Step | Hyp | Ref
| Expression |
1 | | tchcph.1 |
. . . 4
⊢ (𝜑 → 𝑊 ∈ PreHil) |
2 | | tchval.n |
. . . . 5
⊢ 𝐺 = (toℂHil‘𝑊) |
3 | 2 | tchphl 23026 |
. . . 4
⊢ (𝑊 ∈ PreHil ↔ 𝐺 ∈ PreHil) |
4 | 1, 3 | sylib 208 |
. . 3
⊢ (𝜑 → 𝐺 ∈ PreHil) |
5 | | tchcph.v |
. . . . . . 7
⊢ 𝑉 = (Base‘𝑊) |
6 | | tchcph.h |
. . . . . . 7
⊢ , =
(·𝑖‘𝑊) |
7 | 2, 5, 6 | tchval 23017 |
. . . . . 6
⊢ 𝐺 = (𝑊 toNrmGrp (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))) |
8 | | eqid 2622 |
. . . . . 6
⊢
(-g‘𝑊) = (-g‘𝑊) |
9 | | eqid 2622 |
. . . . . 6
⊢
(0g‘𝑊) = (0g‘𝑊) |
10 | | phllmod 19975 |
. . . . . . . 8
⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LMod) |
11 | 1, 10 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 𝑊 ∈ LMod) |
12 | | lmodgrp 18870 |
. . . . . . 7
⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp) |
13 | 11, 12 | syl 17 |
. . . . . 6
⊢ (𝜑 → 𝑊 ∈ Grp) |
14 | | tchcph.f |
. . . . . . . . 9
⊢ 𝐹 = (Scalar‘𝑊) |
15 | | tchcph.2 |
. . . . . . . . 9
⊢ (𝜑 → 𝐹 = (ℂfld
↾s 𝐾)) |
16 | 2, 5, 14, 1, 15, 6 | tchcphlem3 23032 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (𝑥 , 𝑥) ∈ ℝ) |
17 | | tchcph.4 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 0 ≤ (𝑥 , 𝑥)) |
18 | 16, 17 | resqrtcld 14156 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (√‘(𝑥 , 𝑥)) ∈ ℝ) |
19 | | eqid 2622 |
. . . . . . 7
⊢ (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥))) = (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥))) |
20 | 18, 19 | fmptd 6385 |
. . . . . 6
⊢ (𝜑 → (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥))):𝑉⟶ℝ) |
21 | | oveq12 6659 |
. . . . . . . . . . . 12
⊢ ((𝑥 = 𝑦 ∧ 𝑥 = 𝑦) → (𝑥 , 𝑥) = (𝑦 , 𝑦)) |
22 | 21 | anidms 677 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑦 → (𝑥 , 𝑥) = (𝑦 , 𝑦)) |
23 | 22 | fveq2d 6195 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑦 → (√‘(𝑥 , 𝑥)) = (√‘(𝑦 , 𝑦))) |
24 | | fvex 6201 |
. . . . . . . . . 10
⊢
(√‘(𝑥
, 𝑥)) ∈ V |
25 | 23, 19, 24 | fvmpt3i 6287 |
. . . . . . . . 9
⊢ (𝑦 ∈ 𝑉 → ((𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))‘𝑦) = (√‘(𝑦 , 𝑦))) |
26 | 25 | adantl 482 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → ((𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))‘𝑦) = (√‘(𝑦 , 𝑦))) |
27 | 26 | eqeq1d 2624 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → (((𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))‘𝑦) = 0 ↔ (√‘(𝑦 , 𝑦)) = 0)) |
28 | | eqid 2622 |
. . . . . . . . . . . . . . 15
⊢
(Base‘𝐹) =
(Base‘𝐹) |
29 | | phllvec 19974 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LVec) |
30 | 1, 29 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝑊 ∈ LVec) |
31 | 14 | lvecdrng 19105 |
. . . . . . . . . . . . . . . 16
⊢ (𝑊 ∈ LVec → 𝐹 ∈
DivRing) |
32 | 30, 31 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝐹 ∈ DivRing) |
33 | 28, 15, 32 | cphsubrglem 22977 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝐹 = (ℂfld
↾s (Base‘𝐹)) ∧ (Base‘𝐹) = (𝐾 ∩ ℂ) ∧ (Base‘𝐹) ∈
(SubRing‘ℂfld))) |
34 | 33 | simp2d 1074 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (Base‘𝐹) = (𝐾 ∩ ℂ)) |
35 | | inss2 3834 |
. . . . . . . . . . . . 13
⊢ (𝐾 ∩ ℂ) ⊆
ℂ |
36 | 34, 35 | syl6eqss 3655 |
. . . . . . . . . . . 12
⊢ (𝜑 → (Base‘𝐹) ⊆
ℂ) |
37 | 36 | adantr 481 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → (Base‘𝐹) ⊆ ℂ) |
38 | 14, 6, 5, 28 | ipcl 19978 |
. . . . . . . . . . . . 13
⊢ ((𝑊 ∈ PreHil ∧ 𝑦 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑦 , 𝑦) ∈ (Base‘𝐹)) |
39 | 38 | 3anidm23 1385 |
. . . . . . . . . . . 12
⊢ ((𝑊 ∈ PreHil ∧ 𝑦 ∈ 𝑉) → (𝑦 , 𝑦) ∈ (Base‘𝐹)) |
40 | 1, 39 | sylan 488 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → (𝑦 , 𝑦) ∈ (Base‘𝐹)) |
41 | 37, 40 | sseldd 3604 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → (𝑦 , 𝑦) ∈ ℂ) |
42 | 41 | sqrtcld 14176 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → (√‘(𝑦 , 𝑦)) ∈ ℂ) |
43 | | sqeq0 12927 |
. . . . . . . . 9
⊢
((√‘(𝑦
, 𝑦)) ∈ ℂ →
(((√‘(𝑦 , 𝑦))↑2) = 0 ↔
(√‘(𝑦 , 𝑦)) = 0)) |
44 | 42, 43 | syl 17 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → (((√‘(𝑦 , 𝑦))↑2) = 0 ↔ (√‘(𝑦 , 𝑦)) = 0)) |
45 | 41 | sqsqrtd 14178 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → ((√‘(𝑦 , 𝑦))↑2) = (𝑦 , 𝑦)) |
46 | 2, 5, 14, 1, 15 | tchclm 23031 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑊 ∈ ℂMod) |
47 | 14 | clm0 22872 |
. . . . . . . . . . 11
⊢ (𝑊 ∈ ℂMod → 0 =
(0g‘𝐹)) |
48 | 46, 47 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → 0 =
(0g‘𝐹)) |
49 | 48 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → 0 = (0g‘𝐹)) |
50 | 45, 49 | eqeq12d 2637 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → (((√‘(𝑦 , 𝑦))↑2) = 0 ↔ (𝑦 , 𝑦) = (0g‘𝐹))) |
51 | 44, 50 | bitr3d 270 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → ((√‘(𝑦 , 𝑦)) = 0 ↔ (𝑦 , 𝑦) = (0g‘𝐹))) |
52 | | eqid 2622 |
. . . . . . . . 9
⊢
(0g‘𝐹) = (0g‘𝐹) |
53 | 14, 6, 5, 52, 9 | ipeq0 19983 |
. . . . . . . 8
⊢ ((𝑊 ∈ PreHil ∧ 𝑦 ∈ 𝑉) → ((𝑦 , 𝑦) = (0g‘𝐹) ↔ 𝑦 = (0g‘𝑊))) |
54 | 1, 53 | sylan 488 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → ((𝑦 , 𝑦) = (0g‘𝐹) ↔ 𝑦 = (0g‘𝑊))) |
55 | 27, 51, 54 | 3bitrd 294 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → (((𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))‘𝑦) = 0 ↔ 𝑦 = (0g‘𝑊))) |
56 | 1 | adantr 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑊 ∈ PreHil) |
57 | 33 | simp1d 1073 |
. . . . . . . . 9
⊢ (𝜑 → 𝐹 = (ℂfld
↾s (Base‘𝐹))) |
58 | 57 | adantr 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝐹 = (ℂfld
↾s (Base‘𝐹))) |
59 | | 3anass 1042 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ (Base‘𝐹) ∧ 𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) ↔ (𝑥 ∈ (Base‘𝐹) ∧ (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))) |
60 | | tchcph.3 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)) → (√‘𝑥) ∈ 𝐾) |
61 | | simpr2 1068 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)) → 𝑥 ∈ ℝ) |
62 | 61 | recnd 10068 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)) → 𝑥 ∈ ℂ) |
63 | 62 | sqrtcld 14176 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)) → (√‘𝑥) ∈
ℂ) |
64 | 60, 63 | jca 554 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)) → ((√‘𝑥) ∈ 𝐾 ∧ (√‘𝑥) ∈ ℂ)) |
65 | 64 | ex 450 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((𝑥 ∈ 𝐾 ∧ 𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → ((√‘𝑥) ∈ 𝐾 ∧ (√‘𝑥) ∈ ℂ))) |
66 | 34 | eleq2d 2687 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (𝑥 ∈ (Base‘𝐹) ↔ 𝑥 ∈ (𝐾 ∩ ℂ))) |
67 | | recn 10026 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 ∈ ℝ → 𝑥 ∈
ℂ) |
68 | | elin 3796 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑥 ∈ (𝐾 ∩ ℂ) ↔ (𝑥 ∈ 𝐾 ∧ 𝑥 ∈ ℂ)) |
69 | 68 | rbaib 947 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 ∈ ℂ → (𝑥 ∈ (𝐾 ∩ ℂ) ↔ 𝑥 ∈ 𝐾)) |
70 | 67, 69 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 ∈ ℝ → (𝑥 ∈ (𝐾 ∩ ℂ) ↔ 𝑥 ∈ 𝐾)) |
71 | 66, 70 | sylan9bb 736 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑥 ∈ (Base‘𝐹) ↔ 𝑥 ∈ 𝐾)) |
72 | 71 | adantrr 753 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)) → (𝑥 ∈ (Base‘𝐹) ↔ 𝑥 ∈ 𝐾)) |
73 | 72 | ex 450 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → (𝑥 ∈ (Base‘𝐹) ↔ 𝑥 ∈ 𝐾))) |
74 | 73 | pm5.32rd 672 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((𝑥 ∈ (Base‘𝐹) ∧ (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)) ↔ (𝑥 ∈ 𝐾 ∧ (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)))) |
75 | | 3anass 1042 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ 𝐾 ∧ 𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) ↔ (𝑥 ∈ 𝐾 ∧ (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))) |
76 | 74, 75 | syl6bbr 278 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((𝑥 ∈ (Base‘𝐹) ∧ (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)) ↔ (𝑥 ∈ 𝐾 ∧ 𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))) |
77 | 34 | eleq2d 2687 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((√‘𝑥) ∈ (Base‘𝐹) ↔ (√‘𝑥) ∈ (𝐾 ∩ ℂ))) |
78 | | elin 3796 |
. . . . . . . . . . . . 13
⊢
((√‘𝑥)
∈ (𝐾 ∩ ℂ)
↔ ((√‘𝑥)
∈ 𝐾 ∧
(√‘𝑥) ∈
ℂ)) |
79 | 77, 78 | syl6bb 276 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((√‘𝑥) ∈ (Base‘𝐹) ↔ ((√‘𝑥) ∈ 𝐾 ∧ (√‘𝑥) ∈ ℂ))) |
80 | 65, 76, 79 | 3imtr4d 283 |
. . . . . . . . . . 11
⊢ (𝜑 → ((𝑥 ∈ (Base‘𝐹) ∧ (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)) → (√‘𝑥) ∈ (Base‘𝐹))) |
81 | 59, 80 | syl5bi 232 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝑥 ∈ (Base‘𝐹) ∧ 𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → (√‘𝑥) ∈ (Base‘𝐹))) |
82 | 81 | imp 445 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)) → (√‘𝑥) ∈ (Base‘𝐹)) |
83 | 82 | adantlr 751 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)) → (√‘𝑥) ∈ (Base‘𝐹)) |
84 | 17 | adantlr 751 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ 𝑥 ∈ 𝑉) → 0 ≤ (𝑥 , 𝑥)) |
85 | | simprl 794 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑦 ∈ 𝑉) |
86 | | simprr 796 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑧 ∈ 𝑉) |
87 | 2, 5, 14, 56, 58, 6, 83, 84, 28, 8, 85, 86 | tchcphlem1 23034 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (√‘((𝑦(-g‘𝑊)𝑧) , (𝑦(-g‘𝑊)𝑧))) ≤ ((√‘(𝑦 , 𝑦)) + (√‘(𝑧 , 𝑧)))) |
88 | 5, 8 | grpsubcl 17495 |
. . . . . . . . . 10
⊢ ((𝑊 ∈ Grp ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → (𝑦(-g‘𝑊)𝑧) ∈ 𝑉) |
89 | 88 | 3expb 1266 |
. . . . . . . . 9
⊢ ((𝑊 ∈ Grp ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑦(-g‘𝑊)𝑧) ∈ 𝑉) |
90 | 13, 89 | sylan 488 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑦(-g‘𝑊)𝑧) ∈ 𝑉) |
91 | | oveq12 6659 |
. . . . . . . . . . 11
⊢ ((𝑥 = (𝑦(-g‘𝑊)𝑧) ∧ 𝑥 = (𝑦(-g‘𝑊)𝑧)) → (𝑥 , 𝑥) = ((𝑦(-g‘𝑊)𝑧) , (𝑦(-g‘𝑊)𝑧))) |
92 | 91 | anidms 677 |
. . . . . . . . . 10
⊢ (𝑥 = (𝑦(-g‘𝑊)𝑧) → (𝑥 , 𝑥) = ((𝑦(-g‘𝑊)𝑧) , (𝑦(-g‘𝑊)𝑧))) |
93 | 92 | fveq2d 6195 |
. . . . . . . . 9
⊢ (𝑥 = (𝑦(-g‘𝑊)𝑧) → (√‘(𝑥 , 𝑥)) = (√‘((𝑦(-g‘𝑊)𝑧) , (𝑦(-g‘𝑊)𝑧)))) |
94 | 93, 19, 24 | fvmpt3i 6287 |
. . . . . . . 8
⊢ ((𝑦(-g‘𝑊)𝑧) ∈ 𝑉 → ((𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))‘(𝑦(-g‘𝑊)𝑧)) = (√‘((𝑦(-g‘𝑊)𝑧) , (𝑦(-g‘𝑊)𝑧)))) |
95 | 90, 94 | syl 17 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))‘(𝑦(-g‘𝑊)𝑧)) = (√‘((𝑦(-g‘𝑊)𝑧) , (𝑦(-g‘𝑊)𝑧)))) |
96 | | oveq12 6659 |
. . . . . . . . . . . 12
⊢ ((𝑥 = 𝑧 ∧ 𝑥 = 𝑧) → (𝑥 , 𝑥) = (𝑧 , 𝑧)) |
97 | 96 | anidms 677 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑧 → (𝑥 , 𝑥) = (𝑧 , 𝑧)) |
98 | 97 | fveq2d 6195 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑧 → (√‘(𝑥 , 𝑥)) = (√‘(𝑧 , 𝑧))) |
99 | 98, 19, 24 | fvmpt3i 6287 |
. . . . . . . . 9
⊢ (𝑧 ∈ 𝑉 → ((𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))‘𝑧) = (√‘(𝑧 , 𝑧))) |
100 | 25, 99 | oveqan12d 6669 |
. . . . . . . 8
⊢ ((𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → (((𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))‘𝑦) + ((𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))‘𝑧)) = ((√‘(𝑦 , 𝑦)) + (√‘(𝑧 , 𝑧)))) |
101 | 100 | adantl 482 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (((𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))‘𝑦) + ((𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))‘𝑧)) = ((√‘(𝑦 , 𝑦)) + (√‘(𝑧 , 𝑧)))) |
102 | 87, 95, 101 | 3brtr4d 4685 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))‘(𝑦(-g‘𝑊)𝑧)) ≤ (((𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))‘𝑦) + ((𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))‘𝑧))) |
103 | 7, 5, 8, 9, 13, 20, 55, 102 | tngngpd 22457 |
. . . . 5
⊢ (𝜑 → 𝐺 ∈ NrmGrp) |
104 | | phllmod 19975 |
. . . . . 6
⊢ (𝐺 ∈ PreHil → 𝐺 ∈ LMod) |
105 | 4, 104 | syl 17 |
. . . . 5
⊢ (𝜑 → 𝐺 ∈ LMod) |
106 | | cnnrg 22584 |
. . . . . . 7
⊢
ℂfld ∈ NrmRing |
107 | 33 | simp3d 1075 |
. . . . . . 7
⊢ (𝜑 → (Base‘𝐹) ∈
(SubRing‘ℂfld)) |
108 | | eqid 2622 |
. . . . . . . 8
⊢
(ℂfld ↾s (Base‘𝐹)) = (ℂfld
↾s (Base‘𝐹)) |
109 | 108 | subrgnrg 22477 |
. . . . . . 7
⊢
((ℂfld ∈ NrmRing ∧ (Base‘𝐹) ∈
(SubRing‘ℂfld)) → (ℂfld
↾s (Base‘𝐹)) ∈ NrmRing) |
110 | 106, 107,
109 | sylancr 695 |
. . . . . 6
⊢ (𝜑 → (ℂfld
↾s (Base‘𝐹)) ∈ NrmRing) |
111 | 57, 110 | eqeltrd 2701 |
. . . . 5
⊢ (𝜑 → 𝐹 ∈ NrmRing) |
112 | 103, 105,
111 | 3jca 1242 |
. . . 4
⊢ (𝜑 → (𝐺 ∈ NrmGrp ∧ 𝐺 ∈ LMod ∧ 𝐹 ∈ NrmRing)) |
113 | 1 | adantr 481 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝐹) ∧ 𝑧 ∈ 𝑉)) → 𝑊 ∈ PreHil) |
114 | 57 | adantr 481 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝐹) ∧ 𝑧 ∈ 𝑉)) → 𝐹 = (ℂfld
↾s (Base‘𝐹))) |
115 | 82 | adantlr 751 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑦 ∈ (Base‘𝐹) ∧ 𝑧 ∈ 𝑉)) ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)) → (√‘𝑥) ∈ (Base‘𝐹)) |
116 | 17 | adantlr 751 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑦 ∈ (Base‘𝐹) ∧ 𝑧 ∈ 𝑉)) ∧ 𝑥 ∈ 𝑉) → 0 ≤ (𝑥 , 𝑥)) |
117 | | eqid 2622 |
. . . . . . 7
⊢ (
·𝑠 ‘𝑊) = ( ·𝑠
‘𝑊) |
118 | | simprl 794 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝐹) ∧ 𝑧 ∈ 𝑉)) → 𝑦 ∈ (Base‘𝐹)) |
119 | | simprr 796 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝐹) ∧ 𝑧 ∈ 𝑉)) → 𝑧 ∈ 𝑉) |
120 | 2, 5, 14, 113, 114, 6, 115, 116, 28, 117, 118, 119 | tchcphlem2 23035 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝐹) ∧ 𝑧 ∈ 𝑉)) → (√‘((𝑦(
·𝑠 ‘𝑊)𝑧) , (𝑦( ·𝑠
‘𝑊)𝑧))) = ((abs‘𝑦) · (√‘(𝑧 , 𝑧)))) |
121 | 13 | adantr 481 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝐹) ∧ 𝑧 ∈ 𝑉)) → 𝑊 ∈ Grp) |
122 | 5, 14, 117, 28 | lmodvscl 18880 |
. . . . . . . . 9
⊢ ((𝑊 ∈ LMod ∧ 𝑦 ∈ (Base‘𝐹) ∧ 𝑧 ∈ 𝑉) → (𝑦( ·𝑠
‘𝑊)𝑧) ∈ 𝑉) |
123 | 122 | 3expb 1266 |
. . . . . . . 8
⊢ ((𝑊 ∈ LMod ∧ (𝑦 ∈ (Base‘𝐹) ∧ 𝑧 ∈ 𝑉)) → (𝑦( ·𝑠
‘𝑊)𝑧) ∈ 𝑉) |
124 | 11, 123 | sylan 488 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝐹) ∧ 𝑧 ∈ 𝑉)) → (𝑦( ·𝑠
‘𝑊)𝑧) ∈ 𝑉) |
125 | | eqid 2622 |
. . . . . . . 8
⊢
(norm‘𝐺) =
(norm‘𝐺) |
126 | 2, 125, 5, 6 | tchnmval 23028 |
. . . . . . 7
⊢ ((𝑊 ∈ Grp ∧ (𝑦(
·𝑠 ‘𝑊)𝑧) ∈ 𝑉) → ((norm‘𝐺)‘(𝑦( ·𝑠
‘𝑊)𝑧)) = (√‘((𝑦( ·𝑠
‘𝑊)𝑧) , (𝑦( ·𝑠
‘𝑊)𝑧)))) |
127 | 121, 124,
126 | syl2anc 693 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝐹) ∧ 𝑧 ∈ 𝑉)) → ((norm‘𝐺)‘(𝑦( ·𝑠
‘𝑊)𝑧)) = (√‘((𝑦( ·𝑠
‘𝑊)𝑧) , (𝑦( ·𝑠
‘𝑊)𝑧)))) |
128 | 114 | fveq2d 6195 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝐹) ∧ 𝑧 ∈ 𝑉)) → (norm‘𝐹) = (norm‘(ℂfld
↾s (Base‘𝐹)))) |
129 | 128 | fveq1d 6193 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝐹) ∧ 𝑧 ∈ 𝑉)) → ((norm‘𝐹)‘𝑦) = ((norm‘(ℂfld
↾s (Base‘𝐹)))‘𝑦)) |
130 | | subrgsubg 18786 |
. . . . . . . . . . 11
⊢
((Base‘𝐹)
∈ (SubRing‘ℂfld) → (Base‘𝐹) ∈
(SubGrp‘ℂfld)) |
131 | 107, 130 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → (Base‘𝐹) ∈
(SubGrp‘ℂfld)) |
132 | 131 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝐹) ∧ 𝑧 ∈ 𝑉)) → (Base‘𝐹) ∈
(SubGrp‘ℂfld)) |
133 | | cnfldnm 22582 |
. . . . . . . . . 10
⊢ abs =
(norm‘ℂfld) |
134 | | eqid 2622 |
. . . . . . . . . 10
⊢
(norm‘(ℂfld ↾s (Base‘𝐹))) =
(norm‘(ℂfld ↾s (Base‘𝐹))) |
135 | 108, 133,
134 | subgnm2 22438 |
. . . . . . . . 9
⊢
(((Base‘𝐹)
∈ (SubGrp‘ℂfld) ∧ 𝑦 ∈ (Base‘𝐹)) →
((norm‘(ℂfld ↾s (Base‘𝐹)))‘𝑦) = (abs‘𝑦)) |
136 | 132, 118,
135 | syl2anc 693 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝐹) ∧ 𝑧 ∈ 𝑉)) →
((norm‘(ℂfld ↾s (Base‘𝐹)))‘𝑦) = (abs‘𝑦)) |
137 | 129, 136 | eqtrd 2656 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝐹) ∧ 𝑧 ∈ 𝑉)) → ((norm‘𝐹)‘𝑦) = (abs‘𝑦)) |
138 | 2, 125, 5, 6 | tchnmval 23028 |
. . . . . . . 8
⊢ ((𝑊 ∈ Grp ∧ 𝑧 ∈ 𝑉) → ((norm‘𝐺)‘𝑧) = (√‘(𝑧 , 𝑧))) |
139 | 121, 119,
138 | syl2anc 693 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝐹) ∧ 𝑧 ∈ 𝑉)) → ((norm‘𝐺)‘𝑧) = (√‘(𝑧 , 𝑧))) |
140 | 137, 139 | oveq12d 6668 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝐹) ∧ 𝑧 ∈ 𝑉)) → (((norm‘𝐹)‘𝑦) · ((norm‘𝐺)‘𝑧)) = ((abs‘𝑦) · (√‘(𝑧 , 𝑧)))) |
141 | 120, 127,
140 | 3eqtr4d 2666 |
. . . . 5
⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝐹) ∧ 𝑧 ∈ 𝑉)) → ((norm‘𝐺)‘(𝑦( ·𝑠
‘𝑊)𝑧)) = (((norm‘𝐹)‘𝑦) · ((norm‘𝐺)‘𝑧))) |
142 | 141 | ralrimivva 2971 |
. . . 4
⊢ (𝜑 → ∀𝑦 ∈ (Base‘𝐹)∀𝑧 ∈ 𝑉 ((norm‘𝐺)‘(𝑦( ·𝑠
‘𝑊)𝑧)) = (((norm‘𝐹)‘𝑦) · ((norm‘𝐺)‘𝑧))) |
143 | 2, 5 | tchbas 23018 |
. . . . 5
⊢ 𝑉 = (Base‘𝐺) |
144 | 2, 117 | tchvsca 23023 |
. . . . 5
⊢ (
·𝑠 ‘𝑊) = ( ·𝑠
‘𝐺) |
145 | 2, 14 | tchsca 23022 |
. . . . 5
⊢ 𝐹 = (Scalar‘𝐺) |
146 | | eqid 2622 |
. . . . 5
⊢
(norm‘𝐹) =
(norm‘𝐹) |
147 | 143, 125,
144, 145, 28, 146 | isnlm 22479 |
. . . 4
⊢ (𝐺 ∈ NrmMod ↔ ((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ LMod ∧ 𝐹 ∈ NrmRing) ∧
∀𝑦 ∈
(Base‘𝐹)∀𝑧 ∈ 𝑉 ((norm‘𝐺)‘(𝑦( ·𝑠
‘𝑊)𝑧)) = (((norm‘𝐹)‘𝑦) · ((norm‘𝐺)‘𝑧)))) |
148 | 112, 142,
147 | sylanbrc 698 |
. . 3
⊢ (𝜑 → 𝐺 ∈ NrmMod) |
149 | 4, 148, 57 | 3jca 1242 |
. 2
⊢ (𝜑 → (𝐺 ∈ PreHil ∧ 𝐺 ∈ NrmMod ∧ 𝐹 = (ℂfld
↾s (Base‘𝐹)))) |
150 | | elin 3796 |
. . . . . 6
⊢ (𝑥 ∈ ((Base‘𝐹) ∩ (0[,)+∞)) ↔
(𝑥 ∈ (Base‘𝐹) ∧ 𝑥 ∈ (0[,)+∞))) |
151 | | elrege0 12278 |
. . . . . . 7
⊢ (𝑥 ∈ (0[,)+∞) ↔
(𝑥 ∈ ℝ ∧ 0
≤ 𝑥)) |
152 | 151 | anbi2i 730 |
. . . . . 6
⊢ ((𝑥 ∈ (Base‘𝐹) ∧ 𝑥 ∈ (0[,)+∞)) ↔ (𝑥 ∈ (Base‘𝐹) ∧ (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))) |
153 | 150, 152 | bitri 264 |
. . . . 5
⊢ (𝑥 ∈ ((Base‘𝐹) ∩ (0[,)+∞)) ↔
(𝑥 ∈ (Base‘𝐹) ∧ (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))) |
154 | 153, 80 | syl5bi 232 |
. . . 4
⊢ (𝜑 → (𝑥 ∈ ((Base‘𝐹) ∩ (0[,)+∞)) →
(√‘𝑥) ∈
(Base‘𝐹))) |
155 | 154 | ralrimiv 2965 |
. . 3
⊢ (𝜑 → ∀𝑥 ∈ ((Base‘𝐹) ∩ (0[,)+∞))(√‘𝑥) ∈ (Base‘𝐹)) |
156 | | sqrtf 14103 |
. . . . 5
⊢
√:ℂ⟶ℂ |
157 | | ffun 6048 |
. . . . 5
⊢
(√:ℂ⟶ℂ → Fun √) |
158 | 156, 157 | ax-mp 5 |
. . . 4
⊢ Fun
√ |
159 | | inss1 3833 |
. . . . . 6
⊢
((Base‘𝐹)
∩ (0[,)+∞)) ⊆ (Base‘𝐹) |
160 | 159, 36 | syl5ss 3614 |
. . . . 5
⊢ (𝜑 → ((Base‘𝐹) ∩ (0[,)+∞)) ⊆
ℂ) |
161 | 156 | fdmi 6052 |
. . . . 5
⊢ dom
√ = ℂ |
162 | 160, 161 | syl6sseqr 3652 |
. . . 4
⊢ (𝜑 → ((Base‘𝐹) ∩ (0[,)+∞)) ⊆
dom √) |
163 | | funimass4 6247 |
. . . 4
⊢ ((Fun
√ ∧ ((Base‘𝐹) ∩ (0[,)+∞)) ⊆ dom √)
→ ((√ “ ((Base‘𝐹) ∩ (0[,)+∞))) ⊆
(Base‘𝐹) ↔
∀𝑥 ∈
((Base‘𝐹) ∩
(0[,)+∞))(√‘𝑥) ∈ (Base‘𝐹))) |
164 | 158, 162,
163 | sylancr 695 |
. . 3
⊢ (𝜑 → ((√ “
((Base‘𝐹) ∩
(0[,)+∞))) ⊆ (Base‘𝐹) ↔ ∀𝑥 ∈ ((Base‘𝐹) ∩ (0[,)+∞))(√‘𝑥) ∈ (Base‘𝐹))) |
165 | 155, 164 | mpbird 247 |
. 2
⊢ (𝜑 → (√ “
((Base‘𝐹) ∩
(0[,)+∞))) ⊆ (Base‘𝐹)) |
166 | | eqid 2622 |
. . . . 5
⊢ (𝑦 ∈ 𝑉 ↦ (√‘(𝑦 , 𝑦))) = (𝑦 ∈ 𝑉 ↦ (√‘(𝑦 , 𝑦))) |
167 | 42, 166 | fmptd 6385 |
. . . 4
⊢ (𝜑 → (𝑦 ∈ 𝑉 ↦ (√‘(𝑦 , 𝑦))):𝑉⟶ℂ) |
168 | 2, 5, 6 | tchval 23017 |
. . . . 5
⊢ 𝐺 = (𝑊 toNrmGrp (𝑦 ∈ 𝑉 ↦ (√‘(𝑦 , 𝑦)))) |
169 | | cnex 10017 |
. . . . 5
⊢ ℂ
∈ V |
170 | 168, 5, 169 | tngnm 22455 |
. . . 4
⊢ ((𝑊 ∈ Grp ∧ (𝑦 ∈ 𝑉 ↦ (√‘(𝑦 , 𝑦))):𝑉⟶ℂ) → (𝑦 ∈ 𝑉 ↦ (√‘(𝑦 , 𝑦))) = (norm‘𝐺)) |
171 | 13, 167, 170 | syl2anc 693 |
. . 3
⊢ (𝜑 → (𝑦 ∈ 𝑉 ↦ (√‘(𝑦 , 𝑦))) = (norm‘𝐺)) |
172 | 171 | eqcomd 2628 |
. 2
⊢ (𝜑 → (norm‘𝐺) = (𝑦 ∈ 𝑉 ↦ (√‘(𝑦 , 𝑦)))) |
173 | 2, 6 | tchip 23024 |
. . 3
⊢ , =
(·𝑖‘𝐺) |
174 | 143, 173,
125, 145, 28 | iscph 22970 |
. 2
⊢ (𝐺 ∈ ℂPreHil ↔
((𝐺 ∈ PreHil ∧
𝐺 ∈ NrmMod ∧ 𝐹 = (ℂfld
↾s (Base‘𝐹))) ∧ (√ “
((Base‘𝐹) ∩
(0[,)+∞))) ⊆ (Base‘𝐹) ∧ (norm‘𝐺) = (𝑦 ∈ 𝑉 ↦ (√‘(𝑦 , 𝑦))))) |
175 | 149, 165,
172, 174 | syl3anbrc 1246 |
1
⊢ (𝜑 → 𝐺 ∈ ℂPreHil) |