| Step | Hyp | Ref
| Expression |
| 1 | | 4re 12350 |
. . . . . 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 25459 |
. . . . . . 7
⊢ (𝜑 → 𝑆 ∈ ℝ) |
| 13 | 12 | resqcld 14165 |
. . . . . 6
⊢ (𝜑 → (𝑆↑2) ∈ ℝ) |
| 14 | | remulcl 11240 |
. . . . . 6
⊢ ((4
∈ ℝ ∧ (𝑆↑2) ∈ ℝ) → (4 ·
(𝑆↑2)) ∈
ℝ) |
| 15 | 1, 13, 14 | sylancr 587 |
. . . . 5
⊢ (𝜑 → (4 · (𝑆↑2)) ∈
ℝ) |
| 16 | | cphngp 25207 |
. . . . . . . . . 10
⊢ (𝑈 ∈ ℂPreHil →
𝑈 ∈
NrmGrp) |
| 17 | 5, 16 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝑈 ∈ NrmGrp) |
| 18 | | ngpms 24613 |
. . . . . . . . 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 24467 |
. . . . . . . 8
⊢ (𝑈 ∈ MetSp → 𝐷 ∈ (Met‘𝑋)) |
| 22 | 19, 21 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋)) |
| 23 | | eqid 2737 |
. . . . . . . . . 10
⊢
(LSubSp‘𝑈) =
(LSubSp‘𝑈) |
| 24 | 2, 23 | lssss 20934 |
. . . . . . . . 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 3984 |
. . . . . . 7
⊢ (𝜑 → 𝐾 ∈ 𝑋) |
| 28 | | minveclem2.4 |
. . . . . . . 8
⊢ (𝜑 → 𝐿 ∈ 𝑌) |
| 29 | 25, 28 | sseldd 3984 |
. . . . . . 7
⊢ (𝜑 → 𝐿 ∈ 𝑋) |
| 30 | | metcl 24342 |
. . . . . . 7
⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐾 ∈ 𝑋 ∧ 𝐿 ∈ 𝑋) → (𝐾𝐷𝐿) ∈ ℝ) |
| 31 | 22, 27, 29, 30 | syl3anc 1373 |
. . . . . 6
⊢ (𝜑 → (𝐾𝐷𝐿) ∈ ℝ) |
| 32 | 31 | resqcld 14165 |
. . . . 5
⊢ (𝜑 → ((𝐾𝐷𝐿)↑2) ∈ ℝ) |
| 33 | 15, 32 | readdcld 11290 |
. . . 4
⊢ (𝜑 → ((4 · (𝑆↑2)) + ((𝐾𝐷𝐿)↑2)) ∈ ℝ) |
| 34 | | cphlmod 25208 |
. . . . . . . . . 10
⊢ (𝑈 ∈ ℂPreHil →
𝑈 ∈
LMod) |
| 35 | 5, 34 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝑈 ∈ LMod) |
| 36 | | cphclm 25223 |
. . . . . . . . . . . . . . 15
⊢ (𝑈 ∈ ℂPreHil →
𝑈 ∈
ℂMod) |
| 37 | 5, 36 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑈 ∈ ℂMod) |
| 38 | | eqid 2737 |
. . . . . . . . . . . . . . 15
⊢
(Scalar‘𝑈) =
(Scalar‘𝑈) |
| 39 | | eqid 2737 |
. . . . . . . . . . . . . . 15
⊢
(Base‘(Scalar‘𝑈)) = (Base‘(Scalar‘𝑈)) |
| 40 | 38, 39 | clmzss 25111 |
. . . . . . . . . . . . . 14
⊢ (𝑈 ∈ ℂMod →
ℤ ⊆ (Base‘(Scalar‘𝑈))) |
| 41 | 37, 40 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ℤ ⊆
(Base‘(Scalar‘𝑈))) |
| 42 | | 2z 12649 |
. . . . . . . . . . . . . 14
⊢ 2 ∈
ℤ |
| 43 | 42 | a1i 11 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 2 ∈
ℤ) |
| 44 | 41, 43 | sseldd 3984 |
. . . . . . . . . . . 12
⊢ (𝜑 → 2 ∈
(Base‘(Scalar‘𝑈))) |
| 45 | | 2ne0 12370 |
. . . . . . . . . . . . 13
⊢ 2 ≠
0 |
| 46 | 45 | a1i 11 |
. . . . . . . . . . . 12
⊢ (𝜑 → 2 ≠ 0) |
| 47 | 38, 39 | cphreccl 25215 |
. . . . . . . . . . . 12
⊢ ((𝑈 ∈ ℂPreHil ∧ 2
∈ (Base‘(Scalar‘𝑈)) ∧ 2 ≠ 0) → (1 / 2) ∈
(Base‘(Scalar‘𝑈))) |
| 48 | 5, 44, 46, 47 | syl3anc 1373 |
. . . . . . . . . . 11
⊢ (𝜑 → (1 / 2) ∈
(Base‘(Scalar‘𝑈))) |
| 49 | | eqid 2737 |
. . . . . . . . . . . . 13
⊢
(+g‘𝑈) = (+g‘𝑈) |
| 50 | 49, 23 | lssvacl 20941 |
. . . . . . . . . . . 12
⊢ (((𝑈 ∈ LMod ∧ 𝑌 ∈ (LSubSp‘𝑈)) ∧ (𝐾 ∈ 𝑌 ∧ 𝐿 ∈ 𝑌)) → (𝐾(+g‘𝑈)𝐿) ∈ 𝑌) |
| 51 | 35, 6, 26, 28, 50 | syl22anc 839 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐾(+g‘𝑈)𝐿) ∈ 𝑌) |
| 52 | | eqid 2737 |
. . . . . . . . . . . 12
⊢ (
·𝑠 ‘𝑈) = ( ·𝑠
‘𝑈) |
| 53 | 38, 52, 39, 23 | lssvscl 20953 |
. . . . . . . . . . 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 839 |
. . . . . . . . . 10
⊢ (𝜑 → ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿)) ∈ 𝑌) |
| 55 | 25, 54 | sseldd 3984 |
. . . . . . . . 9
⊢ (𝜑 → ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿)) ∈ 𝑋) |
| 56 | 2, 3 | lmodvsubcl 20905 |
. . . . . . . . 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 1373 |
. . . . . . . 8
⊢ (𝜑 → (𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿))) ∈ 𝑋) |
| 58 | 2, 4 | nmcl 24629 |
. . . . . . . 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 584 |
. . . . . . 7
⊢ (𝜑 → (𝑁‘(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿)))) ∈ ℝ) |
| 60 | 59 | resqcld 14165 |
. . . . . 6
⊢ (𝜑 → ((𝑁‘(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿))))↑2) ∈
ℝ) |
| 61 | | remulcl 11240 |
. . . . . 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 587 |
. . . . 5
⊢ (𝜑 → (4 · ((𝑁‘(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿))))↑2)) ∈
ℝ) |
| 63 | 62, 32 | readdcld 11290 |
. . . 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 11290 |
. . . . 5
⊢ (𝜑 → ((𝑆↑2) + 𝐵) ∈ ℝ) |
| 66 | | remulcl 11240 |
. . . . 5
⊢ ((4
∈ ℝ ∧ ((𝑆↑2) + 𝐵) ∈ ℝ) → (4 · ((𝑆↑2) + 𝐵)) ∈ ℝ) |
| 67 | 1, 65, 66 | sylancr 587 |
. . . 4
⊢ (𝜑 → (4 · ((𝑆↑2) + 𝐵)) ∈ ℝ) |
| 68 | 2, 3, 4, 5, 6, 7, 8, 9, 10 | minveclem1 25458 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∀𝑤 ∈ 𝑅 0 ≤ 𝑤)) |
| 69 | 68 | simp3d 1145 |
. . . . . . . . 9
⊢ (𝜑 → ∀𝑤 ∈ 𝑅 0 ≤ 𝑤) |
| 70 | 68 | simp1d 1143 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑅 ⊆ ℝ) |
| 71 | 68 | simp2d 1144 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑅 ≠ ∅) |
| 72 | | 0re 11263 |
. . . . . . . . . . 11
⊢ 0 ∈
ℝ |
| 73 | | breq1 5146 |
. . . . . . . . . . . . 13
⊢ (𝑥 = 0 → (𝑥 ≤ 𝑤 ↔ 0 ≤ 𝑤)) |
| 74 | 73 | ralbidv 3178 |
. . . . . . . . . . . 12
⊢ (𝑥 = 0 → (∀𝑤 ∈ 𝑅 𝑥 ≤ 𝑤 ↔ ∀𝑤 ∈ 𝑅 0 ≤ 𝑤)) |
| 75 | 74 | rspcev 3622 |
. . . . . . . . . . 11
⊢ ((0
∈ ℝ ∧ ∀𝑤 ∈ 𝑅 0 ≤ 𝑤) → ∃𝑥 ∈ ℝ ∀𝑤 ∈ 𝑅 𝑥 ≤ 𝑤) |
| 76 | 72, 69, 75 | sylancr 587 |
. . . . . . . . . 10
⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑤 ∈ 𝑅 𝑥 ≤ 𝑤) |
| 77 | 72 | a1i 11 |
. . . . . . . . . 10
⊢ (𝜑 → 0 ∈
ℝ) |
| 78 | | infregelb 12252 |
. . . . . . . . . 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 1375 |
. . . . . . . . 9
⊢ (𝜑 → (0 ≤ inf(𝑅, ℝ, < ) ↔
∀𝑤 ∈ 𝑅 0 ≤ 𝑤)) |
| 80 | 69, 79 | mpbird 257 |
. . . . . . . 8
⊢ (𝜑 → 0 ≤ inf(𝑅, ℝ, <
)) |
| 81 | 80, 11 | breqtrrdi 5185 |
. . . . . . 7
⊢ (𝜑 → 0 ≤ 𝑆) |
| 82 | | eqid 2737 |
. . . . . . . . . . . 12
⊢ (𝑁‘(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿)))) = (𝑁‘(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿)))) |
| 83 | | oveq2 7439 |
. . . . . . . . . . . . . 14
⊢ (𝑦 = ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿)) → (𝐴 − 𝑦) = (𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿)))) |
| 84 | 83 | fveq2d 6910 |
. . . . . . . . . . . . 13
⊢ (𝑦 = ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿)) → (𝑁‘(𝐴 − 𝑦)) = (𝑁‘(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿))))) |
| 85 | 84 | rspceeqv 3645 |
. . . . . . . . . . . 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))) |
| 86 | 54, 82, 85 | sylancl 586 |
. . . . . . . . . . 11
⊢ (𝜑 → ∃𝑦 ∈ 𝑌 (𝑁‘(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿)))) = (𝑁‘(𝐴 − 𝑦))) |
| 87 | | eqid 2737 |
. . . . . . . . . . . 12
⊢ (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))) = (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))) |
| 88 | | fvex 6919 |
. . . . . . . . . . . 12
⊢ (𝑁‘(𝐴 − 𝑦)) ∈ V |
| 89 | 87, 88 | elrnmpti 5973 |
. . . . . . . . . . 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))) |
| 90 | 86, 89 | sylibr 234 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑁‘(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿)))) ∈ ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦)))) |
| 91 | 90, 10 | eleqtrrdi 2852 |
. . . . . . . . 9
⊢ (𝜑 → (𝑁‘(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿)))) ∈ 𝑅) |
| 92 | | infrelb 12253 |
. . . . . . . . 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))))) |
| 93 | 70, 76, 91, 92 | syl3anc 1373 |
. . . . . . . 8
⊢ (𝜑 → inf(𝑅, ℝ, < ) ≤ (𝑁‘(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿))))) |
| 94 | 11, 93 | eqbrtrid 5178 |
. . . . . . 7
⊢ (𝜑 → 𝑆 ≤ (𝑁‘(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿))))) |
| 95 | | le2sq2 14175 |
. . . . . . 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)) |
| 96 | 12, 81, 59, 94, 95 | syl22anc 839 |
. . . . . 6
⊢ (𝜑 → (𝑆↑2) ≤ ((𝑁‘(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿))))↑2)) |
| 97 | | 4pos 12373 |
. . . . . . . . 9
⊢ 0 <
4 |
| 98 | 1, 97 | pm3.2i 470 |
. . . . . . . 8
⊢ (4 ∈
ℝ ∧ 0 < 4) |
| 99 | | lemul2 12120 |
. . . . . . . 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)))) |
| 100 | 98, 99 | mp3an3 1452 |
. . . . . . 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)))) |
| 101 | 13, 60, 100 | syl2anc 584 |
. . . . . 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)))) |
| 102 | 96, 101 | mpbid 232 |
. . . . 5
⊢ (𝜑 → (4 · (𝑆↑2)) ≤ (4 ·
((𝑁‘(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿))))↑2))) |
| 103 | 15, 62, 32, 102 | leadd1dd 11877 |
. . . 4
⊢ (𝜑 → ((4 · (𝑆↑2)) + ((𝐾𝐷𝐿)↑2)) ≤ ((4 · ((𝑁‘(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿))))↑2)) + ((𝐾𝐷𝐿)↑2))) |
| 104 | | metcl 24342 |
. . . . . . . . . 10
⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐾 ∈ 𝑋) → (𝐴𝐷𝐾) ∈ ℝ) |
| 105 | 22, 8, 27, 104 | syl3anc 1373 |
. . . . . . . . 9
⊢ (𝜑 → (𝐴𝐷𝐾) ∈ ℝ) |
| 106 | 105 | resqcld 14165 |
. . . . . . . 8
⊢ (𝜑 → ((𝐴𝐷𝐾)↑2) ∈ ℝ) |
| 107 | | metcl 24342 |
. . . . . . . . . 10
⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐿 ∈ 𝑋) → (𝐴𝐷𝐿) ∈ ℝ) |
| 108 | 22, 8, 29, 107 | syl3anc 1373 |
. . . . . . . . 9
⊢ (𝜑 → (𝐴𝐷𝐿) ∈ ℝ) |
| 109 | 108 | resqcld 14165 |
. . . . . . . 8
⊢ (𝜑 → ((𝐴𝐷𝐿)↑2) ∈ ℝ) |
| 110 | | minveclem2.5 |
. . . . . . . 8
⊢ (𝜑 → ((𝐴𝐷𝐾)↑2) ≤ ((𝑆↑2) + 𝐵)) |
| 111 | | minveclem2.6 |
. . . . . . . 8
⊢ (𝜑 → ((𝐴𝐷𝐿)↑2) ≤ ((𝑆↑2) + 𝐵)) |
| 112 | 106, 109,
65, 65, 110, 111 | le2addd 11882 |
. . . . . . 7
⊢ (𝜑 → (((𝐴𝐷𝐾)↑2) + ((𝐴𝐷𝐿)↑2)) ≤ (((𝑆↑2) + 𝐵) + ((𝑆↑2) + 𝐵))) |
| 113 | 65 | recnd 11289 |
. . . . . . . 8
⊢ (𝜑 → ((𝑆↑2) + 𝐵) ∈ ℂ) |
| 114 | 113 | 2timesd 12509 |
. . . . . . 7
⊢ (𝜑 → (2 · ((𝑆↑2) + 𝐵)) = (((𝑆↑2) + 𝐵) + ((𝑆↑2) + 𝐵))) |
| 115 | 112, 114 | breqtrrd 5171 |
. . . . . 6
⊢ (𝜑 → (((𝐴𝐷𝐾)↑2) + ((𝐴𝐷𝐿)↑2)) ≤ (2 · ((𝑆↑2) + 𝐵))) |
| 116 | 106, 109 | readdcld 11290 |
. . . . . . 7
⊢ (𝜑 → (((𝐴𝐷𝐾)↑2) + ((𝐴𝐷𝐿)↑2)) ∈ ℝ) |
| 117 | | 2re 12340 |
. . . . . . . 8
⊢ 2 ∈
ℝ |
| 118 | | remulcl 11240 |
. . . . . . . 8
⊢ ((2
∈ ℝ ∧ ((𝑆↑2) + 𝐵) ∈ ℝ) → (2 · ((𝑆↑2) + 𝐵)) ∈ ℝ) |
| 119 | 117, 65, 118 | sylancr 587 |
. . . . . . 7
⊢ (𝜑 → (2 · ((𝑆↑2) + 𝐵)) ∈ ℝ) |
| 120 | | 2pos 12369 |
. . . . . . . . 9
⊢ 0 <
2 |
| 121 | 117, 120 | pm3.2i 470 |
. . . . . . . 8
⊢ (2 ∈
ℝ ∧ 0 < 2) |
| 122 | | lemul2 12120 |
. . . . . . . 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))))) |
| 123 | 121, 122 | mp3an3 1452 |
. . . . . . 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))))) |
| 124 | 116, 119,
123 | syl2anc 584 |
. . . . . 6
⊢ (𝜑 → ((((𝐴𝐷𝐾)↑2) + ((𝐴𝐷𝐿)↑2)) ≤ (2 · ((𝑆↑2) + 𝐵)) ↔ (2 · (((𝐴𝐷𝐾)↑2) + ((𝐴𝐷𝐿)↑2))) ≤ (2 · (2 ·
((𝑆↑2) + 𝐵))))) |
| 125 | 115, 124 | mpbid 232 |
. . . . 5
⊢ (𝜑 → (2 · (((𝐴𝐷𝐾)↑2) + ((𝐴𝐷𝐿)↑2))) ≤ (2 · (2 ·
((𝑆↑2) + 𝐵)))) |
| 126 | 2, 3 | lmodvsubcl 20905 |
. . . . . . . 8
⊢ ((𝑈 ∈ LMod ∧ 𝐴 ∈ 𝑋 ∧ 𝐾 ∈ 𝑋) → (𝐴 − 𝐾) ∈ 𝑋) |
| 127 | 35, 8, 27, 126 | syl3anc 1373 |
. . . . . . 7
⊢ (𝜑 → (𝐴 − 𝐾) ∈ 𝑋) |
| 128 | 2, 3 | lmodvsubcl 20905 |
. . . . . . . 8
⊢ ((𝑈 ∈ LMod ∧ 𝐴 ∈ 𝑋 ∧ 𝐿 ∈ 𝑋) → (𝐴 − 𝐿) ∈ 𝑋) |
| 129 | 35, 8, 29, 128 | syl3anc 1373 |
. . . . . . 7
⊢ (𝜑 → (𝐴 − 𝐿) ∈ 𝑋) |
| 130 | 2, 49, 3, 4 | nmpar 25274 |
. . . . . . 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)))) |
| 131 | 5, 127, 129, 130 | syl3anc 1373 |
. . . . . 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)))) |
| 132 | | 2cn 12341 |
. . . . . . . . . 10
⊢ 2 ∈
ℂ |
| 133 | 59 | recnd 11289 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑁‘(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿)))) ∈ ℂ) |
| 134 | | sqmul 14159 |
. . . . . . . . . 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))) |
| 135 | 132, 133,
134 | sylancr 587 |
. . . . . . . . 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))) |
| 136 | | sq2 14236 |
. . . . . . . . . 10
⊢
(2↑2) = 4 |
| 137 | 136 | oveq1i 7441 |
. . . . . . . . 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)) |
| 138 | 135, 137 | eqtrdi 2793 |
. . . . . . . 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))) |
| 139 | 2, 4, 52, 38, 39 | cphnmvs 25224 |
. . . . . . . . . . . 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)))))) |
| 140 | 5, 44, 57, 139 | syl3anc 1373 |
. . . . . . . . . . 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)))))) |
| 141 | | 0le2 12368 |
. . . . . . . . . . . . 13
⊢ 0 ≤
2 |
| 142 | | absid 15335 |
. . . . . . . . . . . . 13
⊢ ((2
∈ ℝ ∧ 0 ≤ 2) → (abs‘2) = 2) |
| 143 | 117, 141,
142 | mp2an 692 |
. . . . . . . . . . . 12
⊢
(abs‘2) = 2 |
| 144 | 143 | oveq1i 7441 |
. . . . . . . . . . 11
⊢
((abs‘2) · (𝑁‘(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿))))) = (2 · (𝑁‘(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿))))) |
| 145 | 140, 144 | eqtrdi 2793 |
. . . . . . . . . 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)))))) |
| 146 | 2, 52, 38, 39, 3, 35, 44, 8, 55 | lmodsubdi 20917 |
. . . . . . . . . . . 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))))) |
| 147 | | eqid 2737 |
. . . . . . . . . . . . . . . 16
⊢
(.g‘𝑈) = (.g‘𝑈) |
| 148 | 2, 147, 49 | mulg2 19101 |
. . . . . . . . . . . . . . 15
⊢ (𝐴 ∈ 𝑋 → (2(.g‘𝑈)𝐴) = (𝐴(+g‘𝑈)𝐴)) |
| 149 | 8, 148 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝜑 →
(2(.g‘𝑈)𝐴) = (𝐴(+g‘𝑈)𝐴)) |
| 150 | 2, 147, 52 | clmmulg 25134 |
. . . . . . . . . . . . . . 15
⊢ ((𝑈 ∈ ℂMod ∧ 2
∈ ℤ ∧ 𝐴
∈ 𝑋) →
(2(.g‘𝑈)𝐴) = (2( ·𝑠
‘𝑈)𝐴)) |
| 151 | 37, 43, 8, 150 | syl3anc 1373 |
. . . . . . . . . . . . . 14
⊢ (𝜑 →
(2(.g‘𝑈)𝐴) = (2( ·𝑠
‘𝑈)𝐴)) |
| 152 | 149, 151 | eqtr3d 2779 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝐴(+g‘𝑈)𝐴) = (2( ·𝑠
‘𝑈)𝐴)) |
| 153 | 2, 49 | lmodvacl 20873 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑈 ∈ LMod ∧ 𝐾 ∈ 𝑋 ∧ 𝐿 ∈ 𝑋) → (𝐾(+g‘𝑈)𝐿) ∈ 𝑋) |
| 154 | 35, 27, 29, 153 | syl3anc 1373 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝐾(+g‘𝑈)𝐿) ∈ 𝑋) |
| 155 | 2, 52 | clmvs1 25126 |
. . . . . . . . . . . . . . 15
⊢ ((𝑈 ∈ ℂMod ∧ (𝐾(+g‘𝑈)𝐿) ∈ 𝑋) → (1(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿)) = (𝐾(+g‘𝑈)𝐿)) |
| 156 | 37, 154, 155 | syl2anc 584 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (1(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿)) = (𝐾(+g‘𝑈)𝐿)) |
| 157 | 132, 45 | recidi 11998 |
. . . . . . . . . . . . . . . 16
⊢ (2
· (1 / 2)) = 1 |
| 158 | 157 | oveq1i 7441 |
. . . . . . . . . . . . . . 15
⊢ ((2
· (1 / 2))( ·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿)) = (1( ·𝑠
‘𝑈)(𝐾(+g‘𝑈)𝐿)) |
| 159 | 2, 38, 52, 39 | clmvsass 25122 |
. . . . . . . . . . . . . . . 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)))) |
| 160 | 37, 44, 48, 154, 159 | syl13anc 1374 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ((2 · (1 / 2))(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿)) = (2( ·𝑠
‘𝑈)((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿)))) |
| 161 | 158, 160 | eqtr3id 2791 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (1(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿)) = (2( ·𝑠
‘𝑈)((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿)))) |
| 162 | 156, 161 | eqtr3d 2779 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝐾(+g‘𝑈)𝐿) = (2( ·𝑠
‘𝑈)((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿)))) |
| 163 | 152, 162 | oveq12d 7449 |
. . . . . . . . . . . 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))))) |
| 164 | | lmodabl 20907 |
. . . . . . . . . . . . . 14
⊢ (𝑈 ∈ LMod → 𝑈 ∈ Abel) |
| 165 | 35, 164 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑈 ∈ Abel) |
| 166 | 2, 49, 3 | ablsub4 19828 |
. . . . . . . . . . . . 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))) |
| 167 | 165, 8, 8, 27, 29, 166 | syl122anc 1381 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((𝐴(+g‘𝑈)𝐴) − (𝐾(+g‘𝑈)𝐿)) = ((𝐴 − 𝐾)(+g‘𝑈)(𝐴 − 𝐿))) |
| 168 | 146, 163,
167 | 3eqtr2d 2783 |
. . . . . . . . . . 11
⊢ (𝜑 → (2(
·𝑠 ‘𝑈)(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿)))) = ((𝐴 − 𝐾)(+g‘𝑈)(𝐴 − 𝐿))) |
| 169 | 168 | fveq2d 6910 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑁‘(2(
·𝑠 ‘𝑈)(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿))))) = (𝑁‘((𝐴 − 𝐾)(+g‘𝑈)(𝐴 − 𝐿)))) |
| 170 | 145, 169 | eqtr3d 2779 |
. . . . . . . . 9
⊢ (𝜑 → (2 · (𝑁‘(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿))))) = (𝑁‘((𝐴 − 𝐾)(+g‘𝑈)(𝐴 − 𝐿)))) |
| 171 | 170 | oveq1d 7446 |
. . . . . . . 8
⊢ (𝜑 → ((2 · (𝑁‘(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿)))))↑2) = ((𝑁‘((𝐴 − 𝐾)(+g‘𝑈)(𝐴 − 𝐿)))↑2)) |
| 172 | 138, 171 | eqtr3d 2779 |
. . . . . . 7
⊢ (𝜑 → (4 · ((𝑁‘(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿))))↑2)) = ((𝑁‘((𝐴 − 𝐾)(+g‘𝑈)(𝐴 − 𝐿)))↑2)) |
| 173 | | eqid 2737 |
. . . . . . . . . . 11
⊢
(dist‘𝑈) =
(dist‘𝑈) |
| 174 | 4, 2, 3, 173 | ngpdsr 24618 |
. . . . . . . . . 10
⊢ ((𝑈 ∈ NrmGrp ∧ 𝐾 ∈ 𝑋 ∧ 𝐿 ∈ 𝑋) → (𝐾(dist‘𝑈)𝐿) = (𝑁‘(𝐿 − 𝐾))) |
| 175 | 17, 27, 29, 174 | syl3anc 1373 |
. . . . . . . . 9
⊢ (𝜑 → (𝐾(dist‘𝑈)𝐿) = (𝑁‘(𝐿 − 𝐾))) |
| 176 | 20 | oveqi 7444 |
. . . . . . . . . 10
⊢ (𝐾𝐷𝐿) = (𝐾((dist‘𝑈) ↾ (𝑋 × 𝑋))𝐿) |
| 177 | 27, 29 | ovresd 7600 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐾((dist‘𝑈) ↾ (𝑋 × 𝑋))𝐿) = (𝐾(dist‘𝑈)𝐿)) |
| 178 | 176, 177 | eqtrid 2789 |
. . . . . . . . 9
⊢ (𝜑 → (𝐾𝐷𝐿) = (𝐾(dist‘𝑈)𝐿)) |
| 179 | 2, 3, 165, 8, 27, 29 | ablnnncan1 19841 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝐴 − 𝐾) − (𝐴 − 𝐿)) = (𝐿 − 𝐾)) |
| 180 | 179 | fveq2d 6910 |
. . . . . . . . 9
⊢ (𝜑 → (𝑁‘((𝐴 − 𝐾) − (𝐴 − 𝐿))) = (𝑁‘(𝐿 − 𝐾))) |
| 181 | 175, 178,
180 | 3eqtr4d 2787 |
. . . . . . . 8
⊢ (𝜑 → (𝐾𝐷𝐿) = (𝑁‘((𝐴 − 𝐾) − (𝐴 − 𝐿)))) |
| 182 | 181 | oveq1d 7446 |
. . . . . . 7
⊢ (𝜑 → ((𝐾𝐷𝐿)↑2) = ((𝑁‘((𝐴 − 𝐾) − (𝐴 − 𝐿)))↑2)) |
| 183 | 172, 182 | oveq12d 7449 |
. . . . . 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))) |
| 184 | 20 | oveqi 7444 |
. . . . . . . . . . 11
⊢ (𝐴𝐷𝐾) = (𝐴((dist‘𝑈) ↾ (𝑋 × 𝑋))𝐾) |
| 185 | 8, 27 | ovresd 7600 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐴((dist‘𝑈) ↾ (𝑋 × 𝑋))𝐾) = (𝐴(dist‘𝑈)𝐾)) |
| 186 | 184, 185 | eqtrid 2789 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐴𝐷𝐾) = (𝐴(dist‘𝑈)𝐾)) |
| 187 | 4, 2, 3, 173 | ngpds 24617 |
. . . . . . . . . . 11
⊢ ((𝑈 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐾 ∈ 𝑋) → (𝐴(dist‘𝑈)𝐾) = (𝑁‘(𝐴 − 𝐾))) |
| 188 | 17, 8, 27, 187 | syl3anc 1373 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐴(dist‘𝑈)𝐾) = (𝑁‘(𝐴 − 𝐾))) |
| 189 | 186, 188 | eqtrd 2777 |
. . . . . . . . 9
⊢ (𝜑 → (𝐴𝐷𝐾) = (𝑁‘(𝐴 − 𝐾))) |
| 190 | 189 | oveq1d 7446 |
. . . . . . . 8
⊢ (𝜑 → ((𝐴𝐷𝐾)↑2) = ((𝑁‘(𝐴 − 𝐾))↑2)) |
| 191 | 20 | oveqi 7444 |
. . . . . . . . . . 11
⊢ (𝐴𝐷𝐿) = (𝐴((dist‘𝑈) ↾ (𝑋 × 𝑋))𝐿) |
| 192 | 8, 29 | ovresd 7600 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐴((dist‘𝑈) ↾ (𝑋 × 𝑋))𝐿) = (𝐴(dist‘𝑈)𝐿)) |
| 193 | 191, 192 | eqtrid 2789 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐴𝐷𝐿) = (𝐴(dist‘𝑈)𝐿)) |
| 194 | 4, 2, 3, 173 | ngpds 24617 |
. . . . . . . . . . 11
⊢ ((𝑈 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐿 ∈ 𝑋) → (𝐴(dist‘𝑈)𝐿) = (𝑁‘(𝐴 − 𝐿))) |
| 195 | 17, 8, 29, 194 | syl3anc 1373 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐴(dist‘𝑈)𝐿) = (𝑁‘(𝐴 − 𝐿))) |
| 196 | 193, 195 | eqtrd 2777 |
. . . . . . . . 9
⊢ (𝜑 → (𝐴𝐷𝐿) = (𝑁‘(𝐴 − 𝐿))) |
| 197 | 196 | oveq1d 7446 |
. . . . . . . 8
⊢ (𝜑 → ((𝐴𝐷𝐿)↑2) = ((𝑁‘(𝐴 − 𝐿))↑2)) |
| 198 | 190, 197 | oveq12d 7449 |
. . . . . . 7
⊢ (𝜑 → (((𝐴𝐷𝐾)↑2) + ((𝐴𝐷𝐿)↑2)) = (((𝑁‘(𝐴 − 𝐾))↑2) + ((𝑁‘(𝐴 − 𝐿))↑2))) |
| 199 | 198 | oveq2d 7447 |
. . . . . 6
⊢ (𝜑 → (2 · (((𝐴𝐷𝐾)↑2) + ((𝐴𝐷𝐿)↑2))) = (2 · (((𝑁‘(𝐴 − 𝐾))↑2) + ((𝑁‘(𝐴 − 𝐿))↑2)))) |
| 200 | 131, 183,
199 | 3eqtr4d 2787 |
. . . . 5
⊢ (𝜑 → ((4 · ((𝑁‘(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿))))↑2)) + ((𝐾𝐷𝐿)↑2)) = (2 · (((𝐴𝐷𝐾)↑2) + ((𝐴𝐷𝐿)↑2)))) |
| 201 | | 2t2e4 12430 |
. . . . . . 7
⊢ (2
· 2) = 4 |
| 202 | 201 | oveq1i 7441 |
. . . . . 6
⊢ ((2
· 2) · ((𝑆↑2) + 𝐵)) = (4 · ((𝑆↑2) + 𝐵)) |
| 203 | | 2cnd 12344 |
. . . . . . 7
⊢ (𝜑 → 2 ∈
ℂ) |
| 204 | 203, 203,
113 | mulassd 11284 |
. . . . . 6
⊢ (𝜑 → ((2 · 2) ·
((𝑆↑2) + 𝐵)) = (2 · (2 ·
((𝑆↑2) + 𝐵)))) |
| 205 | 202, 204 | eqtr3id 2791 |
. . . . 5
⊢ (𝜑 → (4 · ((𝑆↑2) + 𝐵)) = (2 · (2 · ((𝑆↑2) + 𝐵)))) |
| 206 | 125, 200,
205 | 3brtr4d 5175 |
. . . 4
⊢ (𝜑 → ((4 · ((𝑁‘(𝐴 − ((1 / 2)(
·𝑠 ‘𝑈)(𝐾(+g‘𝑈)𝐿))))↑2)) + ((𝐾𝐷𝐿)↑2)) ≤ (4 · ((𝑆↑2) + 𝐵))) |
| 207 | 33, 63, 67, 103, 206 | letrd 11418 |
. . 3
⊢ (𝜑 → ((4 · (𝑆↑2)) + ((𝐾𝐷𝐿)↑2)) ≤ (4 · ((𝑆↑2) + 𝐵))) |
| 208 | | 4cn 12351 |
. . . . 5
⊢ 4 ∈
ℂ |
| 209 | 208 | a1i 11 |
. . . 4
⊢ (𝜑 → 4 ∈
ℂ) |
| 210 | 13 | recnd 11289 |
. . . 4
⊢ (𝜑 → (𝑆↑2) ∈ ℂ) |
| 211 | 64 | recnd 11289 |
. . . 4
⊢ (𝜑 → 𝐵 ∈ ℂ) |
| 212 | 209, 210,
211 | adddid 11285 |
. . 3
⊢ (𝜑 → (4 · ((𝑆↑2) + 𝐵)) = ((4 · (𝑆↑2)) + (4 · 𝐵))) |
| 213 | 207, 212 | breqtrd 5169 |
. 2
⊢ (𝜑 → ((4 · (𝑆↑2)) + ((𝐾𝐷𝐿)↑2)) ≤ ((4 · (𝑆↑2)) + (4 · 𝐵))) |
| 214 | | remulcl 11240 |
. . . 4
⊢ ((4
∈ ℝ ∧ 𝐵
∈ ℝ) → (4 · 𝐵) ∈ ℝ) |
| 215 | 1, 64, 214 | sylancr 587 |
. . 3
⊢ (𝜑 → (4 · 𝐵) ∈
ℝ) |
| 216 | 32, 215, 15 | leadd2d 11858 |
. 2
⊢ (𝜑 → (((𝐾𝐷𝐿)↑2) ≤ (4 · 𝐵) ↔ ((4 · (𝑆↑2)) + ((𝐾𝐷𝐿)↑2)) ≤ ((4 · (𝑆↑2)) + (4 · 𝐵)))) |
| 217 | 213, 216 | mpbird 257 |
1
⊢ (𝜑 → ((𝐾𝐷𝐿)↑2) ≤ (4 · 𝐵)) |