
𝑊 ∈
LMod) |
48 | 3, 47 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑊 ∈ LMod) |
49 | | pjthlem.8 |
. . . . . . . . . . . . . 14
⊢ 𝑇 = ((𝐴 , 𝐵) / ((𝐵 , 𝐵) + 1)) |
50 | | hlphl 24267 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ PreHil) |
51 | 1, 50 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝑊 ∈ PreHil) |
52 | | eqid 2737 |
. . . . . . . . . . . . . . . . 17
⊢
(Scalar‘𝑊) =
(Scalar‘𝑊) |
53 | | eqid 2737 |
. . . . . . . . . . . . . . . . 17
⊢
(Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) |
54 | 52, 12, 6, 53 | ipcl 20600 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴 , 𝐵) ∈ (Base‘(Scalar‘𝑊))) |
55 | 51, 4, 11, 54 | syl3anc 1373 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝐴 , 𝐵) ∈ (Base‘(Scalar‘𝑊))) |
56 | 52, 53 | hlress 24270 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑊 ∈ ℂHil →
ℝ ⊆ (Base‘(Scalar‘𝑊))) |
57 | 1, 56 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ℝ ⊆
(Base‘(Scalar‘𝑊))) |
58 | 57, 26 | sseldd 3907 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ((𝐵 , 𝐵) + 1) ∈
(Base‘(Scalar‘𝑊))) |
59 | 20, 28 | ge0p1rpd 12663 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((𝐵 , 𝐵) + 1) ∈
ℝ+) |
60 | 59 | rpne0d 12638 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ((𝐵 , 𝐵) + 1) ≠ 0) |
61 | 52, 53 | cphdivcl 24084 |
. . . . . . . . . . . . . . 15
⊢ ((𝑊 ∈ ℂPreHil ∧
((𝐴 , 𝐵) ∈ (Base‘(Scalar‘𝑊)) ∧ ((𝐵 , 𝐵) + 1) ∈
(Base‘(Scalar‘𝑊)) ∧ ((𝐵 , 𝐵) + 1) ≠ 0)) → ((𝐴 , 𝐵) / ((𝐵 , 𝐵) + 1)) ∈
(Base‘(Scalar‘𝑊))) |
62 | 3, 55, 58, 60, 61 | syl13anc 1374 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((𝐴 , 𝐵) / ((𝐵 , 𝐵) + 1)) ∈
(Base‘(Scalar‘𝑊))) |
63 | 49, 62 | eqeltrid 2842 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑇 ∈ (Base‘(Scalar‘𝑊))) |
64 | | eqid 2737 |
. . . . . . . . . . . . . 14
⊢ (
·𝑠 ‘𝑊) = ( ·𝑠
‘𝑊) |
65 | 52, 64, 53, 7 | lssvscl 19997 |
. . . . . . . . . . . . 13
⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿) ∧ (𝑇 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝐵 ∈ 𝑈)) → (𝑇( ·𝑠
‘𝑊)𝐵) ∈ 𝑈) |
66 | 48, 5, 63, 10, 65 | syl22anc 839 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑇( ·𝑠
‘𝑊)𝐵) ∈ 𝑈) |
67 | 45, 46, 66 | rspcdva 3544 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑁‘𝐴) ≤ (𝑁‘(𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵)))) |
68 | | cphngp 24075 |
. . . . . . . . . . . . . 14
⊢ (𝑊 ∈ ℂPreHil →
𝑊 ∈
NrmGrp) |
69 | 3, 68 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑊 ∈ NrmGrp) |
70 | | pjthlem.n |
. . . . . . . . . . . . . 14
⊢ 𝑁 = (norm‘𝑊) |
71 | 6, 70 | nmcl 23519 |
. . . . . . . . . . . . 13
⊢ ((𝑊 ∈ NrmGrp ∧ 𝐴 ∈ 𝑉) → (𝑁‘𝐴) ∈ ℝ) |
72 | 69, 4, 71 | syl2anc 587 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑁‘𝐴) ∈ ℝ) |
73 | 6, 52, 64, 53 | lmodvscl 19921 |
. . . . . . . . . . . . . . 15
⊢ ((𝑊 ∈ LMod ∧ 𝑇 ∈
(Base‘(Scalar‘𝑊)) ∧ 𝐵 ∈ 𝑉) → (𝑇( ·𝑠
‘𝑊)𝐵) ∈ 𝑉) |
74 | 48, 63, 11, 73 | syl3anc 1373 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑇( ·𝑠
‘𝑊)𝐵) ∈ 𝑉) |
75 | | pjthlem.m |
. . . . . . . . . . . . . . 15
⊢ − =
(-g‘𝑊) |
76 | 6, 75 | lmodvsubcl 19949 |
. . . . . . . . . . . . . 14
⊢ ((𝑊 ∈ LMod ∧ 𝐴 ∈ 𝑉 ∧ (𝑇( ·𝑠
‘𝑊)𝐵) ∈ 𝑉) → (𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵)) ∈ 𝑉) |
77 | 48, 4, 74, 76 | syl3anc 1373 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵)) ∈ 𝑉) |
78 | 6, 70 | nmcl 23519 |
. . . . . . . . . . . . 13
⊢ ((𝑊 ∈ NrmGrp ∧ (𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵)) ∈ 𝑉) → (𝑁‘(𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵))) ∈ ℝ) |
79 | 69, 77, 78 | syl2anc 587 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑁‘(𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵))) ∈ ℝ) |
80 | 6, 70 | nmge0 23520 |
. . . . . . . . . . . . 13
⊢ ((𝑊 ∈ NrmGrp ∧ 𝐴 ∈ 𝑉) → 0 ≤ (𝑁‘𝐴)) |
81 | 69, 4, 80 | syl2anc 587 |
. . . . . . . . . . . 12
⊢ (𝜑 → 0 ≤ (𝑁‘𝐴)) |
82 | 6, 70 | nmge0 23520 |
. . . . . . . . . . . . 13
⊢ ((𝑊 ∈ NrmGrp ∧ (𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵)) ∈ 𝑉) → 0 ≤ (𝑁‘(𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵)))) |
83 | 69, 77, 82 | syl2anc 587 |
. . . . . . . . . . . 12
⊢ (𝜑 → 0 ≤ (𝑁‘(𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵)))) |
84 | 72, 79, 81, 83 | le2sqd 13831 |
. . . . . . . . . . 11
⊢ (𝜑 → ((𝑁‘𝐴) ≤ (𝑁‘(𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵))) ↔ ((𝑁‘𝐴)↑2) ≤ ((𝑁‘(𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵)))↑2))) |
85 | 67, 84 | mpbid 235 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝑁‘𝐴)↑2) ≤ ((𝑁‘(𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵)))↑2)) |
86 | 79 | resqcld 13822 |
. . . . . . . . . . 11
⊢ (𝜑 → ((𝑁‘(𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵)))↑2) ∈ ℝ) |
87 | 72 | resqcld 13822 |
. . . . . . . . . . 11
⊢ (𝜑 → ((𝑁‘𝐴)↑2) ∈ ℝ) |
88 | 86, 87 | subge0d 11427 |
. . . . . . . . . 10
⊢ (𝜑 → (0 ≤ (((𝑁‘(𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵)))↑2) − ((𝑁‘𝐴)↑2)) ↔ ((𝑁‘𝐴)↑2) ≤ ((𝑁‘(𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵)))↑2))) |
89 | 85, 88 | mpbird 260 |
. . . . . . . . 9
⊢ (𝜑 → 0 ≤ (((𝑁‘(𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵)))↑2) − ((𝑁‘𝐴)↑2))) |
90 | | 2z 12214 |
. . . . . . . . . . . . . . . 16
⊢ 2 ∈
ℤ |
91 | | rpexpcl 13659 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝐵 , 𝐵) + 1) ∈ ℝ+ ∧ 2
∈ ℤ) → (((𝐵
, 𝐵) + 1)↑2) ∈
ℝ+) |
92 | 59, 90, 91 | sylancl 589 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (((𝐵 , 𝐵) + 1)↑2) ∈
ℝ+) |
93 | 17, 92 | rerpdivcld 12664 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) ∈
ℝ) |
94 | 93, 23 | remulcld 10868 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · ((𝐵 , 𝐵) + 2)) ∈ ℝ) |
95 | 94 | recnd 10866 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · ((𝐵 , 𝐵) + 2)) ∈ ℂ) |
96 | 95 | negcld 11181 |
. . . . . . . . . . 11
⊢ (𝜑 → -((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · ((𝐵 , 𝐵) + 2)) ∈ ℂ) |
97 | 6, 12 | cphipcl 24093 |
. . . . . . . . . . . 12
⊢ ((𝑊 ∈ ℂPreHil ∧
𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → (𝐴 , 𝐴) ∈ ℂ) |
98 | 3, 4, 4, 97 | syl3anc 1373 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐴 , 𝐴) ∈ ℂ) |
99 | 96, 98 | pncand 11195 |
. . . . . . . . . 10
⊢ (𝜑 → ((-((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · ((𝐵 , 𝐵) + 2)) + (𝐴 , 𝐴)) − (𝐴 , 𝐴)) = -((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · ((𝐵 , 𝐵) + 2))) |
100 | 6, 12, 70 | nmsq 24096 |
. . . . . . . . . . . . . 14
⊢ ((𝑊 ∈ ℂPreHil ∧
(𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵)) ∈ 𝑉) → ((𝑁‘(𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵)))↑2) = ((𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵)) , (𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵)))) |
101 | 3, 77, 100 | syl2anc 587 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((𝑁‘(𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵)))↑2) = ((𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵)) , (𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵)))) |
102 | 12, 6, 75 | cphsubdir 24110 |
. . . . . . . . . . . . . 14
⊢ ((𝑊 ∈ ℂPreHil ∧
(𝐴 ∈ 𝑉 ∧ (𝑇( ·𝑠
‘𝑊)𝐵) ∈ 𝑉 ∧ (𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵)) ∈ 𝑉)) → ((𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵)) , (𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵))) = ((𝐴 , (𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵))) − ((𝑇( ·𝑠
‘𝑊)𝐵) , (𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵))))) |
103 | 3, 4, 74, 77, 102 | syl13anc 1374 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵)) , (𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵))) = ((𝐴 , (𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵))) − ((𝑇( ·𝑠
‘𝑊)𝐵) , (𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵))))) |
104 | 12, 6, 75 | cphsubdi 24111 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑊 ∈ ℂPreHil ∧
(𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉 ∧ (𝑇( ·𝑠
‘𝑊)𝐵) ∈ 𝑉)) → (𝐴 , (𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵))) = ((𝐴 , 𝐴) − (𝐴 , (𝑇( ·𝑠
‘𝑊)𝐵)))) |
105 | 3, 4, 4, 74, 104 | syl13anc 1374 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝐴 , (𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵))) = ((𝐴 , 𝐴) − (𝐴 , (𝑇( ·𝑠
‘𝑊)𝐵)))) |
106 | 105 | oveq1d 7233 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((𝐴 , (𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵))) − ((𝑇( ·𝑠
‘𝑊)𝐵) , (𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵)))) = (((𝐴 , 𝐴) − (𝐴 , (𝑇( ·𝑠
‘𝑊)𝐵))) − ((𝑇( ·𝑠
‘𝑊)𝐵) , (𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵))))) |
107 | 6, 12 | cphipcl 24093 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑊 ∈ ℂPreHil ∧
𝐴 ∈ 𝑉 ∧ (𝑇( ·𝑠
‘𝑊)𝐵) ∈ 𝑉) → (𝐴 , (𝑇( ·𝑠
‘𝑊)𝐵)) ∈ ℂ) |
108 | 3, 4, 74, 107 | syl3anc 1373 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝐴 , (𝑇( ·𝑠
‘𝑊)𝐵)) ∈ ℂ) |
109 | 12, 6, 75 | cphsubdi 24111 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑊 ∈ ℂPreHil ∧
((𝑇(
·𝑠 ‘𝑊)𝐵) ∈ 𝑉 ∧ 𝐴 ∈ 𝑉 ∧ (𝑇( ·𝑠
‘𝑊)𝐵) ∈ 𝑉)) → ((𝑇( ·𝑠
‘𝑊)𝐵) , (𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵))) = (((𝑇( ·𝑠
‘𝑊)𝐵) , 𝐴) − ((𝑇( ·𝑠
‘𝑊)𝐵) , (𝑇( ·𝑠
‘𝑊)𝐵)))) |
110 | 3, 74, 4, 74, 109 | syl13anc 1374 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((𝑇( ·𝑠
‘𝑊)𝐵) , (𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵))) = (((𝑇( ·𝑠
‘𝑊)𝐵) , 𝐴) − ((𝑇( ·𝑠
‘𝑊)𝐵) , (𝑇( ·𝑠
‘𝑊)𝐵)))) |
111 | 6, 12 | cphipcl 24093 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑊 ∈ ℂPreHil ∧
(𝑇(
·𝑠 ‘𝑊)𝐵) ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → ((𝑇( ·𝑠
‘𝑊)𝐵) , 𝐴) ∈ ℂ) |
112 | 3, 74, 4, 111 | syl3anc 1373 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → ((𝑇( ·𝑠
‘𝑊)𝐵) , 𝐴) ∈ ℂ) |
113 | 6, 12 | cphipcl 24093 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑊 ∈ ℂPreHil ∧
(𝑇(
·𝑠 ‘𝑊)𝐵) ∈ 𝑉 ∧ (𝑇( ·𝑠
‘𝑊)𝐵) ∈ 𝑉) → ((𝑇( ·𝑠
‘𝑊)𝐵) , (𝑇( ·𝑠
‘𝑊)𝐵)) ∈ ℂ) |
114 | 3, 74, 74, 113 | syl3anc 1373 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → ((𝑇( ·𝑠
‘𝑊)𝐵) , (𝑇( ·𝑠
‘𝑊)𝐵)) ∈ ℂ) |
115 | 112, 114 | subcld 11194 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (((𝑇( ·𝑠
‘𝑊)𝐵) , 𝐴) − ((𝑇( ·𝑠
‘𝑊)𝐵) , (𝑇( ·𝑠
‘𝑊)𝐵))) ∈ ℂ) |
116 | 110, 115 | eqeltrd 2838 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ((𝑇( ·𝑠
‘𝑊)𝐵) , (𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵))) ∈ ℂ) |
117 | 98, 108, 116 | subsub4d 11225 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (((𝐴 , 𝐴) − (𝐴 , (𝑇( ·𝑠
‘𝑊)𝐵))) − ((𝑇( ·𝑠
‘𝑊)𝐵) , (𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵)))) = ((𝐴 , 𝐴) − ((𝐴 , (𝑇( ·𝑠
‘𝑊)𝐵)) + ((𝑇( ·𝑠
‘𝑊)𝐵) , (𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵)))))) |
118 | 93 | recnd 10866 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) ∈
ℂ) |
119 | 26 | recnd 10866 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → ((𝐵 , 𝐵) + 1) ∈ ℂ) |
120 | | 1cnd 10833 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 1 ∈
ℂ) |
121 | 118, 119,
120 | adddid 10862 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · (((𝐵 , 𝐵) + 1) + 1)) = (((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · ((𝐵 , 𝐵) + 1)) + ((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) ·
1))) |
122 | 39 | oveq2d 7234 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · ((𝐵 , 𝐵) + 2)) = ((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · (((𝐵 , 𝐵) + 1) + 1))) |
123 | 12, 6, 52, 53, 64 | cphassr 24114 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑊 ∈ ℂPreHil ∧
(𝑇 ∈
(Base‘(Scalar‘𝑊)) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → (𝐴 , (𝑇( ·𝑠
‘𝑊)𝐵)) = ((∗‘𝑇) · (𝐴 , 𝐵))) |
124 | 3, 63, 4, 11, 123 | syl13anc 1374 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (𝐴 , (𝑇( ·𝑠
‘𝑊)𝐵)) = ((∗‘𝑇) · (𝐴 , 𝐵))) |
125 | 14, 119, 60 | divcld 11613 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → ((𝐴 , 𝐵) / ((𝐵 , 𝐵) + 1)) ∈ ℂ) |
126 | 49, 125 | eqeltrid 2842 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → 𝑇 ∈ ℂ) |
127 | 126 | cjcld 14764 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (∗‘𝑇) ∈
ℂ) |
128 | 127, 14 | mulcomd 10859 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → ((∗‘𝑇) · (𝐴 , 𝐵)) = ((𝐴 , 𝐵) · (∗‘𝑇))) |
129 | 14 | cjcld 14764 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (∗‘(𝐴 , 𝐵)) ∈ ℂ) |
130 | 14, 129, 119, 60 | divassd 11648 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (((𝐴 , 𝐵) · (∗‘(𝐴 , 𝐵))) / ((𝐵 , 𝐵) + 1)) = ((𝐴 , 𝐵) · ((∗‘(𝐴 , 𝐵)) / ((𝐵 , 𝐵) + 1)))) |
131 | 14 | absvalsqd 15011 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → ((abs‘(𝐴 , 𝐵))↑2) = ((𝐴 , 𝐵) · (∗‘(𝐴 , 𝐵)))) |
132 | 131 | oveq1d 7233 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (((abs‘(𝐴 , 𝐵))↑2) / ((𝐵 , 𝐵) + 1)) = (((𝐴 , 𝐵) · (∗‘(𝐴 , 𝐵))) / ((𝐵 , 𝐵) + 1))) |
133 | 49 | fveq2i 6725 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(∗‘𝑇) =
(∗‘((𝐴 , 𝐵) / ((𝐵 , 𝐵) + 1))) |
134 | 14, 119, 60 | cjdivd 14791 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → (∗‘((𝐴 , 𝐵) / ((𝐵 , 𝐵) + 1))) = ((∗‘(𝐴 , 𝐵)) / (∗‘((𝐵 , 𝐵) + 1)))) |
135 | 26 | cjred 14794 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → (∗‘((𝐵 , 𝐵) + 1)) = ((𝐵 , 𝐵) + 1)) |
136 | 135 | oveq2d 7234 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → ((∗‘(𝐴 , 𝐵)) / (∗‘((𝐵 , 𝐵) + 1))) = ((∗‘(𝐴 , 𝐵)) / ((𝐵 , 𝐵) + 1))) |
137 | 134, 136 | eqtrd 2777 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → (∗‘((𝐴 , 𝐵) / ((𝐵 , 𝐵) + 1))) = ((∗‘(𝐴 , 𝐵)) / ((𝐵 , 𝐵) + 1))) |
138 | 133, 137 | syl5eq 2790 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (∗‘𝑇) = ((∗‘(𝐴 , 𝐵)) / ((𝐵 , 𝐵) + 1))) |
139 | 138 | oveq2d 7234 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → ((𝐴 , 𝐵) · (∗‘𝑇)) = ((𝐴 , 𝐵) · ((∗‘(𝐴 , 𝐵)) / ((𝐵 , 𝐵) + 1)))) |
140 | 130, 132,
139 | 3eqtr4rd 2788 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → ((𝐴 , 𝐵) · (∗‘𝑇)) = (((abs‘(𝐴 , 𝐵))↑2) / ((𝐵 , 𝐵) + 1))) |
141 | 124, 128,
140 | 3eqtrd 2781 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (𝐴 , (𝑇( ·𝑠
‘𝑊)𝐵)) = (((abs‘(𝐴 , 𝐵))↑2) / ((𝐵 , 𝐵) + 1))) |
142 | 17 | recnd 10866 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → ((abs‘(𝐴 , 𝐵))↑2) ∈ ℂ) |
143 | 142, 119 | mulcomd 10859 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (((abs‘(𝐴 , 𝐵))↑2) · ((𝐵 , 𝐵) + 1)) = (((𝐵 , 𝐵) + 1) · ((abs‘(𝐴 , 𝐵))↑2))) |
144 | 119 | sqvald 13718 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (((𝐵 , 𝐵) + 1)↑2) = (((𝐵 , 𝐵) + 1) · ((𝐵 , 𝐵) + 1))) |
145 | 143, 144 | oveq12d 7236 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → ((((abs‘(𝐴 , 𝐵))↑2) · ((𝐵 , 𝐵) + 1)) / (((𝐵 , 𝐵) + 1)↑2)) = ((((𝐵 , 𝐵) + 1) · ((abs‘(𝐴 , 𝐵))↑2)) / (((𝐵 , 𝐵) + 1) · ((𝐵 , 𝐵) + 1)))) |
146 | 142, 119,
119, 60, 60 | divcan5d 11639 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → ((((𝐵 , 𝐵) + 1) · ((abs‘(𝐴 , 𝐵))↑2)) / (((𝐵 , 𝐵) + 1) · ((𝐵 , 𝐵) + 1))) = (((abs‘(𝐴 , 𝐵))↑2) / ((𝐵 , 𝐵) + 1))) |
147 | 145, 146 | eqtr2d 2778 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (((abs‘(𝐴 , 𝐵))↑2) / ((𝐵 , 𝐵) + 1)) = ((((abs‘(𝐴 , 𝐵))↑2) · ((𝐵 , 𝐵) + 1)) / (((𝐵 , 𝐵) + 1)↑2))) |
148 | 92 | rpcnd 12635 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (((𝐵 , 𝐵) + 1)↑2) ∈
ℂ) |
149 | 92 | rpne0d 12638 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (((𝐵 , 𝐵) + 1)↑2) ≠ 0) |
150 | 142, 119,
148, 149 | div23d 11650 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → ((((abs‘(𝐴 , 𝐵))↑2) · ((𝐵 , 𝐵) + 1)) / (((𝐵 , 𝐵) + 1)↑2)) = ((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · ((𝐵 , 𝐵) + 1))) |
151 | 141, 147,
150 | 3eqtrd 2781 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (𝐴 , (𝑇( ·𝑠
‘𝑊)𝐵)) = ((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · ((𝐵 , 𝐵) + 1))) |
152 | 93, 26 | remulcld 10868 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → ((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · ((𝐵 , 𝐵) + 1)) ∈ ℝ) |
153 | 151, 152 | eqeltrd 2838 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (𝐴 , (𝑇( ·𝑠
‘𝑊)𝐵)) ∈ ℝ) |
154 | 153 | cjred 14794 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (∗‘(𝐴 , (𝑇( ·𝑠
‘𝑊)𝐵))) = (𝐴 , (𝑇( ·𝑠
‘𝑊)𝐵))) |
155 | 12, 6 | cphipcj 24101 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑊 ∈ ℂPreHil ∧
𝐴 ∈ 𝑉 ∧ (𝑇( ·𝑠
‘𝑊)𝐵) ∈ 𝑉) → (∗‘(𝐴 , (𝑇( ·𝑠
‘𝑊)𝐵))) = ((𝑇( ·𝑠
‘𝑊)𝐵) , 𝐴)) |
156 | 3, 4, 74, 155 | syl3anc 1373 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (∗‘(𝐴 , (𝑇( ·𝑠
‘𝑊)𝐵))) = ((𝑇( ·𝑠
‘𝑊)𝐵) , 𝐴)) |
157 | 154, 156,
151 | 3eqtr3d 2785 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → ((𝑇( ·𝑠
‘𝑊)𝐵) , 𝐴) = ((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · ((𝐵 , 𝐵) + 1))) |
158 | 12, 6, 52, 53, 64 | cph2ass 24115 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑊 ∈ ℂPreHil ∧
(𝑇 ∈
(Base‘(Scalar‘𝑊)) ∧ 𝑇 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝐵 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → ((𝑇( ·𝑠
‘𝑊)𝐵) , (𝑇( ·𝑠
‘𝑊)𝐵)) = ((𝑇 · (∗‘𝑇)) · (𝐵 , 𝐵))) |
159 | 3, 63, 63, 11, 11, 158 | syl122anc 1381 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → ((𝑇( ·𝑠
‘𝑊)𝐵) , (𝑇( ·𝑠
‘𝑊)𝐵)) = ((𝑇 · (∗‘𝑇)) · (𝐵 , 𝐵))) |
160 | 49 | fveq2i 6725 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(abs‘𝑇) =
(abs‘((𝐴 , 𝐵) / ((𝐵 , 𝐵) + 1))) |
161 | 14, 119, 60 | absdivd 15024 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝜑 → (abs‘((𝐴 , 𝐵) / ((𝐵 , 𝐵) + 1))) = ((abs‘(𝐴 , 𝐵)) / (abs‘((𝐵 , 𝐵) + 1)))) |
162 | 59 | rpge0d 12637 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝜑 → 0 ≤ ((𝐵 , 𝐵) + 1)) |
163 | 26, 162 | absidd 14991 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝜑 → (abs‘((𝐵 , 𝐵) + 1)) = ((𝐵 , 𝐵) + 1)) |
164 | 163 | oveq2d 7234 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝜑 → ((abs‘(𝐴 , 𝐵)) / (abs‘((𝐵 , 𝐵) + 1))) = ((abs‘(𝐴 , 𝐵)) / ((𝐵 , 𝐵) + 1))) |
165 | 161, 164 | eqtrd 2777 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → (abs‘((𝐴 , 𝐵) / ((𝐵 , 𝐵) + 1))) = ((abs‘(𝐴 , 𝐵)) / ((𝐵 , 𝐵) + 1))) |
166 | 160, 165 | syl5eq 2790 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → (abs‘𝑇) = ((abs‘(𝐴 , 𝐵)) / ((𝐵 , 𝐵) + 1))) |
167 | 166 | oveq1d 7233 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → ((abs‘𝑇)↑2) = (((abs‘(𝐴 , 𝐵)) / ((𝐵 , 𝐵) + 1))↑2)) |
168 | 126 | absvalsqd 15011 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → ((abs‘𝑇)↑2) = (𝑇 · (∗‘𝑇))) |
169 | 16, 119, 60 | sqdivd 13734 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → (((abs‘(𝐴 , 𝐵)) / ((𝐵 , 𝐵) + 1))↑2) = (((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2))) |
170 | 167, 168,
169 | 3eqtr3d 2785 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (𝑇 · (∗‘𝑇)) = (((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2))) |
171 | 170 | oveq1d 7233 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → ((𝑇 · (∗‘𝑇)) · (𝐵 , 𝐵)) = ((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · (𝐵 , 𝐵))) |
172 | 159, 171 | eqtrd 2777 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → ((𝑇( ·𝑠
‘𝑊)𝐵) , (𝑇( ·𝑠
‘𝑊)𝐵)) = ((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · (𝐵 , 𝐵))) |
173 | 157, 172 | oveq12d 7236 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (((𝑇( ·𝑠
‘𝑊)𝐵) , 𝐴) − ((𝑇( ·𝑠
‘𝑊)𝐵) , (𝑇( ·𝑠
‘𝑊)𝐵))) = (((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · ((𝐵 , 𝐵) + 1)) − ((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · (𝐵 , 𝐵)))) |
174 | | pncan2 11090 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝐵 , 𝐵) ∈ ℂ ∧ 1 ∈ ℂ)
→ (((𝐵 , 𝐵) + 1) − (𝐵 , 𝐵)) = 1) |
175 | 34, 35, 174 | sylancl 589 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (((𝐵 , 𝐵) + 1) − (𝐵 , 𝐵)) = 1) |
176 | 175 | oveq2d 7234 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → ((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · (((𝐵 , 𝐵) + 1) − (𝐵 , 𝐵))) = ((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) ·
1)) |
177 | 118, 119,
34 | subdid 11293 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → ((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · (((𝐵 , 𝐵) + 1) − (𝐵 , 𝐵))) = (((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · ((𝐵 , 𝐵) + 1)) − ((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · (𝐵 , 𝐵)))) |
178 | 176, 177 | eqtr3d 2779 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → ((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · 1) =
(((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · ((𝐵 , 𝐵) + 1)) − ((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · (𝐵 , 𝐵)))) |
179 | 173, 110,
178 | 3eqtr4d 2787 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → ((𝑇( ·𝑠
‘𝑊)𝐵) , (𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵))) = ((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) ·
1)) |
180 | 151, 179 | oveq12d 7236 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((𝐴 , (𝑇( ·𝑠
‘𝑊)𝐵)) + ((𝑇( ·𝑠
‘𝑊)𝐵) , (𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵)))) = (((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · ((𝐵 , 𝐵) + 1)) + ((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) ·
1))) |
181 | 121, 122,
180 | 3eqtr4rd 2788 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ((𝐴 , (𝑇( ·𝑠
‘𝑊)𝐵)) + ((𝑇( ·𝑠
‘𝑊)𝐵) , (𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵)))) = ((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · ((𝐵 , 𝐵) + 2))) |
182 | 181 | oveq2d 7234 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((𝐴 , 𝐴) − ((𝐴 , (𝑇( ·𝑠
‘𝑊)𝐵)) + ((𝑇( ·𝑠
‘𝑊)𝐵) , (𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵))))) = ((𝐴 , 𝐴) − ((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · ((𝐵 , 𝐵) + 2)))) |
183 | 106, 117,
182 | 3eqtrd 2781 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((𝐴 , (𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵))) − ((𝑇( ·𝑠
‘𝑊)𝐵) , (𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵)))) = ((𝐴 , 𝐴) − ((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · ((𝐵 , 𝐵) + 2)))) |
184 | 101, 103,
183 | 3eqtrd 2781 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((𝑁‘(𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵)))↑2) = ((𝐴 , 𝐴) − ((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · ((𝐵 , 𝐵) + 2)))) |
185 | 98, 95 | negsubd 11200 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((𝐴 , 𝐴) + -((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · ((𝐵 , 𝐵) + 2))) = ((𝐴 , 𝐴) − ((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · ((𝐵 , 𝐵) + 2)))) |
186 | 98, 96 | addcomd 11039 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((𝐴 , 𝐴) + -((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · ((𝐵 , 𝐵) + 2))) = (-((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · ((𝐵 , 𝐵) + 2)) + (𝐴 , 𝐴))) |
187 | 184, 185,
186 | 3eqtr2d 2783 |
. . . . . . . . . . 11
⊢ (𝜑 → ((𝑁‘(𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵)))↑2) = (-((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · ((𝐵 , 𝐵) + 2)) + (𝐴 , 𝐴))) |
188 | 6, 12, 70 | nmsq 24096 |
. . . . . . . . . . . 12
⊢ ((𝑊 ∈ ℂPreHil ∧
𝐴 ∈ 𝑉) → ((𝑁‘𝐴)↑2) = (𝐴 , 𝐴)) |
189 | 3, 4, 188 | syl2anc 587 |
. . . . . . . . . . 11
⊢ (𝜑 → ((𝑁‘𝐴)↑2) = (𝐴 , 𝐴)) |
190 | 187, 189 | oveq12d 7236 |
. . . . . . . . . 10
⊢ (𝜑 → (((𝑁‘(𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵)))↑2) − ((𝑁‘𝐴)↑2)) = ((-((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · ((𝐵 , 𝐵) + 2)) + (𝐴 , 𝐴)) − (𝐴 , 𝐴))) |
191 | 23 | renegcld 11264 |
. . . . . . . . . . . . 13
⊢ (𝜑 → -((𝐵 , 𝐵) + 2) ∈ ℝ) |
192 | 191 | recnd 10866 |
. . . . . . . . . . . 12
⊢ (𝜑 → -((𝐵 , 𝐵) + 2) ∈ ℂ) |
193 | 142, 192,
148, 149 | div23d 11650 |
. . . . . . . . . . 11
⊢ (𝜑 → ((((abs‘(𝐴 , 𝐵))↑2) · -((𝐵 , 𝐵) + 2)) / (((𝐵 , 𝐵) + 1)↑2)) = ((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · -((𝐵 , 𝐵) + 2))) |
194 | 23 | recnd 10866 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((𝐵 , 𝐵) + 2) ∈ ℂ) |
195 | 118, 194 | mulneg2d 11291 |
. . . . . . . . . . 11
⊢ (𝜑 → ((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · -((𝐵 , 𝐵) + 2)) = -((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · ((𝐵 , 𝐵) + 2))) |
196 | 193, 195 | eqtrd 2777 |
. . . . . . . . . 10
⊢ (𝜑 → ((((abs‘(𝐴 , 𝐵))↑2) · -((𝐵 , 𝐵) + 2)) / (((𝐵 , 𝐵) + 1)↑2)) = -((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · ((𝐵 , 𝐵) + 2))) |
197 | 99, 190, 196 | 3eqtr4rd 2788 |
. . . . . . . . 9
⊢ (𝜑 → ((((abs‘(𝐴 , 𝐵))↑2) · -((𝐵 , 𝐵) + 2)) / (((𝐵 , 𝐵) + 1)↑2)) = (((𝑁‘(𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵)))↑2) − ((𝑁‘𝐴)↑2))) |
198 | 89, 197 | breqtrrd 5086 |
. . . . . . . 8
⊢ (𝜑 → 0 ≤
((((abs‘(𝐴 , 𝐵))↑2) · -((𝐵 , 𝐵) + 2)) / (((𝐵 , 𝐵) + 1)↑2))) |
199 | 17, 191 | remulcld 10868 |
. . . . . . . . 9
⊢ (𝜑 → (((abs‘(𝐴 , 𝐵))↑2) · -((𝐵 , 𝐵) + 2)) ∈ ℝ) |
200 | 199, 92 | ge0divd 12671 |
. . . . . . . 8
⊢ (𝜑 → (0 ≤
(((abs‘(𝐴 , 𝐵))↑2) · -((𝐵 , 𝐵) + 2)) ↔ 0 ≤ ((((abs‘(𝐴 , 𝐵))↑2) · -((𝐵 , 𝐵) + 2)) / (((𝐵 , 𝐵) + 1)↑2)))) |
201 | 198, 200 | mpbird 260 |
. . . . . . 7
⊢ (𝜑 → 0 ≤ (((abs‘(𝐴 , 𝐵))↑2) · -((𝐵 , 𝐵) + 2))) |
202 | | mulneg12 11275 |
. . . . . . . 8
⊢
((((abs‘(𝐴
, 𝐵))↑2) ∈ ℂ ∧
((𝐵 , 𝐵) + 2) ∈ ℂ) →
(-((abs‘(𝐴 , 𝐵))↑2) · ((𝐵 , 𝐵) + 2)) = (((abs‘(𝐴 , 𝐵))↑2) · -((𝐵 , 𝐵) + 2))) |
203 | 142, 194,
202 | syl2anc 587 |
. . . . . . 7
⊢ (𝜑 → (-((abs‘(𝐴 , 𝐵))↑2) · ((𝐵 , 𝐵) + 2)) = (((abs‘(𝐴 , 𝐵))↑2) · -((𝐵 , 𝐵) + 2))) |
204 | 201, 203 | breqtrrd 5086 |
. . . . . 6
⊢ (𝜑 → 0 ≤
(-((abs‘(𝐴 , 𝐵))↑2) · ((𝐵 , 𝐵) + 2))) |
205 | 18, 42, 204 | prodge0ld 12699 |
. . . . 5
⊢ (𝜑 → 0 ≤ -((abs‘(𝐴 , 𝐵))↑2)) |
206 | 17 | le0neg1d 11408 |
. . . . 5
⊢ (𝜑 → (((abs‘(𝐴 , 𝐵))↑2) ≤ 0 ↔ 0 ≤
-((abs‘(𝐴 , 𝐵))↑2))) |
207 | 205, 206 | mpbird 260 |
. . . 4
⊢ (𝜑 → ((abs‘(𝐴 , 𝐵))↑2) ≤ 0) |
208 | 15 | sqge0d 13823 |
. . . 4
⊢ (𝜑 → 0 ≤ ((abs‘(𝐴 , 𝐵))↑2)) |
209 | | 0re 10840 |
. . . . 5
⊢ 0 ∈
ℝ |
210 | | letri3 10923 |
. . . . 5
⊢
((((abs‘(𝐴
, 𝐵))↑2) ∈ ℝ ∧
0 ∈ ℝ) → (((abs‘(𝐴 , 𝐵))↑2) = 0 ↔ (((abs‘(𝐴 , 𝐵))↑2) ≤ 0 ∧ 0 ≤
((abs‘(𝐴 , 𝐵))↑2)))) |
211 | 17, 209, 210 | sylancl 589 |
. . . 4
⊢ (𝜑 → (((abs‘(𝐴 , 𝐵))↑2) = 0 ↔ (((abs‘(𝐴 , 𝐵))↑2) ≤ 0 ∧ 0 ≤
((abs‘(𝐴 , 𝐵))↑2)))) |
212 | 207, 208,
211 | mpbir2and 713 |
. . 3
⊢ (𝜑 → ((abs‘(𝐴 , 𝐵))↑2) = 0) |
213 | 16, 212 | sqeq0d 13720 |
. 2
⊢ (𝜑 → (abs‘(𝐴 , 𝐵)) = 0) |
214 | 14, 213 | abs00d 15015 |
1
⊢ (𝜑 → (𝐴 , 𝐵) = 0) |