Proof of Theorem pjthlem1
Step | Hyp | Ref
| Expression |
1 | | pjthlem.1 |
. . . 4
⊢ (𝜑 → 𝑊 ∈ ℂHil) |
2 | | hlcph 22968 |
. . . 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 18758 |
. . . . 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 3569 |
. . 3
⊢ (𝜑 → 𝐵 ∈ 𝑉) |
12 | | pjthlem.h |
. . . 4
⊢ , =
(·𝑖‘𝑊) |
13 | 6, 12 | cphipcl 22799 |
. . 3
⊢ ((𝑊 ∈ ℂPreHil ∧
𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴 , 𝐵) ∈ ℂ) |
14 | 3, 4, 11, 13 | syl3anc 1318 |
. 2
⊢ (𝜑 → (𝐴 , 𝐵) ∈ ℂ) |
15 | 14 | abscld 14023 |
. . . 4
⊢ (𝜑 → (abs‘(𝐴 , 𝐵)) ∈ ℝ) |
16 | 15 | recnd 9947 |
. . 3
⊢ (𝜑 → (abs‘(𝐴 , 𝐵)) ∈ ℂ) |
17 | 15 | resqcld 12897 |
. . . . . . 7
⊢ (𝜑 → ((abs‘(𝐴 , 𝐵))↑2) ∈ ℝ) |
18 | 17 | renegcld 10336 |
. . . . . 6
⊢ (𝜑 → -((abs‘(𝐴 , 𝐵))↑2) ∈ ℝ) |
19 | 6, 12 | reipcl 22805 |
. . . . . . . 8
⊢ ((𝑊 ∈ ℂPreHil ∧
𝐵 ∈ 𝑉) → (𝐵 , 𝐵) ∈ ℝ) |
20 | 3, 11, 19 | syl2anc 691 |
. . . . . . 7
⊢ (𝜑 → (𝐵 , 𝐵) ∈ ℝ) |
21 | | 2re 10967 |
. . . . . . 7
⊢ 2 ∈
ℝ |
22 | | readdcl 9898 |
. . . . . . 7
⊢ (((𝐵 , 𝐵) ∈ ℝ ∧ 2 ∈ ℝ)
→ ((𝐵 , 𝐵) + 2) ∈
ℝ) |
23 | 20, 21, 22 | sylancl 693 |
. . . . . 6
⊢ (𝜑 → ((𝐵 , 𝐵) + 2) ∈ ℝ) |
24 | | 0red 9920 |
. . . . . . 7
⊢ (𝜑 → 0 ∈
ℝ) |
25 | | peano2re 10088 |
. . . . . . . 8
⊢ ((𝐵 , 𝐵) ∈ ℝ → ((𝐵 , 𝐵) + 1) ∈ ℝ) |
26 | 20, 25 | syl 17 |
. . . . . . 7
⊢ (𝜑 → ((𝐵 , 𝐵) + 1) ∈ ℝ) |
27 | 6, 12 | ipge0 22806 |
. . . . . . . . 9
⊢ ((𝑊 ∈ ℂPreHil ∧
𝐵 ∈ 𝑉) → 0 ≤ (𝐵 , 𝐵)) |
28 | 3, 11, 27 | syl2anc 691 |
. . . . . . . 8
⊢ (𝜑 → 0 ≤ (𝐵 , 𝐵)) |
29 | 20 | ltp1d 10833 |
. . . . . . . 8
⊢ (𝜑 → (𝐵 , 𝐵) < ((𝐵 , 𝐵) + 1)) |
30 | 24, 20, 26, 28, 29 | lelttrd 10074 |
. . . . . . 7
⊢ (𝜑 → 0 < ((𝐵 , 𝐵) + 1)) |
31 | 26 | ltp1d 10833 |
. . . . . . . 8
⊢ (𝜑 → ((𝐵 , 𝐵) + 1) < (((𝐵 , 𝐵) + 1) + 1)) |
32 | 20 | recnd 9947 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐵 , 𝐵) ∈ ℂ) |
33 | | ax-1cn 9873 |
. . . . . . . . . . 11
⊢ 1 ∈
ℂ |
34 | | addass 9902 |
. . . . . . . . . . 11
⊢ (((𝐵 , 𝐵) ∈ ℂ ∧ 1 ∈ ℂ
∧ 1 ∈ ℂ) → (((𝐵 , 𝐵) + 1) + 1) = ((𝐵 , 𝐵) + (1 + 1))) |
35 | 33, 33, 34 | mp3an23 1408 |
. . . . . . . . . 10
⊢ ((𝐵 , 𝐵) ∈ ℂ → (((𝐵 , 𝐵) + 1) + 1) = ((𝐵 , 𝐵) + (1 + 1))) |
36 | 32, 35 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → (((𝐵 , 𝐵) + 1) + 1) = ((𝐵 , 𝐵) + (1 + 1))) |
37 | | df-2 10956 |
. . . . . . . . . 10
⊢ 2 = (1 +
1) |
38 | 37 | oveq2i 6560 |
. . . . . . . . 9
⊢ ((𝐵 , 𝐵) + 2) = ((𝐵 , 𝐵) + (1 + 1)) |
39 | 36, 38 | syl6reqr 2663 |
. . . . . . . 8
⊢ (𝜑 → ((𝐵 , 𝐵) + 2) = (((𝐵 , 𝐵) + 1) + 1)) |
40 | 31, 39 | breqtrrd 4611 |
. . . . . . 7
⊢ (𝜑 → ((𝐵 , 𝐵) + 1) < ((𝐵 , 𝐵) + 2)) |
41 | 24, 26, 23, 30, 40 | lttrd 10077 |
. . . . . 6
⊢ (𝜑 → 0 < ((𝐵 , 𝐵) + 2)) |
42 | | cphlmod 22782 |
. . . . . . . . . . . . . 14
⊢ (𝑊 ∈ ℂPreHil →
𝑊 ∈
LMod) |
43 | 3, 42 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑊 ∈ LMod) |
44 | | pjthlem.8 |
. . . . . . . . . . . . . 14
⊢ 𝑇 = ((𝐴 , 𝐵) / ((𝐵 , 𝐵) + 1)) |
45 | | hlphl 22969 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ PreHil) |
46 | 1, 45 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝑊 ∈ PreHil) |
47 | | eqid 2610 |
. . . . . . . . . . . . . . . . 17
⊢
(Scalar‘𝑊) =
(Scalar‘𝑊) |
48 | | eqid 2610 |
. . . . . . . . . . . . . . . . 17
⊢
(Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) |
49 | 47, 12, 6, 48 | ipcl 19797 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴 , 𝐵) ∈ (Base‘(Scalar‘𝑊))) |
50 | 46, 4, 11, 49 | syl3anc 1318 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝐴 , 𝐵) ∈ (Base‘(Scalar‘𝑊))) |
51 | 47, 48 | hlress 22972 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑊 ∈ ℂHil →
ℝ ⊆ (Base‘(Scalar‘𝑊))) |
52 | 1, 51 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ℝ ⊆
(Base‘(Scalar‘𝑊))) |
53 | 52, 26 | sseldd 3569 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ((𝐵 , 𝐵) + 1) ∈
(Base‘(Scalar‘𝑊))) |
54 | 20, 28 | ge0p1rpd 11778 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((𝐵 , 𝐵) + 1) ∈
ℝ+) |
55 | 54 | rpne0d 11753 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ((𝐵 , 𝐵) + 1) ≠ 0) |
56 | 47, 48 | cphdivcl 22790 |
. . . . . . . . . . . . . . 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))) |
57 | 3, 50, 53, 55, 56 | syl13anc 1320 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((𝐴 , 𝐵) / ((𝐵 , 𝐵) + 1)) ∈
(Base‘(Scalar‘𝑊))) |
58 | 44, 57 | syl5eqel 2692 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑇 ∈ (Base‘(Scalar‘𝑊))) |
59 | | eqid 2610 |
. . . . . . . . . . . . . 14
⊢ (
·𝑠 ‘𝑊) = ( ·𝑠
‘𝑊) |
60 | 47, 59, 48, 7 | lssvscl 18776 |
. . . . . . . . . . . . 13
⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿) ∧ (𝑇 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝐵 ∈ 𝑈)) → (𝑇( ·𝑠
‘𝑊)𝐵) ∈ 𝑈) |
61 | 43, 5, 58, 10, 60 | syl22anc 1319 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑇( ·𝑠
‘𝑊)𝐵) ∈ 𝑈) |
62 | | pjthlem.7 |
. . . . . . . . . . . 12
⊢ (𝜑 → ∀𝑥 ∈ 𝑈 (𝑁‘𝐴) ≤ (𝑁‘(𝐴 − 𝑥))) |
63 | | oveq2 6557 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 = (𝑇( ·𝑠
‘𝑊)𝐵) → (𝐴 − 𝑥) = (𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵))) |
64 | 63 | fveq2d 6107 |
. . . . . . . . . . . . . 14
⊢ (𝑥 = (𝑇( ·𝑠
‘𝑊)𝐵) → (𝑁‘(𝐴 − 𝑥)) = (𝑁‘(𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵)))) |
65 | 64 | breq2d 4595 |
. . . . . . . . . . . . 13
⊢ (𝑥 = (𝑇( ·𝑠
‘𝑊)𝐵) → ((𝑁‘𝐴) ≤ (𝑁‘(𝐴 − 𝑥)) ↔ (𝑁‘𝐴) ≤ (𝑁‘(𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵))))) |
66 | 65 | rspcv 3278 |
. . . . . . . . . . . 12
⊢ ((𝑇(
·𝑠 ‘𝑊)𝐵) ∈ 𝑈 → (∀𝑥 ∈ 𝑈 (𝑁‘𝐴) ≤ (𝑁‘(𝐴 − 𝑥)) → (𝑁‘𝐴) ≤ (𝑁‘(𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵))))) |
67 | 61, 62, 66 | sylc 63 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑁‘𝐴) ≤ (𝑁‘(𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵)))) |
68 | | cphngp 22781 |
. . . . . . . . . . . . . 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 22230 |
. . . . . . . . . . . . 13
⊢ ((𝑊 ∈ NrmGrp ∧ 𝐴 ∈ 𝑉) → (𝑁‘𝐴) ∈ ℝ) |
72 | 69, 4, 71 | syl2anc 691 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑁‘𝐴) ∈ ℝ) |
73 | 6, 47, 59, 48 | lmodvscl 18703 |
. . . . . . . . . . . . . . 15
⊢ ((𝑊 ∈ LMod ∧ 𝑇 ∈
(Base‘(Scalar‘𝑊)) ∧ 𝐵 ∈ 𝑉) → (𝑇( ·𝑠
‘𝑊)𝐵) ∈ 𝑉) |
74 | 43, 58, 11, 73 | syl3anc 1318 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑇( ·𝑠
‘𝑊)𝐵) ∈ 𝑉) |
75 | | pjthlem.m |
. . . . . . . . . . . . . . 15
⊢ − =
(-g‘𝑊) |
76 | 6, 75 | lmodvsubcl 18731 |
. . . . . . . . . . . . . 14
⊢ ((𝑊 ∈ LMod ∧ 𝐴 ∈ 𝑉 ∧ (𝑇( ·𝑠
‘𝑊)𝐵) ∈ 𝑉) → (𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵)) ∈ 𝑉) |
77 | 43, 4, 74, 76 | syl3anc 1318 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵)) ∈ 𝑉) |
78 | 6, 70 | nmcl 22230 |
. . . . . . . . . . . . 13
⊢ ((𝑊 ∈ NrmGrp ∧ (𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵)) ∈ 𝑉) → (𝑁‘(𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵))) ∈ ℝ) |
79 | 69, 77, 78 | syl2anc 691 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑁‘(𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵))) ∈ ℝ) |
80 | 6, 70 | nmge0 22231 |
. . . . . . . . . . . . 13
⊢ ((𝑊 ∈ NrmGrp ∧ 𝐴 ∈ 𝑉) → 0 ≤ (𝑁‘𝐴)) |
81 | 69, 4, 80 | syl2anc 691 |
. . . . . . . . . . . 12
⊢ (𝜑 → 0 ≤ (𝑁‘𝐴)) |
82 | 6, 70 | nmge0 22231 |
. . . . . . . . . . . . 13
⊢ ((𝑊 ∈ NrmGrp ∧ (𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵)) ∈ 𝑉) → 0 ≤ (𝑁‘(𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵)))) |
83 | 69, 77, 82 | syl2anc 691 |
. . . . . . . . . . . 12
⊢ (𝜑 → 0 ≤ (𝑁‘(𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵)))) |
84 | 72, 79, 81, 83 | le2sqd 12906 |
. . . . . . . . . . 11
⊢ (𝜑 → ((𝑁‘𝐴) ≤ (𝑁‘(𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵))) ↔ ((𝑁‘𝐴)↑2) ≤ ((𝑁‘(𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵)))↑2))) |
85 | 67, 84 | mpbid 221 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝑁‘𝐴)↑2) ≤ ((𝑁‘(𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵)))↑2)) |
86 | 79 | resqcld 12897 |
. . . . . . . . . . 11
⊢ (𝜑 → ((𝑁‘(𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵)))↑2) ∈ ℝ) |
87 | 72 | resqcld 12897 |
. . . . . . . . . . 11
⊢ (𝜑 → ((𝑁‘𝐴)↑2) ∈ ℝ) |
88 | 86, 87 | subge0d 10496 |
. . . . . . . . . 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 246 |
. . . . . . . . 9
⊢ (𝜑 → 0 ≤ (((𝑁‘(𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵)))↑2) − ((𝑁‘𝐴)↑2))) |
90 | | 2z 11286 |
. . . . . . . . . . . . . . . 16
⊢ 2 ∈
ℤ |
91 | | rpexpcl 12741 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝐵 , 𝐵) + 1) ∈ ℝ+ ∧ 2
∈ ℤ) → (((𝐵
, 𝐵) + 1)↑2) ∈
ℝ+) |
92 | 54, 90, 91 | sylancl 693 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (((𝐵 , 𝐵) + 1)↑2) ∈
ℝ+) |
93 | 17, 92 | rerpdivcld 11779 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) ∈
ℝ) |
94 | 93, 23 | remulcld 9949 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · ((𝐵 , 𝐵) + 2)) ∈ ℝ) |
95 | 94 | recnd 9947 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · ((𝐵 , 𝐵) + 2)) ∈ ℂ) |
96 | 95 | negcld 10258 |
. . . . . . . . . . 11
⊢ (𝜑 → -((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · ((𝐵 , 𝐵) + 2)) ∈ ℂ) |
97 | 6, 12 | cphipcl 22799 |
. . . . . . . . . . . 12
⊢ ((𝑊 ∈ ℂPreHil ∧
𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → (𝐴 , 𝐴) ∈ ℂ) |
98 | 3, 4, 4, 97 | syl3anc 1318 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐴 , 𝐴) ∈ ℂ) |
99 | 96, 98 | pncand 10272 |
. . . . . . . . . 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 22802 |
. . . . . . . . . . . . . 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 691 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((𝑁‘(𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵)))↑2) = ((𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵)) , (𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵)))) |
102 | 12, 6, 75 | cphsubdir 22816 |
. . . . . . . . . . . . . 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 1320 |
. . . . . . . . . . . . 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 22817 |
. . . . . . . . . . . . . . . 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 1320 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝐴 , (𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵))) = ((𝐴 , 𝐴) − (𝐴 , (𝑇( ·𝑠
‘𝑊)𝐵)))) |
106 | 105 | oveq1d 6564 |
. . . . . . . . . . . . . 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 22799 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑊 ∈ ℂPreHil ∧
𝐴 ∈ 𝑉 ∧ (𝑇( ·𝑠
‘𝑊)𝐵) ∈ 𝑉) → (𝐴 , (𝑇( ·𝑠
‘𝑊)𝐵)) ∈ ℂ) |
108 | 3, 4, 74, 107 | syl3anc 1318 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝐴 , (𝑇( ·𝑠
‘𝑊)𝐵)) ∈ ℂ) |
109 | 12, 6, 75 | cphsubdi 22817 |
. . . . . . . . . . . . . . . . 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 1320 |
. . . . . . . . . . . . . . . 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 22799 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑊 ∈ ℂPreHil ∧
(𝑇(
·𝑠 ‘𝑊)𝐵) ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → ((𝑇( ·𝑠
‘𝑊)𝐵) , 𝐴) ∈ ℂ) |
112 | 3, 74, 4, 111 | syl3anc 1318 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → ((𝑇( ·𝑠
‘𝑊)𝐵) , 𝐴) ∈ ℂ) |
113 | 6, 12 | cphipcl 22799 |
. . . . . . . . . . . . . . . . . 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 1318 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → ((𝑇( ·𝑠
‘𝑊)𝐵) , (𝑇( ·𝑠
‘𝑊)𝐵)) ∈ ℂ) |
115 | 112, 114 | subcld 10271 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (((𝑇( ·𝑠
‘𝑊)𝐵) , 𝐴) − ((𝑇( ·𝑠
‘𝑊)𝐵) , (𝑇( ·𝑠
‘𝑊)𝐵))) ∈ ℂ) |
116 | 110, 115 | eqeltrd 2688 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ((𝑇( ·𝑠
‘𝑊)𝐵) , (𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵))) ∈ ℂ) |
117 | 98, 108, 116 | subsub4d 10302 |
. . . . . . . . . . . . . 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 9947 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) ∈
ℂ) |
119 | 26 | recnd 9947 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → ((𝐵 , 𝐵) + 1) ∈ ℂ) |
120 | | 1cnd 9935 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 1 ∈
ℂ) |
121 | 118, 119,
120 | adddid 9943 |
. . . . . . . . . . . . . . . 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 6565 |
. . . . . . . . . . . . . . . 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, 47, 48, 59 | cphassr 22820 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑊 ∈ ℂPreHil ∧
(𝑇 ∈
(Base‘(Scalar‘𝑊)) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → (𝐴 , (𝑇( ·𝑠
‘𝑊)𝐵)) = ((∗‘𝑇) · (𝐴 , 𝐵))) |
124 | 3, 58, 4, 11, 123 | syl13anc 1320 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (𝐴 , (𝑇( ·𝑠
‘𝑊)𝐵)) = ((∗‘𝑇) · (𝐴 , 𝐵))) |
125 | 14, 119, 55 | divcld 10680 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → ((𝐴 , 𝐵) / ((𝐵 , 𝐵) + 1)) ∈ ℂ) |
126 | 44, 125 | syl5eqel 2692 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → 𝑇 ∈ ℂ) |
127 | 126 | cjcld 13784 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (∗‘𝑇) ∈
ℂ) |
128 | 127, 14 | mulcomd 9940 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → ((∗‘𝑇) · (𝐴 , 𝐵)) = ((𝐴 , 𝐵) · (∗‘𝑇))) |
129 | 14 | cjcld 13784 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (∗‘(𝐴 , 𝐵)) ∈ ℂ) |
130 | 14, 129, 119, 55 | divassd 10715 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (((𝐴 , 𝐵) · (∗‘(𝐴 , 𝐵))) / ((𝐵 , 𝐵) + 1)) = ((𝐴 , 𝐵) · ((∗‘(𝐴 , 𝐵)) / ((𝐵 , 𝐵) + 1)))) |
131 | 14 | absvalsqd 14029 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → ((abs‘(𝐴 , 𝐵))↑2) = ((𝐴 , 𝐵) · (∗‘(𝐴 , 𝐵)))) |
132 | 131 | oveq1d 6564 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (((abs‘(𝐴 , 𝐵))↑2) / ((𝐵 , 𝐵) + 1)) = (((𝐴 , 𝐵) · (∗‘(𝐴 , 𝐵))) / ((𝐵 , 𝐵) + 1))) |
133 | 44 | fveq2i 6106 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(∗‘𝑇) =
(∗‘((𝐴 , 𝐵) / ((𝐵 , 𝐵) + 1))) |
134 | 14, 119, 55 | cjdivd 13811 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → (∗‘((𝐴 , 𝐵) / ((𝐵 , 𝐵) + 1))) = ((∗‘(𝐴 , 𝐵)) / (∗‘((𝐵 , 𝐵) + 1)))) |
135 | 26 | cjred 13814 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → (∗‘((𝐵 , 𝐵) + 1)) = ((𝐵 , 𝐵) + 1)) |
136 | 135 | oveq2d 6565 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → ((∗‘(𝐴 , 𝐵)) / (∗‘((𝐵 , 𝐵) + 1))) = ((∗‘(𝐴 , 𝐵)) / ((𝐵 , 𝐵) + 1))) |
137 | 134, 136 | eqtrd 2644 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → (∗‘((𝐴 , 𝐵) / ((𝐵 , 𝐵) + 1))) = ((∗‘(𝐴 , 𝐵)) / ((𝐵 , 𝐵) + 1))) |
138 | 133, 137 | syl5eq 2656 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (∗‘𝑇) = ((∗‘(𝐴 , 𝐵)) / ((𝐵 , 𝐵) + 1))) |
139 | 138 | oveq2d 6565 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → ((𝐴 , 𝐵) · (∗‘𝑇)) = ((𝐴 , 𝐵) · ((∗‘(𝐴 , 𝐵)) / ((𝐵 , 𝐵) + 1)))) |
140 | 130, 132,
139 | 3eqtr4rd 2655 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → ((𝐴 , 𝐵) · (∗‘𝑇)) = (((abs‘(𝐴 , 𝐵))↑2) / ((𝐵 , 𝐵) + 1))) |
141 | 124, 128,
140 | 3eqtrd 2648 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (𝐴 , (𝑇( ·𝑠
‘𝑊)𝐵)) = (((abs‘(𝐴 , 𝐵))↑2) / ((𝐵 , 𝐵) + 1))) |
142 | 17 | recnd 9947 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → ((abs‘(𝐴 , 𝐵))↑2) ∈ ℂ) |
143 | 142, 119 | mulcomd 9940 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (((abs‘(𝐴 , 𝐵))↑2) · ((𝐵 , 𝐵) + 1)) = (((𝐵 , 𝐵) + 1) · ((abs‘(𝐴 , 𝐵))↑2))) |
144 | 119 | sqvald 12867 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (((𝐵 , 𝐵) + 1)↑2) = (((𝐵 , 𝐵) + 1) · ((𝐵 , 𝐵) + 1))) |
145 | 143, 144 | oveq12d 6567 |
. . . . . . . . . . . . . . . . . . 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, 55, 55 | divcan5d 10706 |
. . . . . . . . . . . . . . . . . . 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 2645 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (((abs‘(𝐴 , 𝐵))↑2) / ((𝐵 , 𝐵) + 1)) = ((((abs‘(𝐴 , 𝐵))↑2) · ((𝐵 , 𝐵) + 1)) / (((𝐵 , 𝐵) + 1)↑2))) |
148 | 92 | rpcnd 11750 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (((𝐵 , 𝐵) + 1)↑2) ∈
ℂ) |
149 | 92 | rpne0d 11753 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (((𝐵 , 𝐵) + 1)↑2) ≠ 0) |
150 | 142, 119,
148, 149 | div23d 10717 |
. . . . . . . . . . . . . . . . . 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 2648 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (𝐴 , (𝑇( ·𝑠
‘𝑊)𝐵)) = ((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · ((𝐵 , 𝐵) + 1))) |
152 | 93, 26 | remulcld 9949 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → ((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · ((𝐵 , 𝐵) + 1)) ∈ ℝ) |
153 | 151, 152 | eqeltrd 2688 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (𝐴 , (𝑇( ·𝑠
‘𝑊)𝐵)) ∈ ℝ) |
154 | 153 | cjred 13814 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (∗‘(𝐴 , (𝑇( ·𝑠
‘𝑊)𝐵))) = (𝐴 , (𝑇( ·𝑠
‘𝑊)𝐵))) |
155 | 12, 6 | cphipcj 22807 |
. . . . . . . . . . . . . . . . . . . . 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 1318 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (∗‘(𝐴 , (𝑇( ·𝑠
‘𝑊)𝐵))) = ((𝑇( ·𝑠
‘𝑊)𝐵) , 𝐴)) |
157 | 154, 156,
151 | 3eqtr3d 2652 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → ((𝑇( ·𝑠
‘𝑊)𝐵) , 𝐴) = ((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · ((𝐵 , 𝐵) + 1))) |
158 | 12, 6, 47, 48, 59 | cph2ass 22821 |
. . . . . . . . . . . . . . . . . . . . 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, 58, 58, 11, 11, 158 | syl122anc 1327 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → ((𝑇( ·𝑠
‘𝑊)𝐵) , (𝑇( ·𝑠
‘𝑊)𝐵)) = ((𝑇 · (∗‘𝑇)) · (𝐵 , 𝐵))) |
160 | 44 | fveq2i 6106 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(abs‘𝑇) =
(abs‘((𝐴 , 𝐵) / ((𝐵 , 𝐵) + 1))) |
161 | 14, 119, 55 | absdivd 14042 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝜑 → (abs‘((𝐴 , 𝐵) / ((𝐵 , 𝐵) + 1))) = ((abs‘(𝐴 , 𝐵)) / (abs‘((𝐵 , 𝐵) + 1)))) |
162 | 54 | rpge0d 11752 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝜑 → 0 ≤ ((𝐵 , 𝐵) + 1)) |
163 | 26, 162 | absidd 14009 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝜑 → (abs‘((𝐵 , 𝐵) + 1)) = ((𝐵 , 𝐵) + 1)) |
164 | 163 | oveq2d 6565 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝜑 → ((abs‘(𝐴 , 𝐵)) / (abs‘((𝐵 , 𝐵) + 1))) = ((abs‘(𝐴 , 𝐵)) / ((𝐵 , 𝐵) + 1))) |
165 | 161, 164 | eqtrd 2644 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → (abs‘((𝐴 , 𝐵) / ((𝐵 , 𝐵) + 1))) = ((abs‘(𝐴 , 𝐵)) / ((𝐵 , 𝐵) + 1))) |
166 | 160, 165 | syl5eq 2656 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → (abs‘𝑇) = ((abs‘(𝐴 , 𝐵)) / ((𝐵 , 𝐵) + 1))) |
167 | 166 | oveq1d 6564 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → ((abs‘𝑇)↑2) = (((abs‘(𝐴 , 𝐵)) / ((𝐵 , 𝐵) + 1))↑2)) |
168 | 126 | absvalsqd 14029 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → ((abs‘𝑇)↑2) = (𝑇 · (∗‘𝑇))) |
169 | 16, 119, 55 | sqdivd 12883 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → (((abs‘(𝐴 , 𝐵)) / ((𝐵 , 𝐵) + 1))↑2) = (((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2))) |
170 | 167, 168,
169 | 3eqtr3d 2652 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (𝑇 · (∗‘𝑇)) = (((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2))) |
171 | 170 | oveq1d 6564 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → ((𝑇 · (∗‘𝑇)) · (𝐵 , 𝐵)) = ((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · (𝐵 , 𝐵))) |
172 | 159, 171 | eqtrd 2644 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → ((𝑇( ·𝑠
‘𝑊)𝐵) , (𝑇( ·𝑠
‘𝑊)𝐵)) = ((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · (𝐵 , 𝐵))) |
173 | 157, 172 | oveq12d 6567 |
. . . . . . . . . . . . . . . . . 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 10167 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝐵 , 𝐵) ∈ ℂ ∧ 1 ∈ ℂ)
→ (((𝐵 , 𝐵) + 1) − (𝐵 , 𝐵)) = 1) |
175 | 32, 33, 174 | sylancl 693 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (((𝐵 , 𝐵) + 1) − (𝐵 , 𝐵)) = 1) |
176 | 175 | oveq2d 6565 |
. . . . . . . . . . . . . . . . . . 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,
32 | subdid 10365 |
. . . . . . . . . . . . . . . . . . 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 2646 |
. . . . . . . . . . . . . . . . . 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 2654 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → ((𝑇( ·𝑠
‘𝑊)𝐵) , (𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵))) = ((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) ·
1)) |
180 | 151, 179 | oveq12d 6567 |
. . . . . . . . . . . . . . . 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 2655 |
. . . . . . . . . . . . . . 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 6565 |
. . . . . . . . . . . . . 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 2648 |
. . . . . . . . . . . . 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 2648 |
. . . . . . . . . . . 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 10277 |
. . . . . . . . . . . 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 10117 |
. . . . . . . . . . . 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 2650 |
. . . . . . . . . . 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 22802 |
. . . . . . . . . . . 12
⊢ ((𝑊 ∈ ℂPreHil ∧
𝐴 ∈ 𝑉) → ((𝑁‘𝐴)↑2) = (𝐴 , 𝐴)) |
189 | 3, 4, 188 | syl2anc 691 |
. . . . . . . . . . 11
⊢ (𝜑 → ((𝑁‘𝐴)↑2) = (𝐴 , 𝐴)) |
190 | 187, 189 | oveq12d 6567 |
. . . . . . . . . 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 10336 |
. . . . . . . . . . . . 13
⊢ (𝜑 → -((𝐵 , 𝐵) + 2) ∈ ℝ) |
192 | 191 | recnd 9947 |
. . . . . . . . . . . 12
⊢ (𝜑 → -((𝐵 , 𝐵) + 2) ∈ ℂ) |
193 | 142, 192,
148, 149 | div23d 10717 |
. . . . . . . . . . 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 9947 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((𝐵 , 𝐵) + 2) ∈ ℂ) |
195 | 118, 194 | mulneg2d 10363 |
. . . . . . . . . . 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 2644 |
. . . . . . . . . 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 2655 |
. . . . . . . . 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 4611 |
. . . . . . . 8
⊢ (𝜑 → 0 ≤
((((abs‘(𝐴 , 𝐵))↑2) · -((𝐵 , 𝐵) + 2)) / (((𝐵 , 𝐵) + 1)↑2))) |
199 | 17, 191 | remulcld 9949 |
. . . . . . . . 9
⊢ (𝜑 → (((abs‘(𝐴 , 𝐵))↑2) · -((𝐵 , 𝐵) + 2)) ∈ ℝ) |
200 | 199, 92 | ge0divd 11786 |
. . . . . . . 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 246 |
. . . . . . 7
⊢ (𝜑 → 0 ≤ (((abs‘(𝐴 , 𝐵))↑2) · -((𝐵 , 𝐵) + 2))) |
202 | | mulneg12 10347 |
. . . . . . . 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 691 |
. . . . . . 7
⊢ (𝜑 → (-((abs‘(𝐴 , 𝐵))↑2) · ((𝐵 , 𝐵) + 2)) = (((abs‘(𝐴 , 𝐵))↑2) · -((𝐵 , 𝐵) + 2))) |
204 | 201, 203 | breqtrrd 4611 |
. . . . . 6
⊢ (𝜑 → 0 ≤
(-((abs‘(𝐴 , 𝐵))↑2) · ((𝐵 , 𝐵) + 2))) |
205 | | prodge02 10750 |
. . . . . 6
⊢
(((-((abs‘(𝐴
, 𝐵))↑2) ∈ ℝ ∧
((𝐵 , 𝐵) + 2) ∈ ℝ) ∧ (0 < ((𝐵 , 𝐵) + 2) ∧ 0 ≤ (-((abs‘(𝐴 , 𝐵))↑2) · ((𝐵 , 𝐵) + 2)))) → 0 ≤ -((abs‘(𝐴 , 𝐵))↑2)) |
206 | 18, 23, 41, 204, 205 | syl22anc 1319 |
. . . . 5
⊢ (𝜑 → 0 ≤ -((abs‘(𝐴 , 𝐵))↑2)) |
207 | 17 | le0neg1d 10478 |
. . . . 5
⊢ (𝜑 → (((abs‘(𝐴 , 𝐵))↑2) ≤ 0 ↔ 0 ≤
-((abs‘(𝐴 , 𝐵))↑2))) |
208 | 206, 207 | mpbird 246 |
. . . 4
⊢ (𝜑 → ((abs‘(𝐴 , 𝐵))↑2) ≤ 0) |
209 | 15 | sqge0d 12898 |
. . . 4
⊢ (𝜑 → 0 ≤ ((abs‘(𝐴 , 𝐵))↑2)) |
210 | | 0re 9919 |
. . . . 5
⊢ 0 ∈
ℝ |
211 | | letri3 10002 |
. . . . 5
⊢
((((abs‘(𝐴
, 𝐵))↑2) ∈ ℝ ∧
0 ∈ ℝ) → (((abs‘(𝐴 , 𝐵))↑2) = 0 ↔ (((abs‘(𝐴 , 𝐵))↑2) ≤ 0 ∧ 0 ≤
((abs‘(𝐴 , 𝐵))↑2)))) |
212 | 17, 210, 211 | sylancl 693 |
. . . 4
⊢ (𝜑 → (((abs‘(𝐴 , 𝐵))↑2) = 0 ↔ (((abs‘(𝐴 , 𝐵))↑2) ≤ 0 ∧ 0 ≤
((abs‘(𝐴 , 𝐵))↑2)))) |
213 | 208, 209,
212 | mpbir2and 959 |
. . 3
⊢ (𝜑 → ((abs‘(𝐴 , 𝐵))↑2) = 0) |
214 | 16, 213 | sqeq0d 12869 |
. 2
⊢ (𝜑 → (abs‘(𝐴 , 𝐵)) = 0) |
215 | 14, 214 | abs00d 14033 |
1
⊢ (𝜑 → (𝐴 , 𝐵) = 0) |