Step | Hyp | Ref
| Expression |
1 | | 4re 10974 |
. . . . . 6
⊢ 4 ∈
ℝ |
2 | | minvec.x |
. . . . . . . 8
⊢ 𝑋 = (Base‘𝑈) |
3 | | minvec.m |
. . . . . . . 8
⊢ − =
(-g‘𝑈) |
4 | | minvec.n |
. . . . . . . 8
⊢ 𝑁 = (norm‘𝑈) |
5 | | minvec.u |
. . . . . . . 8
⊢ (𝜑 → 𝑈 ∈ ℂPreHil) |
6 | | minvec.y |
. . . . . . . 8
⊢ (𝜑 → 𝑌 ∈ (LSubSp‘𝑈)) |
7 | | minvec.w |
. . . . . . . 8
⊢ (𝜑 → (𝑈 ↾s 𝑌) ∈ CMetSp) |
8 | | minvec.a |
. . . . . . . 8
⊢ (𝜑 → 𝐴 ∈ 𝑋) |
9 | | minvec.j |
. . . . . . . 8
⊢ 𝐽 = (TopOpen‘𝑈) |
10 | | minvec.r |
. . . . . . . 8
⊢ 𝑅 = ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))) |
11 | | minvec.s |
. . . . . . . 8
⊢ 𝑆 = inf(𝑅, ℝ, < ) |
12 | 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 | minveclem4c 23004 |
. . . . . . 7
⊢ (𝜑 → 𝑆 ∈ ℝ) |
13 | 12 | resqcld 12897 |
. . . . . 6
⊢ (𝜑 → (𝑆↑2) ∈ ℝ) |
14 | | remulcl 9900 |
. . . . . 6
⊢ ((4
∈ ℝ ∧ (𝑆↑2) ∈ ℝ) → (4 ·
(𝑆↑2)) ∈
ℝ) |
15 | 1, 13, 14 | sylancr 694 |
. . . . 5
⊢ (𝜑 → (4 · (𝑆↑2)) ∈
ℝ) |
16 | | cphngp 22781 |
. . . . . . . . . 10
⊢ (𝑈 ∈ ℂPreHil →
𝑈 ∈
NrmGrp) |
17 | 5, 16 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝑈 ∈ NrmGrp) |
18 | | ngpms 22214 |
. . . . . . . . 9
⊢ (𝑈 ∈ NrmGrp → 𝑈 ∈ MetSp) |
19 | 17, 18 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 𝑈 ∈ MetSp) |
20 | | minvec.d |
. . . . . . . . 9
⊢ 𝐷 = ((dist‘𝑈) ↾ (𝑋 × 𝑋)) |
21 | 2, 20 | msmet 22072 |
. . . . . . . 8
⊢ (𝑈 ∈ MetSp → 𝐷 ∈ (Met‘𝑋)) |
22 | 19, 21 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋)) |
23 | | eqid 2610 |
. . . . . . . . . 10
⊢
(LSubSp‘𝑈) =
(LSubSp‘𝑈) |
24 | 2, 23 | lssss 18758 |
. . . . . . . . 9
⊢ (𝑌 ∈ (LSubSp‘𝑈) → 𝑌 ⊆ 𝑋) |
25 | 6, 24 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 𝑌 ⊆ 𝑋) |
26 | | minveclem2.3 |
. . . . . . . 8
⊢ (𝜑 → 𝐾 ∈ 𝑌) |
27 | 25, 26 | sseldd 3569 |
. . . . . . 7
⊢ (𝜑 → 𝐾 ∈ 𝑋) |
28 | | minveclem2.4 |
. . . . . . . 8
⊢ (𝜑 → 𝐿 ∈ 𝑌) |
29 | 25, 28 | sseldd 3569 |
. . . . . . 7
⊢ (𝜑 → 𝐿 ∈ 𝑋) |
30 | | metcl 21947 |
. . . . . . 7
⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐾 ∈ 𝑋 ∧ 𝐿 ∈ 𝑋) → (𝐾𝐷𝐿) ∈ ℝ) |
31 | 22, 27, 29, 30 | syl3anc 1318 |
. . . . . 6
⊢ (𝜑 → (𝐾𝐷𝐿) ∈ ℝ) |
32 | 31 | resqcld 12897 |
. . . . 5
⊢ (𝜑 → ((𝐾𝐷𝐿)↑2) ∈ ℝ) |
33 | 15, 32 | readdcld 9948 |
. . . 4
⊢ (𝜑 → ((4 · (𝑆↑2)) + ((𝐾𝐷𝐿)↑2)) ∈ ℝ) |
34 | | cphlmod 22782 |
. . . . . . . . . 10
⊢ (𝑈 ∈ ℂPreHil →
𝑈 ∈
LMod) |
35 | 5, 34 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝑈 ∈ LMod) |
36 | | cphclm 22797 |
. . . . . . . . . . . . . . 15
⊢ (𝑈 ∈ ℂPreHil →
𝑈 ∈
ℂMod) |
37 | 5, 36 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑈 ∈ ℂMod) |
38 | | eqid 2610 |
. . . . . . . . . . . . . . 15
⊢
(Scalar‘𝑈) =
(Scalar‘𝑈) |
39 | | eqid 2610 |
. . . . . . . . . . . . . . 15
⊢
(Base‘(Scalar‘𝑈)) = (Base‘(Scalar‘𝑈)) |
40 | 38, 39 | clmzss 22686 |
. . . . . . . . . . . . . 14
⊢ (𝑈 ∈ ℂMod →
ℤ ⊆ (Base‘(Scalar‘𝑈))) |
41 | 37, 40 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ℤ ⊆
(Base‘(Scalar‘𝑈))) |
42 | | 2z 11286 |
. . . . . . . . . . . . . 14
⊢ 2 ∈
ℤ |
43 | 42 | a1i 11 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 2 ∈
ℤ) |
44 | 41, 43 | sseldd 3569 |
. . . . . . . . . . . 12
⊢ (𝜑 → 2 ∈
(Base‘(Scalar‘𝑈))) |
45 | | 2ne0 10990 |
. . . . . . . . . . . . 13
⊢ 2 ≠
0 |
46 | 45 | a1i 11 |
. . . . . . . . . . . 12
⊢ (𝜑 → 2 ≠ 0) |
47 | 38, 39 | cphreccl 22789 |
. . . . . . . . . . . 12
⊢ ((𝑈 ∈ ℂPreHil ∧ 2
∈ (Base‘(Scalar‘𝑈)) ∧ 2 ≠ 0) → (1 / 2) ∈
(Base‘(Scalar‘𝑈))) |
48 | 5, 44, 46, 47 | syl3anc 1318 |
. . . . . . . . . . 11
⊢ (𝜑 → (1 / 2) ∈
(Base‘(Scalar‘𝑈))) |
49 | | eqid 2610 |
. . . . . . . . . . . . 13
⊢
(+g‘𝑈) = (+g‘𝑈) |
50 | 49, 23 | lssvacl 18775 |
. . . . . . . . . . . 12
⊢ (((𝑈 ∈ LMod ∧ 𝑌 ∈ (LSubSp‘𝑈)) ∧ (𝐾 ∈ 𝑌 ∧ 𝐿 ∈ 𝑌)) → (𝐾(+g‘𝑈)𝐿) ∈ 𝑌) |
51 | 35, 6, 26, 28, 50 | syl22anc 1319 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐾(+g‘𝑈)𝐿) ∈ 𝑌) |
52 | | eqid 2610 |
. . . . . . . . . . . 12
⊢ (
·𝑠 ‘𝑈) = ( ·𝑠
‘𝑈) |
53 | 38, 52, 39, 23 | lssvscl 18776 |
. . . . . . . . . . 11
⊢ (((𝑈 ∈ LMod ∧ 𝑌 ∈ (LSubSp‘𝑈)) ∧ ((1 / 2) ∈
(Base‘(Scalar‘𝑈)) ∧ (𝐾(+g‘𝑈)𝐿) ∈ 𝑌)) → ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿)) ∈ 𝑌) |
54 | 35, 6, 48, 51, 53 | syl22anc 1319 |
. . . . . . . . . 10
⊢ (𝜑 → ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿)) ∈ 𝑌) |
55 | 25, 54 | sseldd 3569 |
. . . . . . . . 9
⊢ (𝜑 → ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿)) ∈ 𝑋) |
56 | 2, 3 | lmodvsubcl 18731 |
. . . . . . . . 9
⊢ ((𝑈 ∈ LMod ∧ 𝐴 ∈ 𝑋 ∧ ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿)) ∈ 𝑋) → (𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿))) ∈ 𝑋) |
57 | 35, 8, 55, 56 | syl3anc 1318 |
. . . . . . . 8
⊢ (𝜑 → (𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿))) ∈ 𝑋) |
58 | 2, 4 | nmcl 22230 |
. . . . . . . 8
⊢ ((𝑈 ∈ NrmGrp ∧ (𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿))) ∈ 𝑋) → (𝑁‘(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿)))) ∈ ℝ) |
59 | 17, 57, 58 | syl2anc 691 |
. . . . . . 7
⊢ (𝜑 → (𝑁‘(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿)))) ∈ ℝ) |
60 | 59 | resqcld 12897 |
. . . . . 6
⊢ (𝜑 → ((𝑁‘(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿))))↑2) ∈
ℝ) |
61 | | remulcl 9900 |
. . . . . 6
⊢ ((4
∈ ℝ ∧ ((𝑁‘(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿))))↑2) ∈ ℝ) → (4
· ((𝑁‘(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿))))↑2)) ∈
ℝ) |
62 | 1, 60, 61 | sylancr 694 |
. . . . 5
⊢ (𝜑 → (4 · ((𝑁‘(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿))))↑2)) ∈
ℝ) |
63 | 62, 32 | readdcld 9948 |
. . . 4
⊢ (𝜑 → ((4 · ((𝑁‘(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿))))↑2)) + ((𝐾𝐷𝐿)↑2)) ∈ ℝ) |
64 | | minveclem2.1 |
. . . . . 6
⊢ (𝜑 → 𝐵 ∈ ℝ) |
65 | 13, 64 | readdcld 9948 |
. . . . 5
⊢ (𝜑 → ((𝑆↑2) + 𝐵) ∈ ℝ) |
66 | | remulcl 9900 |
. . . . 5
⊢ ((4
∈ ℝ ∧ ((𝑆↑2) + 𝐵) ∈ ℝ) → (4 · ((𝑆↑2) + 𝐵)) ∈ ℝ) |
67 | 1, 65, 66 | sylancr 694 |
. . . 4
⊢ (𝜑 → (4 · ((𝑆↑2) + 𝐵)) ∈ ℝ) |
68 | 2, 3, 4, 5, 6, 7, 8, 9, 10 | minveclem1 23003 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∀𝑤 ∈ 𝑅 0 ≤ 𝑤)) |
69 | 68 | simp3d 1068 |
. . . . . . . . 9
⊢ (𝜑 → ∀𝑤 ∈ 𝑅 0 ≤ 𝑤) |
70 | 68 | simp1d 1066 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑅 ⊆ ℝ) |
71 | 68 | simp2d 1067 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑅 ≠ ∅) |
72 | | 0re 9919 |
. . . . . . . . . . 11
⊢ 0 ∈
ℝ |
73 | | breq1 4586 |
. . . . . . . . . . . . 13
⊢ (𝑥 = 0 → (𝑥 ≤ 𝑤 ↔ 0 ≤ 𝑤)) |
74 | 73 | ralbidv 2969 |
. . . . . . . . . . . 12
⊢ (𝑥 = 0 → (∀𝑤 ∈ 𝑅 𝑥 ≤ 𝑤 ↔ ∀𝑤 ∈ 𝑅 0 ≤ 𝑤)) |
75 | 74 | rspcev 3282 |
. . . . . . . . . . 11
⊢ ((0
∈ ℝ ∧ ∀𝑤 ∈ 𝑅 0 ≤ 𝑤) → ∃𝑥 ∈ ℝ ∀𝑤 ∈ 𝑅 𝑥 ≤ 𝑤) |
76 | 72, 69, 75 | sylancr 694 |
. . . . . . . . . 10
⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑤 ∈ 𝑅 𝑥 ≤ 𝑤) |
77 | 72 | 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 | 70, 71, 76, 77, 78 | syl31anc 1321 |
. . . . . . . . 9
⊢ (𝜑 → (0 ≤ inf(𝑅, ℝ, < ) ↔
∀𝑤 ∈ 𝑅 0 ≤ 𝑤)) |
80 | 69, 79 | mpbird 246 |
. . . . . . . 8
⊢ (𝜑 → 0 ≤ inf(𝑅, ℝ, <
)) |
81 | 80, 11 | syl6breqr 4625 |
. . . . . . 7
⊢ (𝜑 → 0 ≤ 𝑆) |
82 | | eqid 2610 |
. . . . . . . . . . . 12
⊢ (𝑁‘(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿)))) = (𝑁‘(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿)))) |
83 | | oveq2 6557 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 = ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿)) → (𝐴 − 𝑦) = (𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿)))) |
84 | 83 | fveq2d 6107 |
. . . . . . . . . . . . . 14
⊢ (𝑦 = ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿)) → (𝑁‘(𝐴 − 𝑦)) = (𝑁‘(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿))))) |
85 | 84 | eqeq2d 2620 |
. . . . . . . . . . . . 13
⊢ (𝑦 = ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿)) → ((𝑁‘(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿)))) = (𝑁‘(𝐴 − 𝑦)) ↔ (𝑁‘(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿)))) = (𝑁‘(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿)))))) |
86 | 85 | rspcev 3282 |
. . . . . . . . . . . 12
⊢ ((((1 /
2)( ·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿)) ∈ 𝑌 ∧ (𝑁‘(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿)))) = (𝑁‘(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿))))) → ∃𝑦 ∈ 𝑌 (𝑁‘(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿)))) = (𝑁‘(𝐴 − 𝑦))) |
87 | 54, 82, 86 | sylancl 693 |
. . . . . . . . . . 11
⊢ (𝜑 → ∃𝑦 ∈ 𝑌 (𝑁‘(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿)))) = (𝑁‘(𝐴 − 𝑦))) |
88 | | eqid 2610 |
. . . . . . . . . . . 12
⊢ (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))) = (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))) |
89 | | fvex 6113 |
. . . . . . . . . . . 12
⊢ (𝑁‘(𝐴 − 𝑦)) ∈ V |
90 | 88, 89 | elrnmpti 5297 |
. . . . . . . . . . 11
⊢ ((𝑁‘(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿)))) ∈ ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))) ↔ ∃𝑦 ∈ 𝑌 (𝑁‘(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿)))) = (𝑁‘(𝐴 − 𝑦))) |
91 | 87, 90 | sylibr 223 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑁‘(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿)))) ∈ ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦)))) |
92 | 91, 10 | syl6eleqr 2699 |
. . . . . . . . 9
⊢ (𝜑 → (𝑁‘(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿)))) ∈ 𝑅) |
93 | | infrelb 10885 |
. . . . . . . . 9
⊢ ((𝑅 ⊆ ℝ ∧
∃𝑥 ∈ ℝ
∀𝑤 ∈ 𝑅 𝑥 ≤ 𝑤 ∧ (𝑁‘(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿)))) ∈ 𝑅) → inf(𝑅, ℝ, < ) ≤ (𝑁‘(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿))))) |
94 | 70, 76, 92, 93 | syl3anc 1318 |
. . . . . . . 8
⊢ (𝜑 → inf(𝑅, ℝ, < ) ≤ (𝑁‘(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿))))) |
95 | 11, 94 | syl5eqbr 4618 |
. . . . . . 7
⊢ (𝜑 → 𝑆 ≤ (𝑁‘(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿))))) |
96 | | le2sq2 12801 |
. . . . . . 7
⊢ (((𝑆 ∈ ℝ ∧ 0 ≤
𝑆) ∧ ((𝑁‘(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿)))) ∈ ℝ ∧ 𝑆 ≤ (𝑁‘(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿)))))) → (𝑆↑2) ≤ ((𝑁‘(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿))))↑2)) |
97 | 12, 81, 59, 95, 96 | syl22anc 1319 |
. . . . . 6
⊢ (𝜑 → (𝑆↑2) ≤ ((𝑁‘(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘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‘(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿))))↑2) ∈ ℝ ∧ (4 ∈
ℝ ∧ 0 < 4)) → ((𝑆↑2) ≤ ((𝑁‘(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿))))↑2) ↔ (4 · (𝑆↑2)) ≤ (4 ·
((𝑁‘(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿))))↑2)))) |
101 | 99, 100 | mp3an3 1405 |
. . . . . . 7
⊢ (((𝑆↑2) ∈ ℝ ∧
((𝑁‘(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿))))↑2) ∈ ℝ) → ((𝑆↑2) ≤ ((𝑁‘(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿))))↑2) ↔ (4 · (𝑆↑2)) ≤ (4 ·
((𝑁‘(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿))))↑2)))) |
102 | 13, 60, 101 | syl2anc 691 |
. . . . . 6
⊢ (𝜑 → ((𝑆↑2) ≤ ((𝑁‘(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿))))↑2) ↔ (4 · (𝑆↑2)) ≤ (4 ·
((𝑁‘(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿))))↑2)))) |
103 | 97, 102 | mpbid 221 |
. . . . 5
⊢ (𝜑 → (4 · (𝑆↑2)) ≤ (4 ·
((𝑁‘(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿))))↑2))) |
104 | 15, 62, 32, 103 | leadd1dd 10520 |
. . . 4
⊢ (𝜑 → ((4 · (𝑆↑2)) + ((𝐾𝐷𝐿)↑2)) ≤ ((4 · ((𝑁‘(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿))))↑2)) + ((𝐾𝐷𝐿)↑2))) |
105 | | metcl 21947 |
. . . . . . . . . 10
⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐾 ∈ 𝑋) → (𝐴𝐷𝐾) ∈ ℝ) |
106 | 22, 8, 27, 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 | 22, 8, 29, 108 | syl3anc 1318 |
. . . . . . . . 9
⊢ (𝜑 → (𝐴𝐷𝐿) ∈ ℝ) |
110 | 109 | resqcld 12897 |
. . . . . . . 8
⊢ (𝜑 → ((𝐴𝐷𝐿)↑2) ∈ ℝ) |
111 | | minveclem2.5 |
. . . . . . . 8
⊢ (𝜑 → ((𝐴𝐷𝐾)↑2) ≤ ((𝑆↑2) + 𝐵)) |
112 | | minveclem2.6 |
. . . . . . . 8
⊢ (𝜑 → ((𝐴𝐷𝐿)↑2) ≤ ((𝑆↑2) + 𝐵)) |
113 | 107, 110,
65, 65, 111, 112 | le2addd 10525 |
. . . . . . 7
⊢ (𝜑 → (((𝐴𝐷𝐾)↑2) + ((𝐴𝐷𝐿)↑2)) ≤ (((𝑆↑2) + 𝐵) + ((𝑆↑2) + 𝐵))) |
114 | 65 | 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, 65, 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 | 2, 3 | lmodvsubcl 18731 |
. . . . . . . 8
⊢ ((𝑈 ∈ LMod ∧ 𝐴 ∈ 𝑋 ∧ 𝐾 ∈ 𝑋) → (𝐴 − 𝐾) ∈ 𝑋) |
128 | 35, 8, 27, 127 | syl3anc 1318 |
. . . . . . 7
⊢ (𝜑 → (𝐴 − 𝐾) ∈ 𝑋) |
129 | 2, 3 | lmodvsubcl 18731 |
. . . . . . . 8
⊢ ((𝑈 ∈ LMod ∧ 𝐴 ∈ 𝑋 ∧ 𝐿 ∈ 𝑋) → (𝐴 − 𝐿) ∈ 𝑋) |
130 | 35, 8, 29, 129 | syl3anc 1318 |
. . . . . . 7
⊢ (𝜑 → (𝐴 − 𝐿) ∈ 𝑋) |
131 | 2, 49, 3, 4 | nmpar 22847 |
. . . . . . 7
⊢ ((𝑈 ∈ ℂPreHil ∧
(𝐴 − 𝐾) ∈ 𝑋 ∧ (𝐴 − 𝐿) ∈ 𝑋) → (((𝑁‘((𝐴 − 𝐾)(+g‘𝑈)(𝐴 − 𝐿)))↑2) + ((𝑁‘((𝐴 − 𝐾) − (𝐴 − 𝐿)))↑2)) = (2 · (((𝑁‘(𝐴 − 𝐾))↑2) + ((𝑁‘(𝐴 − 𝐿))↑2)))) |
132 | 5, 128, 130, 131 | syl3anc 1318 |
. . . . . 6
⊢ (𝜑 → (((𝑁‘((𝐴 − 𝐾)(+g‘𝑈)(𝐴 − 𝐿)))↑2) + ((𝑁‘((𝐴 − 𝐾) − (𝐴 − 𝐿)))↑2)) = (2 · (((𝑁‘(𝐴 − 𝐾))↑2) + ((𝑁‘(𝐴 − 𝐿))↑2)))) |
133 | | 2cn 10968 |
. . . . . . . . . 10
⊢ 2 ∈
ℂ |
134 | 59 | recnd 9947 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑁‘(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿)))) ∈ ℂ) |
135 | | sqmul 12788 |
. . . . . . . . . 10
⊢ ((2
∈ ℂ ∧ (𝑁‘(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿)))) ∈ ℂ) → ((2 ·
(𝑁‘(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿)))))↑2) = ((2↑2) · ((𝑁‘(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿))))↑2))) |
136 | 133, 134,
135 | sylancr 694 |
. . . . . . . . 9
⊢ (𝜑 → ((2 · (𝑁‘(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿)))))↑2) = ((2↑2) · ((𝑁‘(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿))))↑2))) |
137 | | sq2 12822 |
. . . . . . . . . 10
⊢
(2↑2) = 4 |
138 | 137 | oveq1i 6559 |
. . . . . . . . 9
⊢
((2↑2) · ((𝑁‘(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿))))↑2)) = (4 · ((𝑁‘(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿))))↑2)) |
139 | 136, 138 | syl6eq 2660 |
. . . . . . . 8
⊢ (𝜑 → ((2 · (𝑁‘(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿)))))↑2) = (4 · ((𝑁‘(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿))))↑2))) |
140 | 2, 4, 52, 38, 39 | cphnmvs 22798 |
. . . . . . . . . . . 12
⊢ ((𝑈 ∈ ℂPreHil ∧ 2
∈ (Base‘(Scalar‘𝑈)) ∧ (𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿))) ∈ 𝑋) → (𝑁‘(2(
·𝑠 ‘𝑈)(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿))))) = ((abs‘2) · (𝑁‘(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿)))))) |
141 | 5, 44, 57, 140 | syl3anc 1318 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑁‘(2(
·𝑠 ‘𝑈)(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿))))) = ((abs‘2) · (𝑁‘(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿)))))) |
142 | | 0le2 10988 |
. . . . . . . . . . . . 13
⊢ 0 ≤
2 |
143 | | absid 13884 |
. . . . . . . . . . . . 13
⊢ ((2
∈ ℝ ∧ 0 ≤ 2) → (abs‘2) = 2) |
144 | 118, 142,
143 | mp2an 704 |
. . . . . . . . . . . 12
⊢
(abs‘2) = 2 |
145 | 144 | oveq1i 6559 |
. . . . . . . . . . 11
⊢
((abs‘2) · (𝑁‘(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿))))) = (2 · (𝑁‘(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿))))) |
146 | 141, 145 | syl6eq 2660 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑁‘(2(
·𝑠 ‘𝑈)(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿))))) = (2 · (𝑁‘(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿)))))) |
147 | 2, 52, 38, 39, 3, 35, 44, 8, 55 | lmodsubdi 18743 |
. . . . . . . . . . . 12
⊢ (𝜑 → (2(
·𝑠 ‘𝑈)(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿)))) = ((2(
·𝑠 ‘𝑈)𝐴) − (2(
·𝑠 ‘𝑈)((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿))))) |
148 | | eqid 2610 |
. . . . . . . . . . . . . . . 16
⊢
(.g‘𝑈) = (.g‘𝑈) |
149 | 2, 148, 49 | mulg2 17373 |
. . . . . . . . . . . . . . 15
⊢ (𝐴 ∈ 𝑋 → (2(.g‘𝑈)𝐴) = (𝐴(+g‘𝑈)𝐴)) |
150 | 8, 149 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝜑 →
(2(.g‘𝑈)𝐴) = (𝐴(+g‘𝑈)𝐴)) |
151 | 2, 148, 52 | clmmulg 22709 |
. . . . . . . . . . . . . . 15
⊢ ((𝑈 ∈ ℂMod ∧ 2
∈ ℤ ∧ 𝐴
∈ 𝑋) →
(2(.g‘𝑈)𝐴) = (2( ·𝑠
‘𝑈)𝐴)) |
152 | 37, 43, 8, 151 | syl3anc 1318 |
. . . . . . . . . . . . . 14
⊢ (𝜑 →
(2(.g‘𝑈)𝐴) = (2( ·𝑠
‘𝑈)𝐴)) |
153 | 150, 152 | eqtr3d 2646 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝐴(+g‘𝑈)𝐴) = (2( ·𝑠
‘𝑈)𝐴)) |
154 | 2, 49 | lmodvacl 18700 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑈 ∈ LMod ∧ 𝐾 ∈ 𝑋 ∧ 𝐿 ∈ 𝑋) → (𝐾(+g‘𝑈)𝐿) ∈ 𝑋) |
155 | 35, 27, 29, 154 | syl3anc 1318 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝐾(+g‘𝑈)𝐿) ∈ 𝑋) |
156 | 2, 52 | clmvs1 22701 |
. . . . . . . . . . . . . . 15
⊢ ((𝑈 ∈ ℂMod ∧ (𝐾(+g‘𝑈)𝐿) ∈ 𝑋) → (1(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿)) = (𝐾(+g‘𝑈)𝐿)) |
157 | 37, 155, 156 | syl2anc 691 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (1(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿)) = (𝐾(+g‘𝑈)𝐿)) |
158 | 133, 45 | recidi 10635 |
. . . . . . . . . . . . . . . 16
⊢ (2
· (1 / 2)) = 1 |
159 | 158 | oveq1i 6559 |
. . . . . . . . . . . . . . 15
⊢ ((2
· (1 / 2))( ·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿)) = (1( ·𝑠
‘𝑈)(𝐾(+g‘𝑈)𝐿)) |
160 | 2, 38, 52, 39 | clmvsass 22697 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑈 ∈ ℂMod ∧ (2
∈ (Base‘(Scalar‘𝑈)) ∧ (1 / 2) ∈
(Base‘(Scalar‘𝑈)) ∧ (𝐾(+g‘𝑈)𝐿) ∈ 𝑋)) → ((2 · (1 / 2))(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿)) = (2( ·𝑠
‘𝑈)((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿)))) |
161 | 37, 44, 48, 155, 160 | syl13anc 1320 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ((2 · (1 / 2))(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿)) = (2( ·𝑠
‘𝑈)((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿)))) |
162 | 159, 161 | syl5eqr 2658 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (1(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿)) = (2( ·𝑠
‘𝑈)((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿)))) |
163 | 157, 162 | eqtr3d 2646 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝐾(+g‘𝑈)𝐿) = (2( ·𝑠
‘𝑈)((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿)))) |
164 | 153, 163 | oveq12d 6567 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((𝐴(+g‘𝑈)𝐴) − (𝐾(+g‘𝑈)𝐿)) = ((2(
·𝑠 ‘𝑈)𝐴) − (2(
·𝑠 ‘𝑈)((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿))))) |
165 | | lmodabl 18733 |
. . . . . . . . . . . . . 14
⊢ (𝑈 ∈ LMod → 𝑈 ∈ Abel) |
166 | 35, 165 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑈 ∈ Abel) |
167 | 2, 49, 3 | ablsub4 18041 |
. . . . . . . . . . . . 13
⊢ ((𝑈 ∈ Abel ∧ (𝐴 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) ∧ (𝐾 ∈ 𝑋 ∧ 𝐿 ∈ 𝑋)) → ((𝐴(+g‘𝑈)𝐴) − (𝐾(+g‘𝑈)𝐿)) = ((𝐴 − 𝐾)(+g‘𝑈)(𝐴 − 𝐿))) |
168 | 166, 8, 8, 27, 29, 167 | syl122anc 1327 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((𝐴(+g‘𝑈)𝐴) − (𝐾(+g‘𝑈)𝐿)) = ((𝐴 − 𝐾)(+g‘𝑈)(𝐴 − 𝐿))) |
169 | 147, 164,
168 | 3eqtr2d 2650 |
. . . . . . . . . . 11
⊢ (𝜑 → (2(
·𝑠 ‘𝑈)(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿)))) = ((𝐴 − 𝐾)(+g‘𝑈)(𝐴 − 𝐿))) |
170 | 169 | fveq2d 6107 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑁‘(2(
·𝑠 ‘𝑈)(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿))))) = (𝑁‘((𝐴 − 𝐾)(+g‘𝑈)(𝐴 − 𝐿)))) |
171 | 146, 170 | eqtr3d 2646 |
. . . . . . . . 9
⊢ (𝜑 → (2 · (𝑁‘(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿))))) = (𝑁‘((𝐴 − 𝐾)(+g‘𝑈)(𝐴 − 𝐿)))) |
172 | 171 | oveq1d 6564 |
. . . . . . . 8
⊢ (𝜑 → ((2 · (𝑁‘(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿)))))↑2) = ((𝑁‘((𝐴 − 𝐾)(+g‘𝑈)(𝐴 − 𝐿)))↑2)) |
173 | 139, 172 | eqtr3d 2646 |
. . . . . . 7
⊢ (𝜑 → (4 · ((𝑁‘(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿))))↑2)) = ((𝑁‘((𝐴 − 𝐾)(+g‘𝑈)(𝐴 − 𝐿)))↑2)) |
174 | | eqid 2610 |
. . . . . . . . . . 11
⊢
(dist‘𝑈) =
(dist‘𝑈) |
175 | 4, 2, 3, 174 | ngpdsr 22219 |
. . . . . . . . . 10
⊢ ((𝑈 ∈ NrmGrp ∧ 𝐾 ∈ 𝑋 ∧ 𝐿 ∈ 𝑋) → (𝐾(dist‘𝑈)𝐿) = (𝑁‘(𝐿 − 𝐾))) |
176 | 17, 27, 29, 175 | syl3anc 1318 |
. . . . . . . . 9
⊢ (𝜑 → (𝐾(dist‘𝑈)𝐿) = (𝑁‘(𝐿 − 𝐾))) |
177 | 20 | oveqi 6562 |
. . . . . . . . . 10
⊢ (𝐾𝐷𝐿) = (𝐾((dist‘𝑈) ↾ (𝑋 × 𝑋))𝐿) |
178 | 27, 29 | ovresd 6699 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐾((dist‘𝑈) ↾ (𝑋 × 𝑋))𝐿) = (𝐾(dist‘𝑈)𝐿)) |
179 | 177, 178 | syl5eq 2656 |
. . . . . . . . 9
⊢ (𝜑 → (𝐾𝐷𝐿) = (𝐾(dist‘𝑈)𝐿)) |
180 | 2, 3, 166, 8, 27, 29 | ablnnncan1 18052 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝐴 − 𝐾) − (𝐴 − 𝐿)) = (𝐿 − 𝐾)) |
181 | 180 | fveq2d 6107 |
. . . . . . . . 9
⊢ (𝜑 → (𝑁‘((𝐴 − 𝐾) − (𝐴 − 𝐿))) = (𝑁‘(𝐿 − 𝐾))) |
182 | 176, 179,
181 | 3eqtr4d 2654 |
. . . . . . . 8
⊢ (𝜑 → (𝐾𝐷𝐿) = (𝑁‘((𝐴 − 𝐾) − (𝐴 − 𝐿)))) |
183 | 182 | oveq1d 6564 |
. . . . . . 7
⊢ (𝜑 → ((𝐾𝐷𝐿)↑2) = ((𝑁‘((𝐴 − 𝐾) − (𝐴 − 𝐿)))↑2)) |
184 | 173, 183 | oveq12d 6567 |
. . . . . 6
⊢ (𝜑 → ((4 · ((𝑁‘(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿))))↑2)) + ((𝐾𝐷𝐿)↑2)) = (((𝑁‘((𝐴 − 𝐾)(+g‘𝑈)(𝐴 − 𝐿)))↑2) + ((𝑁‘((𝐴 − 𝐾) − (𝐴 − 𝐿)))↑2))) |
185 | 20 | oveqi 6562 |
. . . . . . . . . . 11
⊢ (𝐴𝐷𝐾) = (𝐴((dist‘𝑈) ↾ (𝑋 × 𝑋))𝐾) |
186 | 8, 27 | ovresd 6699 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐴((dist‘𝑈) ↾ (𝑋 × 𝑋))𝐾) = (𝐴(dist‘𝑈)𝐾)) |
187 | 185, 186 | syl5eq 2656 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐴𝐷𝐾) = (𝐴(dist‘𝑈)𝐾)) |
188 | 4, 2, 3, 174 | ngpds 22218 |
. . . . . . . . . . 11
⊢ ((𝑈 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐾 ∈ 𝑋) → (𝐴(dist‘𝑈)𝐾) = (𝑁‘(𝐴 − 𝐾))) |
189 | 17, 8, 27, 188 | syl3anc 1318 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐴(dist‘𝑈)𝐾) = (𝑁‘(𝐴 − 𝐾))) |
190 | 187, 189 | eqtrd 2644 |
. . . . . . . . 9
⊢ (𝜑 → (𝐴𝐷𝐾) = (𝑁‘(𝐴 − 𝐾))) |
191 | 190 | oveq1d 6564 |
. . . . . . . 8
⊢ (𝜑 → ((𝐴𝐷𝐾)↑2) = ((𝑁‘(𝐴 − 𝐾))↑2)) |
192 | 20 | oveqi 6562 |
. . . . . . . . . . 11
⊢ (𝐴𝐷𝐿) = (𝐴((dist‘𝑈) ↾ (𝑋 × 𝑋))𝐿) |
193 | 8, 29 | ovresd 6699 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐴((dist‘𝑈) ↾ (𝑋 × 𝑋))𝐿) = (𝐴(dist‘𝑈)𝐿)) |
194 | 192, 193 | syl5eq 2656 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐴𝐷𝐿) = (𝐴(dist‘𝑈)𝐿)) |
195 | 4, 2, 3, 174 | ngpds 22218 |
. . . . . . . . . . 11
⊢ ((𝑈 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐿 ∈ 𝑋) → (𝐴(dist‘𝑈)𝐿) = (𝑁‘(𝐴 − 𝐿))) |
196 | 17, 8, 29, 195 | syl3anc 1318 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐴(dist‘𝑈)𝐿) = (𝑁‘(𝐴 − 𝐿))) |
197 | 194, 196 | eqtrd 2644 |
. . . . . . . . 9
⊢ (𝜑 → (𝐴𝐷𝐿) = (𝑁‘(𝐴 − 𝐿))) |
198 | 197 | oveq1d 6564 |
. . . . . . . 8
⊢ (𝜑 → ((𝐴𝐷𝐿)↑2) = ((𝑁‘(𝐴 − 𝐿))↑2)) |
199 | 191, 198 | oveq12d 6567 |
. . . . . . 7
⊢ (𝜑 → (((𝐴𝐷𝐾)↑2) + ((𝐴𝐷𝐿)↑2)) = (((𝑁‘(𝐴 − 𝐾))↑2) + ((𝑁‘(𝐴 − 𝐿))↑2))) |
200 | 199 | oveq2d 6565 |
. . . . . 6
⊢ (𝜑 → (2 · (((𝐴𝐷𝐾)↑2) + ((𝐴𝐷𝐿)↑2))) = (2 · (((𝑁‘(𝐴 − 𝐾))↑2) + ((𝑁‘(𝐴 − 𝐿))↑2)))) |
201 | 132, 184,
200 | 3eqtr4d 2654 |
. . . . 5
⊢ (𝜑 → ((4 · ((𝑁‘(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿))))↑2)) + ((𝐾𝐷𝐿)↑2)) = (2 · (((𝐴𝐷𝐾)↑2) + ((𝐴𝐷𝐿)↑2)))) |
202 | | 2t2e4 11054 |
. . . . . . 7
⊢ (2
· 2) = 4 |
203 | 202 | oveq1i 6559 |
. . . . . 6
⊢ ((2
· 2) · ((𝑆↑2) + 𝐵)) = (4 · ((𝑆↑2) + 𝐵)) |
204 | | 2cnd 10970 |
. . . . . . 7
⊢ (𝜑 → 2 ∈
ℂ) |
205 | 204, 204,
114 | mulassd 9942 |
. . . . . 6
⊢ (𝜑 → ((2 · 2) ·
((𝑆↑2) + 𝐵)) = (2 · (2 ·
((𝑆↑2) + 𝐵)))) |
206 | 203, 205 | syl5eqr 2658 |
. . . . 5
⊢ (𝜑 → (4 · ((𝑆↑2) + 𝐵)) = (2 · (2 · ((𝑆↑2) + 𝐵)))) |
207 | 126, 201,
206 | 3brtr4d 4615 |
. . . 4
⊢ (𝜑 → ((4 · ((𝑁‘(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿))))↑2)) + ((𝐾𝐷𝐿)↑2)) ≤ (4 · ((𝑆↑2) + 𝐵))) |
208 | 33, 63, 67, 104, 207 | letrd 10073 |
. . 3
⊢ (𝜑 → ((4 · (𝑆↑2)) + ((𝐾𝐷𝐿)↑2)) ≤ (4 · ((𝑆↑2) + 𝐵))) |
209 | | 4cn 10975 |
. . . . 5
⊢ 4 ∈
ℂ |
210 | 209 | a1i 11 |
. . . 4
⊢ (𝜑 → 4 ∈
ℂ) |
211 | 13 | recnd 9947 |
. . . 4
⊢ (𝜑 → (𝑆↑2) ∈ ℂ) |
212 | 64 | recnd 9947 |
. . . 4
⊢ (𝜑 → 𝐵 ∈ ℂ) |
213 | 210, 211,
212 | adddid 9943 |
. . 3
⊢ (𝜑 → (4 · ((𝑆↑2) + 𝐵)) = ((4 · (𝑆↑2)) + (4 · 𝐵))) |
214 | 208, 213 | breqtrd 4609 |
. 2
⊢ (𝜑 → ((4 · (𝑆↑2)) + ((𝐾𝐷𝐿)↑2)) ≤ ((4 · (𝑆↑2)) + (4 · 𝐵))) |
215 | | remulcl 9900 |
. . . 4
⊢ ((4
∈ ℝ ∧ 𝐵
∈ ℝ) → (4 · 𝐵) ∈ ℝ) |
216 | 1, 64, 215 | sylancr 694 |
. . 3
⊢ (𝜑 → (4 · 𝐵) ∈
ℝ) |
217 | 32, 216, 15 | leadd2d 10501 |
. 2
⊢ (𝜑 → (((𝐾𝐷𝐿)↑2) ≤ (4 · 𝐵) ↔ ((4 · (𝑆↑2)) + ((𝐾𝐷𝐿)↑2)) ≤ ((4 · (𝑆↑2)) + (4 · 𝐵)))) |
218 | 214, 217 | mpbird 246 |
1
⊢ (𝜑 → ((𝐾𝐷𝐿)↑2) ≤ (4 · 𝐵)) |