Proof of Theorem tchcphlem1
Step | Hyp | Ref
| Expression |
1 | | tchcph.1 |
. . . . . . 7
⊢ (𝜑 → 𝑊 ∈ PreHil) |
2 | | phllmod 19975 |
. . . . . . 7
⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LMod) |
3 | | lmodgrp 18870 |
. . . . . . 7
⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp) |
4 | 1, 2, 3 | 3syl 18 |
. . . . . 6
⊢ (𝜑 → 𝑊 ∈ Grp) |
5 | | tchcphlem1.3 |
. . . . . 6
⊢ (𝜑 → 𝑋 ∈ 𝑉) |
6 | | tchcphlem1.4 |
. . . . . 6
⊢ (𝜑 → 𝑌 ∈ 𝑉) |
7 | | tchcph.v |
. . . . . . 7
⊢ 𝑉 = (Base‘𝑊) |
8 | | tchcph.m |
. . . . . . 7
⊢ − =
(-g‘𝑊) |
9 | 7, 8 | grpsubcl 17495 |
. . . . . 6
⊢ ((𝑊 ∈ Grp ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 − 𝑌) ∈ 𝑉) |
10 | 4, 5, 6, 9 | syl3anc 1326 |
. . . . 5
⊢ (𝜑 → (𝑋 − 𝑌) ∈ 𝑉) |
11 | | tchval.n |
. . . . . 6
⊢ 𝐺 = (toℂHil‘𝑊) |
12 | | tchcph.f |
. . . . . 6
⊢ 𝐹 = (Scalar‘𝑊) |
13 | | tchcph.2 |
. . . . . 6
⊢ (𝜑 → 𝐹 = (ℂfld
↾s 𝐾)) |
14 | | tchcph.h |
. . . . . 6
⊢ , =
(·𝑖‘𝑊) |
15 | 11, 7, 12, 1, 13, 14 | tchcphlem3 23032 |
. . . . 5
⊢ ((𝜑 ∧ (𝑋 − 𝑌) ∈ 𝑉) → ((𝑋 − 𝑌) , (𝑋 − 𝑌)) ∈ ℝ) |
16 | 10, 15 | mpdan 702 |
. . . 4
⊢ (𝜑 → ((𝑋 − 𝑌) , (𝑋 − 𝑌)) ∈ ℝ) |
17 | 11, 7, 12, 1, 13, 14 | tchcphlem3 23032 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉) → (𝑋 , 𝑋) ∈ ℝ) |
18 | 5, 17 | mpdan 702 |
. . . . . 6
⊢ (𝜑 → (𝑋 , 𝑋) ∈ ℝ) |
19 | 11, 7, 12, 1, 13, 14 | tchcphlem3 23032 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑌 ∈ 𝑉) → (𝑌 , 𝑌) ∈ ℝ) |
20 | 6, 19 | mpdan 702 |
. . . . . 6
⊢ (𝜑 → (𝑌 , 𝑌) ∈ ℝ) |
21 | 18, 20 | readdcld 10069 |
. . . . 5
⊢ (𝜑 → ((𝑋 , 𝑋) + (𝑌 , 𝑌)) ∈ ℝ) |
22 | 11, 7, 12, 1, 13 | tchclm 23031 |
. . . . . . . . 9
⊢ (𝜑 → 𝑊 ∈ ℂMod) |
23 | | tchcph.k |
. . . . . . . . . 10
⊢ 𝐾 = (Base‘𝐹) |
24 | 12, 23 | clmsscn 22879 |
. . . . . . . . 9
⊢ (𝑊 ∈ ℂMod → 𝐾 ⊆
ℂ) |
25 | 22, 24 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 𝐾 ⊆ ℂ) |
26 | 12, 14, 7, 23 | ipcl 19978 |
. . . . . . . . 9
⊢ ((𝑊 ∈ PreHil ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 , 𝑌) ∈ 𝐾) |
27 | 1, 5, 6, 26 | syl3anc 1326 |
. . . . . . . 8
⊢ (𝜑 → (𝑋 , 𝑌) ∈ 𝐾) |
28 | 25, 27 | sseldd 3604 |
. . . . . . 7
⊢ (𝜑 → (𝑋 , 𝑌) ∈ ℂ) |
29 | 12, 14, 7, 23 | ipcl 19978 |
. . . . . . . . 9
⊢ ((𝑊 ∈ PreHil ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) → (𝑌 , 𝑋) ∈ 𝐾) |
30 | 1, 6, 5, 29 | syl3anc 1326 |
. . . . . . . 8
⊢ (𝜑 → (𝑌 , 𝑋) ∈ 𝐾) |
31 | 25, 30 | sseldd 3604 |
. . . . . . 7
⊢ (𝜑 → (𝑌 , 𝑋) ∈ ℂ) |
32 | 28, 31 | addcld 10059 |
. . . . . 6
⊢ (𝜑 → ((𝑋 , 𝑌) + (𝑌 , 𝑋)) ∈ ℂ) |
33 | 32 | abscld 14175 |
. . . . 5
⊢ (𝜑 → (abs‘((𝑋 , 𝑌) + (𝑌 , 𝑋))) ∈ ℝ) |
34 | 21, 33 | readdcld 10069 |
. . . 4
⊢ (𝜑 → (((𝑋 , 𝑋) + (𝑌 , 𝑌)) + (abs‘((𝑋 , 𝑌) + (𝑌 , 𝑋)))) ∈ ℝ) |
35 | 18 | recnd 10068 |
. . . . . 6
⊢ (𝜑 → (𝑋 , 𝑋) ∈ ℂ) |
36 | | 2re 11090 |
. . . . . . . 8
⊢ 2 ∈
ℝ |
37 | | tchcph.4 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 0 ≤ (𝑥 , 𝑥)) |
38 | 37 | ralrimiva 2966 |
. . . . . . . . . . 11
⊢ (𝜑 → ∀𝑥 ∈ 𝑉 0 ≤ (𝑥 , 𝑥)) |
39 | | oveq12 6659 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 = 𝑋 ∧ 𝑥 = 𝑋) → (𝑥 , 𝑥) = (𝑋 , 𝑋)) |
40 | 39 | anidms 677 |
. . . . . . . . . . . . 13
⊢ (𝑥 = 𝑋 → (𝑥 , 𝑥) = (𝑋 , 𝑋)) |
41 | 40 | breq2d 4665 |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝑋 → (0 ≤ (𝑥 , 𝑥) ↔ 0 ≤ (𝑋 , 𝑋))) |
42 | 41 | rspcv 3305 |
. . . . . . . . . . 11
⊢ (𝑋 ∈ 𝑉 → (∀𝑥 ∈ 𝑉 0 ≤ (𝑥 , 𝑥) → 0 ≤ (𝑋 , 𝑋))) |
43 | 5, 38, 42 | sylc 65 |
. . . . . . . . . 10
⊢ (𝜑 → 0 ≤ (𝑋 , 𝑋)) |
44 | 18, 43 | resqrtcld 14156 |
. . . . . . . . 9
⊢ (𝜑 → (√‘(𝑋 , 𝑋)) ∈ ℝ) |
45 | | oveq12 6659 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 = 𝑌 ∧ 𝑥 = 𝑌) → (𝑥 , 𝑥) = (𝑌 , 𝑌)) |
46 | 45 | anidms 677 |
. . . . . . . . . . . . 13
⊢ (𝑥 = 𝑌 → (𝑥 , 𝑥) = (𝑌 , 𝑌)) |
47 | 46 | breq2d 4665 |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝑌 → (0 ≤ (𝑥 , 𝑥) ↔ 0 ≤ (𝑌 , 𝑌))) |
48 | 47 | rspcv 3305 |
. . . . . . . . . . 11
⊢ (𝑌 ∈ 𝑉 → (∀𝑥 ∈ 𝑉 0 ≤ (𝑥 , 𝑥) → 0 ≤ (𝑌 , 𝑌))) |
49 | 6, 38, 48 | sylc 65 |
. . . . . . . . . 10
⊢ (𝜑 → 0 ≤ (𝑌 , 𝑌)) |
50 | 20, 49 | resqrtcld 14156 |
. . . . . . . . 9
⊢ (𝜑 → (√‘(𝑌 , 𝑌)) ∈ ℝ) |
51 | 44, 50 | remulcld 10070 |
. . . . . . . 8
⊢ (𝜑 → ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))) ∈ ℝ) |
52 | | remulcl 10021 |
. . . . . . . 8
⊢ ((2
∈ ℝ ∧ ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))) ∈ ℝ) → (2 ·
((√‘(𝑋 , 𝑋)) ·
(√‘(𝑌 , 𝑌)))) ∈
ℝ) |
53 | 36, 51, 52 | sylancr 695 |
. . . . . . 7
⊢ (𝜑 → (2 ·
((√‘(𝑋 , 𝑋)) ·
(√‘(𝑌 , 𝑌)))) ∈
ℝ) |
54 | 53 | recnd 10068 |
. . . . . 6
⊢ (𝜑 → (2 ·
((√‘(𝑋 , 𝑋)) ·
(√‘(𝑌 , 𝑌)))) ∈
ℂ) |
55 | 20 | recnd 10068 |
. . . . . 6
⊢ (𝜑 → (𝑌 , 𝑌) ∈ ℂ) |
56 | 35, 54, 55 | add32d 10263 |
. . . . 5
⊢ (𝜑 → (((𝑋 , 𝑋) + (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))))) + (𝑌 , 𝑌)) = (((𝑋 , 𝑋) + (𝑌 , 𝑌)) + (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌)))))) |
57 | 21, 53 | readdcld 10069 |
. . . . 5
⊢ (𝜑 → (((𝑋 , 𝑋) + (𝑌 , 𝑌)) + (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))))) ∈ ℝ) |
58 | 56, 57 | eqeltrd 2701 |
. . . 4
⊢ (𝜑 → (((𝑋 , 𝑋) + (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))))) + (𝑌 , 𝑌)) ∈ ℝ) |
59 | | oveq12 6659 |
. . . . . . . . . . . 12
⊢ ((𝑥 = (𝑋 − 𝑌) ∧ 𝑥 = (𝑋 − 𝑌)) → (𝑥 , 𝑥) = ((𝑋 − 𝑌) , (𝑋 − 𝑌))) |
60 | 59 | anidms 677 |
. . . . . . . . . . 11
⊢ (𝑥 = (𝑋 − 𝑌) → (𝑥 , 𝑥) = ((𝑋 − 𝑌) , (𝑋 − 𝑌))) |
61 | 60 | breq2d 4665 |
. . . . . . . . . 10
⊢ (𝑥 = (𝑋 − 𝑌) → (0 ≤ (𝑥 , 𝑥) ↔ 0 ≤ ((𝑋 − 𝑌) , (𝑋 − 𝑌)))) |
62 | 61 | rspcv 3305 |
. . . . . . . . 9
⊢ ((𝑋 − 𝑌) ∈ 𝑉 → (∀𝑥 ∈ 𝑉 0 ≤ (𝑥 , 𝑥) → 0 ≤ ((𝑋 − 𝑌) , (𝑋 − 𝑌)))) |
63 | 10, 38, 62 | sylc 65 |
. . . . . . . 8
⊢ (𝜑 → 0 ≤ ((𝑋 − 𝑌) , (𝑋 − 𝑌))) |
64 | 16, 63 | absidd 14161 |
. . . . . . 7
⊢ (𝜑 → (abs‘((𝑋 − 𝑌) , (𝑋 − 𝑌))) = ((𝑋 − 𝑌) , (𝑋 − 𝑌))) |
65 | 12 | clmadd 22874 |
. . . . . . . . . . . 12
⊢ (𝑊 ∈ ℂMod → + =
(+g‘𝐹)) |
66 | 22, 65 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → + =
(+g‘𝐹)) |
67 | 66 | oveqd 6667 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝑋 , 𝑋) + (𝑌 , 𝑌)) = ((𝑋 , 𝑋)(+g‘𝐹)(𝑌 , 𝑌))) |
68 | 66 | oveqd 6667 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝑋 , 𝑌) + (𝑌 , 𝑋)) = ((𝑋 , 𝑌)(+g‘𝐹)(𝑌 , 𝑋))) |
69 | 67, 68 | oveq12d 6668 |
. . . . . . . . 9
⊢ (𝜑 → (((𝑋 , 𝑋) + (𝑌 , 𝑌))(-g‘𝐹)((𝑋 , 𝑌) + (𝑌 , 𝑋))) = (((𝑋 , 𝑋)(+g‘𝐹)(𝑌 , 𝑌))(-g‘𝐹)((𝑋 , 𝑌)(+g‘𝐹)(𝑌 , 𝑋)))) |
70 | 12, 14, 7, 23 | ipcl 19978 |
. . . . . . . . . . . 12
⊢ ((𝑊 ∈ PreHil ∧ 𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) → (𝑋 , 𝑋) ∈ 𝐾) |
71 | 1, 5, 5, 70 | syl3anc 1326 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑋 , 𝑋) ∈ 𝐾) |
72 | 12, 14, 7, 23 | ipcl 19978 |
. . . . . . . . . . . 12
⊢ ((𝑊 ∈ PreHil ∧ 𝑌 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑌 , 𝑌) ∈ 𝐾) |
73 | 1, 6, 6, 72 | syl3anc 1326 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑌 , 𝑌) ∈ 𝐾) |
74 | 12, 23 | clmacl 22884 |
. . . . . . . . . . 11
⊢ ((𝑊 ∈ ℂMod ∧ (𝑋 , 𝑋) ∈ 𝐾 ∧ (𝑌 , 𝑌) ∈ 𝐾) → ((𝑋 , 𝑋) + (𝑌 , 𝑌)) ∈ 𝐾) |
75 | 22, 71, 73, 74 | syl3anc 1326 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝑋 , 𝑋) + (𝑌 , 𝑌)) ∈ 𝐾) |
76 | 12, 23 | clmacl 22884 |
. . . . . . . . . . 11
⊢ ((𝑊 ∈ ℂMod ∧ (𝑋 , 𝑌) ∈ 𝐾 ∧ (𝑌 , 𝑋) ∈ 𝐾) → ((𝑋 , 𝑌) + (𝑌 , 𝑋)) ∈ 𝐾) |
77 | 22, 27, 30, 76 | syl3anc 1326 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝑋 , 𝑌) + (𝑌 , 𝑋)) ∈ 𝐾) |
78 | 12, 23 | clmsub 22880 |
. . . . . . . . . 10
⊢ ((𝑊 ∈ ℂMod ∧ ((𝑋 , 𝑋) + (𝑌 , 𝑌)) ∈ 𝐾 ∧ ((𝑋 , 𝑌) + (𝑌 , 𝑋)) ∈ 𝐾) → (((𝑋 , 𝑋) + (𝑌 , 𝑌)) − ((𝑋 , 𝑌) + (𝑌 , 𝑋))) = (((𝑋 , 𝑋) + (𝑌 , 𝑌))(-g‘𝐹)((𝑋 , 𝑌) + (𝑌 , 𝑋)))) |
79 | 22, 75, 77, 78 | syl3anc 1326 |
. . . . . . . . 9
⊢ (𝜑 → (((𝑋 , 𝑋) + (𝑌 , 𝑌)) − ((𝑋 , 𝑌) + (𝑌 , 𝑋))) = (((𝑋 , 𝑋) + (𝑌 , 𝑌))(-g‘𝐹)((𝑋 , 𝑌) + (𝑌 , 𝑋)))) |
80 | | eqid 2622 |
. . . . . . . . . 10
⊢
(-g‘𝐹) = (-g‘𝐹) |
81 | | eqid 2622 |
. . . . . . . . . 10
⊢
(+g‘𝐹) = (+g‘𝐹) |
82 | 12, 14, 7, 8, 80, 81, 1, 5, 6,
5, 6 | ip2subdi 19989 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑋 − 𝑌) , (𝑋 − 𝑌)) = (((𝑋 , 𝑋)(+g‘𝐹)(𝑌 , 𝑌))(-g‘𝐹)((𝑋 , 𝑌)(+g‘𝐹)(𝑌 , 𝑋)))) |
83 | 69, 79, 82 | 3eqtr4rd 2667 |
. . . . . . . 8
⊢ (𝜑 → ((𝑋 − 𝑌) , (𝑋 − 𝑌)) = (((𝑋 , 𝑋) + (𝑌 , 𝑌)) − ((𝑋 , 𝑌) + (𝑌 , 𝑋)))) |
84 | 83 | fveq2d 6195 |
. . . . . . 7
⊢ (𝜑 → (abs‘((𝑋 − 𝑌) , (𝑋 − 𝑌))) = (abs‘(((𝑋 , 𝑋) + (𝑌 , 𝑌)) − ((𝑋 , 𝑌) + (𝑌 , 𝑋))))) |
85 | 64, 84 | eqtr3d 2658 |
. . . . . 6
⊢ (𝜑 → ((𝑋 − 𝑌) , (𝑋 − 𝑌)) = (abs‘(((𝑋 , 𝑋) + (𝑌 , 𝑌)) − ((𝑋 , 𝑌) + (𝑌 , 𝑋))))) |
86 | 25, 75 | sseldd 3604 |
. . . . . . 7
⊢ (𝜑 → ((𝑋 , 𝑋) + (𝑌 , 𝑌)) ∈ ℂ) |
87 | 86, 32 | abs2dif2d 14197 |
. . . . . 6
⊢ (𝜑 → (abs‘(((𝑋 , 𝑋) + (𝑌 , 𝑌)) − ((𝑋 , 𝑌) + (𝑌 , 𝑋)))) ≤ ((abs‘((𝑋 , 𝑋) + (𝑌 , 𝑌))) + (abs‘((𝑋 , 𝑌) + (𝑌 , 𝑋))))) |
88 | 85, 87 | eqbrtrd 4675 |
. . . . 5
⊢ (𝜑 → ((𝑋 − 𝑌) , (𝑋 − 𝑌)) ≤ ((abs‘((𝑋 , 𝑋) + (𝑌 , 𝑌))) + (abs‘((𝑋 , 𝑌) + (𝑌 , 𝑋))))) |
89 | 18, 20, 43, 49 | addge0d 10603 |
. . . . . . 7
⊢ (𝜑 → 0 ≤ ((𝑋 , 𝑋) + (𝑌 , 𝑌))) |
90 | 21, 89 | absidd 14161 |
. . . . . 6
⊢ (𝜑 → (abs‘((𝑋 , 𝑋) + (𝑌 , 𝑌))) = ((𝑋 , 𝑋) + (𝑌 , 𝑌))) |
91 | 90 | oveq1d 6665 |
. . . . 5
⊢ (𝜑 → ((abs‘((𝑋 , 𝑋) + (𝑌 , 𝑌))) + (abs‘((𝑋 , 𝑌) + (𝑌 , 𝑋)))) = (((𝑋 , 𝑋) + (𝑌 , 𝑌)) + (abs‘((𝑋 , 𝑌) + (𝑌 , 𝑋))))) |
92 | 88, 91 | breqtrd 4679 |
. . . 4
⊢ (𝜑 → ((𝑋 − 𝑌) , (𝑋 − 𝑌)) ≤ (((𝑋 , 𝑋) + (𝑌 , 𝑌)) + (abs‘((𝑋 , 𝑌) + (𝑌 , 𝑋))))) |
93 | 28 | abscld 14175 |
. . . . . . . 8
⊢ (𝜑 → (abs‘(𝑋 , 𝑌)) ∈ ℝ) |
94 | | remulcl 10021 |
. . . . . . . 8
⊢ ((2
∈ ℝ ∧ (abs‘(𝑋 , 𝑌)) ∈ ℝ) → (2 ·
(abs‘(𝑋 , 𝑌))) ∈
ℝ) |
95 | 36, 93, 94 | sylancr 695 |
. . . . . . 7
⊢ (𝜑 → (2 ·
(abs‘(𝑋 , 𝑌))) ∈
ℝ) |
96 | 28, 31 | abstrid 14195 |
. . . . . . . 8
⊢ (𝜑 → (abs‘((𝑋 , 𝑌) + (𝑌 , 𝑋))) ≤ ((abs‘(𝑋 , 𝑌)) + (abs‘(𝑌 , 𝑋)))) |
97 | 93 | recnd 10068 |
. . . . . . . . . 10
⊢ (𝜑 → (abs‘(𝑋 , 𝑌)) ∈ ℂ) |
98 | 97 | 2timesd 11275 |
. . . . . . . . 9
⊢ (𝜑 → (2 ·
(abs‘(𝑋 , 𝑌))) = ((abs‘(𝑋 , 𝑌)) + (abs‘(𝑋 , 𝑌)))) |
99 | 28 | abscjd 14189 |
. . . . . . . . . . 11
⊢ (𝜑 →
(abs‘(∗‘(𝑋 , 𝑌))) = (abs‘(𝑋 , 𝑌))) |
100 | 12 | clmcj 22876 |
. . . . . . . . . . . . . . 15
⊢ (𝑊 ∈ ℂMod →
∗ = (*𝑟‘𝐹)) |
101 | 22, 100 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ∗ =
(*𝑟‘𝐹)) |
102 | 101 | fveq1d 6193 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (∗‘(𝑋 , 𝑌)) = ((*𝑟‘𝐹)‘(𝑋 , 𝑌))) |
103 | | eqid 2622 |
. . . . . . . . . . . . . . 15
⊢
(*𝑟‘𝐹) = (*𝑟‘𝐹) |
104 | 12, 14, 7, 103 | ipcj 19979 |
. . . . . . . . . . . . . 14
⊢ ((𝑊 ∈ PreHil ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ((*𝑟‘𝐹)‘(𝑋 , 𝑌)) = (𝑌 , 𝑋)) |
105 | 1, 5, 6, 104 | syl3anc 1326 |
. . . . . . . . . . . . 13
⊢ (𝜑 →
((*𝑟‘𝐹)‘(𝑋 , 𝑌)) = (𝑌 , 𝑋)) |
106 | 102, 105 | eqtrd 2656 |
. . . . . . . . . . . 12
⊢ (𝜑 → (∗‘(𝑋 , 𝑌)) = (𝑌 , 𝑋)) |
107 | 106 | fveq2d 6195 |
. . . . . . . . . . 11
⊢ (𝜑 →
(abs‘(∗‘(𝑋 , 𝑌))) = (abs‘(𝑌 , 𝑋))) |
108 | 99, 107 | eqtr3d 2658 |
. . . . . . . . . 10
⊢ (𝜑 → (abs‘(𝑋 , 𝑌)) = (abs‘(𝑌 , 𝑋))) |
109 | 108 | oveq2d 6666 |
. . . . . . . . 9
⊢ (𝜑 → ((abs‘(𝑋 , 𝑌)) + (abs‘(𝑋 , 𝑌))) = ((abs‘(𝑋 , 𝑌)) + (abs‘(𝑌 , 𝑋)))) |
110 | 98, 109 | eqtrd 2656 |
. . . . . . . 8
⊢ (𝜑 → (2 ·
(abs‘(𝑋 , 𝑌))) = ((abs‘(𝑋 , 𝑌)) + (abs‘(𝑌 , 𝑋)))) |
111 | 96, 110 | breqtrrd 4681 |
. . . . . . 7
⊢ (𝜑 → (abs‘((𝑋 , 𝑌) + (𝑌 , 𝑋))) ≤ (2 · (abs‘(𝑋 , 𝑌)))) |
112 | | tchcph.3 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)) → (√‘𝑥) ∈ 𝐾) |
113 | | eqid 2622 |
. . . . . . . . . 10
⊢
(norm‘𝐺) =
(norm‘𝐺) |
114 | | eqid 2622 |
. . . . . . . . . 10
⊢ ((𝑌 , 𝑋) / (𝑌 , 𝑌)) = ((𝑌 , 𝑋) / (𝑌 , 𝑌)) |
115 | 11, 7, 12, 1, 13, 14, 112, 37, 23, 113, 114, 5, 6 | ipcau2 23033 |
. . . . . . . . 9
⊢ (𝜑 → (abs‘(𝑋 , 𝑌)) ≤ (((norm‘𝐺)‘𝑋) · ((norm‘𝐺)‘𝑌))) |
116 | 11, 113, 7, 14 | tchnmval 23028 |
. . . . . . . . . . 11
⊢ ((𝑊 ∈ Grp ∧ 𝑋 ∈ 𝑉) → ((norm‘𝐺)‘𝑋) = (√‘(𝑋 , 𝑋))) |
117 | 4, 5, 116 | syl2anc 693 |
. . . . . . . . . 10
⊢ (𝜑 → ((norm‘𝐺)‘𝑋) = (√‘(𝑋 , 𝑋))) |
118 | 11, 113, 7, 14 | tchnmval 23028 |
. . . . . . . . . . 11
⊢ ((𝑊 ∈ Grp ∧ 𝑌 ∈ 𝑉) → ((norm‘𝐺)‘𝑌) = (√‘(𝑌 , 𝑌))) |
119 | 4, 6, 118 | syl2anc 693 |
. . . . . . . . . 10
⊢ (𝜑 → ((norm‘𝐺)‘𝑌) = (√‘(𝑌 , 𝑌))) |
120 | 117, 119 | oveq12d 6668 |
. . . . . . . . 9
⊢ (𝜑 → (((norm‘𝐺)‘𝑋) · ((norm‘𝐺)‘𝑌)) = ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌)))) |
121 | 115, 120 | breqtrd 4679 |
. . . . . . . 8
⊢ (𝜑 → (abs‘(𝑋 , 𝑌)) ≤ ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌)))) |
122 | 36 | a1i 11 |
. . . . . . . . 9
⊢ (𝜑 → 2 ∈
ℝ) |
123 | | 2pos 11112 |
. . . . . . . . . 10
⊢ 0 <
2 |
124 | 123 | a1i 11 |
. . . . . . . . 9
⊢ (𝜑 → 0 < 2) |
125 | | lemul2 10876 |
. . . . . . . . 9
⊢
(((abs‘(𝑋
, 𝑌)) ∈ ℝ ∧
((√‘(𝑋 , 𝑋)) ·
(√‘(𝑌 , 𝑌))) ∈ ℝ ∧ (2
∈ ℝ ∧ 0 < 2)) → ((abs‘(𝑋 , 𝑌)) ≤ ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))) ↔ (2 · (abs‘(𝑋 , 𝑌))) ≤ (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌)))))) |
126 | 93, 51, 122, 124, 125 | syl112anc 1330 |
. . . . . . . 8
⊢ (𝜑 → ((abs‘(𝑋 , 𝑌)) ≤ ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))) ↔ (2 · (abs‘(𝑋 , 𝑌))) ≤ (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌)))))) |
127 | 121, 126 | mpbid 222 |
. . . . . . 7
⊢ (𝜑 → (2 ·
(abs‘(𝑋 , 𝑌))) ≤ (2 ·
((√‘(𝑋 , 𝑋)) ·
(√‘(𝑌 , 𝑌))))) |
128 | 33, 95, 53, 111, 127 | letrd 10194 |
. . . . . 6
⊢ (𝜑 → (abs‘((𝑋 , 𝑌) + (𝑌 , 𝑋))) ≤ (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))))) |
129 | 33, 53, 21, 128 | leadd2dd 10642 |
. . . . 5
⊢ (𝜑 → (((𝑋 , 𝑋) + (𝑌 , 𝑌)) + (abs‘((𝑋 , 𝑌) + (𝑌 , 𝑋)))) ≤ (((𝑋 , 𝑋) + (𝑌 , 𝑌)) + (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌)))))) |
130 | 129, 56 | breqtrrd 4681 |
. . . 4
⊢ (𝜑 → (((𝑋 , 𝑋) + (𝑌 , 𝑌)) + (abs‘((𝑋 , 𝑌) + (𝑌 , 𝑋)))) ≤ (((𝑋 , 𝑋) + (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))))) + (𝑌 , 𝑌))) |
131 | 16, 34, 58, 92, 130 | letrd 10194 |
. . 3
⊢ (𝜑 → ((𝑋 − 𝑌) , (𝑋 − 𝑌)) ≤ (((𝑋 , 𝑋) + (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))))) + (𝑌 , 𝑌))) |
132 | 16 | recnd 10068 |
. . . 4
⊢ (𝜑 → ((𝑋 − 𝑌) , (𝑋 − 𝑌)) ∈ ℂ) |
133 | 132 | sqsqrtd 14178 |
. . 3
⊢ (𝜑 → ((√‘((𝑋 − 𝑌) , (𝑋 − 𝑌)))↑2) = ((𝑋 − 𝑌) , (𝑋 − 𝑌))) |
134 | 35 | sqrtcld 14176 |
. . . . 5
⊢ (𝜑 → (√‘(𝑋 , 𝑋)) ∈ ℂ) |
135 | 50 | recnd 10068 |
. . . . 5
⊢ (𝜑 → (√‘(𝑌 , 𝑌)) ∈ ℂ) |
136 | | binom2 12979 |
. . . . 5
⊢
(((√‘(𝑋
, 𝑋)) ∈ ℂ ∧
(√‘(𝑌 , 𝑌)) ∈ ℂ) →
(((√‘(𝑋 , 𝑋)) + (√‘(𝑌 , 𝑌)))↑2) = ((((√‘(𝑋 , 𝑋))↑2) + (2 ·
((√‘(𝑋 , 𝑋)) ·
(√‘(𝑌 , 𝑌))))) + ((√‘(𝑌 , 𝑌))↑2))) |
137 | 134, 135,
136 | syl2anc 693 |
. . . 4
⊢ (𝜑 → (((√‘(𝑋 , 𝑋)) + (√‘(𝑌 , 𝑌)))↑2) = ((((√‘(𝑋 , 𝑋))↑2) + (2 ·
((√‘(𝑋 , 𝑋)) ·
(√‘(𝑌 , 𝑌))))) + ((√‘(𝑌 , 𝑌))↑2))) |
138 | 35 | sqsqrtd 14178 |
. . . . . 6
⊢ (𝜑 → ((√‘(𝑋 , 𝑋))↑2) = (𝑋 , 𝑋)) |
139 | 138 | oveq1d 6665 |
. . . . 5
⊢ (𝜑 → (((√‘(𝑋 , 𝑋))↑2) + (2 ·
((√‘(𝑋 , 𝑋)) ·
(√‘(𝑌 , 𝑌))))) = ((𝑋 , 𝑋) + (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌)))))) |
140 | 55 | sqsqrtd 14178 |
. . . . 5
⊢ (𝜑 → ((√‘(𝑌 , 𝑌))↑2) = (𝑌 , 𝑌)) |
141 | 139, 140 | oveq12d 6668 |
. . . 4
⊢ (𝜑 → ((((√‘(𝑋 , 𝑋))↑2) + (2 ·
((√‘(𝑋 , 𝑋)) ·
(√‘(𝑌 , 𝑌))))) + ((√‘(𝑌 , 𝑌))↑2)) = (((𝑋 , 𝑋) + (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))))) + (𝑌 , 𝑌))) |
142 | 137, 141 | eqtrd 2656 |
. . 3
⊢ (𝜑 → (((√‘(𝑋 , 𝑋)) + (√‘(𝑌 , 𝑌)))↑2) = (((𝑋 , 𝑋) + (2 · ((√‘(𝑋 , 𝑋)) · (√‘(𝑌 , 𝑌))))) + (𝑌 , 𝑌))) |
143 | 131, 133,
142 | 3brtr4d 4685 |
. 2
⊢ (𝜑 → ((√‘((𝑋 − 𝑌) , (𝑋 − 𝑌)))↑2) ≤ (((√‘(𝑋 , 𝑋)) + (√‘(𝑌 , 𝑌)))↑2)) |
144 | 16, 63 | resqrtcld 14156 |
. . 3
⊢ (𝜑 → (√‘((𝑋 − 𝑌) , (𝑋 − 𝑌))) ∈ ℝ) |
145 | 44, 50 | readdcld 10069 |
. . 3
⊢ (𝜑 → ((√‘(𝑋 , 𝑋)) + (√‘(𝑌 , 𝑌))) ∈ ℝ) |
146 | 16, 63 | sqrtge0d 14159 |
. . 3
⊢ (𝜑 → 0 ≤
(√‘((𝑋 − 𝑌) , (𝑋 − 𝑌)))) |
147 | 18, 43 | sqrtge0d 14159 |
. . . 4
⊢ (𝜑 → 0 ≤
(√‘(𝑋 , 𝑋))) |
148 | 20, 49 | sqrtge0d 14159 |
. . . 4
⊢ (𝜑 → 0 ≤
(√‘(𝑌 , 𝑌))) |
149 | 44, 50, 147, 148 | addge0d 10603 |
. . 3
⊢ (𝜑 → 0 ≤
((√‘(𝑋 , 𝑋)) + (√‘(𝑌 , 𝑌)))) |
150 | 144, 145,
146, 149 | le2sqd 13044 |
. 2
⊢ (𝜑 → ((√‘((𝑋 − 𝑌) , (𝑋 − 𝑌))) ≤ ((√‘(𝑋 , 𝑋)) + (√‘(𝑌 , 𝑌))) ↔ ((√‘((𝑋 − 𝑌) , (𝑋 − 𝑌)))↑2) ≤ (((√‘(𝑋 , 𝑋)) + (√‘(𝑌 , 𝑌)))↑2))) |
151 | 143, 150 | mpbird 247 |
1
⊢ (𝜑 → (√‘((𝑋 − 𝑌) , (𝑋 − 𝑌))) ≤ ((√‘(𝑋 , 𝑋)) + (√‘(𝑌 , 𝑌)))) |