| Step | Hyp | Ref
| Expression |
| 1 | | 4re 12350 |
. . . . . 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 30893 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∀𝑤 ∈ 𝑅 0 ≤ 𝑤)) |
| 14 | 13 | simp1d 1143 |
. . . . . . . . 9
⊢ (𝜑 → 𝑅 ⊆ ℝ) |
| 15 | 13 | simp2d 1144 |
. . . . . . . . 9
⊢ (𝜑 → 𝑅 ≠ ∅) |
| 16 | | 0re 11263 |
. . . . . . . . . 10
⊢ 0 ∈
ℝ |
| 17 | 13 | simp3d 1145 |
. . . . . . . . . 10
⊢ (𝜑 → ∀𝑤 ∈ 𝑅 0 ≤ 𝑤) |
| 18 | | breq1 5146 |
. . . . . . . . . . . 12
⊢ (𝑥 = 0 → (𝑥 ≤ 𝑤 ↔ 0 ≤ 𝑤)) |
| 19 | 18 | ralbidv 3178 |
. . . . . . . . . . 11
⊢ (𝑥 = 0 → (∀𝑤 ∈ 𝑅 𝑥 ≤ 𝑤 ↔ ∀𝑤 ∈ 𝑅 0 ≤ 𝑤)) |
| 20 | 19 | rspcev 3622 |
. . . . . . . . . 10
⊢ ((0
∈ ℝ ∧ ∀𝑤 ∈ 𝑅 0 ≤ 𝑤) → ∃𝑥 ∈ ℝ ∀𝑤 ∈ 𝑅 𝑥 ≤ 𝑤) |
| 21 | 16, 17, 20 | sylancr 587 |
. . . . . . . . 9
⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑤 ∈ 𝑅 𝑥 ≤ 𝑤) |
| 22 | | infrecl 12250 |
. . . . . . . . 9
⊢ ((𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑤 ∈ 𝑅 𝑥 ≤ 𝑤) → inf(𝑅, ℝ, < ) ∈
ℝ) |
| 23 | 14, 15, 21, 22 | syl3anc 1373 |
. . . . . . . 8
⊢ (𝜑 → inf(𝑅, ℝ, < ) ∈
ℝ) |
| 24 | 2, 23 | eqeltrid 2845 |
. . . . . . 7
⊢ (𝜑 → 𝑆 ∈ ℝ) |
| 25 | 24 | resqcld 14165 |
. . . . . 6
⊢ (𝜑 → (𝑆↑2) ∈ ℝ) |
| 26 | | remulcl 11240 |
. . . . . 6
⊢ ((4
∈ ℝ ∧ (𝑆↑2) ∈ ℝ) → (4 ·
(𝑆↑2)) ∈
ℝ) |
| 27 | 1, 25, 26 | sylancr 587 |
. . . . 5
⊢ (𝜑 → (4 · (𝑆↑2)) ∈
ℝ) |
| 28 | | phnv 30833 |
. . . . . . . . 9
⊢ (𝑈 ∈ CPreHilOLD
→ 𝑈 ∈
NrmCVec) |
| 29 | 7, 28 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 𝑈 ∈ NrmCVec) |
| 30 | 3, 10 | imsmet 30710 |
. . . . . . . 8
⊢ (𝑈 ∈ NrmCVec → 𝐷 ∈ (Met‘𝑋)) |
| 31 | 29, 30 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋)) |
| 32 | | inss1 4237 |
. . . . . . . . . 10
⊢
((SubSp‘𝑈)
∩ CBan) ⊆ (SubSp‘𝑈) |
| 33 | 32, 8 | sselid 3981 |
. . . . . . . . 9
⊢ (𝜑 → 𝑊 ∈ (SubSp‘𝑈)) |
| 34 | | eqid 2737 |
. . . . . . . . . 10
⊢
(SubSp‘𝑈) =
(SubSp‘𝑈) |
| 35 | 3, 6, 34 | sspba 30746 |
. . . . . . . . 9
⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ (SubSp‘𝑈)) → 𝑌 ⊆ 𝑋) |
| 36 | 29, 33, 35 | syl2anc 584 |
. . . . . . . 8
⊢ (𝜑 → 𝑌 ⊆ 𝑋) |
| 37 | | minvecolem2.3 |
. . . . . . . 8
⊢ (𝜑 → 𝐾 ∈ 𝑌) |
| 38 | 36, 37 | sseldd 3984 |
. . . . . . 7
⊢ (𝜑 → 𝐾 ∈ 𝑋) |
| 39 | | minvecolem2.4 |
. . . . . . . 8
⊢ (𝜑 → 𝐿 ∈ 𝑌) |
| 40 | 36, 39 | sseldd 3984 |
. . . . . . 7
⊢ (𝜑 → 𝐿 ∈ 𝑋) |
| 41 | | metcl 24342 |
. . . . . . 7
⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐾 ∈ 𝑋 ∧ 𝐿 ∈ 𝑋) → (𝐾𝐷𝐿) ∈ ℝ) |
| 42 | 31, 38, 40, 41 | syl3anc 1373 |
. . . . . 6
⊢ (𝜑 → (𝐾𝐷𝐿) ∈ ℝ) |
| 43 | 42 | resqcld 14165 |
. . . . 5
⊢ (𝜑 → ((𝐾𝐷𝐿)↑2) ∈ ℝ) |
| 44 | 27, 43 | readdcld 11290 |
. . . 4
⊢ (𝜑 → ((4 · (𝑆↑2)) + ((𝐾𝐷𝐿)↑2)) ∈ ℝ) |
| 45 | | ax-1cn 11213 |
. . . . . . . . . . . . 13
⊢ 1 ∈
ℂ |
| 46 | | halfcl 12491 |
. . . . . . . . . . . . 13
⊢ (1 ∈
ℂ → (1 / 2) ∈ ℂ) |
| 47 | 45, 46 | mp1i 13 |
. . . . . . . . . . . 12
⊢ (𝜑 → (1 / 2) ∈
ℂ) |
| 48 | | eqid 2737 |
. . . . . . . . . . . . . . 15
⊢ (
+𝑣 ‘𝑈) = ( +𝑣 ‘𝑈) |
| 49 | | eqid 2737 |
. . . . . . . . . . . . . . 15
⊢ (
+𝑣 ‘𝑊) = ( +𝑣 ‘𝑊) |
| 50 | 6, 48, 49, 34 | sspgval 30748 |
. . . . . . . . . . . . . 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 839 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝐾( +𝑣 ‘𝑊)𝐿) = (𝐾( +𝑣 ‘𝑈)𝐿)) |
| 52 | 34 | sspnv 30745 |
. . . . . . . . . . . . . . 15
⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ (SubSp‘𝑈)) → 𝑊 ∈ NrmCVec) |
| 53 | 29, 33, 52 | syl2anc 584 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑊 ∈ NrmCVec) |
| 54 | 6, 49 | nvgcl 30639 |
. . . . . . . . . . . . . 14
⊢ ((𝑊 ∈ NrmCVec ∧ 𝐾 ∈ 𝑌 ∧ 𝐿 ∈ 𝑌) → (𝐾( +𝑣 ‘𝑊)𝐿) ∈ 𝑌) |
| 55 | 53, 37, 39, 54 | syl3anc 1373 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝐾( +𝑣 ‘𝑊)𝐿) ∈ 𝑌) |
| 56 | 51, 55 | eqeltrrd 2842 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝐾( +𝑣 ‘𝑈)𝐿) ∈ 𝑌) |
| 57 | | eqid 2737 |
. . . . . . . . . . . . 13
⊢ (
·𝑠OLD ‘𝑈) = ( ·𝑠OLD
‘𝑈) |
| 58 | | eqid 2737 |
. . . . . . . . . . . . 13
⊢ (
·𝑠OLD ‘𝑊) = ( ·𝑠OLD
‘𝑊) |
| 59 | 6, 57, 58, 34 | sspsval 30750 |
. . . . . . . . . . . 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 839 |
. . . . . . . . . . 11
⊢ (𝜑 → ((1 / 2)(
·𝑠OLD ‘𝑊)(𝐾( +𝑣 ‘𝑈)𝐿)) = ((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿))) |
| 61 | 6, 58 | nvscl 30645 |
. . . . . . . . . . . 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 1373 |
. . . . . . . . . . 11
⊢ (𝜑 → ((1 / 2)(
·𝑠OLD ‘𝑊)(𝐾( +𝑣 ‘𝑈)𝐿)) ∈ 𝑌) |
| 63 | 60, 62 | eqeltrrd 2842 |
. . . . . . . . . 10
⊢ (𝜑 → ((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿)) ∈ 𝑌) |
| 64 | 36, 63 | sseldd 3984 |
. . . . . . . . 9
⊢ (𝜑 → ((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿)) ∈ 𝑋) |
| 65 | 3, 4 | nvmcl 30665 |
. . . . . . . . 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 1373 |
. . . . . . . 8
⊢ (𝜑 → (𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿))) ∈ 𝑋) |
| 67 | 3, 5 | nvcl 30680 |
. . . . . . . 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 584 |
. . . . . . 7
⊢ (𝜑 → (𝑁‘(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿)))) ∈ ℝ) |
| 69 | 68 | resqcld 14165 |
. . . . . 6
⊢ (𝜑 → ((𝑁‘(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿))))↑2) ∈
ℝ) |
| 70 | | remulcl 11240 |
. . . . . 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 587 |
. . . . 5
⊢ (𝜑 → (4 · ((𝑁‘(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿))))↑2)) ∈
ℝ) |
| 72 | 71, 43 | readdcld 11290 |
. . . 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 11290 |
. . . . 5
⊢ (𝜑 → ((𝑆↑2) + 𝐵) ∈ ℝ) |
| 75 | | remulcl 11240 |
. . . . 5
⊢ ((4
∈ ℝ ∧ ((𝑆↑2) + 𝐵) ∈ ℝ) → (4 · ((𝑆↑2) + 𝐵)) ∈ ℝ) |
| 76 | 1, 74, 75 | sylancr 587 |
. . . 4
⊢ (𝜑 → (4 · ((𝑆↑2) + 𝐵)) ∈ ℝ) |
| 77 | 16 | 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 | 14, 15, 21, 77, 78 | syl31anc 1375 |
. . . . . . . . 9
⊢ (𝜑 → (0 ≤ inf(𝑅, ℝ, < ) ↔
∀𝑤 ∈ 𝑅 0 ≤ 𝑤)) |
| 80 | 17, 79 | mpbird 257 |
. . . . . . . 8
⊢ (𝜑 → 0 ≤ inf(𝑅, ℝ, <
)) |
| 81 | 80, 2 | breqtrrdi 5185 |
. . . . . . 7
⊢ (𝜑 → 0 ≤ 𝑆) |
| 82 | | eqid 2737 |
. . . . . . . . . . . 12
⊢ (𝑁‘(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿)))) = (𝑁‘(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿)))) |
| 83 | | oveq2 7439 |
. . . . . . . . . . . . . 14
⊢ (𝑦 = ((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿)) → (𝐴𝑀𝑦) = (𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿)))) |
| 84 | 83 | fveq2d 6910 |
. . . . . . . . . . . . 13
⊢ (𝑦 = ((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿)) → (𝑁‘(𝐴𝑀𝑦)) = (𝑁‘(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿))))) |
| 85 | 84 | rspceeqv 3645 |
. . . . . . . . . . . 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))) |
| 86 | 63, 82, 85 | sylancl 586 |
. . . . . . . . . . 11
⊢ (𝜑 → ∃𝑦 ∈ 𝑌 (𝑁‘(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿)))) = (𝑁‘(𝐴𝑀𝑦))) |
| 87 | | eqid 2737 |
. . . . . . . . . . . 12
⊢ (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) = (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) |
| 88 | | fvex 6919 |
. . . . . . . . . . . 12
⊢ (𝑁‘(𝐴𝑀𝑦)) ∈ V |
| 89 | 87, 88 | elrnmpti 5973 |
. . . . . . . . . . 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))) |
| 90 | 86, 89 | sylibr 234 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑁‘(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿)))) ∈ ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦)))) |
| 91 | 90, 12 | eleqtrrdi 2852 |
. . . . . . . . 9
⊢ (𝜑 → (𝑁‘(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿)))) ∈ 𝑅) |
| 92 | | infrelb 12253 |
. . . . . . . . 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))))) |
| 93 | 14, 21, 91, 92 | syl3anc 1373 |
. . . . . . . 8
⊢ (𝜑 → inf(𝑅, ℝ, < ) ≤ (𝑁‘(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿))))) |
| 94 | 2, 93 | eqbrtrid 5178 |
. . . . . . 7
⊢ (𝜑 → 𝑆 ≤ (𝑁‘(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿))))) |
| 95 | | le2sq2 14175 |
. . . . . . 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)) |
| 96 | 24, 81, 68, 94, 95 | syl22anc 839 |
. . . . . 6
⊢ (𝜑 → (𝑆↑2) ≤ ((𝑁‘(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘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‘(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)))) |
| 100 | 98, 99 | mp3an3 1452 |
. . . . . . 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)))) |
| 101 | 25, 69, 100 | syl2anc 584 |
. . . . . 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)))) |
| 102 | 96, 101 | mpbid 232 |
. . . . 5
⊢ (𝜑 → (4 · (𝑆↑2)) ≤ (4 ·
((𝑁‘(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿))))↑2))) |
| 103 | 27, 71, 43, 102 | leadd1dd 11877 |
. . . 4
⊢ (𝜑 → ((4 · (𝑆↑2)) + ((𝐾𝐷𝐿)↑2)) ≤ ((4 · ((𝑁‘(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿))))↑2)) + ((𝐾𝐷𝐿)↑2))) |
| 104 | | metcl 24342 |
. . . . . . . . . 10
⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐾 ∈ 𝑋) → (𝐴𝐷𝐾) ∈ ℝ) |
| 105 | 31, 9, 38, 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 | 31, 9, 40, 107 | syl3anc 1373 |
. . . . . . . . 9
⊢ (𝜑 → (𝐴𝐷𝐿) ∈ ℝ) |
| 109 | 108 | resqcld 14165 |
. . . . . . . 8
⊢ (𝜑 → ((𝐴𝐷𝐿)↑2) ∈ ℝ) |
| 110 | | minvecolem2.5 |
. . . . . . . 8
⊢ (𝜑 → ((𝐴𝐷𝐾)↑2) ≤ ((𝑆↑2) + 𝐵)) |
| 111 | | minvecolem2.6 |
. . . . . . . 8
⊢ (𝜑 → ((𝐴𝐷𝐿)↑2) ≤ ((𝑆↑2) + 𝐵)) |
| 112 | 106, 109,
74, 74, 110, 111 | le2addd 11882 |
. . . . . . 7
⊢ (𝜑 → (((𝐴𝐷𝐾)↑2) + ((𝐴𝐷𝐿)↑2)) ≤ (((𝑆↑2) + 𝐵) + ((𝑆↑2) + 𝐵))) |
| 113 | 74 | 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, 74, 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 | 3, 4 | nvmcl 30665 |
. . . . . . . 8
⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐾 ∈ 𝑋) → (𝐴𝑀𝐾) ∈ 𝑋) |
| 127 | 29, 9, 38, 126 | syl3anc 1373 |
. . . . . . 7
⊢ (𝜑 → (𝐴𝑀𝐾) ∈ 𝑋) |
| 128 | 3, 4 | nvmcl 30665 |
. . . . . . . 8
⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐿 ∈ 𝑋) → (𝐴𝑀𝐿) ∈ 𝑋) |
| 129 | 29, 9, 40, 128 | syl3anc 1373 |
. . . . . . 7
⊢ (𝜑 → (𝐴𝑀𝐿) ∈ 𝑋) |
| 130 | 3, 48, 4, 5 | phpar2 30842 |
. . . . . . 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)))) |
| 131 | 7, 127, 129, 130 | syl3anc 1373 |
. . . . . 6
⊢ (𝜑 → (((𝑁‘((𝐴𝑀𝐾)( +𝑣 ‘𝑈)(𝐴𝑀𝐿)))↑2) + ((𝑁‘((𝐴𝑀𝐾)𝑀(𝐴𝑀𝐿)))↑2)) = (2 · (((𝑁‘(𝐴𝑀𝐾))↑2) + ((𝑁‘(𝐴𝑀𝐿))↑2)))) |
| 132 | | 2cn 12341 |
. . . . . . . . . 10
⊢ 2 ∈
ℂ |
| 133 | 68 | recnd 11289 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑁‘(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿)))) ∈ ℂ) |
| 134 | | sqmul 14159 |
. . . . . . . . . 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))) |
| 135 | 132, 133,
134 | sylancr 587 |
. . . . . . . . 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))) |
| 136 | | sq2 14236 |
. . . . . . . . . 10
⊢
(2↑2) = 4 |
| 137 | 136 | oveq1i 7441 |
. . . . . . . . 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)) |
| 138 | 135, 137 | eqtrdi 2793 |
. . . . . . . 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))) |
| 139 | 132 | a1i 11 |
. . . . . . . . . . . 12
⊢ (𝜑 → 2 ∈
ℂ) |
| 140 | 3, 57, 5 | nvs 30682 |
. . . . . . . . . . . 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)))))) |
| 141 | 29, 139, 66, 140 | syl3anc 1373 |
. . . . . . . . . . 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)))))) |
| 142 | | 0le2 12368 |
. . . . . . . . . . . . 13
⊢ 0 ≤
2 |
| 143 | | absid 15335 |
. . . . . . . . . . . . 13
⊢ ((2
∈ ℝ ∧ 0 ≤ 2) → (abs‘2) = 2) |
| 144 | 117, 142,
143 | mp2an 692 |
. . . . . . . . . . . 12
⊢
(abs‘2) = 2 |
| 145 | 144 | oveq1i 7441 |
. . . . . . . . . . 11
⊢
((abs‘2) · (𝑁‘(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿))))) = (2 · (𝑁‘(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿))))) |
| 146 | 141, 145 | eqtrdi 2793 |
. . . . . . . . . 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)))))) |
| 147 | 3, 4, 57 | nvmdi 30667 |
. . . . . . . . . . . . 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))))) |
| 148 | 29, 139, 9, 64, 147 | syl13anc 1374 |
. . . . . . . . . . . 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))))) |
| 149 | 3, 48, 57 | nv2 30651 |
. . . . . . . . . . . . . 14
⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴( +𝑣 ‘𝑈)𝐴) = (2(
·𝑠OLD ‘𝑈)𝐴)) |
| 150 | 29, 9, 149 | syl2anc 584 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝐴( +𝑣 ‘𝑈)𝐴) = (2(
·𝑠OLD ‘𝑈)𝐴)) |
| 151 | | 2ne0 12370 |
. . . . . . . . . . . . . . . . 17
⊢ 2 ≠
0 |
| 152 | 132, 151 | recidi 11998 |
. . . . . . . . . . . . . . . 16
⊢ (2
· (1 / 2)) = 1 |
| 153 | 152 | oveq1i 7441 |
. . . . . . . . . . . . . . 15
⊢ ((2
· (1 / 2))( ·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿)) = (1(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿)) |
| 154 | 3, 48 | nvgcl 30639 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑈 ∈ NrmCVec ∧ 𝐾 ∈ 𝑋 ∧ 𝐿 ∈ 𝑋) → (𝐾( +𝑣 ‘𝑈)𝐿) ∈ 𝑋) |
| 155 | 29, 38, 40, 154 | syl3anc 1373 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝐾( +𝑣 ‘𝑈)𝐿) ∈ 𝑋) |
| 156 | 3, 57 | nvsid 30646 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑈 ∈ NrmCVec ∧ (𝐾( +𝑣
‘𝑈)𝐿) ∈ 𝑋) → (1(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿)) = (𝐾( +𝑣 ‘𝑈)𝐿)) |
| 157 | 29, 155, 156 | syl2anc 584 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (1(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿)) = (𝐾( +𝑣 ‘𝑈)𝐿)) |
| 158 | 153, 157 | eqtrid 2789 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((2 · (1 / 2))(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿)) = (𝐾( +𝑣 ‘𝑈)𝐿)) |
| 159 | 3, 57 | nvsass 30647 |
. . . . . . . . . . . . . . 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)))) |
| 160 | 29, 139, 47, 155, 159 | syl13anc 1374 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((2 · (1 / 2))(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿)) = (2(
·𝑠OLD ‘𝑈)((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿)))) |
| 161 | 158, 160 | eqtr3d 2779 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝐾( +𝑣 ‘𝑈)𝐿) = (2(
·𝑠OLD ‘𝑈)((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿)))) |
| 162 | 150, 161 | oveq12d 7449 |
. . . . . . . . . . . 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))))) |
| 163 | 3, 48, 4 | nvaddsub4 30676 |
. . . . . . . . . . . . 13
⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) ∧ (𝐾 ∈ 𝑋 ∧ 𝐿 ∈ 𝑋)) → ((𝐴( +𝑣 ‘𝑈)𝐴)𝑀(𝐾( +𝑣 ‘𝑈)𝐿)) = ((𝐴𝑀𝐾)( +𝑣 ‘𝑈)(𝐴𝑀𝐿))) |
| 164 | 29, 9, 9, 38, 40, 163 | syl122anc 1381 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((𝐴( +𝑣 ‘𝑈)𝐴)𝑀(𝐾( +𝑣 ‘𝑈)𝐿)) = ((𝐴𝑀𝐾)( +𝑣 ‘𝑈)(𝐴𝑀𝐿))) |
| 165 | 148, 162,
164 | 3eqtr2d 2783 |
. . . . . . . . . . 11
⊢ (𝜑 → (2(
·𝑠OLD ‘𝑈)(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿)))) = ((𝐴𝑀𝐾)( +𝑣 ‘𝑈)(𝐴𝑀𝐿))) |
| 166 | 165 | fveq2d 6910 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑁‘(2(
·𝑠OLD ‘𝑈)(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿))))) = (𝑁‘((𝐴𝑀𝐾)( +𝑣 ‘𝑈)(𝐴𝑀𝐿)))) |
| 167 | 146, 166 | eqtr3d 2779 |
. . . . . . . . 9
⊢ (𝜑 → (2 · (𝑁‘(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿))))) = (𝑁‘((𝐴𝑀𝐾)( +𝑣 ‘𝑈)(𝐴𝑀𝐿)))) |
| 168 | 167 | oveq1d 7446 |
. . . . . . . 8
⊢ (𝜑 → ((2 · (𝑁‘(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿)))))↑2) = ((𝑁‘((𝐴𝑀𝐾)( +𝑣 ‘𝑈)(𝐴𝑀𝐿)))↑2)) |
| 169 | 138, 168 | eqtr3d 2779 |
. . . . . . 7
⊢ (𝜑 → (4 · ((𝑁‘(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿))))↑2)) = ((𝑁‘((𝐴𝑀𝐾)( +𝑣 ‘𝑈)(𝐴𝑀𝐿)))↑2)) |
| 170 | 3, 4, 5, 10 | imsdval 30705 |
. . . . . . . . . 10
⊢ ((𝑈 ∈ NrmCVec ∧ 𝐿 ∈ 𝑋 ∧ 𝐾 ∈ 𝑋) → (𝐿𝐷𝐾) = (𝑁‘(𝐿𝑀𝐾))) |
| 171 | 29, 40, 38, 170 | syl3anc 1373 |
. . . . . . . . 9
⊢ (𝜑 → (𝐿𝐷𝐾) = (𝑁‘(𝐿𝑀𝐾))) |
| 172 | | metsym 24360 |
. . . . . . . . . 10
⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐾 ∈ 𝑋 ∧ 𝐿 ∈ 𝑋) → (𝐾𝐷𝐿) = (𝐿𝐷𝐾)) |
| 173 | 31, 38, 40, 172 | syl3anc 1373 |
. . . . . . . . 9
⊢ (𝜑 → (𝐾𝐷𝐿) = (𝐿𝐷𝐾)) |
| 174 | 3, 4 | nvnnncan1 30666 |
. . . . . . . . . . 11
⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐾 ∈ 𝑋 ∧ 𝐿 ∈ 𝑋)) → ((𝐴𝑀𝐾)𝑀(𝐴𝑀𝐿)) = (𝐿𝑀𝐾)) |
| 175 | 29, 9, 38, 40, 174 | syl13anc 1374 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝐴𝑀𝐾)𝑀(𝐴𝑀𝐿)) = (𝐿𝑀𝐾)) |
| 176 | 175 | fveq2d 6910 |
. . . . . . . . 9
⊢ (𝜑 → (𝑁‘((𝐴𝑀𝐾)𝑀(𝐴𝑀𝐿))) = (𝑁‘(𝐿𝑀𝐾))) |
| 177 | 171, 173,
176 | 3eqtr4d 2787 |
. . . . . . . 8
⊢ (𝜑 → (𝐾𝐷𝐿) = (𝑁‘((𝐴𝑀𝐾)𝑀(𝐴𝑀𝐿)))) |
| 178 | 177 | oveq1d 7446 |
. . . . . . 7
⊢ (𝜑 → ((𝐾𝐷𝐿)↑2) = ((𝑁‘((𝐴𝑀𝐾)𝑀(𝐴𝑀𝐿)))↑2)) |
| 179 | 169, 178 | oveq12d 7449 |
. . . . . 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))) |
| 180 | 3, 4, 5, 10 | imsdval 30705 |
. . . . . . . . . 10
⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐾 ∈ 𝑋) → (𝐴𝐷𝐾) = (𝑁‘(𝐴𝑀𝐾))) |
| 181 | 29, 9, 38, 180 | syl3anc 1373 |
. . . . . . . . 9
⊢ (𝜑 → (𝐴𝐷𝐾) = (𝑁‘(𝐴𝑀𝐾))) |
| 182 | 181 | oveq1d 7446 |
. . . . . . . 8
⊢ (𝜑 → ((𝐴𝐷𝐾)↑2) = ((𝑁‘(𝐴𝑀𝐾))↑2)) |
| 183 | 3, 4, 5, 10 | imsdval 30705 |
. . . . . . . . . 10
⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐿 ∈ 𝑋) → (𝐴𝐷𝐿) = (𝑁‘(𝐴𝑀𝐿))) |
| 184 | 29, 9, 40, 183 | syl3anc 1373 |
. . . . . . . . 9
⊢ (𝜑 → (𝐴𝐷𝐿) = (𝑁‘(𝐴𝑀𝐿))) |
| 185 | 184 | oveq1d 7446 |
. . . . . . . 8
⊢ (𝜑 → ((𝐴𝐷𝐿)↑2) = ((𝑁‘(𝐴𝑀𝐿))↑2)) |
| 186 | 182, 185 | oveq12d 7449 |
. . . . . . 7
⊢ (𝜑 → (((𝐴𝐷𝐾)↑2) + ((𝐴𝐷𝐿)↑2)) = (((𝑁‘(𝐴𝑀𝐾))↑2) + ((𝑁‘(𝐴𝑀𝐿))↑2))) |
| 187 | 186 | oveq2d 7447 |
. . . . . 6
⊢ (𝜑 → (2 · (((𝐴𝐷𝐾)↑2) + ((𝐴𝐷𝐿)↑2))) = (2 · (((𝑁‘(𝐴𝑀𝐾))↑2) + ((𝑁‘(𝐴𝑀𝐿))↑2)))) |
| 188 | 131, 179,
187 | 3eqtr4d 2787 |
. . . . 5
⊢ (𝜑 → ((4 · ((𝑁‘(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿))))↑2)) + ((𝐾𝐷𝐿)↑2)) = (2 · (((𝐴𝐷𝐾)↑2) + ((𝐴𝐷𝐿)↑2)))) |
| 189 | | 2t2e4 12430 |
. . . . . . 7
⊢ (2
· 2) = 4 |
| 190 | 189 | oveq1i 7441 |
. . . . . 6
⊢ ((2
· 2) · ((𝑆↑2) + 𝐵)) = (4 · ((𝑆↑2) + 𝐵)) |
| 191 | 139, 139,
113 | mulassd 11284 |
. . . . . 6
⊢ (𝜑 → ((2 · 2) ·
((𝑆↑2) + 𝐵)) = (2 · (2 ·
((𝑆↑2) + 𝐵)))) |
| 192 | 190, 191 | eqtr3id 2791 |
. . . . 5
⊢ (𝜑 → (4 · ((𝑆↑2) + 𝐵)) = (2 · (2 · ((𝑆↑2) + 𝐵)))) |
| 193 | 125, 188,
192 | 3brtr4d 5175 |
. . . 4
⊢ (𝜑 → ((4 · ((𝑁‘(𝐴𝑀((1 / 2)(
·𝑠OLD ‘𝑈)(𝐾( +𝑣 ‘𝑈)𝐿))))↑2)) + ((𝐾𝐷𝐿)↑2)) ≤ (4 · ((𝑆↑2) + 𝐵))) |
| 194 | 44, 72, 76, 103, 193 | letrd 11418 |
. . 3
⊢ (𝜑 → ((4 · (𝑆↑2)) + ((𝐾𝐷𝐿)↑2)) ≤ (4 · ((𝑆↑2) + 𝐵))) |
| 195 | | 4cn 12351 |
. . . . 5
⊢ 4 ∈
ℂ |
| 196 | 195 | a1i 11 |
. . . 4
⊢ (𝜑 → 4 ∈
ℂ) |
| 197 | 25 | recnd 11289 |
. . . 4
⊢ (𝜑 → (𝑆↑2) ∈ ℂ) |
| 198 | 73 | recnd 11289 |
. . . 4
⊢ (𝜑 → 𝐵 ∈ ℂ) |
| 199 | 196, 197,
198 | adddid 11285 |
. . 3
⊢ (𝜑 → (4 · ((𝑆↑2) + 𝐵)) = ((4 · (𝑆↑2)) + (4 · 𝐵))) |
| 200 | 194, 199 | breqtrd 5169 |
. 2
⊢ (𝜑 → ((4 · (𝑆↑2)) + ((𝐾𝐷𝐿)↑2)) ≤ ((4 · (𝑆↑2)) + (4 · 𝐵))) |
| 201 | | remulcl 11240 |
. . . 4
⊢ ((4
∈ ℝ ∧ 𝐵
∈ ℝ) → (4 · 𝐵) ∈ ℝ) |
| 202 | 1, 73, 201 | sylancr 587 |
. . 3
⊢ (𝜑 → (4 · 𝐵) ∈
ℝ) |
| 203 | 43, 202, 27 | leadd2d 11858 |
. 2
⊢ (𝜑 → (((𝐾𝐷𝐿)↑2) ≤ (4 · 𝐵) ↔ ((4 · (𝑆↑2)) + ((𝐾𝐷𝐿)↑2)) ≤ ((4 · (𝑆↑2)) + (4 · 𝐵)))) |
| 204 | 200, 203 | mpbird 257 |
1
⊢ (𝜑 → ((𝐾𝐷𝐿)↑2) ≤ (4 · 𝐵)) |