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Theorem minvecolem2 27619
Description: Lemma for minveco 27628. Any two points 𝐾 and 𝐿 in 𝑌 are close to each other if they are close to the infimum of distance to 𝐴. (Contributed by Mario Carneiro, 9-May-2014.) (Revised by AV, 4-Oct-2020.) (New usage is discouraged.)
Hypotheses
Ref Expression
minveco.x 𝑋 = (BaseSet‘𝑈)
minveco.m 𝑀 = ( −𝑣𝑈)
minveco.n 𝑁 = (normCV𝑈)
minveco.y 𝑌 = (BaseSet‘𝑊)
minveco.u (𝜑𝑈 ∈ CPreHilOLD)
minveco.w (𝜑𝑊 ∈ ((SubSp‘𝑈) ∩ CBan))
minveco.a (𝜑𝐴𝑋)
minveco.d 𝐷 = (IndMet‘𝑈)
minveco.j 𝐽 = (MetOpen‘𝐷)
minveco.r 𝑅 = ran (𝑦𝑌 ↦ (𝑁‘(𝐴𝑀𝑦)))
minveco.s 𝑆 = inf(𝑅, ℝ, < )
minvecolem2.1 (𝜑𝐵 ∈ ℝ)
minvecolem2.2 (𝜑 → 0 ≤ 𝐵)
minvecolem2.3 (𝜑𝐾𝑌)
minvecolem2.4 (𝜑𝐿𝑌)
minvecolem2.5 (𝜑 → ((𝐴𝐷𝐾)↑2) ≤ ((𝑆↑2) + 𝐵))
minvecolem2.6 (𝜑 → ((𝐴𝐷𝐿)↑2) ≤ ((𝑆↑2) + 𝐵))
Assertion
Ref Expression
minvecolem2 (𝜑 → ((𝐾𝐷𝐿)↑2) ≤ (4 · 𝐵))
Distinct variable groups:   𝑦,𝐽   𝑦,𝐾   𝑦,𝐿   𝑦,𝑀   𝑦,𝑁   𝜑,𝑦   𝑦,𝑆   𝑦,𝐴   𝑦,𝐷   𝑦,𝑈   𝑦,𝑊   𝑦,𝑌
Allowed substitution hints:   𝐵(𝑦)   𝑅(𝑦)   𝑋(𝑦)

Proof of Theorem minvecolem2
Dummy variables 𝑥 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 4re 11057 . . . . . 6 4 ∈ ℝ
2 minveco.s . . . . . . . 8 𝑆 = inf(𝑅, ℝ, < )
3 minveco.x . . . . . . . . . . 11 𝑋 = (BaseSet‘𝑈)
4 minveco.m . . . . . . . . . . 11 𝑀 = ( −𝑣𝑈)
5 minveco.n . . . . . . . . . . 11 𝑁 = (normCV𝑈)
6 minveco.y . . . . . . . . . . 11 𝑌 = (BaseSet‘𝑊)
7 minveco.u . . . . . . . . . . 11 (𝜑𝑈 ∈ CPreHilOLD)
8 minveco.w . . . . . . . . . . 11 (𝜑𝑊 ∈ ((SubSp‘𝑈) ∩ CBan))
9 minveco.a . . . . . . . . . . 11 (𝜑𝐴𝑋)
10 minveco.d . . . . . . . . . . 11 𝐷 = (IndMet‘𝑈)
11 minveco.j . . . . . . . . . . 11 𝐽 = (MetOpen‘𝐷)
12 minveco.r . . . . . . . . . . 11 𝑅 = ran (𝑦𝑌 ↦ (𝑁‘(𝐴𝑀𝑦)))
133, 4, 5, 6, 7, 8, 9, 10, 11, 12minvecolem1 27618 . . . . . . . . . 10 (𝜑 → (𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∀𝑤𝑅 0 ≤ 𝑤))
1413simp1d 1071 . . . . . . . . 9 (𝜑𝑅 ⊆ ℝ)
1513simp2d 1072 . . . . . . . . 9 (𝜑𝑅 ≠ ∅)
16 0re 10000 . . . . . . . . . 10 0 ∈ ℝ
1713simp3d 1073 . . . . . . . . . 10 (𝜑 → ∀𝑤𝑅 0 ≤ 𝑤)
18 breq1 4626 . . . . . . . . . . . 12 (𝑥 = 0 → (𝑥𝑤 ↔ 0 ≤ 𝑤))
1918ralbidv 2982 . . . . . . . . . . 11 (𝑥 = 0 → (∀𝑤𝑅 𝑥𝑤 ↔ ∀𝑤𝑅 0 ≤ 𝑤))
2019rspcev 3299 . . . . . . . . . 10 ((0 ∈ ℝ ∧ ∀𝑤𝑅 0 ≤ 𝑤) → ∃𝑥 ∈ ℝ ∀𝑤𝑅 𝑥𝑤)
2116, 17, 20sylancr 694 . . . . . . . . 9 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑤𝑅 𝑥𝑤)
22 infrecl 10965 . . . . . . . . 9 ((𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑤𝑅 𝑥𝑤) → inf(𝑅, ℝ, < ) ∈ ℝ)
2314, 15, 21, 22syl3anc 1323 . . . . . . . 8 (𝜑 → inf(𝑅, ℝ, < ) ∈ ℝ)
242, 23syl5eqel 2702 . . . . . . 7 (𝜑𝑆 ∈ ℝ)
2524resqcld 12991 . . . . . 6 (𝜑 → (𝑆↑2) ∈ ℝ)
26 remulcl 9981 . . . . . 6 ((4 ∈ ℝ ∧ (𝑆↑2) ∈ ℝ) → (4 · (𝑆↑2)) ∈ ℝ)
271, 25, 26sylancr 694 . . . . 5 (𝜑 → (4 · (𝑆↑2)) ∈ ℝ)
28 phnv 27557 . . . . . . . . 9 (𝑈 ∈ CPreHilOLD𝑈 ∈ NrmCVec)
297, 28syl 17 . . . . . . . 8 (𝜑𝑈 ∈ NrmCVec)
303, 10imsmet 27434 . . . . . . . 8 (𝑈 ∈ NrmCVec → 𝐷 ∈ (Met‘𝑋))
3129, 30syl 17 . . . . . . 7 (𝜑𝐷 ∈ (Met‘𝑋))
32 inss1 3817 . . . . . . . . . 10 ((SubSp‘𝑈) ∩ CBan) ⊆ (SubSp‘𝑈)
3332, 8sseldi 3586 . . . . . . . . 9 (𝜑𝑊 ∈ (SubSp‘𝑈))
34 eqid 2621 . . . . . . . . . 10 (SubSp‘𝑈) = (SubSp‘𝑈)
353, 6, 34sspba 27470 . . . . . . . . 9 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ (SubSp‘𝑈)) → 𝑌𝑋)
3629, 33, 35syl2anc 692 . . . . . . . 8 (𝜑𝑌𝑋)
37 minvecolem2.3 . . . . . . . 8 (𝜑𝐾𝑌)
3836, 37sseldd 3589 . . . . . . 7 (𝜑𝐾𝑋)
39 minvecolem2.4 . . . . . . . 8 (𝜑𝐿𝑌)
4036, 39sseldd 3589 . . . . . . 7 (𝜑𝐿𝑋)
41 metcl 22077 . . . . . . 7 ((𝐷 ∈ (Met‘𝑋) ∧ 𝐾𝑋𝐿𝑋) → (𝐾𝐷𝐿) ∈ ℝ)
4231, 38, 40, 41syl3anc 1323 . . . . . 6 (𝜑 → (𝐾𝐷𝐿) ∈ ℝ)
4342resqcld 12991 . . . . 5 (𝜑 → ((𝐾𝐷𝐿)↑2) ∈ ℝ)
4427, 43readdcld 10029 . . . 4 (𝜑 → ((4 · (𝑆↑2)) + ((𝐾𝐷𝐿)↑2)) ∈ ℝ)
45 ax-1cn 9954 . . . . . . . . . . . . 13 1 ∈ ℂ
46 halfcl 11217 . . . . . . . . . . . . 13 (1 ∈ ℂ → (1 / 2) ∈ ℂ)
4745, 46mp1i 13 . . . . . . . . . . . 12 (𝜑 → (1 / 2) ∈ ℂ)
48 eqid 2621 . . . . . . . . . . . . . . 15 ( +𝑣𝑈) = ( +𝑣𝑈)
49 eqid 2621 . . . . . . . . . . . . . . 15 ( +𝑣𝑊) = ( +𝑣𝑊)
506, 48, 49, 34sspgval 27472 . . . . . . . . . . . . . 14 (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ (SubSp‘𝑈)) ∧ (𝐾𝑌𝐿𝑌)) → (𝐾( +𝑣𝑊)𝐿) = (𝐾( +𝑣𝑈)𝐿))
5129, 33, 37, 39, 50syl22anc 1324 . . . . . . . . . . . . 13 (𝜑 → (𝐾( +𝑣𝑊)𝐿) = (𝐾( +𝑣𝑈)𝐿))
5234sspnv 27469 . . . . . . . . . . . . . . 15 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ (SubSp‘𝑈)) → 𝑊 ∈ NrmCVec)
5329, 33, 52syl2anc 692 . . . . . . . . . . . . . 14 (𝜑𝑊 ∈ NrmCVec)
546, 49nvgcl 27363 . . . . . . . . . . . . . 14 ((𝑊 ∈ NrmCVec ∧ 𝐾𝑌𝐿𝑌) → (𝐾( +𝑣𝑊)𝐿) ∈ 𝑌)
5553, 37, 39, 54syl3anc 1323 . . . . . . . . . . . . 13 (𝜑 → (𝐾( +𝑣𝑊)𝐿) ∈ 𝑌)
5651, 55eqeltrrd 2699 . . . . . . . . . . . 12 (𝜑 → (𝐾( +𝑣𝑈)𝐿) ∈ 𝑌)
57 eqid 2621 . . . . . . . . . . . . 13 ( ·𝑠OLD𝑈) = ( ·𝑠OLD𝑈)
58 eqid 2621 . . . . . . . . . . . . 13 ( ·𝑠OLD𝑊) = ( ·𝑠OLD𝑊)
596, 57, 58, 34sspsval 27474 . . . . . . . . . . . 12 (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ (SubSp‘𝑈)) ∧ ((1 / 2) ∈ ℂ ∧ (𝐾( +𝑣𝑈)𝐿) ∈ 𝑌)) → ((1 / 2)( ·𝑠OLD𝑊)(𝐾( +𝑣𝑈)𝐿)) = ((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)))
6029, 33, 47, 56, 59syl22anc 1324 . . . . . . . . . . 11 (𝜑 → ((1 / 2)( ·𝑠OLD𝑊)(𝐾( +𝑣𝑈)𝐿)) = ((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)))
616, 58nvscl 27369 . . . . . . . . . . . 12 ((𝑊 ∈ NrmCVec ∧ (1 / 2) ∈ ℂ ∧ (𝐾( +𝑣𝑈)𝐿) ∈ 𝑌) → ((1 / 2)( ·𝑠OLD𝑊)(𝐾( +𝑣𝑈)𝐿)) ∈ 𝑌)
6253, 47, 56, 61syl3anc 1323 . . . . . . . . . . 11 (𝜑 → ((1 / 2)( ·𝑠OLD𝑊)(𝐾( +𝑣𝑈)𝐿)) ∈ 𝑌)
6360, 62eqeltrrd 2699 . . . . . . . . . 10 (𝜑 → ((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)) ∈ 𝑌)
6436, 63sseldd 3589 . . . . . . . . 9 (𝜑 → ((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)) ∈ 𝑋)
653, 4nvmcl 27389 . . . . . . . . 9 ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋 ∧ ((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)) ∈ 𝑋) → (𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))) ∈ 𝑋)
6629, 9, 64, 65syl3anc 1323 . . . . . . . 8 (𝜑 → (𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))) ∈ 𝑋)
673, 5nvcl 27404 . . . . . . . 8 ((𝑈 ∈ NrmCVec ∧ (𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))) ∈ 𝑋) → (𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)))) ∈ ℝ)
6829, 66, 67syl2anc 692 . . . . . . 7 (𝜑 → (𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)))) ∈ ℝ)
6968resqcld 12991 . . . . . 6 (𝜑 → ((𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))↑2) ∈ ℝ)
70 remulcl 9981 . . . . . 6 ((4 ∈ ℝ ∧ ((𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))↑2) ∈ ℝ) → (4 · ((𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))↑2)) ∈ ℝ)
711, 69, 70sylancr 694 . . . . 5 (𝜑 → (4 · ((𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))↑2)) ∈ ℝ)
7271, 43readdcld 10029 . . . 4 (𝜑 → ((4 · ((𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))↑2)) + ((𝐾𝐷𝐿)↑2)) ∈ ℝ)
73 minvecolem2.1 . . . . . 6 (𝜑𝐵 ∈ ℝ)
7425, 73readdcld 10029 . . . . 5 (𝜑 → ((𝑆↑2) + 𝐵) ∈ ℝ)
75 remulcl 9981 . . . . 5 ((4 ∈ ℝ ∧ ((𝑆↑2) + 𝐵) ∈ ℝ) → (4 · ((𝑆↑2) + 𝐵)) ∈ ℝ)
761, 74, 75sylancr 694 . . . 4 (𝜑 → (4 · ((𝑆↑2) + 𝐵)) ∈ ℝ)
7716a1i 11 . . . . . . . . . 10 (𝜑 → 0 ∈ ℝ)
78 infregelb 10967 . . . . . . . . . 10 (((𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑤𝑅 𝑥𝑤) ∧ 0 ∈ ℝ) → (0 ≤ inf(𝑅, ℝ, < ) ↔ ∀𝑤𝑅 0 ≤ 𝑤))
7914, 15, 21, 77, 78syl31anc 1326 . . . . . . . . 9 (𝜑 → (0 ≤ inf(𝑅, ℝ, < ) ↔ ∀𝑤𝑅 0 ≤ 𝑤))
8017, 79mpbird 247 . . . . . . . 8 (𝜑 → 0 ≤ inf(𝑅, ℝ, < ))
8180, 2syl6breqr 4665 . . . . . . 7 (𝜑 → 0 ≤ 𝑆)
82 eqid 2621 . . . . . . . . . . . 12 (𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)))) = (𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))
83 oveq2 6623 . . . . . . . . . . . . . . 15 (𝑦 = ((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)) → (𝐴𝑀𝑦) = (𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))
8483fveq2d 6162 . . . . . . . . . . . . . 14 (𝑦 = ((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)) → (𝑁‘(𝐴𝑀𝑦)) = (𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)))))
8584eqeq2d 2631 . . . . . . . . . . . . 13 (𝑦 = ((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)) → ((𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)))) = (𝑁‘(𝐴𝑀𝑦)) ↔ (𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)))) = (𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))))
8685rspcev 3299 . . . . . . . . . . . 12 ((((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)) ∈ 𝑌 ∧ (𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)))) = (𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))) → ∃𝑦𝑌 (𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)))) = (𝑁‘(𝐴𝑀𝑦)))
8763, 82, 86sylancl 693 . . . . . . . . . . 11 (𝜑 → ∃𝑦𝑌 (𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)))) = (𝑁‘(𝐴𝑀𝑦)))
88 eqid 2621 . . . . . . . . . . . 12 (𝑦𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) = (𝑦𝑌 ↦ (𝑁‘(𝐴𝑀𝑦)))
89 fvex 6168 . . . . . . . . . . . 12 (𝑁‘(𝐴𝑀𝑦)) ∈ V
9088, 89elrnmpti 5346 . . . . . . . . . . 11 ((𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)))) ∈ ran (𝑦𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) ↔ ∃𝑦𝑌 (𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)))) = (𝑁‘(𝐴𝑀𝑦)))
9187, 90sylibr 224 . . . . . . . . . 10 (𝜑 → (𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)))) ∈ ran (𝑦𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))))
9291, 12syl6eleqr 2709 . . . . . . . . 9 (𝜑 → (𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)))) ∈ 𝑅)
93 infrelb 10968 . . . . . . . . 9 ((𝑅 ⊆ ℝ ∧ ∃𝑥 ∈ ℝ ∀𝑤𝑅 𝑥𝑤 ∧ (𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)))) ∈ 𝑅) → inf(𝑅, ℝ, < ) ≤ (𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)))))
9414, 21, 92, 93syl3anc 1323 . . . . . . . 8 (𝜑 → inf(𝑅, ℝ, < ) ≤ (𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)))))
952, 94syl5eqbr 4658 . . . . . . 7 (𝜑𝑆 ≤ (𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)))))
96 le2sq2 12895 . . . . . . 7 (((𝑆 ∈ ℝ ∧ 0 ≤ 𝑆) ∧ ((𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)))) ∈ ℝ ∧ 𝑆 ≤ (𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)))))) → (𝑆↑2) ≤ ((𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))↑2))
9724, 81, 68, 95, 96syl22anc 1324 . . . . . 6 (𝜑 → (𝑆↑2) ≤ ((𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))↑2))
98 4pos 11076 . . . . . . . . 9 0 < 4
991, 98pm3.2i 471 . . . . . . . 8 (4 ∈ ℝ ∧ 0 < 4)
100 lemul2 10836 . . . . . . . 8 (((𝑆↑2) ∈ ℝ ∧ ((𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))↑2) ∈ ℝ ∧ (4 ∈ ℝ ∧ 0 < 4)) → ((𝑆↑2) ≤ ((𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))↑2) ↔ (4 · (𝑆↑2)) ≤ (4 · ((𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))↑2))))
10199, 100mp3an3 1410 . . . . . . 7 (((𝑆↑2) ∈ ℝ ∧ ((𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))↑2) ∈ ℝ) → ((𝑆↑2) ≤ ((𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))↑2) ↔ (4 · (𝑆↑2)) ≤ (4 · ((𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))↑2))))
10225, 69, 101syl2anc 692 . . . . . 6 (𝜑 → ((𝑆↑2) ≤ ((𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))↑2) ↔ (4 · (𝑆↑2)) ≤ (4 · ((𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))↑2))))
10397, 102mpbid 222 . . . . 5 (𝜑 → (4 · (𝑆↑2)) ≤ (4 · ((𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))↑2)))
10427, 71, 43, 103leadd1dd 10601 . . . 4 (𝜑 → ((4 · (𝑆↑2)) + ((𝐾𝐷𝐿)↑2)) ≤ ((4 · ((𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))↑2)) + ((𝐾𝐷𝐿)↑2)))
105 metcl 22077 . . . . . . . . . 10 ((𝐷 ∈ (Met‘𝑋) ∧ 𝐴𝑋𝐾𝑋) → (𝐴𝐷𝐾) ∈ ℝ)
10631, 9, 38, 105syl3anc 1323 . . . . . . . . 9 (𝜑 → (𝐴𝐷𝐾) ∈ ℝ)
107106resqcld 12991 . . . . . . . 8 (𝜑 → ((𝐴𝐷𝐾)↑2) ∈ ℝ)
108 metcl 22077 . . . . . . . . . 10 ((𝐷 ∈ (Met‘𝑋) ∧ 𝐴𝑋𝐿𝑋) → (𝐴𝐷𝐿) ∈ ℝ)
10931, 9, 40, 108syl3anc 1323 . . . . . . . . 9 (𝜑 → (𝐴𝐷𝐿) ∈ ℝ)
110109resqcld 12991 . . . . . . . 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))
113107, 110, 74, 74, 111, 112le2addd 10606 . . . . . . 7 (𝜑 → (((𝐴𝐷𝐾)↑2) + ((𝐴𝐷𝐿)↑2)) ≤ (((𝑆↑2) + 𝐵) + ((𝑆↑2) + 𝐵)))
11474recnd 10028 . . . . . . . 8 (𝜑 → ((𝑆↑2) + 𝐵) ∈ ℂ)
1151142timesd 11235 . . . . . . 7 (𝜑 → (2 · ((𝑆↑2) + 𝐵)) = (((𝑆↑2) + 𝐵) + ((𝑆↑2) + 𝐵)))
116113, 115breqtrrd 4651 . . . . . 6 (𝜑 → (((𝐴𝐷𝐾)↑2) + ((𝐴𝐷𝐿)↑2)) ≤ (2 · ((𝑆↑2) + 𝐵)))
117107, 110readdcld 10029 . . . . . . 7 (𝜑 → (((𝐴𝐷𝐾)↑2) + ((𝐴𝐷𝐿)↑2)) ∈ ℝ)
118 2re 11050 . . . . . . . 8 2 ∈ ℝ
119 remulcl 9981 . . . . . . . 8 ((2 ∈ ℝ ∧ ((𝑆↑2) + 𝐵) ∈ ℝ) → (2 · ((𝑆↑2) + 𝐵)) ∈ ℝ)
120118, 74, 119sylancr 694 . . . . . . 7 (𝜑 → (2 · ((𝑆↑2) + 𝐵)) ∈ ℝ)
121 2pos 11072 . . . . . . . . 9 0 < 2
122118, 121pm3.2i 471 . . . . . . . 8 (2 ∈ ℝ ∧ 0 < 2)
123 lemul2 10836 . . . . . . . 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)))))
124122, 123mp3an3 1410 . . . . . . 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)))))
125117, 120, 124syl2anc 692 . . . . . 6 (𝜑 → ((((𝐴𝐷𝐾)↑2) + ((𝐴𝐷𝐿)↑2)) ≤ (2 · ((𝑆↑2) + 𝐵)) ↔ (2 · (((𝐴𝐷𝐾)↑2) + ((𝐴𝐷𝐿)↑2))) ≤ (2 · (2 · ((𝑆↑2) + 𝐵)))))
126116, 125mpbid 222 . . . . 5 (𝜑 → (2 · (((𝐴𝐷𝐾)↑2) + ((𝐴𝐷𝐿)↑2))) ≤ (2 · (2 · ((𝑆↑2) + 𝐵))))
1273, 4nvmcl 27389 . . . . . . . 8 ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐾𝑋) → (𝐴𝑀𝐾) ∈ 𝑋)
12829, 9, 38, 127syl3anc 1323 . . . . . . 7 (𝜑 → (𝐴𝑀𝐾) ∈ 𝑋)
1293, 4nvmcl 27389 . . . . . . . 8 ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐿𝑋) → (𝐴𝑀𝐿) ∈ 𝑋)
13029, 9, 40, 129syl3anc 1323 . . . . . . 7 (𝜑 → (𝐴𝑀𝐿) ∈ 𝑋)
1313, 48, 4, 5phpar2 27566 . . . . . . 7 ((𝑈 ∈ CPreHilOLD ∧ (𝐴𝑀𝐾) ∈ 𝑋 ∧ (𝐴𝑀𝐿) ∈ 𝑋) → (((𝑁‘((𝐴𝑀𝐾)( +𝑣𝑈)(𝐴𝑀𝐿)))↑2) + ((𝑁‘((𝐴𝑀𝐾)𝑀(𝐴𝑀𝐿)))↑2)) = (2 · (((𝑁‘(𝐴𝑀𝐾))↑2) + ((𝑁‘(𝐴𝑀𝐿))↑2))))
1327, 128, 130, 131syl3anc 1323 . . . . . 6 (𝜑 → (((𝑁‘((𝐴𝑀𝐾)( +𝑣𝑈)(𝐴𝑀𝐿)))↑2) + ((𝑁‘((𝐴𝑀𝐾)𝑀(𝐴𝑀𝐿)))↑2)) = (2 · (((𝑁‘(𝐴𝑀𝐾))↑2) + ((𝑁‘(𝐴𝑀𝐿))↑2))))
133 2cn 11051 . . . . . . . . . 10 2 ∈ ℂ
13468recnd 10028 . . . . . . . . . 10 (𝜑 → (𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)))) ∈ ℂ)
135 sqmul 12882 . . . . . . . . . 10 ((2 ∈ ℂ ∧ (𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)))) ∈ ℂ) → ((2 · (𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)))))↑2) = ((2↑2) · ((𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))↑2)))
136133, 134, 135sylancr 694 . . . . . . . . 9 (𝜑 → ((2 · (𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)))))↑2) = ((2↑2) · ((𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))↑2)))
137 sq2 12916 . . . . . . . . . 10 (2↑2) = 4
138137oveq1i 6625 . . . . . . . . 9 ((2↑2) · ((𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))↑2)) = (4 · ((𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))↑2))
139136, 138syl6eq 2671 . . . . . . . 8 (𝜑 → ((2 · (𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)))))↑2) = (4 · ((𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))↑2)))
140133a1i 11 . . . . . . . . . . . 12 (𝜑 → 2 ∈ ℂ)
1413, 57, 5nvs 27406 . . . . . . . . . . . 12 ((𝑈 ∈ NrmCVec ∧ 2 ∈ ℂ ∧ (𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))) ∈ 𝑋) → (𝑁‘(2( ·𝑠OLD𝑈)(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))) = ((abs‘2) · (𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))))
14229, 140, 66, 141syl3anc 1323 . . . . . . . . . . 11 (𝜑 → (𝑁‘(2( ·𝑠OLD𝑈)(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))) = ((abs‘2) · (𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))))
143 0le2 11071 . . . . . . . . . . . . 13 0 ≤ 2
144 absid 13986 . . . . . . . . . . . . 13 ((2 ∈ ℝ ∧ 0 ≤ 2) → (abs‘2) = 2)
145118, 143, 144mp2an 707 . . . . . . . . . . . 12 (abs‘2) = 2
146145oveq1i 6625 . . . . . . . . . . 11 ((abs‘2) · (𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))) = (2 · (𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)))))
147142, 146syl6eq 2671 . . . . . . . . . 10 (𝜑 → (𝑁‘(2( ·𝑠OLD𝑈)(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))) = (2 · (𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))))
1483, 4, 57nvmdi 27391 . . . . . . . . . . . . 13 ((𝑈 ∈ NrmCVec ∧ (2 ∈ ℂ ∧ 𝐴𝑋 ∧ ((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)) ∈ 𝑋)) → (2( ·𝑠OLD𝑈)(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)))) = ((2( ·𝑠OLD𝑈)𝐴)𝑀(2( ·𝑠OLD𝑈)((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)))))
14929, 140, 9, 64, 148syl13anc 1325 . . . . . . . . . . . 12 (𝜑 → (2( ·𝑠OLD𝑈)(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)))) = ((2( ·𝑠OLD𝑈)𝐴)𝑀(2( ·𝑠OLD𝑈)((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)))))
1503, 48, 57nv2 27375 . . . . . . . . . . . . . 14 ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋) → (𝐴( +𝑣𝑈)𝐴) = (2( ·𝑠OLD𝑈)𝐴))
15129, 9, 150syl2anc 692 . . . . . . . . . . . . 13 (𝜑 → (𝐴( +𝑣𝑈)𝐴) = (2( ·𝑠OLD𝑈)𝐴))
152 2ne0 11073 . . . . . . . . . . . . . . . . 17 2 ≠ 0
153133, 152recidi 10716 . . . . . . . . . . . . . . . 16 (2 · (1 / 2)) = 1
154153oveq1i 6625 . . . . . . . . . . . . . . 15 ((2 · (1 / 2))( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)) = (1( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))
1553, 48nvgcl 27363 . . . . . . . . . . . . . . . . 17 ((𝑈 ∈ NrmCVec ∧ 𝐾𝑋𝐿𝑋) → (𝐾( +𝑣𝑈)𝐿) ∈ 𝑋)
15629, 38, 40, 155syl3anc 1323 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐾( +𝑣𝑈)𝐿) ∈ 𝑋)
1573, 57nvsid 27370 . . . . . . . . . . . . . . . 16 ((𝑈 ∈ NrmCVec ∧ (𝐾( +𝑣𝑈)𝐿) ∈ 𝑋) → (1( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)) = (𝐾( +𝑣𝑈)𝐿))
15829, 156, 157syl2anc 692 . . . . . . . . . . . . . . 15 (𝜑 → (1( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)) = (𝐾( +𝑣𝑈)𝐿))
159154, 158syl5eq 2667 . . . . . . . . . . . . . 14 (𝜑 → ((2 · (1 / 2))( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)) = (𝐾( +𝑣𝑈)𝐿))
1603, 57nvsass 27371 . . . . . . . . . . . . . . 15 ((𝑈 ∈ NrmCVec ∧ (2 ∈ ℂ ∧ (1 / 2) ∈ ℂ ∧ (𝐾( +𝑣𝑈)𝐿) ∈ 𝑋)) → ((2 · (1 / 2))( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)) = (2( ·𝑠OLD𝑈)((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))
16129, 140, 47, 156, 160syl13anc 1325 . . . . . . . . . . . . . 14 (𝜑 → ((2 · (1 / 2))( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)) = (2( ·𝑠OLD𝑈)((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))
162159, 161eqtr3d 2657 . . . . . . . . . . . . 13 (𝜑 → (𝐾( +𝑣𝑈)𝐿) = (2( ·𝑠OLD𝑈)((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))
163151, 162oveq12d 6633 . . . . . . . . . . . 12 (𝜑 → ((𝐴( +𝑣𝑈)𝐴)𝑀(𝐾( +𝑣𝑈)𝐿)) = ((2( ·𝑠OLD𝑈)𝐴)𝑀(2( ·𝑠OLD𝑈)((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)))))
1643, 48, 4nvaddsub4 27400 . . . . . . . . . . . . 13 ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐴𝑋) ∧ (𝐾𝑋𝐿𝑋)) → ((𝐴( +𝑣𝑈)𝐴)𝑀(𝐾( +𝑣𝑈)𝐿)) = ((𝐴𝑀𝐾)( +𝑣𝑈)(𝐴𝑀𝐿)))
16529, 9, 9, 38, 40, 164syl122anc 1332 . . . . . . . . . . . 12 (𝜑 → ((𝐴( +𝑣𝑈)𝐴)𝑀(𝐾( +𝑣𝑈)𝐿)) = ((𝐴𝑀𝐾)( +𝑣𝑈)(𝐴𝑀𝐿)))
166149, 163, 1653eqtr2d 2661 . . . . . . . . . . 11 (𝜑 → (2( ·𝑠OLD𝑈)(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)))) = ((𝐴𝑀𝐾)( +𝑣𝑈)(𝐴𝑀𝐿)))
167166fveq2d 6162 . . . . . . . . . 10 (𝜑 → (𝑁‘(2( ·𝑠OLD𝑈)(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))) = (𝑁‘((𝐴𝑀𝐾)( +𝑣𝑈)(𝐴𝑀𝐿))))
168147, 167eqtr3d 2657 . . . . . . . . 9 (𝜑 → (2 · (𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))) = (𝑁‘((𝐴𝑀𝐾)( +𝑣𝑈)(𝐴𝑀𝐿))))
169168oveq1d 6630 . . . . . . . 8 (𝜑 → ((2 · (𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)))))↑2) = ((𝑁‘((𝐴𝑀𝐾)( +𝑣𝑈)(𝐴𝑀𝐿)))↑2))
170139, 169eqtr3d 2657 . . . . . . 7 (𝜑 → (4 · ((𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))↑2)) = ((𝑁‘((𝐴𝑀𝐾)( +𝑣𝑈)(𝐴𝑀𝐿)))↑2))
1713, 4, 5, 10imsdval 27429 . . . . . . . . . 10 ((𝑈 ∈ NrmCVec ∧ 𝐿𝑋𝐾𝑋) → (𝐿𝐷𝐾) = (𝑁‘(𝐿𝑀𝐾)))
17229, 40, 38, 171syl3anc 1323 . . . . . . . . 9 (𝜑 → (𝐿𝐷𝐾) = (𝑁‘(𝐿𝑀𝐾)))
173 metsym 22095 . . . . . . . . . 10 ((𝐷 ∈ (Met‘𝑋) ∧ 𝐾𝑋𝐿𝑋) → (𝐾𝐷𝐿) = (𝐿𝐷𝐾))
17431, 38, 40, 173syl3anc 1323 . . . . . . . . 9 (𝜑 → (𝐾𝐷𝐿) = (𝐿𝐷𝐾))
1753, 4nvnnncan1 27390 . . . . . . . . . . 11 ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐾𝑋𝐿𝑋)) → ((𝐴𝑀𝐾)𝑀(𝐴𝑀𝐿)) = (𝐿𝑀𝐾))
17629, 9, 38, 40, 175syl13anc 1325 . . . . . . . . . 10 (𝜑 → ((𝐴𝑀𝐾)𝑀(𝐴𝑀𝐿)) = (𝐿𝑀𝐾))
177176fveq2d 6162 . . . . . . . . 9 (𝜑 → (𝑁‘((𝐴𝑀𝐾)𝑀(𝐴𝑀𝐿))) = (𝑁‘(𝐿𝑀𝐾)))
178172, 174, 1773eqtr4d 2665 . . . . . . . 8 (𝜑 → (𝐾𝐷𝐿) = (𝑁‘((𝐴𝑀𝐾)𝑀(𝐴𝑀𝐿))))
179178oveq1d 6630 . . . . . . 7 (𝜑 → ((𝐾𝐷𝐿)↑2) = ((𝑁‘((𝐴𝑀𝐾)𝑀(𝐴𝑀𝐿)))↑2))
180170, 179oveq12d 6633 . . . . . 6 (𝜑 → ((4 · ((𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))↑2)) + ((𝐾𝐷𝐿)↑2)) = (((𝑁‘((𝐴𝑀𝐾)( +𝑣𝑈)(𝐴𝑀𝐿)))↑2) + ((𝑁‘((𝐴𝑀𝐾)𝑀(𝐴𝑀𝐿)))↑2)))
1813, 4, 5, 10imsdval 27429 . . . . . . . . . 10 ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐾𝑋) → (𝐴𝐷𝐾) = (𝑁‘(𝐴𝑀𝐾)))
18229, 9, 38, 181syl3anc 1323 . . . . . . . . 9 (𝜑 → (𝐴𝐷𝐾) = (𝑁‘(𝐴𝑀𝐾)))
183182oveq1d 6630 . . . . . . . 8 (𝜑 → ((𝐴𝐷𝐾)↑2) = ((𝑁‘(𝐴𝑀𝐾))↑2))
1843, 4, 5, 10imsdval 27429 . . . . . . . . . 10 ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐿𝑋) → (𝐴𝐷𝐿) = (𝑁‘(𝐴𝑀𝐿)))
18529, 9, 40, 184syl3anc 1323 . . . . . . . . 9 (𝜑 → (𝐴𝐷𝐿) = (𝑁‘(𝐴𝑀𝐿)))
186185oveq1d 6630 . . . . . . . 8 (𝜑 → ((𝐴𝐷𝐿)↑2) = ((𝑁‘(𝐴𝑀𝐿))↑2))
187183, 186oveq12d 6633 . . . . . . 7 (𝜑 → (((𝐴𝐷𝐾)↑2) + ((𝐴𝐷𝐿)↑2)) = (((𝑁‘(𝐴𝑀𝐾))↑2) + ((𝑁‘(𝐴𝑀𝐿))↑2)))
188187oveq2d 6631 . . . . . 6 (𝜑 → (2 · (((𝐴𝐷𝐾)↑2) + ((𝐴𝐷𝐿)↑2))) = (2 · (((𝑁‘(𝐴𝑀𝐾))↑2) + ((𝑁‘(𝐴𝑀𝐿))↑2))))
189132, 180, 1883eqtr4d 2665 . . . . 5 (𝜑 → ((4 · ((𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))↑2)) + ((𝐾𝐷𝐿)↑2)) = (2 · (((𝐴𝐷𝐾)↑2) + ((𝐴𝐷𝐿)↑2))))
190 2t2e4 11137 . . . . . . 7 (2 · 2) = 4
191190oveq1i 6625 . . . . . 6 ((2 · 2) · ((𝑆↑2) + 𝐵)) = (4 · ((𝑆↑2) + 𝐵))
192140, 140, 114mulassd 10023 . . . . . 6 (𝜑 → ((2 · 2) · ((𝑆↑2) + 𝐵)) = (2 · (2 · ((𝑆↑2) + 𝐵))))
193191, 192syl5eqr 2669 . . . . 5 (𝜑 → (4 · ((𝑆↑2) + 𝐵)) = (2 · (2 · ((𝑆↑2) + 𝐵))))
194126, 189, 1933brtr4d 4655 . . . 4 (𝜑 → ((4 · ((𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))↑2)) + ((𝐾𝐷𝐿)↑2)) ≤ (4 · ((𝑆↑2) + 𝐵)))
19544, 72, 76, 104, 194letrd 10154 . . 3 (𝜑 → ((4 · (𝑆↑2)) + ((𝐾𝐷𝐿)↑2)) ≤ (4 · ((𝑆↑2) + 𝐵)))
196 4cn 11058 . . . . 5 4 ∈ ℂ
197196a1i 11 . . . 4 (𝜑 → 4 ∈ ℂ)
19825recnd 10028 . . . 4 (𝜑 → (𝑆↑2) ∈ ℂ)
19973recnd 10028 . . . 4 (𝜑𝐵 ∈ ℂ)
200197, 198, 199adddid 10024 . . 3 (𝜑 → (4 · ((𝑆↑2) + 𝐵)) = ((4 · (𝑆↑2)) + (4 · 𝐵)))
201195, 200breqtrd 4649 . 2 (𝜑 → ((4 · (𝑆↑2)) + ((𝐾𝐷𝐿)↑2)) ≤ ((4 · (𝑆↑2)) + (4 · 𝐵)))
202 remulcl 9981 . . . 4 ((4 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (4 · 𝐵) ∈ ℝ)
2031, 73, 202sylancr 694 . . 3 (𝜑 → (4 · 𝐵) ∈ ℝ)
20443, 203, 27leadd2d 10582 . 2 (𝜑 → (((𝐾𝐷𝐿)↑2) ≤ (4 · 𝐵) ↔ ((4 · (𝑆↑2)) + ((𝐾𝐷𝐿)↑2)) ≤ ((4 · (𝑆↑2)) + (4 · 𝐵))))
205201, 204mpbird 247 1 (𝜑 → ((𝐾𝐷𝐿)↑2) ≤ (4 · 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wne 2790  wral 2908  wrex 2909  cin 3559  wss 3560  c0 3897   class class class wbr 4623  cmpt 4683  ran crn 5085  cfv 5857  (class class class)co 6615  infcinf 8307  cc 9894  cr 9895  0cc0 9896  1c1 9897   + caddc 9899   · cmul 9901   < clt 10034  cle 10035   / cdiv 10644  2c2 11030  4c4 11032  cexp 12816  abscabs 13924  Metcme 19672  MetOpencmopn 19676  NrmCVeccnv 27327   +𝑣 cpv 27328  BaseSetcba 27329   ·𝑠OLD cns 27330  𝑣 cnsb 27332  normCVcnmcv 27333  IndMetcims 27334  SubSpcss 27464  CPreHilOLDccphlo 27555  CBanccbn 27606
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973  ax-pre-sup 9974  ax-addf 9975  ax-mulf 9976
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-er 7702  df-map 7819  df-en 7916  df-dom 7917  df-sdom 7918  df-sup 8308  df-inf 8309  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-div 10645  df-nn 10981  df-2 11039  df-3 11040  df-4 11041  df-n0 11253  df-z 11338  df-uz 11648  df-rp 11793  df-xadd 11907  df-seq 12758  df-exp 12817  df-cj 13789  df-re 13790  df-im 13791  df-sqrt 13925  df-abs 13926  df-xmet 19679  df-met 19680  df-grpo 27235  df-gid 27236  df-ginv 27237  df-gdiv 27238  df-ablo 27287  df-vc 27302  df-nv 27335  df-va 27338  df-ba 27339  df-sm 27340  df-0v 27341  df-vs 27342  df-nmcv 27343  df-ims 27344  df-ssp 27465  df-ph 27556  df-cbn 27607
This theorem is referenced by:  minvecolem3  27620  minvecolem7  27627
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