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Theorem minvecolem2 27115
Description: Lemma for minveco 27124. 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 10974 . . . . . 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 27114 . . . . . . . . . 10 (𝜑 → (𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∀𝑤𝑅 0 ≤ 𝑤))
1413simp1d 1066 . . . . . . . . 9 (𝜑𝑅 ⊆ ℝ)
1513simp2d 1067 . . . . . . . . 9 (𝜑𝑅 ≠ ∅)
16 0re 9919 . . . . . . . . . 10 0 ∈ ℝ
1713simp3d 1068 . . . . . . . . . 10 (𝜑 → ∀𝑤𝑅 0 ≤ 𝑤)
18 breq1 4586 . . . . . . . . . . . 12 (𝑥 = 0 → (𝑥𝑤 ↔ 0 ≤ 𝑤))
1918ralbidv 2969 . . . . . . . . . . 11 (𝑥 = 0 → (∀𝑤𝑅 𝑥𝑤 ↔ ∀𝑤𝑅 0 ≤ 𝑤))
2019rspcev 3282 . . . . . . . . . 10 ((0 ∈ ℝ ∧ ∀𝑤𝑅 0 ≤ 𝑤) → ∃𝑥 ∈ ℝ ∀𝑤𝑅 𝑥𝑤)
2116, 17, 20sylancr 694 . . . . . . . . 9 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑤𝑅 𝑥𝑤)
22 infrecl 10882 . . . . . . . . 9 ((𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑤𝑅 𝑥𝑤) → inf(𝑅, ℝ, < ) ∈ ℝ)
2314, 15, 21, 22syl3anc 1318 . . . . . . . 8 (𝜑 → inf(𝑅, ℝ, < ) ∈ ℝ)
242, 23syl5eqel 2692 . . . . . . 7 (𝜑𝑆 ∈ ℝ)
2524resqcld 12897 . . . . . 6 (𝜑 → (𝑆↑2) ∈ ℝ)
26 remulcl 9900 . . . . . 6 ((4 ∈ ℝ ∧ (𝑆↑2) ∈ ℝ) → (4 · (𝑆↑2)) ∈ ℝ)
271, 25, 26sylancr 694 . . . . 5 (𝜑 → (4 · (𝑆↑2)) ∈ ℝ)
28 phnv 27053 . . . . . . . . 9 (𝑈 ∈ CPreHilOLD𝑈 ∈ NrmCVec)
297, 28syl 17 . . . . . . . 8 (𝜑𝑈 ∈ NrmCVec)
303, 10imsmet 26930 . . . . . . . 8 (𝑈 ∈ NrmCVec → 𝐷 ∈ (Met‘𝑋))
3129, 30syl 17 . . . . . . 7 (𝜑𝐷 ∈ (Met‘𝑋))
32 inss1 3795 . . . . . . . . . 10 ((SubSp‘𝑈) ∩ CBan) ⊆ (SubSp‘𝑈)
3332, 8sseldi 3566 . . . . . . . . 9 (𝜑𝑊 ∈ (SubSp‘𝑈))
34 eqid 2610 . . . . . . . . . 10 (SubSp‘𝑈) = (SubSp‘𝑈)
353, 6, 34sspba 26966 . . . . . . . . 9 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ (SubSp‘𝑈)) → 𝑌𝑋)
3629, 33, 35syl2anc 691 . . . . . . . 8 (𝜑𝑌𝑋)
37 minvecolem2.3 . . . . . . . 8 (𝜑𝐾𝑌)
3836, 37sseldd 3569 . . . . . . 7 (𝜑𝐾𝑋)
39 minvecolem2.4 . . . . . . . 8 (𝜑𝐿𝑌)
4036, 39sseldd 3569 . . . . . . 7 (𝜑𝐿𝑋)
41 metcl 21947 . . . . . . 7 ((𝐷 ∈ (Met‘𝑋) ∧ 𝐾𝑋𝐿𝑋) → (𝐾𝐷𝐿) ∈ ℝ)
4231, 38, 40, 41syl3anc 1318 . . . . . 6 (𝜑 → (𝐾𝐷𝐿) ∈ ℝ)
4342resqcld 12897 . . . . 5 (𝜑 → ((𝐾𝐷𝐿)↑2) ∈ ℝ)
4427, 43readdcld 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)
4745, 46mp1i 13 . . . . . . . . . . . 12 (𝜑 → (1 / 2) ∈ ℂ)
48 eqid 2610 . . . . . . . . . . . . . . 15 ( +𝑣𝑈) = ( +𝑣𝑈)
49 eqid 2610 . . . . . . . . . . . . . . 15 ( +𝑣𝑊) = ( +𝑣𝑊)
506, 48, 49, 34sspgval 26968 . . . . . . . . . . . . . 14 (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ (SubSp‘𝑈)) ∧ (𝐾𝑌𝐿𝑌)) → (𝐾( +𝑣𝑊)𝐿) = (𝐾( +𝑣𝑈)𝐿))
5129, 33, 37, 39, 50syl22anc 1319 . . . . . . . . . . . . 13 (𝜑 → (𝐾( +𝑣𝑊)𝐿) = (𝐾( +𝑣𝑈)𝐿))
5234sspnv 26965 . . . . . . . . . . . . . . 15 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ (SubSp‘𝑈)) → 𝑊 ∈ NrmCVec)
5329, 33, 52syl2anc 691 . . . . . . . . . . . . . 14 (𝜑𝑊 ∈ NrmCVec)
546, 49nvgcl 26859 . . . . . . . . . . . . . 14 ((𝑊 ∈ NrmCVec ∧ 𝐾𝑌𝐿𝑌) → (𝐾( +𝑣𝑊)𝐿) ∈ 𝑌)
5553, 37, 39, 54syl3anc 1318 . . . . . . . . . . . . 13 (𝜑 → (𝐾( +𝑣𝑊)𝐿) ∈ 𝑌)
5651, 55eqeltrrd 2689 . . . . . . . . . . . 12 (𝜑 → (𝐾( +𝑣𝑈)𝐿) ∈ 𝑌)
57 eqid 2610 . . . . . . . . . . . . 13 ( ·𝑠OLD𝑈) = ( ·𝑠OLD𝑈)
58 eqid 2610 . . . . . . . . . . . . 13 ( ·𝑠OLD𝑊) = ( ·𝑠OLD𝑊)
596, 57, 58, 34sspsval 26970 . . . . . . . . . . . 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 1319 . . . . . . . . . . 11 (𝜑 → ((1 / 2)( ·𝑠OLD𝑊)(𝐾( +𝑣𝑈)𝐿)) = ((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)))
616, 58nvscl 26865 . . . . . . . . . . . 12 ((𝑊 ∈ NrmCVec ∧ (1 / 2) ∈ ℂ ∧ (𝐾( +𝑣𝑈)𝐿) ∈ 𝑌) → ((1 / 2)( ·𝑠OLD𝑊)(𝐾( +𝑣𝑈)𝐿)) ∈ 𝑌)
6253, 47, 56, 61syl3anc 1318 . . . . . . . . . . 11 (𝜑 → ((1 / 2)( ·𝑠OLD𝑊)(𝐾( +𝑣𝑈)𝐿)) ∈ 𝑌)
6360, 62eqeltrrd 2689 . . . . . . . . . 10 (𝜑 → ((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)) ∈ 𝑌)
6436, 63sseldd 3569 . . . . . . . . 9 (𝜑 → ((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)) ∈ 𝑋)
653, 4nvmcl 26885 . . . . . . . . 9 ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋 ∧ ((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)) ∈ 𝑋) → (𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))) ∈ 𝑋)
6629, 9, 64, 65syl3anc 1318 . . . . . . . 8 (𝜑 → (𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))) ∈ 𝑋)
673, 5nvcl 26900 . . . . . . . 8 ((𝑈 ∈ NrmCVec ∧ (𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))) ∈ 𝑋) → (𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)))) ∈ ℝ)
6829, 66, 67syl2anc 691 . . . . . . 7 (𝜑 → (𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)))) ∈ ℝ)
6968resqcld 12897 . . . . . 6 (𝜑 → ((𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))↑2) ∈ ℝ)
70 remulcl 9900 . . . . . 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 9948 . . . 4 (𝜑 → ((4 · ((𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))↑2)) + ((𝐾𝐷𝐿)↑2)) ∈ ℝ)
73 minvecolem2.1 . . . . . 6 (𝜑𝐵 ∈ ℝ)
7425, 73readdcld 9948 . . . . 5 (𝜑 → ((𝑆↑2) + 𝐵) ∈ ℝ)
75 remulcl 9900 . . . . 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 10884 . . . . . . . . . 10 (((𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑤𝑅 𝑥𝑤) ∧ 0 ∈ ℝ) → (0 ≤ inf(𝑅, ℝ, < ) ↔ ∀𝑤𝑅 0 ≤ 𝑤))
7914, 15, 21, 77, 78syl31anc 1321 . . . . . . . . 9 (𝜑 → (0 ≤ inf(𝑅, ℝ, < ) ↔ ∀𝑤𝑅 0 ≤ 𝑤))
8017, 79mpbird 246 . . . . . . . 8 (𝜑 → 0 ≤ inf(𝑅, ℝ, < ))
8180, 2syl6breqr 4625 . . . . . . 7 (𝜑 → 0 ≤ 𝑆)
82 eqid 2610 . . . . . . . . . . . 12 (𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)))) = (𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))
83 oveq2 6557 . . . . . . . . . . . . . . 15 (𝑦 = ((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)) → (𝐴𝑀𝑦) = (𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))
8483fveq2d 6107 . . . . . . . . . . . . . 14 (𝑦 = ((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)) → (𝑁‘(𝐴𝑀𝑦)) = (𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)))))
8584eqeq2d 2620 . . . . . . . . . . . . 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 3282 . . . . . . . . . . . 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 2610 . . . . . . . . . . . 12 (𝑦𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) = (𝑦𝑌 ↦ (𝑁‘(𝐴𝑀𝑦)))
89 fvex 6113 . . . . . . . . . . . 12 (𝑁‘(𝐴𝑀𝑦)) ∈ V
9088, 89elrnmpti 5297 . . . . . . . . . . 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 223 . . . . . . . . . 10 (𝜑 → (𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)))) ∈ ran (𝑦𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))))
9291, 12syl6eleqr 2699 . . . . . . . . 9 (𝜑 → (𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)))) ∈ 𝑅)
93 infrelb 10885 . . . . . . . . 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 1318 . . . . . . . 8 (𝜑 → inf(𝑅, ℝ, < ) ≤ (𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)))))
952, 94syl5eqbr 4618 . . . . . . 7 (𝜑𝑆 ≤ (𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)))))
96 le2sq2 12801 . . . . . . 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 1319 . . . . . 6 (𝜑 → (𝑆↑2) ≤ ((𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))↑2))
98 4pos 10993 . . . . . . . . 9 0 < 4
991, 98pm3.2i 470 . . . . . . . 8 (4 ∈ ℝ ∧ 0 < 4)
100 lemul2 10755 . . . . . . . 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 1405 . . . . . . 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 691 . . . . . 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 221 . . . . 5 (𝜑 → (4 · (𝑆↑2)) ≤ (4 · ((𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))↑2)))
10427, 71, 43, 103leadd1dd 10520 . . . 4 (𝜑 → ((4 · (𝑆↑2)) + ((𝐾𝐷𝐿)↑2)) ≤ ((4 · ((𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))↑2)) + ((𝐾𝐷𝐿)↑2)))
105 metcl 21947 . . . . . . . . . 10 ((𝐷 ∈ (Met‘𝑋) ∧ 𝐴𝑋𝐾𝑋) → (𝐴𝐷𝐾) ∈ ℝ)
10631, 9, 38, 105syl3anc 1318 . . . . . . . . 9 (𝜑 → (𝐴𝐷𝐾) ∈ ℝ)
107106resqcld 12897 . . . . . . . 8 (𝜑 → ((𝐴𝐷𝐾)↑2) ∈ ℝ)
108 metcl 21947 . . . . . . . . . 10 ((𝐷 ∈ (Met‘𝑋) ∧ 𝐴𝑋𝐿𝑋) → (𝐴𝐷𝐿) ∈ ℝ)
10931, 9, 40, 108syl3anc 1318 . . . . . . . . 9 (𝜑 → (𝐴𝐷𝐿) ∈ ℝ)
110109resqcld 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))
113107, 110, 74, 74, 111, 112le2addd 10525 . . . . . . 7 (𝜑 → (((𝐴𝐷𝐾)↑2) + ((𝐴𝐷𝐿)↑2)) ≤ (((𝑆↑2) + 𝐵) + ((𝑆↑2) + 𝐵)))
11474recnd 9947 . . . . . . . 8 (𝜑 → ((𝑆↑2) + 𝐵) ∈ ℂ)
1151142timesd 11152 . . . . . . 7 (𝜑 → (2 · ((𝑆↑2) + 𝐵)) = (((𝑆↑2) + 𝐵) + ((𝑆↑2) + 𝐵)))
116113, 115breqtrrd 4611 . . . . . 6 (𝜑 → (((𝐴𝐷𝐾)↑2) + ((𝐴𝐷𝐿)↑2)) ≤ (2 · ((𝑆↑2) + 𝐵)))
117107, 110readdcld 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)
120118, 74, 119sylancr 694 . . . . . . 7 (𝜑 → (2 · ((𝑆↑2) + 𝐵)) ∈ ℝ)
121 2pos 10989 . . . . . . . . 9 0 < 2
122118, 121pm3.2i 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)))))
124122, 123mp3an3 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)))))
125117, 120, 124syl2anc 691 . . . . . 6 (𝜑 → ((((𝐴𝐷𝐾)↑2) + ((𝐴𝐷𝐿)↑2)) ≤ (2 · ((𝑆↑2) + 𝐵)) ↔ (2 · (((𝐴𝐷𝐾)↑2) + ((𝐴𝐷𝐿)↑2))) ≤ (2 · (2 · ((𝑆↑2) + 𝐵)))))
126116, 125mpbid 221 . . . . 5 (𝜑 → (2 · (((𝐴𝐷𝐾)↑2) + ((𝐴𝐷𝐿)↑2))) ≤ (2 · (2 · ((𝑆↑2) + 𝐵))))
1273, 4nvmcl 26885 . . . . . . . 8 ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐾𝑋) → (𝐴𝑀𝐾) ∈ 𝑋)
12829, 9, 38, 127syl3anc 1318 . . . . . . 7 (𝜑 → (𝐴𝑀𝐾) ∈ 𝑋)
1293, 4nvmcl 26885 . . . . . . . 8 ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐿𝑋) → (𝐴𝑀𝐿) ∈ 𝑋)
13029, 9, 40, 129syl3anc 1318 . . . . . . 7 (𝜑 → (𝐴𝑀𝐿) ∈ 𝑋)
1313, 48, 4, 5phpar2 27062 . . . . . . 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 1318 . . . . . 6 (𝜑 → (((𝑁‘((𝐴𝑀𝐾)( +𝑣𝑈)(𝐴𝑀𝐿)))↑2) + ((𝑁‘((𝐴𝑀𝐾)𝑀(𝐴𝑀𝐿)))↑2)) = (2 · (((𝑁‘(𝐴𝑀𝐾))↑2) + ((𝑁‘(𝐴𝑀𝐿))↑2))))
133 2cn 10968 . . . . . . . . . 10 2 ∈ ℂ
13468recnd 9947 . . . . . . . . . 10 (𝜑 → (𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)))) ∈ ℂ)
135 sqmul 12788 . . . . . . . . . 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 12822 . . . . . . . . . 10 (2↑2) = 4
138137oveq1i 6559 . . . . . . . . 9 ((2↑2) · ((𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))↑2)) = (4 · ((𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))↑2))
139136, 138syl6eq 2660 . . . . . . . 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 26902 . . . . . . . . . . . 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 1318 . . . . . . . . . . 11 (𝜑 → (𝑁‘(2( ·𝑠OLD𝑈)(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))) = ((abs‘2) · (𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))))
143 0le2 10988 . . . . . . . . . . . . 13 0 ≤ 2
144 absid 13884 . . . . . . . . . . . . 13 ((2 ∈ ℝ ∧ 0 ≤ 2) → (abs‘2) = 2)
145118, 143, 144mp2an 704 . . . . . . . . . . . 12 (abs‘2) = 2
146145oveq1i 6559 . . . . . . . . . . 11 ((abs‘2) · (𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))) = (2 · (𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)))))
147142, 146syl6eq 2660 . . . . . . . . . 10 (𝜑 → (𝑁‘(2( ·𝑠OLD𝑈)(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))) = (2 · (𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))))
1483, 4, 57nvmdi 26887 . . . . . . . . . . . . 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 1320 . . . . . . . . . . . 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 26871 . . . . . . . . . . . . . 14 ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋) → (𝐴( +𝑣𝑈)𝐴) = (2( ·𝑠OLD𝑈)𝐴))
15129, 9, 150syl2anc 691 . . . . . . . . . . . . 13 (𝜑 → (𝐴( +𝑣𝑈)𝐴) = (2( ·𝑠OLD𝑈)𝐴))
152 2ne0 10990 . . . . . . . . . . . . . . . . 17 2 ≠ 0
153133, 152recidi 10635 . . . . . . . . . . . . . . . 16 (2 · (1 / 2)) = 1
154153oveq1i 6559 . . . . . . . . . . . . . . 15 ((2 · (1 / 2))( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)) = (1( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))
1553, 48nvgcl 26859 . . . . . . . . . . . . . . . . 17 ((𝑈 ∈ NrmCVec ∧ 𝐾𝑋𝐿𝑋) → (𝐾( +𝑣𝑈)𝐿) ∈ 𝑋)
15629, 38, 40, 155syl3anc 1318 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐾( +𝑣𝑈)𝐿) ∈ 𝑋)
1573, 57nvsid 26866 . . . . . . . . . . . . . . . 16 ((𝑈 ∈ NrmCVec ∧ (𝐾( +𝑣𝑈)𝐿) ∈ 𝑋) → (1( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)) = (𝐾( +𝑣𝑈)𝐿))
15829, 156, 157syl2anc 691 . . . . . . . . . . . . . . 15 (𝜑 → (1( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)) = (𝐾( +𝑣𝑈)𝐿))
159154, 158syl5eq 2656 . . . . . . . . . . . . . 14 (𝜑 → ((2 · (1 / 2))( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)) = (𝐾( +𝑣𝑈)𝐿))
1603, 57nvsass 26867 . . . . . . . . . . . . . . 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 1320 . . . . . . . . . . . . . 14 (𝜑 → ((2 · (1 / 2))( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)) = (2( ·𝑠OLD𝑈)((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))
162159, 161eqtr3d 2646 . . . . . . . . . . . . 13 (𝜑 → (𝐾( +𝑣𝑈)𝐿) = (2( ·𝑠OLD𝑈)((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))
163151, 162oveq12d 6567 . . . . . . . . . . . 12 (𝜑 → ((𝐴( +𝑣𝑈)𝐴)𝑀(𝐾( +𝑣𝑈)𝐿)) = ((2( ·𝑠OLD𝑈)𝐴)𝑀(2( ·𝑠OLD𝑈)((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)))))
1643, 48, 4nvaddsub4 26896 . . . . . . . . . . . . 13 ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐴𝑋) ∧ (𝐾𝑋𝐿𝑋)) → ((𝐴( +𝑣𝑈)𝐴)𝑀(𝐾( +𝑣𝑈)𝐿)) = ((𝐴𝑀𝐾)( +𝑣𝑈)(𝐴𝑀𝐿)))
16529, 9, 9, 38, 40, 164syl122anc 1327 . . . . . . . . . . . 12 (𝜑 → ((𝐴( +𝑣𝑈)𝐴)𝑀(𝐾( +𝑣𝑈)𝐿)) = ((𝐴𝑀𝐾)( +𝑣𝑈)(𝐴𝑀𝐿)))
166149, 163, 1653eqtr2d 2650 . . . . . . . . . . 11 (𝜑 → (2( ·𝑠OLD𝑈)(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)))) = ((𝐴𝑀𝐾)( +𝑣𝑈)(𝐴𝑀𝐿)))
167166fveq2d 6107 . . . . . . . . . 10 (𝜑 → (𝑁‘(2( ·𝑠OLD𝑈)(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))) = (𝑁‘((𝐴𝑀𝐾)( +𝑣𝑈)(𝐴𝑀𝐿))))
168147, 167eqtr3d 2646 . . . . . . . . 9 (𝜑 → (2 · (𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))) = (𝑁‘((𝐴𝑀𝐾)( +𝑣𝑈)(𝐴𝑀𝐿))))
169168oveq1d 6564 . . . . . . . 8 (𝜑 → ((2 · (𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)))))↑2) = ((𝑁‘((𝐴𝑀𝐾)( +𝑣𝑈)(𝐴𝑀𝐿)))↑2))
170139, 169eqtr3d 2646 . . . . . . 7 (𝜑 → (4 · ((𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))↑2)) = ((𝑁‘((𝐴𝑀𝐾)( +𝑣𝑈)(𝐴𝑀𝐿)))↑2))
1713, 4, 5, 10imsdval 26925 . . . . . . . . . 10 ((𝑈 ∈ NrmCVec ∧ 𝐿𝑋𝐾𝑋) → (𝐿𝐷𝐾) = (𝑁‘(𝐿𝑀𝐾)))
17229, 40, 38, 171syl3anc 1318 . . . . . . . . 9 (𝜑 → (𝐿𝐷𝐾) = (𝑁‘(𝐿𝑀𝐾)))
173 metsym 21965 . . . . . . . . . 10 ((𝐷 ∈ (Met‘𝑋) ∧ 𝐾𝑋𝐿𝑋) → (𝐾𝐷𝐿) = (𝐿𝐷𝐾))
17431, 38, 40, 173syl3anc 1318 . . . . . . . . 9 (𝜑 → (𝐾𝐷𝐿) = (𝐿𝐷𝐾))
1753, 4nvnnncan1 26886 . . . . . . . . . . 11 ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐾𝑋𝐿𝑋)) → ((𝐴𝑀𝐾)𝑀(𝐴𝑀𝐿)) = (𝐿𝑀𝐾))
17629, 9, 38, 40, 175syl13anc 1320 . . . . . . . . . 10 (𝜑 → ((𝐴𝑀𝐾)𝑀(𝐴𝑀𝐿)) = (𝐿𝑀𝐾))
177176fveq2d 6107 . . . . . . . . 9 (𝜑 → (𝑁‘((𝐴𝑀𝐾)𝑀(𝐴𝑀𝐿))) = (𝑁‘(𝐿𝑀𝐾)))
178172, 174, 1773eqtr4d 2654 . . . . . . . 8 (𝜑 → (𝐾𝐷𝐿) = (𝑁‘((𝐴𝑀𝐾)𝑀(𝐴𝑀𝐿))))
179178oveq1d 6564 . . . . . . 7 (𝜑 → ((𝐾𝐷𝐿)↑2) = ((𝑁‘((𝐴𝑀𝐾)𝑀(𝐴𝑀𝐿)))↑2))
180170, 179oveq12d 6567 . . . . . 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 26925 . . . . . . . . . 10 ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐾𝑋) → (𝐴𝐷𝐾) = (𝑁‘(𝐴𝑀𝐾)))
18229, 9, 38, 181syl3anc 1318 . . . . . . . . 9 (𝜑 → (𝐴𝐷𝐾) = (𝑁‘(𝐴𝑀𝐾)))
183182oveq1d 6564 . . . . . . . 8 (𝜑 → ((𝐴𝐷𝐾)↑2) = ((𝑁‘(𝐴𝑀𝐾))↑2))
1843, 4, 5, 10imsdval 26925 . . . . . . . . . 10 ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐿𝑋) → (𝐴𝐷𝐿) = (𝑁‘(𝐴𝑀𝐿)))
18529, 9, 40, 184syl3anc 1318 . . . . . . . . 9 (𝜑 → (𝐴𝐷𝐿) = (𝑁‘(𝐴𝑀𝐿)))
186185oveq1d 6564 . . . . . . . 8 (𝜑 → ((𝐴𝐷𝐿)↑2) = ((𝑁‘(𝐴𝑀𝐿))↑2))
187183, 186oveq12d 6567 . . . . . . 7 (𝜑 → (((𝐴𝐷𝐾)↑2) + ((𝐴𝐷𝐿)↑2)) = (((𝑁‘(𝐴𝑀𝐾))↑2) + ((𝑁‘(𝐴𝑀𝐿))↑2)))
188187oveq2d 6565 . . . . . 6 (𝜑 → (2 · (((𝐴𝐷𝐾)↑2) + ((𝐴𝐷𝐿)↑2))) = (2 · (((𝑁‘(𝐴𝑀𝐾))↑2) + ((𝑁‘(𝐴𝑀𝐿))↑2))))
189132, 180, 1883eqtr4d 2654 . . . . 5 (𝜑 → ((4 · ((𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))↑2)) + ((𝐾𝐷𝐿)↑2)) = (2 · (((𝐴𝐷𝐾)↑2) + ((𝐴𝐷𝐿)↑2))))
190 2t2e4 11054 . . . . . . 7 (2 · 2) = 4
191190oveq1i 6559 . . . . . 6 ((2 · 2) · ((𝑆↑2) + 𝐵)) = (4 · ((𝑆↑2) + 𝐵))
192140, 140, 114mulassd 9942 . . . . . 6 (𝜑 → ((2 · 2) · ((𝑆↑2) + 𝐵)) = (2 · (2 · ((𝑆↑2) + 𝐵))))
193191, 192syl5eqr 2658 . . . . 5 (𝜑 → (4 · ((𝑆↑2) + 𝐵)) = (2 · (2 · ((𝑆↑2) + 𝐵))))
194126, 189, 1933brtr4d 4615 . . . 4 (𝜑 → ((4 · ((𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))↑2)) + ((𝐾𝐷𝐿)↑2)) ≤ (4 · ((𝑆↑2) + 𝐵)))
19544, 72, 76, 104, 194letrd 10073 . . 3 (𝜑 → ((4 · (𝑆↑2)) + ((𝐾𝐷𝐿)↑2)) ≤ (4 · ((𝑆↑2) + 𝐵)))
196 4cn 10975 . . . . 5 4 ∈ ℂ
197196a1i 11 . . . 4 (𝜑 → 4 ∈ ℂ)
19825recnd 9947 . . . 4 (𝜑 → (𝑆↑2) ∈ ℂ)
19973recnd 9947 . . . 4 (𝜑𝐵 ∈ ℂ)
200197, 198, 199adddid 9943 . . 3 (𝜑 → (4 · ((𝑆↑2) + 𝐵)) = ((4 · (𝑆↑2)) + (4 · 𝐵)))
201195, 200breqtrd 4609 . 2 (𝜑 → ((4 · (𝑆↑2)) + ((𝐾𝐷𝐿)↑2)) ≤ ((4 · (𝑆↑2)) + (4 · 𝐵)))
202 remulcl 9900 . . . 4 ((4 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (4 · 𝐵) ∈ ℝ)
2031, 73, 202sylancr 694 . . 3 (𝜑 → (4 · 𝐵) ∈ ℝ)
20443, 203, 27leadd2d 10501 . 2 (𝜑 → (((𝐾𝐷𝐿)↑2) ≤ (4 · 𝐵) ↔ ((4 · (𝑆↑2)) + ((𝐾𝐷𝐿)↑2)) ≤ ((4 · (𝑆↑2)) + (4 · 𝐵))))
205201, 204mpbird 246 1 (𝜑 → ((𝐾𝐷𝐿)↑2) ≤ (4 · 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  wne 2780  wral 2896  wrex 2897  cin 3539  wss 3540  c0 3874   class class class wbr 4583  cmpt 4643  ran crn 5039  cfv 5804  (class class class)co 6549  infcinf 8230  cc 9813  cr 9814  0cc0 9815  1c1 9816   + caddc 9818   · cmul 9820   < clt 9953  cle 9954   / cdiv 10563  2c2 10947  4c4 10949  cexp 12722  abscabs 13822  Metcme 19553  MetOpencmopn 19557  NrmCVeccnv 26823   +𝑣 cpv 26824  BaseSetcba 26825   ·𝑠OLD cns 26826  𝑣 cnsb 26828  normCVcnmcv 26829  IndMetcims 26830  SubSpcss 26960  CPreHilOLDccphlo 27051  CBanccbn 27102
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893  ax-addf 9894  ax-mulf 9895
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-sup 8231  df-inf 8232  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-xadd 11823  df-seq 12664  df-exp 12723  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-xmet 19560  df-met 19561  df-grpo 26731  df-gid 26732  df-ginv 26733  df-gdiv 26734  df-ablo 26783  df-vc 26798  df-nv 26831  df-va 26834  df-ba 26835  df-sm 26836  df-0v 26837  df-vs 26838  df-nmcv 26839  df-ims 26840  df-ssp 26961  df-ph 27052  df-cbn 27103
This theorem is referenced by:  minvecolem3  27116  minvecolem7  27123
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