Proof of Theorem pjthlem1
| Step | Hyp | Ref
| Expression |
| 1 | | pjthlem.1 |
. . . 4
⊢ (𝜑 → 𝑊 ∈ ℂHil) |
| 2 | | hlcph 25398 |
. . . 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 20934 |
. . . . 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 3984 |
. . 3
⊢ (𝜑 → 𝐵 ∈ 𝑉) |
| 12 | | pjthlem.h |
. . . 4
⊢ , =
(·𝑖‘𝑊) |
| 13 | 6, 12 | cphipcl 25225 |
. . 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 15475 |
. . . 4
⊢ (𝜑 → (abs‘(𝐴 , 𝐵)) ∈ ℝ) |
| 16 | 15 | recnd 11289 |
. . 3
⊢ (𝜑 → (abs‘(𝐴 , 𝐵)) ∈ ℂ) |
| 17 | 15 | resqcld 14165 |
. . . . . . 7
⊢ (𝜑 → ((abs‘(𝐴 , 𝐵))↑2) ∈ ℝ) |
| 18 | 17 | renegcld 11690 |
. . . . . 6
⊢ (𝜑 → -((abs‘(𝐴 , 𝐵))↑2) ∈ ℝ) |
| 19 | 6, 12 | reipcl 25231 |
. . . . . . . . 9
⊢ ((𝑊 ∈ ℂPreHil ∧
𝐵 ∈ 𝑉) → (𝐵 , 𝐵) ∈ ℝ) |
| 20 | 3, 11, 19 | syl2anc 584 |
. . . . . . . 8
⊢ (𝜑 → (𝐵 , 𝐵) ∈ ℝ) |
| 21 | | 2re 12340 |
. . . . . . . 8
⊢ 2 ∈
ℝ |
| 22 | | readdcl 11238 |
. . . . . . . 8
⊢ (((𝐵 , 𝐵) ∈ ℝ ∧ 2 ∈ ℝ)
→ ((𝐵 , 𝐵) + 2) ∈
ℝ) |
| 23 | 20, 21, 22 | sylancl 586 |
. . . . . . 7
⊢ (𝜑 → ((𝐵 , 𝐵) + 2) ∈ ℝ) |
| 24 | | 0red 11264 |
. . . . . . . 8
⊢ (𝜑 → 0 ∈
ℝ) |
| 25 | | peano2re 11434 |
. . . . . . . . 9
⊢ ((𝐵 , 𝐵) ∈ ℝ → ((𝐵 , 𝐵) + 1) ∈ ℝ) |
| 26 | 20, 25 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → ((𝐵 , 𝐵) + 1) ∈ ℝ) |
| 27 | 6, 12 | ipge0 25232 |
. . . . . . . . . 10
⊢ ((𝑊 ∈ ℂPreHil ∧
𝐵 ∈ 𝑉) → 0 ≤ (𝐵 , 𝐵)) |
| 28 | 3, 11, 27 | syl2anc 584 |
. . . . . . . . 9
⊢ (𝜑 → 0 ≤ (𝐵 , 𝐵)) |
| 29 | 20 | ltp1d 12198 |
. . . . . . . . 9
⊢ (𝜑 → (𝐵 , 𝐵) < ((𝐵 , 𝐵) + 1)) |
| 30 | 24, 20, 26, 28, 29 | lelttrd 11419 |
. . . . . . . 8
⊢ (𝜑 → 0 < ((𝐵 , 𝐵) + 1)) |
| 31 | 26 | ltp1d 12198 |
. . . . . . . . 9
⊢ (𝜑 → ((𝐵 , 𝐵) + 1) < (((𝐵 , 𝐵) + 1) + 1)) |
| 32 | | df-2 12329 |
. . . . . . . . . . 11
⊢ 2 = (1 +
1) |
| 33 | 32 | oveq2i 7442 |
. . . . . . . . . 10
⊢ ((𝐵 , 𝐵) + 2) = ((𝐵 , 𝐵) + (1 + 1)) |
| 34 | 20 | recnd 11289 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐵 , 𝐵) ∈ ℂ) |
| 35 | | ax-1cn 11213 |
. . . . . . . . . . . 12
⊢ 1 ∈
ℂ |
| 36 | | addass 11242 |
. . . . . . . . . . . 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 2796 |
. . . . . . . . 9
⊢ (𝜑 → ((𝐵 , 𝐵) + 2) = (((𝐵 , 𝐵) + 1) + 1)) |
| 40 | 31, 39 | breqtrrd 5171 |
. . . . . . . 8
⊢ (𝜑 → ((𝐵 , 𝐵) + 1) < ((𝐵 , 𝐵) + 2)) |
| 41 | 24, 26, 23, 30, 40 | lttrd 11422 |
. . . . . . 7
⊢ (𝜑 → 0 < ((𝐵 , 𝐵) + 2)) |
| 42 | 23, 41 | elrpd 13074 |
. . . . . 6
⊢ (𝜑 → ((𝐵 , 𝐵) + 2) ∈
ℝ+) |
| 43 | | oveq2 7439 |
. . . . . . . . . . . . . 14
⊢ (𝑥 = (𝑇( ·𝑠
‘𝑊)𝐵) → (𝐴 − 𝑥) = (𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵))) |
| 44 | 43 | fveq2d 6910 |
. . . . . . . . . . . . 13
⊢ (𝑥 = (𝑇( ·𝑠
‘𝑊)𝐵) → (𝑁‘(𝐴 − 𝑥)) = (𝑁‘(𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵)))) |
| 45 | 44 | breq2d 5155 |
. . . . . . . . . . . 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 25208 |
. . . . . . . . . . . . . 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 25399 |
. . . . . . . . . . . . . . . . 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 21651 |
. . . . . . . . . . . . . . . 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 25402 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑊 ∈ ℂHil →
ℝ ⊆ (Base‘(Scalar‘𝑊))) |
| 57 | 1, 56 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ℝ ⊆
(Base‘(Scalar‘𝑊))) |
| 58 | 57, 26 | sseldd 3984 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ((𝐵 , 𝐵) + 1) ∈
(Base‘(Scalar‘𝑊))) |
| 59 | 20, 28 | ge0p1rpd 13107 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((𝐵 , 𝐵) + 1) ∈
ℝ+) |
| 60 | 59 | rpne0d 13082 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ((𝐵 , 𝐵) + 1) ≠ 0) |
| 61 | 52, 53 | cphdivcl 25216 |
. . . . . . . . . . . . . . 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 2845 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑇 ∈ (Base‘(Scalar‘𝑊))) |
| 64 | | eqid 2737 |
. . . . . . . . . . . . . 14
⊢ (
·𝑠 ‘𝑊) = ( ·𝑠
‘𝑊) |
| 65 | 52, 64, 53, 7 | lssvscl 20953 |
. . . . . . . . . . . . 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 3623 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑁‘𝐴) ≤ (𝑁‘(𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵)))) |
| 68 | | cphngp 25207 |
. . . . . . . . . . . . . 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 24629 |
. . . . . . . . . . . . 13
⊢ ((𝑊 ∈ NrmGrp ∧ 𝐴 ∈ 𝑉) → (𝑁‘𝐴) ∈ ℝ) |
| 72 | 69, 4, 71 | syl2anc 584 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑁‘𝐴) ∈ ℝ) |
| 73 | 6, 52, 64, 53 | lmodvscl 20876 |
. . . . . . . . . . . . . . 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 20905 |
. . . . . . . . . . . . . 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 24629 |
. . . . . . . . . . . . 13
⊢ ((𝑊 ∈ NrmGrp ∧ (𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵)) ∈ 𝑉) → (𝑁‘(𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵))) ∈ ℝ) |
| 79 | 69, 77, 78 | syl2anc 584 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑁‘(𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵))) ∈ ℝ) |
| 80 | 6, 70 | nmge0 24630 |
. . . . . . . . . . . . 13
⊢ ((𝑊 ∈ NrmGrp ∧ 𝐴 ∈ 𝑉) → 0 ≤ (𝑁‘𝐴)) |
| 81 | 69, 4, 80 | syl2anc 584 |
. . . . . . . . . . . 12
⊢ (𝜑 → 0 ≤ (𝑁‘𝐴)) |
| 82 | 6, 70 | nmge0 24630 |
. . . . . . . . . . . . 13
⊢ ((𝑊 ∈ NrmGrp ∧ (𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵)) ∈ 𝑉) → 0 ≤ (𝑁‘(𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵)))) |
| 83 | 69, 77, 82 | syl2anc 584 |
. . . . . . . . . . . 12
⊢ (𝜑 → 0 ≤ (𝑁‘(𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵)))) |
| 84 | 72, 79, 81, 83 | le2sqd 14296 |
. . . . . . . . . . 11
⊢ (𝜑 → ((𝑁‘𝐴) ≤ (𝑁‘(𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵))) ↔ ((𝑁‘𝐴)↑2) ≤ ((𝑁‘(𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵)))↑2))) |
| 85 | 67, 84 | mpbid 232 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝑁‘𝐴)↑2) ≤ ((𝑁‘(𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵)))↑2)) |
| 86 | 79 | resqcld 14165 |
. . . . . . . . . . 11
⊢ (𝜑 → ((𝑁‘(𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵)))↑2) ∈ ℝ) |
| 87 | 72 | resqcld 14165 |
. . . . . . . . . . 11
⊢ (𝜑 → ((𝑁‘𝐴)↑2) ∈ ℝ) |
| 88 | 86, 87 | subge0d 11853 |
. . . . . . . . . 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 257 |
. . . . . . . . 9
⊢ (𝜑 → 0 ≤ (((𝑁‘(𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵)))↑2) − ((𝑁‘𝐴)↑2))) |
| 90 | | 2z 12649 |
. . . . . . . . . . . . . . . 16
⊢ 2 ∈
ℤ |
| 91 | | rpexpcl 14121 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝐵 , 𝐵) + 1) ∈ ℝ+ ∧ 2
∈ ℤ) → (((𝐵
, 𝐵) + 1)↑2) ∈
ℝ+) |
| 92 | 59, 90, 91 | sylancl 586 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (((𝐵 , 𝐵) + 1)↑2) ∈
ℝ+) |
| 93 | 17, 92 | rerpdivcld 13108 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) ∈
ℝ) |
| 94 | 93, 23 | remulcld 11291 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · ((𝐵 , 𝐵) + 2)) ∈ ℝ) |
| 95 | 94 | recnd 11289 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · ((𝐵 , 𝐵) + 2)) ∈ ℂ) |
| 96 | 95 | negcld 11607 |
. . . . . . . . . . 11
⊢ (𝜑 → -((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · ((𝐵 , 𝐵) + 2)) ∈ ℂ) |
| 97 | 6, 12 | cphipcl 25225 |
. . . . . . . . . . . 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 11621 |
. . . . . . . . . 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 25228 |
. . . . . . . . . . . . . 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 584 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((𝑁‘(𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵)))↑2) = ((𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵)) , (𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵)))) |
| 102 | 12, 6, 75 | cphsubdir 25242 |
. . . . . . . . . . . . . 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 25243 |
. . . . . . . . . . . . . . . 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 7446 |
. . . . . . . . . . . . . 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 25225 |
. . . . . . . . . . . . . . . 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 25243 |
. . . . . . . . . . . . . . . . 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 25225 |
. . . . . . . . . . . . . . . . . 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 25225 |
. . . . . . . . . . . . . . . . . 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 11620 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (((𝑇( ·𝑠
‘𝑊)𝐵) , 𝐴) − ((𝑇( ·𝑠
‘𝑊)𝐵) , (𝑇( ·𝑠
‘𝑊)𝐵))) ∈ ℂ) |
| 116 | 110, 115 | eqeltrd 2841 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ((𝑇( ·𝑠
‘𝑊)𝐵) , (𝐴 − (𝑇( ·𝑠
‘𝑊)𝐵))) ∈ ℂ) |
| 117 | 98, 108, 116 | subsub4d 11651 |
. . . . . . . . . . . . . 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 11289 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) ∈
ℂ) |
| 119 | 26 | recnd 11289 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → ((𝐵 , 𝐵) + 1) ∈ ℂ) |
| 120 | | 1cnd 11256 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 1 ∈
ℂ) |
| 121 | 118, 119,
120 | adddid 11285 |
. . . . . . . . . . . . . . . 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 7447 |
. . . . . . . . . . . . . . . 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 25246 |
. . . . . . . . . . . . . . . . . . . 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 12043 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → ((𝐴 , 𝐵) / ((𝐵 , 𝐵) + 1)) ∈ ℂ) |
| 126 | 49, 125 | eqeltrid 2845 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → 𝑇 ∈ ℂ) |
| 127 | 126 | cjcld 15235 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (∗‘𝑇) ∈
ℂ) |
| 128 | 127, 14 | mulcomd 11282 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → ((∗‘𝑇) · (𝐴 , 𝐵)) = ((𝐴 , 𝐵) · (∗‘𝑇))) |
| 129 | 14 | cjcld 15235 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (∗‘(𝐴 , 𝐵)) ∈ ℂ) |
| 130 | 14, 129, 119, 60 | divassd 12078 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (((𝐴 , 𝐵) · (∗‘(𝐴 , 𝐵))) / ((𝐵 , 𝐵) + 1)) = ((𝐴 , 𝐵) · ((∗‘(𝐴 , 𝐵)) / ((𝐵 , 𝐵) + 1)))) |
| 131 | 14 | absvalsqd 15481 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → ((abs‘(𝐴 , 𝐵))↑2) = ((𝐴 , 𝐵) · (∗‘(𝐴 , 𝐵)))) |
| 132 | 131 | oveq1d 7446 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (((abs‘(𝐴 , 𝐵))↑2) / ((𝐵 , 𝐵) + 1)) = (((𝐴 , 𝐵) · (∗‘(𝐴 , 𝐵))) / ((𝐵 , 𝐵) + 1))) |
| 133 | 49 | fveq2i 6909 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(∗‘𝑇) =
(∗‘((𝐴 , 𝐵) / ((𝐵 , 𝐵) + 1))) |
| 134 | 14, 119, 60 | cjdivd 15262 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → (∗‘((𝐴 , 𝐵) / ((𝐵 , 𝐵) + 1))) = ((∗‘(𝐴 , 𝐵)) / (∗‘((𝐵 , 𝐵) + 1)))) |
| 135 | 26 | cjred 15265 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → (∗‘((𝐵 , 𝐵) + 1)) = ((𝐵 , 𝐵) + 1)) |
| 136 | 135 | oveq2d 7447 |
. . . . . . . . . . . . . . . . . . . . . . 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 | eqtrid 2789 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (∗‘𝑇) = ((∗‘(𝐴 , 𝐵)) / ((𝐵 , 𝐵) + 1))) |
| 139 | 138 | oveq2d 7447 |
. . . . . . . . . . . . . . . . . . . 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 11289 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → ((abs‘(𝐴 , 𝐵))↑2) ∈ ℂ) |
| 143 | 142, 119 | mulcomd 11282 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (((abs‘(𝐴 , 𝐵))↑2) · ((𝐵 , 𝐵) + 1)) = (((𝐵 , 𝐵) + 1) · ((abs‘(𝐴 , 𝐵))↑2))) |
| 144 | 119 | sqvald 14183 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (((𝐵 , 𝐵) + 1)↑2) = (((𝐵 , 𝐵) + 1) · ((𝐵 , 𝐵) + 1))) |
| 145 | 143, 144 | oveq12d 7449 |
. . . . . . . . . . . . . . . . . . 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 12069 |
. . . . . . . . . . . . . . . . . . 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 13079 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (((𝐵 , 𝐵) + 1)↑2) ∈
ℂ) |
| 149 | 92 | rpne0d 13082 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (((𝐵 , 𝐵) + 1)↑2) ≠ 0) |
| 150 | 142, 119,
148, 149 | div23d 12080 |
. . . . . . . . . . . . . . . . . 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 11291 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → ((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · ((𝐵 , 𝐵) + 1)) ∈ ℝ) |
| 153 | 151, 152 | eqeltrd 2841 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (𝐴 , (𝑇( ·𝑠
‘𝑊)𝐵)) ∈ ℝ) |
| 154 | 153 | cjred 15265 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (∗‘(𝐴 , (𝑇( ·𝑠
‘𝑊)𝐵))) = (𝐴 , (𝑇( ·𝑠
‘𝑊)𝐵))) |
| 155 | 12, 6 | cphipcj 25233 |
. . . . . . . . . . . . . . . . . . . . 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 25247 |
. . . . . . . . . . . . . . . . . . . . 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 6909 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(abs‘𝑇) =
(abs‘((𝐴 , 𝐵) / ((𝐵 , 𝐵) + 1))) |
| 161 | 14, 119, 60 | absdivd 15494 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝜑 → (abs‘((𝐴 , 𝐵) / ((𝐵 , 𝐵) + 1))) = ((abs‘(𝐴 , 𝐵)) / (abs‘((𝐵 , 𝐵) + 1)))) |
| 162 | 59 | rpge0d 13081 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝜑 → 0 ≤ ((𝐵 , 𝐵) + 1)) |
| 163 | 26, 162 | absidd 15461 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝜑 → (abs‘((𝐵 , 𝐵) + 1)) = ((𝐵 , 𝐵) + 1)) |
| 164 | 163 | oveq2d 7447 |
. . . . . . . . . . . . . . . . . . . . . . . . 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 | eqtrid 2789 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → (abs‘𝑇) = ((abs‘(𝐴 , 𝐵)) / ((𝐵 , 𝐵) + 1))) |
| 167 | 166 | oveq1d 7446 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → ((abs‘𝑇)↑2) = (((abs‘(𝐴 , 𝐵)) / ((𝐵 , 𝐵) + 1))↑2)) |
| 168 | 126 | absvalsqd 15481 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → ((abs‘𝑇)↑2) = (𝑇 · (∗‘𝑇))) |
| 169 | 16, 119, 60 | sqdivd 14199 |
. . . . . . . . . . . . . . . . . . . . . 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 7446 |
. . . . . . . . . . . . . . . . . . . 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 7449 |
. . . . . . . . . . . . . . . . . 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 11515 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝐵 , 𝐵) ∈ ℂ ∧ 1 ∈ ℂ)
→ (((𝐵 , 𝐵) + 1) − (𝐵 , 𝐵)) = 1) |
| 175 | 34, 35, 174 | sylancl 586 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (((𝐵 , 𝐵) + 1) − (𝐵 , 𝐵)) = 1) |
| 176 | 175 | oveq2d 7447 |
. . . . . . . . . . . . . . . . . . 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 11719 |
. . . . . . . . . . . . . . . . . . 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 7449 |
. . . . . . . . . . . . . . . 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 7447 |
. . . . . . . . . . . . . 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 11626 |
. . . . . . . . . . . 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 11463 |
. . . . . . . . . . . 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 25228 |
. . . . . . . . . . . 12
⊢ ((𝑊 ∈ ℂPreHil ∧
𝐴 ∈ 𝑉) → ((𝑁‘𝐴)↑2) = (𝐴 , 𝐴)) |
| 189 | 3, 4, 188 | syl2anc 584 |
. . . . . . . . . . 11
⊢ (𝜑 → ((𝑁‘𝐴)↑2) = (𝐴 , 𝐴)) |
| 190 | 187, 189 | oveq12d 7449 |
. . . . . . . . . 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 11690 |
. . . . . . . . . . . . 13
⊢ (𝜑 → -((𝐵 , 𝐵) + 2) ∈ ℝ) |
| 192 | 191 | recnd 11289 |
. . . . . . . . . . . 12
⊢ (𝜑 → -((𝐵 , 𝐵) + 2) ∈ ℂ) |
| 193 | 142, 192,
148, 149 | div23d 12080 |
. . . . . . . . . . 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 11289 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((𝐵 , 𝐵) + 2) ∈ ℂ) |
| 195 | 118, 194 | mulneg2d 11717 |
. . . . . . . . . . 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 5171 |
. . . . . . . 8
⊢ (𝜑 → 0 ≤
((((abs‘(𝐴 , 𝐵))↑2) · -((𝐵 , 𝐵) + 2)) / (((𝐵 , 𝐵) + 1)↑2))) |
| 199 | 17, 191 | remulcld 11291 |
. . . . . . . . 9
⊢ (𝜑 → (((abs‘(𝐴 , 𝐵))↑2) · -((𝐵 , 𝐵) + 2)) ∈ ℝ) |
| 200 | 199, 92 | ge0divd 13115 |
. . . . . . . 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 257 |
. . . . . . 7
⊢ (𝜑 → 0 ≤ (((abs‘(𝐴 , 𝐵))↑2) · -((𝐵 , 𝐵) + 2))) |
| 202 | | mulneg12 11701 |
. . . . . . . 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 584 |
. . . . . . 7
⊢ (𝜑 → (-((abs‘(𝐴 , 𝐵))↑2) · ((𝐵 , 𝐵) + 2)) = (((abs‘(𝐴 , 𝐵))↑2) · -((𝐵 , 𝐵) + 2))) |
| 204 | 201, 203 | breqtrrd 5171 |
. . . . . 6
⊢ (𝜑 → 0 ≤
(-((abs‘(𝐴 , 𝐵))↑2) · ((𝐵 , 𝐵) + 2))) |
| 205 | 18, 42, 204 | prodge0ld 13143 |
. . . . 5
⊢ (𝜑 → 0 ≤ -((abs‘(𝐴 , 𝐵))↑2)) |
| 206 | 17 | le0neg1d 11834 |
. . . . 5
⊢ (𝜑 → (((abs‘(𝐴 , 𝐵))↑2) ≤ 0 ↔ 0 ≤
-((abs‘(𝐴 , 𝐵))↑2))) |
| 207 | 205, 206 | mpbird 257 |
. . . 4
⊢ (𝜑 → ((abs‘(𝐴 , 𝐵))↑2) ≤ 0) |
| 208 | 15 | sqge0d 14177 |
. . . 4
⊢ (𝜑 → 0 ≤ ((abs‘(𝐴 , 𝐵))↑2)) |
| 209 | | 0re 11263 |
. . . . 5
⊢ 0 ∈
ℝ |
| 210 | | letri3 11346 |
. . . . 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 586 |
. . . 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 14185 |
. 2
⊢ (𝜑 → (abs‘(𝐴 , 𝐵)) = 0) |
| 214 | 14, 213 | abs00d 15485 |
1
⊢ (𝜑 → (𝐴 , 𝐵) = 0) |