Step | Hyp | Ref
| Expression |
1 | | 4re 10974 |
. . . . . 6
⊢ 4 ∈
ℝ |
2 | | minveco.s |
. . . . . . . 8
⊢ 𝑆 = inf(𝑅, ℝ, < ) |
3 | | minveco.x |
. . . . . . . . . . 11
⊢ 𝑋 = (BaseSet‘𝑈) |
4 | | minveco.m |
. . . . . . . . . . 11
⊢ 𝑀 = ( −𝑣
‘𝑈) |
5 | | minveco.n |
. . . . . . . . . . 11
⊢ 𝑁 =
(normCV‘𝑈) |
6 | | minveco.y |
. . . . . . . . . . 11
⊢ 𝑌 = (BaseSet‘𝑊) |
7 | | minveco.u |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑈 ∈
CPreHilOLD) |
8 | | minveco.w |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑊 ∈ ((SubSp‘𝑈) ∩ CBan)) |
9 | | minveco.a |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐴 ∈ 𝑋) |
10 | | minveco.d |
. . . . . . . . . . 11
⊢ 𝐷 = (IndMet‘𝑈) |
11 | | minveco.j |
. . . . . . . . . . 11
⊢ 𝐽 = (MetOpen‘𝐷) |
12 | | minveco.r |
. . . . . . . . . . 11
⊢ 𝑅 = ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) |
13 | 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 | minvecolem1 27114 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∀𝑤 ∈ 𝑅 0 ≤ 𝑤)) |
14 | 13 | simp1d 1066 |
. . . . . . . . 9
⊢ (𝜑 → 𝑅 ⊆ ℝ) |
15 | 13 | simp2d 1067 |
. . . . . . . . 9
⊢ (𝜑 → 𝑅 ≠ ∅) |
16 | | 0re 9919 |
. . . . . . . . . 10
⊢ 0 ∈
ℝ |
17 | 13 | simp3d 1068 |
. . . . . . . . . 10
⊢ (𝜑 → ∀𝑤 ∈ 𝑅 0 ≤ 𝑤) |
18 | | breq1 4586 |
. . . . . . . . . . . 12
⊢ (𝑥 = 0 → (𝑥 ≤ 𝑤 ↔ 0 ≤ 𝑤)) |
19 | 18 | ralbidv 2969 |
. . . . . . . . . . 11
⊢ (𝑥 = 0 → (∀𝑤 ∈ 𝑅 𝑥 ≤ 𝑤 ↔ ∀𝑤 ∈ 𝑅 0 ≤ 𝑤)) |
20 | 19 | rspcev 3282 |
. . . . . . . . . 10
⊢ ((0
∈ ℝ ∧ ∀𝑤 ∈ 𝑅 0 ≤ 𝑤) → ∃𝑥 ∈ ℝ ∀𝑤 ∈ 𝑅 𝑥 ≤ 𝑤) |
21 | 16, 17, 20 | sylancr 694 |
. . . . . . . . 9
⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑤 ∈ 𝑅 𝑥 ≤ 𝑤) |
22 | | infrecl 10882 |
. . . . . . . . 9
⊢ ((𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑤 ∈ 𝑅 𝑥 ≤ 𝑤) → inf(𝑅, ℝ, < ) ∈
ℝ) |
23 | 14, 15, 21, 22 | syl3anc 1318 |
. . . . . . . 8
⊢ (𝜑 → inf(𝑅, ℝ, < ) ∈
ℝ) |
24 | 2, 23 | syl5eqel 2692 |
. . . . . . 7
⊢ (𝜑 → 𝑆 ∈ ℝ) |
25 | 24 | resqcld 12897 |
. . . . . 6
⊢ (𝜑 → (𝑆↑2) ∈ ℝ) |
26 | | remulcl 9900 |
. . . . . 6
⊢ ((4
∈ ℝ ∧ (𝑆↑2) ∈ ℝ) → (4 ·
(𝑆↑2)) ∈
ℝ) |
27 | 1, 25, 26 | sylancr 694 |
. . . . 5
⊢ (𝜑 → (4 · (𝑆↑2)) ∈
ℝ) |
28 | | phnv 27053 |
. . . . . . . . 9
⊢ (𝑈 ∈ CPreHilOLD
→ 𝑈 ∈
NrmCVec) |
29 | 7, 28 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 𝑈 ∈ NrmCVec) |
30 | 3, 10 | imsmet 26930 |
. . . . . . . 8
⊢ (𝑈 ∈ NrmCVec → 𝐷 ∈ (Met‘𝑋)) |
31 | 29, 30 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋)) |
32 | | inss1 3795 |
. . . . . . . . . 10
⊢
((SubSp‘𝑈)
∩ CBan) ⊆ (SubSp‘𝑈) |
33 | 32, 8 | sseldi 3566 |
. . . . . . . . 9
⊢ (𝜑 → 𝑊 ∈ (SubSp‘𝑈)) |
34 | | eqid 2610 |
. . . . . . . . . 10
⊢
(SubSp‘𝑈) =
(SubSp‘𝑈) |
35 | 3, 6, 34 | sspba 26966 |
. . . . . . . . 9
⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ (SubSp‘𝑈)) → 𝑌 ⊆ 𝑋) |
36 | 29, 33, 35 | syl2anc 691 |
. . . . . . . 8
⊢ (𝜑 → 𝑌 ⊆ 𝑋) |
37 | | minvecolem2.3 |
. . . . . . . 8
⊢ (𝜑 → 𝐾 ∈ 𝑌) |
38 | 36, 37 | sseldd 3569 |
. . . . . . 7
⊢ (𝜑 → 𝐾 ∈ 𝑋) |
39 | | minvecolem2.4 |
. . . . . . . 8
⊢ (𝜑 → 𝐿 ∈ 𝑌) |
40 | 36, 39 | sseldd 3569 |
. . . . . . 7
⊢ (𝜑 → 𝐿 ∈ 𝑋) |
41 | | metcl 21947 |
. . . . . . 7
⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐾 ∈ 𝑋 ∧ 𝐿 ∈ 𝑋) → (𝐾𝐷𝐿) ∈ ℝ) |
42 | 31, 38, 40, 41 | syl3anc 1318 |
. . . . . 6
⊢ (𝜑 → (𝐾𝐷𝐿) ∈ ℝ) |
43 | 42 | resqcld 12897 |
. . . . 5
⊢ (𝜑 → ((𝐾𝐷𝐿)↑2) ∈ ℝ) |
44 | 27, 43 | readdcld 9948 |
. . . 4
⊢ (𝜑 → ((4 · (𝑆↑2)) + ((𝐾𝐷𝐿)↑2)) ∈ ℝ) |
45 | | ax-1cn 9873 |
. . . . . . . . . . . . 13
⊢ 1 ∈
ℂ |
46 | | halfcl 11134 |
. . . . . . . . . . . . 13
⊢ (1 ∈
ℂ → (1 / 2) ∈ ℂ) |
47 | 45, 46 | mp1i 13 |
. . . . . . . . . . . 12
⊢ (𝜑 → (1 / 2) ∈
ℂ) |
48 | | eqid 2610 |
. . . . . . . . . . . . . . 15
⊢ (
+𝑣 ‘𝑈) = ( +𝑣 ‘𝑈) |
49 | | eqid 2610 |
. . . . . . . . . . . . . . 15
⊢ (
+𝑣 ‘𝑊) = ( +𝑣 ‘𝑊) |
50 | 6, 48, 49, 34 | sspgval 26968 |
. . . . . . . . . . . . . 14
⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ (SubSp‘𝑈)) ∧ (𝐾 ∈ 𝑌 ∧ 𝐿 ∈ 𝑌)) → (𝐾( +𝑣 ‘𝑊)𝐿) = (𝐾( +𝑣 ‘𝑈)𝐿)) |
51 | 29, 33, 37, 39, 50 | syl22anc 1319 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝐾( +𝑣 ‘𝑊)𝐿) = (𝐾( +𝑣 ‘𝑈)𝐿)) |
52 | 34 | sspnv 26965 |
. . . . . . . . . . . . . . 15
⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ (SubSp‘𝑈)) → 𝑊 ∈ NrmCVec) |
53 | 29, 33, 52 | syl2anc 691 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑊 ∈ NrmCVec) |
54 | 6, 49 | nvgcl 26859 |
. . . . . . . . . . . . . 14
⊢ ((𝑊 ∈ NrmCVec ∧ 𝐾 ∈ 𝑌 ∧ 𝐿 ∈ 𝑌) → (𝐾( +𝑣 ‘𝑊)𝐿) ∈ 𝑌) |
55 | 53, 37, 39, 54 | syl3anc 1318 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝐾( +𝑣 ‘𝑊)𝐿) ∈ 𝑌) |
56 | 51, 55 | eqeltrrd 2689 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝐾( +𝑣 ‘𝑈)𝐿) ∈ 𝑌) |
57 | | eqid 2610 |
. . . . . . . . . . . . 13
⊢ (
·𝑠OLD ‘𝑈) = ( ·𝑠OLD
‘𝑈) |
58 | | eqid 2610 |
. . . . . . . . . . . . 13
⊢ (
·𝑠OLD ‘𝑊) = ( ·𝑠OLD
‘𝑊) |
59 | 6, 57, 58, 34 | sspsval 26970 |
. . . . . . . . . . . 12
⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ (SubSp‘𝑈)) ∧ ((1 / 2) ∈ ℂ
∧ (𝐾(
+𝑣 ‘𝑈)𝐿) ∈ 𝑌)) → ((1 / 2)(
·𝑠OLD ‘𝑊)(𝐾( +𝑣 ‘𝑈)𝐿)) = ((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿))) |
60 | 29, 33, 47, 56, 59 | syl22anc 1319 |
. . . . . . . . . . 11
⊢ (𝜑 → ((1 / 2)(
·𝑠OLD ‘𝑊)(𝐾( +𝑣 ‘𝑈)𝐿)) = ((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿))) |
61 | 6, 58 | nvscl 26865 |
. . . . . . . . . . . 12
⊢ ((𝑊 ∈ NrmCVec ∧ (1 / 2)
∈ ℂ ∧ (𝐾(
+𝑣 ‘𝑈)𝐿) ∈ 𝑌) → ((1 / 2)(
·𝑠OLD ‘𝑊)(𝐾( +𝑣 ‘𝑈)𝐿)) ∈ 𝑌) |
62 | 53, 47, 56, 61 | syl3anc 1318 |
. . . . . . . . . . 11
⊢ (𝜑 → ((1 / 2)(
·𝑠OLD ‘𝑊)(𝐾( +𝑣 ‘𝑈)𝐿)) ∈ 𝑌) |
63 | 60, 62 | eqeltrrd 2689 |
. . . . . . . . . 10
⊢ (𝜑 → ((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿)) ∈ 𝑌) |
64 | 36, 63 | sseldd 3569 |
. . . . . . . . 9
⊢ (𝜑 → ((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿)) ∈ 𝑋) |
65 | 3, 4 | nvmcl 26885 |
. . . . . . . . 9
⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ ((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿)) ∈ 𝑋) → (𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿))) ∈ 𝑋) |
66 | 29, 9, 64, 65 | syl3anc 1318 |
. . . . . . . 8
⊢ (𝜑 → (𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿))) ∈ 𝑋) |
67 | 3, 5 | nvcl 26900 |
. . . . . . . 8
⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿))) ∈ 𝑋) → (𝑁‘(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿)))) ∈ ℝ) |
68 | 29, 66, 67 | syl2anc 691 |
. . . . . . 7
⊢ (𝜑 → (𝑁‘(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿)))) ∈ ℝ) |
69 | 68 | resqcld 12897 |
. . . . . 6
⊢ (𝜑 → ((𝑁‘(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿))))↑2) ∈
ℝ) |
70 | | remulcl 9900 |
. . . . . 6
⊢ ((4
∈ ℝ ∧ ((𝑁‘(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿))))↑2) ∈ ℝ) → (4
· ((𝑁‘(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿))))↑2)) ∈
ℝ) |
71 | 1, 69, 70 | sylancr 694 |
. . . . 5
⊢ (𝜑 → (4 · ((𝑁‘(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿))))↑2)) ∈
ℝ) |
72 | 71, 43 | readdcld 9948 |
. . . 4
⊢ (𝜑 → ((4 · ((𝑁‘(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿))))↑2)) + ((𝐾𝐷𝐿)↑2)) ∈ ℝ) |
73 | | minvecolem2.1 |
. . . . . 6
⊢ (𝜑 → 𝐵 ∈ ℝ) |
74 | 25, 73 | readdcld 9948 |
. . . . 5
⊢ (𝜑 → ((𝑆↑2) + 𝐵) ∈ ℝ) |
75 | | remulcl 9900 |
. . . . 5
⊢ ((4
∈ ℝ ∧ ((𝑆↑2) + 𝐵) ∈ ℝ) → (4 · ((𝑆↑2) + 𝐵)) ∈ ℝ) |
76 | 1, 74, 75 | sylancr 694 |
. . . 4
⊢ (𝜑 → (4 · ((𝑆↑2) + 𝐵)) ∈ ℝ) |
77 | 16 | a1i 11 |
. . . . . . . . . 10
⊢ (𝜑 → 0 ∈
ℝ) |
78 | | infregelb 10884 |
. . . . . . . . . 10
⊢ (((𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑤 ∈ 𝑅 𝑥 ≤ 𝑤) ∧ 0 ∈ ℝ) → (0 ≤
inf(𝑅, ℝ, < )
↔ ∀𝑤 ∈
𝑅 0 ≤ 𝑤)) |
79 | 14, 15, 21, 77, 78 | syl31anc 1321 |
. . . . . . . . 9
⊢ (𝜑 → (0 ≤ inf(𝑅, ℝ, < ) ↔
∀𝑤 ∈ 𝑅 0 ≤ 𝑤)) |
80 | 17, 79 | mpbird 246 |
. . . . . . . 8
⊢ (𝜑 → 0 ≤ inf(𝑅, ℝ, <
)) |
81 | 80, 2 | syl6breqr 4625 |
. . . . . . 7
⊢ (𝜑 → 0 ≤ 𝑆) |
82 | | eqid 2610 |
. . . . . . . . . . . 12
⊢ (𝑁‘(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿)))) = (𝑁‘(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿)))) |
83 | | oveq2 6557 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 = ((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿)) → (𝐴𝑀𝑦) = (𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿)))) |
84 | 83 | fveq2d 6107 |
. . . . . . . . . . . . . 14
⊢ (𝑦 = ((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿)) → (𝑁‘(𝐴𝑀𝑦)) = (𝑁‘(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿))))) |
85 | 84 | eqeq2d 2620 |
. . . . . . . . . . . . 13
⊢ (𝑦 = ((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿)) → ((𝑁‘(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿)))) = (𝑁‘(𝐴𝑀𝑦)) ↔ (𝑁‘(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿)))) = (𝑁‘(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿)))))) |
86 | 85 | rspcev 3282 |
. . . . . . . . . . . 12
⊢ ((((1 /
2)( ·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿)) ∈ 𝑌 ∧ (𝑁‘(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿)))) = (𝑁‘(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿))))) → ∃𝑦 ∈ 𝑌 (𝑁‘(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿)))) = (𝑁‘(𝐴𝑀𝑦))) |
87 | 63, 82, 86 | sylancl 693 |
. . . . . . . . . . 11
⊢ (𝜑 → ∃𝑦 ∈ 𝑌 (𝑁‘(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿)))) = (𝑁‘(𝐴𝑀𝑦))) |
88 | | eqid 2610 |
. . . . . . . . . . . 12
⊢ (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) = (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) |
89 | | fvex 6113 |
. . . . . . . . . . . 12
⊢ (𝑁‘(𝐴𝑀𝑦)) ∈ V |
90 | 88, 89 | elrnmpti 5297 |
. . . . . . . . . . 11
⊢ ((𝑁‘(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿)))) ∈ ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) ↔ ∃𝑦 ∈ 𝑌 (𝑁‘(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿)))) = (𝑁‘(𝐴𝑀𝑦))) |
91 | 87, 90 | sylibr 223 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑁‘(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿)))) ∈ ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦)))) |
92 | 91, 12 | syl6eleqr 2699 |
. . . . . . . . 9
⊢ (𝜑 → (𝑁‘(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿)))) ∈ 𝑅) |
93 | | infrelb 10885 |
. . . . . . . . 9
⊢ ((𝑅 ⊆ ℝ ∧
∃𝑥 ∈ ℝ
∀𝑤 ∈ 𝑅 𝑥 ≤ 𝑤 ∧ (𝑁‘(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿)))) ∈ 𝑅) → inf(𝑅, ℝ, < ) ≤ (𝑁‘(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿))))) |
94 | 14, 21, 92, 93 | syl3anc 1318 |
. . . . . . . 8
⊢ (𝜑 → inf(𝑅, ℝ, < ) ≤ (𝑁‘(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿))))) |
95 | 2, 94 | syl5eqbr 4618 |
. . . . . . 7
⊢ (𝜑 → 𝑆 ≤ (𝑁‘(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿))))) |
96 | | le2sq2 12801 |
. . . . . . 7
⊢ (((𝑆 ∈ ℝ ∧ 0 ≤
𝑆) ∧ ((𝑁‘(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿)))) ∈ ℝ ∧ 𝑆 ≤ (𝑁‘(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿)))))) → (𝑆↑2) ≤ ((𝑁‘(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿))))↑2)) |
97 | 24, 81, 68, 95, 96 | syl22anc 1319 |
. . . . . 6
⊢ (𝜑 → (𝑆↑2) ≤ ((𝑁‘(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿))))↑2)) |
98 | | 4pos 10993 |
. . . . . . . . 9
⊢ 0 <
4 |
99 | 1, 98 | pm3.2i 470 |
. . . . . . . 8
⊢ (4 ∈
ℝ ∧ 0 < 4) |
100 | | lemul2 10755 |
. . . . . . . 8
⊢ (((𝑆↑2) ∈ ℝ ∧
((𝑁‘(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿))))↑2) ∈ ℝ ∧ (4 ∈
ℝ ∧ 0 < 4)) → ((𝑆↑2) ≤ ((𝑁‘(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿))))↑2) ↔ (4 · (𝑆↑2)) ≤ (4 ·
((𝑁‘(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿))))↑2)))) |
101 | 99, 100 | mp3an3 1405 |
. . . . . . 7
⊢ (((𝑆↑2) ∈ ℝ ∧
((𝑁‘(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿))))↑2) ∈ ℝ) → ((𝑆↑2) ≤ ((𝑁‘(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿))))↑2) ↔ (4 · (𝑆↑2)) ≤ (4 ·
((𝑁‘(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿))))↑2)))) |
102 | 25, 69, 101 | syl2anc 691 |
. . . . . 6
⊢ (𝜑 → ((𝑆↑2) ≤ ((𝑁‘(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿))))↑2) ↔ (4 · (𝑆↑2)) ≤ (4 ·
((𝑁‘(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿))))↑2)))) |
103 | 97, 102 | mpbid 221 |
. . . . 5
⊢ (𝜑 → (4 · (𝑆↑2)) ≤ (4 ·
((𝑁‘(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿))))↑2))) |
104 | 27, 71, 43, 103 | leadd1dd 10520 |
. . . 4
⊢ (𝜑 → ((4 · (𝑆↑2)) + ((𝐾𝐷𝐿)↑2)) ≤ ((4 · ((𝑁‘(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿))))↑2)) + ((𝐾𝐷𝐿)↑2))) |
105 | | metcl 21947 |
. . . . . . . . . 10
⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐾 ∈ 𝑋) → (𝐴𝐷𝐾) ∈ ℝ) |
106 | 31, 9, 38, 105 | syl3anc 1318 |
. . . . . . . . 9
⊢ (𝜑 → (𝐴𝐷𝐾) ∈ ℝ) |
107 | 106 | resqcld 12897 |
. . . . . . . 8
⊢ (𝜑 → ((𝐴𝐷𝐾)↑2) ∈ ℝ) |
108 | | metcl 21947 |
. . . . . . . . . 10
⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐿 ∈ 𝑋) → (𝐴𝐷𝐿) ∈ ℝ) |
109 | 31, 9, 40, 108 | syl3anc 1318 |
. . . . . . . . 9
⊢ (𝜑 → (𝐴𝐷𝐿) ∈ ℝ) |
110 | 109 | resqcld 12897 |
. . . . . . . 8
⊢ (𝜑 → ((𝐴𝐷𝐿)↑2) ∈ ℝ) |
111 | | minvecolem2.5 |
. . . . . . . 8
⊢ (𝜑 → ((𝐴𝐷𝐾)↑2) ≤ ((𝑆↑2) + 𝐵)) |
112 | | minvecolem2.6 |
. . . . . . . 8
⊢ (𝜑 → ((𝐴𝐷𝐿)↑2) ≤ ((𝑆↑2) + 𝐵)) |
113 | 107, 110,
74, 74, 111, 112 | le2addd 10525 |
. . . . . . 7
⊢ (𝜑 → (((𝐴𝐷𝐾)↑2) + ((𝐴𝐷𝐿)↑2)) ≤ (((𝑆↑2) + 𝐵) + ((𝑆↑2) + 𝐵))) |
114 | 74 | recnd 9947 |
. . . . . . . 8
⊢ (𝜑 → ((𝑆↑2) + 𝐵) ∈ ℂ) |
115 | 114 | 2timesd 11152 |
. . . . . . 7
⊢ (𝜑 → (2 · ((𝑆↑2) + 𝐵)) = (((𝑆↑2) + 𝐵) + ((𝑆↑2) + 𝐵))) |
116 | 113, 115 | breqtrrd 4611 |
. . . . . 6
⊢ (𝜑 → (((𝐴𝐷𝐾)↑2) + ((𝐴𝐷𝐿)↑2)) ≤ (2 · ((𝑆↑2) + 𝐵))) |
117 | 107, 110 | readdcld 9948 |
. . . . . . 7
⊢ (𝜑 → (((𝐴𝐷𝐾)↑2) + ((𝐴𝐷𝐿)↑2)) ∈ ℝ) |
118 | | 2re 10967 |
. . . . . . . 8
⊢ 2 ∈
ℝ |
119 | | remulcl 9900 |
. . . . . . . 8
⊢ ((2
∈ ℝ ∧ ((𝑆↑2) + 𝐵) ∈ ℝ) → (2 · ((𝑆↑2) + 𝐵)) ∈ ℝ) |
120 | 118, 74, 119 | sylancr 694 |
. . . . . . 7
⊢ (𝜑 → (2 · ((𝑆↑2) + 𝐵)) ∈ ℝ) |
121 | | 2pos 10989 |
. . . . . . . . 9
⊢ 0 <
2 |
122 | 118, 121 | pm3.2i 470 |
. . . . . . . 8
⊢ (2 ∈
ℝ ∧ 0 < 2) |
123 | | lemul2 10755 |
. . . . . . . 8
⊢
(((((𝐴𝐷𝐾)↑2) + ((𝐴𝐷𝐿)↑2)) ∈ ℝ ∧ (2 ·
((𝑆↑2) + 𝐵)) ∈ ℝ ∧ (2
∈ ℝ ∧ 0 < 2)) → ((((𝐴𝐷𝐾)↑2) + ((𝐴𝐷𝐿)↑2)) ≤ (2 · ((𝑆↑2) + 𝐵)) ↔ (2 · (((𝐴𝐷𝐾)↑2) + ((𝐴𝐷𝐿)↑2))) ≤ (2 · (2 ·
((𝑆↑2) + 𝐵))))) |
124 | 122, 123 | mp3an3 1405 |
. . . . . . 7
⊢
(((((𝐴𝐷𝐾)↑2) + ((𝐴𝐷𝐿)↑2)) ∈ ℝ ∧ (2 ·
((𝑆↑2) + 𝐵)) ∈ ℝ) →
((((𝐴𝐷𝐾)↑2) + ((𝐴𝐷𝐿)↑2)) ≤ (2 · ((𝑆↑2) + 𝐵)) ↔ (2 · (((𝐴𝐷𝐾)↑2) + ((𝐴𝐷𝐿)↑2))) ≤ (2 · (2 ·
((𝑆↑2) + 𝐵))))) |
125 | 117, 120,
124 | syl2anc 691 |
. . . . . 6
⊢ (𝜑 → ((((𝐴𝐷𝐾)↑2) + ((𝐴𝐷𝐿)↑2)) ≤ (2 · ((𝑆↑2) + 𝐵)) ↔ (2 · (((𝐴𝐷𝐾)↑2) + ((𝐴𝐷𝐿)↑2))) ≤ (2 · (2 ·
((𝑆↑2) + 𝐵))))) |
126 | 116, 125 | mpbid 221 |
. . . . 5
⊢ (𝜑 → (2 · (((𝐴𝐷𝐾)↑2) + ((𝐴𝐷𝐿)↑2))) ≤ (2 · (2 ·
((𝑆↑2) + 𝐵)))) |
127 | 3, 4 | nvmcl 26885 |
. . . . . . . 8
⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐾 ∈ 𝑋) → (𝐴𝑀𝐾) ∈ 𝑋) |
128 | 29, 9, 38, 127 | syl3anc 1318 |
. . . . . . 7
⊢ (𝜑 → (𝐴𝑀𝐾) ∈ 𝑋) |
129 | 3, 4 | nvmcl 26885 |
. . . . . . . 8
⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐿 ∈ 𝑋) → (𝐴𝑀𝐿) ∈ 𝑋) |
130 | 29, 9, 40, 129 | syl3anc 1318 |
. . . . . . 7
⊢ (𝜑 → (𝐴𝑀𝐿) ∈ 𝑋) |
131 | 3, 48, 4, 5 | phpar2 27062 |
. . . . . . 7
⊢ ((𝑈 ∈ CPreHilOLD
∧ (𝐴𝑀𝐾) ∈ 𝑋 ∧ (𝐴𝑀𝐿) ∈ 𝑋) → (((𝑁‘((𝐴𝑀𝐾)( +𝑣 ‘𝑈)(𝐴𝑀𝐿)))↑2) + ((𝑁‘((𝐴𝑀𝐾)𝑀(𝐴𝑀𝐿)))↑2)) = (2 · (((𝑁‘(𝐴𝑀𝐾))↑2) + ((𝑁‘(𝐴𝑀𝐿))↑2)))) |
132 | 7, 128, 130, 131 | syl3anc 1318 |
. . . . . 6
⊢ (𝜑 → (((𝑁‘((𝐴𝑀𝐾)( +𝑣 ‘𝑈)(𝐴𝑀𝐿)))↑2) + ((𝑁‘((𝐴𝑀𝐾)𝑀(𝐴𝑀𝐿)))↑2)) = (2 · (((𝑁‘(𝐴𝑀𝐾))↑2) + ((𝑁‘(𝐴𝑀𝐿))↑2)))) |
133 | | 2cn 10968 |
. . . . . . . . . 10
⊢ 2 ∈
ℂ |
134 | 68 | recnd 9947 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑁‘(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿)))) ∈ ℂ) |
135 | | sqmul 12788 |
. . . . . . . . . 10
⊢ ((2
∈ ℂ ∧ (𝑁‘(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿)))) ∈ ℂ) → ((2 ·
(𝑁‘(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿)))))↑2) = ((2↑2) · ((𝑁‘(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿))))↑2))) |
136 | 133, 134,
135 | sylancr 694 |
. . . . . . . . 9
⊢ (𝜑 → ((2 · (𝑁‘(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿)))))↑2) = ((2↑2) · ((𝑁‘(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿))))↑2))) |
137 | | sq2 12822 |
. . . . . . . . . 10
⊢
(2↑2) = 4 |
138 | 137 | oveq1i 6559 |
. . . . . . . . 9
⊢
((2↑2) · ((𝑁‘(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿))))↑2)) = (4 · ((𝑁‘(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿))))↑2)) |
139 | 136, 138 | syl6eq 2660 |
. . . . . . . 8
⊢ (𝜑 → ((2 · (𝑁‘(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿)))))↑2) = (4 · ((𝑁‘(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿))))↑2))) |
140 | 133 | a1i 11 |
. . . . . . . . . . . 12
⊢ (𝜑 → 2 ∈
ℂ) |
141 | 3, 57, 5 | nvs 26902 |
. . . . . . . . . . . 12
⊢ ((𝑈 ∈ NrmCVec ∧ 2 ∈
ℂ ∧ (𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿))) ∈ 𝑋) → (𝑁‘(2(
·𝑠OLD ‘𝑈)(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿))))) = ((abs‘2) · (𝑁‘(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿)))))) |
142 | 29, 140, 66, 141 | syl3anc 1318 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑁‘(2(
·𝑠OLD ‘𝑈)(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿))))) = ((abs‘2) · (𝑁‘(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿)))))) |
143 | | 0le2 10988 |
. . . . . . . . . . . . 13
⊢ 0 ≤
2 |
144 | | absid 13884 |
. . . . . . . . . . . . 13
⊢ ((2
∈ ℝ ∧ 0 ≤ 2) → (abs‘2) = 2) |
145 | 118, 143,
144 | mp2an 704 |
. . . . . . . . . . . 12
⊢
(abs‘2) = 2 |
146 | 145 | oveq1i 6559 |
. . . . . . . . . . 11
⊢
((abs‘2) · (𝑁‘(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿))))) = (2 · (𝑁‘(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿))))) |
147 | 142, 146 | syl6eq 2660 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑁‘(2(
·𝑠OLD ‘𝑈)(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿))))) = (2 · (𝑁‘(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿)))))) |
148 | 3, 4, 57 | nvmdi 26887 |
. . . . . . . . . . . . 13
⊢ ((𝑈 ∈ NrmCVec ∧ (2 ∈
ℂ ∧ 𝐴 ∈
𝑋 ∧ ((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿)) ∈ 𝑋)) → (2(
·𝑠OLD ‘𝑈)(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿)))) = ((2(
·𝑠OLD ‘𝑈)𝐴)𝑀(2( ·𝑠OLD
‘𝑈)((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿))))) |
149 | 29, 140, 9, 64, 148 | syl13anc 1320 |
. . . . . . . . . . . 12
⊢ (𝜑 → (2(
·𝑠OLD ‘𝑈)(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿)))) = ((2(
·𝑠OLD ‘𝑈)𝐴)𝑀(2( ·𝑠OLD
‘𝑈)((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿))))) |
150 | 3, 48, 57 | nv2 26871 |
. . . . . . . . . . . . . 14
⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴( +𝑣 ‘𝑈)𝐴) = (2(
·𝑠OLD ‘𝑈)𝐴)) |
151 | 29, 9, 150 | syl2anc 691 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝐴( +𝑣 ‘𝑈)𝐴) = (2(
·𝑠OLD ‘𝑈)𝐴)) |
152 | | 2ne0 10990 |
. . . . . . . . . . . . . . . . 17
⊢ 2 ≠
0 |
153 | 133, 152 | recidi 10635 |
. . . . . . . . . . . . . . . 16
⊢ (2
· (1 / 2)) = 1 |
154 | 153 | oveq1i 6559 |
. . . . . . . . . . . . . . 15
⊢ ((2
· (1 / 2))( ·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿)) = (1(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿)) |
155 | 3, 48 | nvgcl 26859 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑈 ∈ NrmCVec ∧ 𝐾 ∈ 𝑋 ∧ 𝐿 ∈ 𝑋) → (𝐾( +𝑣 ‘𝑈)𝐿) ∈ 𝑋) |
156 | 29, 38, 40, 155 | syl3anc 1318 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝐾( +𝑣 ‘𝑈)𝐿) ∈ 𝑋) |
157 | 3, 57 | nvsid 26866 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑈 ∈ NrmCVec ∧ (𝐾( +𝑣
‘𝑈)𝐿) ∈ 𝑋) → (1(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿)) = (𝐾( +𝑣 ‘𝑈)𝐿)) |
158 | 29, 156, 157 | syl2anc 691 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (1(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿)) = (𝐾( +𝑣 ‘𝑈)𝐿)) |
159 | 154, 158 | syl5eq 2656 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((2 · (1 / 2))(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿)) = (𝐾( +𝑣 ‘𝑈)𝐿)) |
160 | 3, 57 | nvsass 26867 |
. . . . . . . . . . . . . . 15
⊢ ((𝑈 ∈ NrmCVec ∧ (2 ∈
ℂ ∧ (1 / 2) ∈ ℂ ∧ (𝐾( +𝑣 ‘𝑈)𝐿) ∈ 𝑋)) → ((2 · (1 / 2))(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿)) = (2(
·𝑠OLD ‘𝑈)((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿)))) |
161 | 29, 140, 47, 156, 160 | syl13anc 1320 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((2 · (1 / 2))(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿)) = (2(
·𝑠OLD ‘𝑈)((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿)))) |
162 | 159, 161 | eqtr3d 2646 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝐾( +𝑣 ‘𝑈)𝐿) = (2(
·𝑠OLD ‘𝑈)((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿)))) |
163 | 151, 162 | oveq12d 6567 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((𝐴( +𝑣 ‘𝑈)𝐴)𝑀(𝐾( +𝑣 ‘𝑈)𝐿)) = ((2(
·𝑠OLD ‘𝑈)𝐴)𝑀(2( ·𝑠OLD
‘𝑈)((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿))))) |
164 | 3, 48, 4 | nvaddsub4 26896 |
. . . . . . . . . . . . 13
⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) ∧ (𝐾 ∈ 𝑋 ∧ 𝐿 ∈ 𝑋)) → ((𝐴( +𝑣 ‘𝑈)𝐴)𝑀(𝐾( +𝑣 ‘𝑈)𝐿)) = ((𝐴𝑀𝐾)( +𝑣 ‘𝑈)(𝐴𝑀𝐿))) |
165 | 29, 9, 9, 38, 40, 164 | syl122anc 1327 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((𝐴( +𝑣 ‘𝑈)𝐴)𝑀(𝐾( +𝑣 ‘𝑈)𝐿)) = ((𝐴𝑀𝐾)( +𝑣 ‘𝑈)(𝐴𝑀𝐿))) |
166 | 149, 163,
165 | 3eqtr2d 2650 |
. . . . . . . . . . 11
⊢ (𝜑 → (2(
·𝑠OLD ‘𝑈)(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿)))) = ((𝐴𝑀𝐾)( +𝑣 ‘𝑈)(𝐴𝑀𝐿))) |
167 | 166 | fveq2d 6107 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑁‘(2(
·𝑠OLD ‘𝑈)(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿))))) = (𝑁‘((𝐴𝑀𝐾)( +𝑣 ‘𝑈)(𝐴𝑀𝐿)))) |
168 | 147, 167 | eqtr3d 2646 |
. . . . . . . . 9
⊢ (𝜑 → (2 · (𝑁‘(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿))))) = (𝑁‘((𝐴𝑀𝐾)( +𝑣 ‘𝑈)(𝐴𝑀𝐿)))) |
169 | 168 | oveq1d 6564 |
. . . . . . . 8
⊢ (𝜑 → ((2 · (𝑁‘(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿)))))↑2) = ((𝑁‘((𝐴𝑀𝐾)( +𝑣 ‘𝑈)(𝐴𝑀𝐿)))↑2)) |
170 | 139, 169 | eqtr3d 2646 |
. . . . . . 7
⊢ (𝜑 → (4 · ((𝑁‘(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿))))↑2)) = ((𝑁‘((𝐴𝑀𝐾)( +𝑣 ‘𝑈)(𝐴𝑀𝐿)))↑2)) |
171 | 3, 4, 5, 10 | imsdval 26925 |
. . . . . . . . . 10
⊢ ((𝑈 ∈ NrmCVec ∧ 𝐿 ∈ 𝑋 ∧ 𝐾 ∈ 𝑋) → (𝐿𝐷𝐾) = (𝑁‘(𝐿𝑀𝐾))) |
172 | 29, 40, 38, 171 | syl3anc 1318 |
. . . . . . . . 9
⊢ (𝜑 → (𝐿𝐷𝐾) = (𝑁‘(𝐿𝑀𝐾))) |
173 | | metsym 21965 |
. . . . . . . . . 10
⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐾 ∈ 𝑋 ∧ 𝐿 ∈ 𝑋) → (𝐾𝐷𝐿) = (𝐿𝐷𝐾)) |
174 | 31, 38, 40, 173 | syl3anc 1318 |
. . . . . . . . 9
⊢ (𝜑 → (𝐾𝐷𝐿) = (𝐿𝐷𝐾)) |
175 | 3, 4 | nvnnncan1 26886 |
. . . . . . . . . . 11
⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐾 ∈ 𝑋 ∧ 𝐿 ∈ 𝑋)) → ((𝐴𝑀𝐾)𝑀(𝐴𝑀𝐿)) = (𝐿𝑀𝐾)) |
176 | 29, 9, 38, 40, 175 | syl13anc 1320 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝐴𝑀𝐾)𝑀(𝐴𝑀𝐿)) = (𝐿𝑀𝐾)) |
177 | 176 | fveq2d 6107 |
. . . . . . . . 9
⊢ (𝜑 → (𝑁‘((𝐴𝑀𝐾)𝑀(𝐴𝑀𝐿))) = (𝑁‘(𝐿𝑀𝐾))) |
178 | 172, 174,
177 | 3eqtr4d 2654 |
. . . . . . . 8
⊢ (𝜑 → (𝐾𝐷𝐿) = (𝑁‘((𝐴𝑀𝐾)𝑀(𝐴𝑀𝐿)))) |
179 | 178 | oveq1d 6564 |
. . . . . . 7
⊢ (𝜑 → ((𝐾𝐷𝐿)↑2) = ((𝑁‘((𝐴𝑀𝐾)𝑀(𝐴𝑀𝐿)))↑2)) |
180 | 170, 179 | oveq12d 6567 |
. . . . . 6
⊢ (𝜑 → ((4 · ((𝑁‘(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿))))↑2)) + ((𝐾𝐷𝐿)↑2)) = (((𝑁‘((𝐴𝑀𝐾)( +𝑣 ‘𝑈)(𝐴𝑀𝐿)))↑2) + ((𝑁‘((𝐴𝑀𝐾)𝑀(𝐴𝑀𝐿)))↑2))) |
181 | 3, 4, 5, 10 | imsdval 26925 |
. . . . . . . . . 10
⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐾 ∈ 𝑋) → (𝐴𝐷𝐾) = (𝑁‘(𝐴𝑀𝐾))) |
182 | 29, 9, 38, 181 | syl3anc 1318 |
. . . . . . . . 9
⊢ (𝜑 → (𝐴𝐷𝐾) = (𝑁‘(𝐴𝑀𝐾))) |
183 | 182 | oveq1d 6564 |
. . . . . . . 8
⊢ (𝜑 → ((𝐴𝐷𝐾)↑2) = ((𝑁‘(𝐴𝑀𝐾))↑2)) |
184 | 3, 4, 5, 10 | imsdval 26925 |
. . . . . . . . . 10
⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐿 ∈ 𝑋) → (𝐴𝐷𝐿) = (𝑁‘(𝐴𝑀𝐿))) |
185 | 29, 9, 40, 184 | syl3anc 1318 |
. . . . . . . . 9
⊢ (𝜑 → (𝐴𝐷𝐿) = (𝑁‘(𝐴𝑀𝐿))) |
186 | 185 | oveq1d 6564 |
. . . . . . . 8
⊢ (𝜑 → ((𝐴𝐷𝐿)↑2) = ((𝑁‘(𝐴𝑀𝐿))↑2)) |
187 | 183, 186 | oveq12d 6567 |
. . . . . . 7
⊢ (𝜑 → (((𝐴𝐷𝐾)↑2) + ((𝐴𝐷𝐿)↑2)) = (((𝑁‘(𝐴𝑀𝐾))↑2) + ((𝑁‘(𝐴𝑀𝐿))↑2))) |
188 | 187 | oveq2d 6565 |
. . . . . 6
⊢ (𝜑 → (2 · (((𝐴𝐷𝐾)↑2) + ((𝐴𝐷𝐿)↑2))) = (2 · (((𝑁‘(𝐴𝑀𝐾))↑2) + ((𝑁‘(𝐴𝑀𝐿))↑2)))) |
189 | 132, 180,
188 | 3eqtr4d 2654 |
. . . . 5
⊢ (𝜑 → ((4 · ((𝑁‘(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿))))↑2)) + ((𝐾𝐷𝐿)↑2)) = (2 · (((𝐴𝐷𝐾)↑2) + ((𝐴𝐷𝐿)↑2)))) |
190 | | 2t2e4 11054 |
. . . . . . 7
⊢ (2
· 2) = 4 |
191 | 190 | oveq1i 6559 |
. . . . . 6
⊢ ((2
· 2) · ((𝑆↑2) + 𝐵)) = (4 · ((𝑆↑2) + 𝐵)) |
192 | 140, 140,
114 | mulassd 9942 |
. . . . . 6
⊢ (𝜑 → ((2 · 2) ·
((𝑆↑2) + 𝐵)) = (2 · (2 ·
((𝑆↑2) + 𝐵)))) |
193 | 191, 192 | syl5eqr 2658 |
. . . . 5
⊢ (𝜑 → (4 · ((𝑆↑2) + 𝐵)) = (2 · (2 · ((𝑆↑2) + 𝐵)))) |
194 | 126, 189,
193 | 3brtr4d 4615 |
. . . 4
⊢ (𝜑 → ((4 · ((𝑁‘(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿))))↑2)) + ((𝐾𝐷𝐿)↑2)) ≤ (4 · ((𝑆↑2) + 𝐵))) |
195 | 44, 72, 76, 104, 194 | letrd 10073 |
. . 3
⊢ (𝜑 → ((4 · (𝑆↑2)) + ((𝐾𝐷𝐿)↑2)) ≤ (4 · ((𝑆↑2) + 𝐵))) |
196 | | 4cn 10975 |
. . . . 5
⊢ 4 ∈
ℂ |
197 | 196 | a1i 11 |
. . . 4
⊢ (𝜑 → 4 ∈
ℂ) |
198 | 25 | recnd 9947 |
. . . 4
⊢ (𝜑 → (𝑆↑2) ∈ ℂ) |
199 | 73 | recnd 9947 |
. . . 4
⊢ (𝜑 → 𝐵 ∈ ℂ) |
200 | 197, 198,
199 | adddid 9943 |
. . 3
⊢ (𝜑 → (4 · ((𝑆↑2) + 𝐵)) = ((4 · (𝑆↑2)) + (4 · 𝐵))) |
201 | 195, 200 | breqtrd 4609 |
. 2
⊢ (𝜑 → ((4 · (𝑆↑2)) + ((𝐾𝐷𝐿)↑2)) ≤ ((4 · (𝑆↑2)) + (4 · 𝐵))) |
202 | | remulcl 9900 |
. . . 4
⊢ ((4
∈ ℝ ∧ 𝐵
∈ ℝ) → (4 · 𝐵) ∈ ℝ) |
203 | 1, 73, 202 | sylancr 694 |
. . 3
⊢ (𝜑 → (4 · 𝐵) ∈
ℝ) |
204 | 43, 203, 27 | leadd2d 10501 |
. 2
⊢ (𝜑 → (((𝐾𝐷𝐿)↑2) ≤ (4 · 𝐵) ↔ ((4 · (𝑆↑2)) + ((𝐾𝐷𝐿)↑2)) ≤ ((4 · (𝑆↑2)) + (4 · 𝐵)))) |
205 | 201, 204 | mpbird 246 |
1
⊢ (𝜑 → ((𝐾𝐷𝐿)↑2) ≤ (4 · 𝐵)) |