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Theorem minvecolem2 29817
Description: Lemma for minveco 29826. 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 12237 . . . . . 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 29816 . . . . . . . . . 10 (𝜑 → (𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∀𝑤𝑅 0 ≤ 𝑤))
1413simp1d 1142 . . . . . . . . 9 (𝜑𝑅 ⊆ ℝ)
1513simp2d 1143 . . . . . . . . 9 (𝜑𝑅 ≠ ∅)
16 0re 11157 . . . . . . . . . 10 0 ∈ ℝ
1713simp3d 1144 . . . . . . . . . 10 (𝜑 → ∀𝑤𝑅 0 ≤ 𝑤)
18 breq1 5108 . . . . . . . . . . . 12 (𝑥 = 0 → (𝑥𝑤 ↔ 0 ≤ 𝑤))
1918ralbidv 3174 . . . . . . . . . . 11 (𝑥 = 0 → (∀𝑤𝑅 𝑥𝑤 ↔ ∀𝑤𝑅 0 ≤ 𝑤))
2019rspcev 3581 . . . . . . . . . 10 ((0 ∈ ℝ ∧ ∀𝑤𝑅 0 ≤ 𝑤) → ∃𝑥 ∈ ℝ ∀𝑤𝑅 𝑥𝑤)
2116, 17, 20sylancr 587 . . . . . . . . 9 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑤𝑅 𝑥𝑤)
22 infrecl 12137 . . . . . . . . 9 ((𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑤𝑅 𝑥𝑤) → inf(𝑅, ℝ, < ) ∈ ℝ)
2314, 15, 21, 22syl3anc 1371 . . . . . . . 8 (𝜑 → inf(𝑅, ℝ, < ) ∈ ℝ)
242, 23eqeltrid 2842 . . . . . . 7 (𝜑𝑆 ∈ ℝ)
2524resqcld 14030 . . . . . 6 (𝜑 → (𝑆↑2) ∈ ℝ)
26 remulcl 11136 . . . . . 6 ((4 ∈ ℝ ∧ (𝑆↑2) ∈ ℝ) → (4 · (𝑆↑2)) ∈ ℝ)
271, 25, 26sylancr 587 . . . . 5 (𝜑 → (4 · (𝑆↑2)) ∈ ℝ)
28 phnv 29756 . . . . . . . . 9 (𝑈 ∈ CPreHilOLD𝑈 ∈ NrmCVec)
297, 28syl 17 . . . . . . . 8 (𝜑𝑈 ∈ NrmCVec)
303, 10imsmet 29633 . . . . . . . 8 (𝑈 ∈ NrmCVec → 𝐷 ∈ (Met‘𝑋))
3129, 30syl 17 . . . . . . 7 (𝜑𝐷 ∈ (Met‘𝑋))
32 inss1 4188 . . . . . . . . . 10 ((SubSp‘𝑈) ∩ CBan) ⊆ (SubSp‘𝑈)
3332, 8sselid 3942 . . . . . . . . 9 (𝜑𝑊 ∈ (SubSp‘𝑈))
34 eqid 2736 . . . . . . . . . 10 (SubSp‘𝑈) = (SubSp‘𝑈)
353, 6, 34sspba 29669 . . . . . . . . 9 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ (SubSp‘𝑈)) → 𝑌𝑋)
3629, 33, 35syl2anc 584 . . . . . . . 8 (𝜑𝑌𝑋)
37 minvecolem2.3 . . . . . . . 8 (𝜑𝐾𝑌)
3836, 37sseldd 3945 . . . . . . 7 (𝜑𝐾𝑋)
39 minvecolem2.4 . . . . . . . 8 (𝜑𝐿𝑌)
4036, 39sseldd 3945 . . . . . . 7 (𝜑𝐿𝑋)
41 metcl 23685 . . . . . . 7 ((𝐷 ∈ (Met‘𝑋) ∧ 𝐾𝑋𝐿𝑋) → (𝐾𝐷𝐿) ∈ ℝ)
4231, 38, 40, 41syl3anc 1371 . . . . . 6 (𝜑 → (𝐾𝐷𝐿) ∈ ℝ)
4342resqcld 14030 . . . . 5 (𝜑 → ((𝐾𝐷𝐿)↑2) ∈ ℝ)
4427, 43readdcld 11184 . . . 4 (𝜑 → ((4 · (𝑆↑2)) + ((𝐾𝐷𝐿)↑2)) ∈ ℝ)
45 ax-1cn 11109 . . . . . . . . . . . . 13 1 ∈ ℂ
46 halfcl 12378 . . . . . . . . . . . . 13 (1 ∈ ℂ → (1 / 2) ∈ ℂ)
4745, 46mp1i 13 . . . . . . . . . . . 12 (𝜑 → (1 / 2) ∈ ℂ)
48 eqid 2736 . . . . . . . . . . . . . . 15 ( +𝑣𝑈) = ( +𝑣𝑈)
49 eqid 2736 . . . . . . . . . . . . . . 15 ( +𝑣𝑊) = ( +𝑣𝑊)
506, 48, 49, 34sspgval 29671 . . . . . . . . . . . . . 14 (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ (SubSp‘𝑈)) ∧ (𝐾𝑌𝐿𝑌)) → (𝐾( +𝑣𝑊)𝐿) = (𝐾( +𝑣𝑈)𝐿))
5129, 33, 37, 39, 50syl22anc 837 . . . . . . . . . . . . 13 (𝜑 → (𝐾( +𝑣𝑊)𝐿) = (𝐾( +𝑣𝑈)𝐿))
5234sspnv 29668 . . . . . . . . . . . . . . 15 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ (SubSp‘𝑈)) → 𝑊 ∈ NrmCVec)
5329, 33, 52syl2anc 584 . . . . . . . . . . . . . 14 (𝜑𝑊 ∈ NrmCVec)
546, 49nvgcl 29562 . . . . . . . . . . . . . 14 ((𝑊 ∈ NrmCVec ∧ 𝐾𝑌𝐿𝑌) → (𝐾( +𝑣𝑊)𝐿) ∈ 𝑌)
5553, 37, 39, 54syl3anc 1371 . . . . . . . . . . . . 13 (𝜑 → (𝐾( +𝑣𝑊)𝐿) ∈ 𝑌)
5651, 55eqeltrrd 2839 . . . . . . . . . . . 12 (𝜑 → (𝐾( +𝑣𝑈)𝐿) ∈ 𝑌)
57 eqid 2736 . . . . . . . . . . . . 13 ( ·𝑠OLD𝑈) = ( ·𝑠OLD𝑈)
58 eqid 2736 . . . . . . . . . . . . 13 ( ·𝑠OLD𝑊) = ( ·𝑠OLD𝑊)
596, 57, 58, 34sspsval 29673 . . . . . . . . . . . 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 837 . . . . . . . . . . 11 (𝜑 → ((1 / 2)( ·𝑠OLD𝑊)(𝐾( +𝑣𝑈)𝐿)) = ((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)))
616, 58nvscl 29568 . . . . . . . . . . . 12 ((𝑊 ∈ NrmCVec ∧ (1 / 2) ∈ ℂ ∧ (𝐾( +𝑣𝑈)𝐿) ∈ 𝑌) → ((1 / 2)( ·𝑠OLD𝑊)(𝐾( +𝑣𝑈)𝐿)) ∈ 𝑌)
6253, 47, 56, 61syl3anc 1371 . . . . . . . . . . 11 (𝜑 → ((1 / 2)( ·𝑠OLD𝑊)(𝐾( +𝑣𝑈)𝐿)) ∈ 𝑌)
6360, 62eqeltrrd 2839 . . . . . . . . . 10 (𝜑 → ((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)) ∈ 𝑌)
6436, 63sseldd 3945 . . . . . . . . 9 (𝜑 → ((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)) ∈ 𝑋)
653, 4nvmcl 29588 . . . . . . . . 9 ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋 ∧ ((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)) ∈ 𝑋) → (𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))) ∈ 𝑋)
6629, 9, 64, 65syl3anc 1371 . . . . . . . 8 (𝜑 → (𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))) ∈ 𝑋)
673, 5nvcl 29603 . . . . . . . 8 ((𝑈 ∈ NrmCVec ∧ (𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))) ∈ 𝑋) → (𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)))) ∈ ℝ)
6829, 66, 67syl2anc 584 . . . . . . 7 (𝜑 → (𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)))) ∈ ℝ)
6968resqcld 14030 . . . . . 6 (𝜑 → ((𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))↑2) ∈ ℝ)
70 remulcl 11136 . . . . . 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 587 . . . . 5 (𝜑 → (4 · ((𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))↑2)) ∈ ℝ)
7271, 43readdcld 11184 . . . 4 (𝜑 → ((4 · ((𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))↑2)) + ((𝐾𝐷𝐿)↑2)) ∈ ℝ)
73 minvecolem2.1 . . . . . 6 (𝜑𝐵 ∈ ℝ)
7425, 73readdcld 11184 . . . . 5 (𝜑 → ((𝑆↑2) + 𝐵) ∈ ℝ)
75 remulcl 11136 . . . . 5 ((4 ∈ ℝ ∧ ((𝑆↑2) + 𝐵) ∈ ℝ) → (4 · ((𝑆↑2) + 𝐵)) ∈ ℝ)
761, 74, 75sylancr 587 . . . 4 (𝜑 → (4 · ((𝑆↑2) + 𝐵)) ∈ ℝ)
7716a1i 11 . . . . . . . . . 10 (𝜑 → 0 ∈ ℝ)
78 infregelb 12139 . . . . . . . . . 10 (((𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑤𝑅 𝑥𝑤) ∧ 0 ∈ ℝ) → (0 ≤ inf(𝑅, ℝ, < ) ↔ ∀𝑤𝑅 0 ≤ 𝑤))
7914, 15, 21, 77, 78syl31anc 1373 . . . . . . . . 9 (𝜑 → (0 ≤ inf(𝑅, ℝ, < ) ↔ ∀𝑤𝑅 0 ≤ 𝑤))
8017, 79mpbird 256 . . . . . . . 8 (𝜑 → 0 ≤ inf(𝑅, ℝ, < ))
8180, 2breqtrrdi 5147 . . . . . . 7 (𝜑 → 0 ≤ 𝑆)
82 eqid 2736 . . . . . . . . . . . 12 (𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)))) = (𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))
83 oveq2 7365 . . . . . . . . . . . . . 14 (𝑦 = ((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)) → (𝐴𝑀𝑦) = (𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))
8483fveq2d 6846 . . . . . . . . . . . . 13 (𝑦 = ((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)) → (𝑁‘(𝐴𝑀𝑦)) = (𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)))))
8584rspceeqv 3595 . . . . . . . . . . . 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)))
8663, 82, 85sylancl 586 . . . . . . . . . . 11 (𝜑 → ∃𝑦𝑌 (𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)))) = (𝑁‘(𝐴𝑀𝑦)))
87 eqid 2736 . . . . . . . . . . . 12 (𝑦𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) = (𝑦𝑌 ↦ (𝑁‘(𝐴𝑀𝑦)))
88 fvex 6855 . . . . . . . . . . . 12 (𝑁‘(𝐴𝑀𝑦)) ∈ V
8987, 88elrnmpti 5915 . . . . . . . . . . 11 ((𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)))) ∈ ran (𝑦𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) ↔ ∃𝑦𝑌 (𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)))) = (𝑁‘(𝐴𝑀𝑦)))
9086, 89sylibr 233 . . . . . . . . . 10 (𝜑 → (𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)))) ∈ ran (𝑦𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))))
9190, 12eleqtrrdi 2849 . . . . . . . . 9 (𝜑 → (𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)))) ∈ 𝑅)
92 infrelb 12140 . . . . . . . . 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)))))
9314, 21, 91, 92syl3anc 1371 . . . . . . . 8 (𝜑 → inf(𝑅, ℝ, < ) ≤ (𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)))))
942, 93eqbrtrid 5140 . . . . . . 7 (𝜑𝑆 ≤ (𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)))))
95 le2sq2 14040 . . . . . . 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))
9624, 81, 68, 94, 95syl22anc 837 . . . . . 6 (𝜑 → (𝑆↑2) ≤ ((𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))↑2))
97 4pos 12260 . . . . . . . . 9 0 < 4
981, 97pm3.2i 471 . . . . . . . 8 (4 ∈ ℝ ∧ 0 < 4)
99 lemul2 12008 . . . . . . . 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))))
10098, 99mp3an3 1450 . . . . . . 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))))
10125, 69, 100syl2anc 584 . . . . . 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))))
10296, 101mpbid 231 . . . . 5 (𝜑 → (4 · (𝑆↑2)) ≤ (4 · ((𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))↑2)))
10327, 71, 43, 102leadd1dd 11769 . . . 4 (𝜑 → ((4 · (𝑆↑2)) + ((𝐾𝐷𝐿)↑2)) ≤ ((4 · ((𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))↑2)) + ((𝐾𝐷𝐿)↑2)))
104 metcl 23685 . . . . . . . . . 10 ((𝐷 ∈ (Met‘𝑋) ∧ 𝐴𝑋𝐾𝑋) → (𝐴𝐷𝐾) ∈ ℝ)
10531, 9, 38, 104syl3anc 1371 . . . . . . . . 9 (𝜑 → (𝐴𝐷𝐾) ∈ ℝ)
106105resqcld 14030 . . . . . . . 8 (𝜑 → ((𝐴𝐷𝐾)↑2) ∈ ℝ)
107 metcl 23685 . . . . . . . . . 10 ((𝐷 ∈ (Met‘𝑋) ∧ 𝐴𝑋𝐿𝑋) → (𝐴𝐷𝐿) ∈ ℝ)
10831, 9, 40, 107syl3anc 1371 . . . . . . . . 9 (𝜑 → (𝐴𝐷𝐿) ∈ ℝ)
109108resqcld 14030 . . . . . . . 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))
112106, 109, 74, 74, 110, 111le2addd 11774 . . . . . . 7 (𝜑 → (((𝐴𝐷𝐾)↑2) + ((𝐴𝐷𝐿)↑2)) ≤ (((𝑆↑2) + 𝐵) + ((𝑆↑2) + 𝐵)))
11374recnd 11183 . . . . . . . 8 (𝜑 → ((𝑆↑2) + 𝐵) ∈ ℂ)
1141132timesd 12396 . . . . . . 7 (𝜑 → (2 · ((𝑆↑2) + 𝐵)) = (((𝑆↑2) + 𝐵) + ((𝑆↑2) + 𝐵)))
115112, 114breqtrrd 5133 . . . . . 6 (𝜑 → (((𝐴𝐷𝐾)↑2) + ((𝐴𝐷𝐿)↑2)) ≤ (2 · ((𝑆↑2) + 𝐵)))
116106, 109readdcld 11184 . . . . . . 7 (𝜑 → (((𝐴𝐷𝐾)↑2) + ((𝐴𝐷𝐿)↑2)) ∈ ℝ)
117 2re 12227 . . . . . . . 8 2 ∈ ℝ
118 remulcl 11136 . . . . . . . 8 ((2 ∈ ℝ ∧ ((𝑆↑2) + 𝐵) ∈ ℝ) → (2 · ((𝑆↑2) + 𝐵)) ∈ ℝ)
119117, 74, 118sylancr 587 . . . . . . 7 (𝜑 → (2 · ((𝑆↑2) + 𝐵)) ∈ ℝ)
120 2pos 12256 . . . . . . . . 9 0 < 2
121117, 120pm3.2i 471 . . . . . . . 8 (2 ∈ ℝ ∧ 0 < 2)
122 lemul2 12008 . . . . . . . 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)))))
123121, 122mp3an3 1450 . . . . . . 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)))))
124116, 119, 123syl2anc 584 . . . . . 6 (𝜑 → ((((𝐴𝐷𝐾)↑2) + ((𝐴𝐷𝐿)↑2)) ≤ (2 · ((𝑆↑2) + 𝐵)) ↔ (2 · (((𝐴𝐷𝐾)↑2) + ((𝐴𝐷𝐿)↑2))) ≤ (2 · (2 · ((𝑆↑2) + 𝐵)))))
125115, 124mpbid 231 . . . . 5 (𝜑 → (2 · (((𝐴𝐷𝐾)↑2) + ((𝐴𝐷𝐿)↑2))) ≤ (2 · (2 · ((𝑆↑2) + 𝐵))))
1263, 4nvmcl 29588 . . . . . . . 8 ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐾𝑋) → (𝐴𝑀𝐾) ∈ 𝑋)
12729, 9, 38, 126syl3anc 1371 . . . . . . 7 (𝜑 → (𝐴𝑀𝐾) ∈ 𝑋)
1283, 4nvmcl 29588 . . . . . . . 8 ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐿𝑋) → (𝐴𝑀𝐿) ∈ 𝑋)
12929, 9, 40, 128syl3anc 1371 . . . . . . 7 (𝜑 → (𝐴𝑀𝐿) ∈ 𝑋)
1303, 48, 4, 5phpar2 29765 . . . . . . 7 ((𝑈 ∈ CPreHilOLD ∧ (𝐴𝑀𝐾) ∈ 𝑋 ∧ (𝐴𝑀𝐿) ∈ 𝑋) → (((𝑁‘((𝐴𝑀𝐾)( +𝑣𝑈)(𝐴𝑀𝐿)))↑2) + ((𝑁‘((𝐴𝑀𝐾)𝑀(𝐴𝑀𝐿)))↑2)) = (2 · (((𝑁‘(𝐴𝑀𝐾))↑2) + ((𝑁‘(𝐴𝑀𝐿))↑2))))
1317, 127, 129, 130syl3anc 1371 . . . . . 6 (𝜑 → (((𝑁‘((𝐴𝑀𝐾)( +𝑣𝑈)(𝐴𝑀𝐿)))↑2) + ((𝑁‘((𝐴𝑀𝐾)𝑀(𝐴𝑀𝐿)))↑2)) = (2 · (((𝑁‘(𝐴𝑀𝐾))↑2) + ((𝑁‘(𝐴𝑀𝐿))↑2))))
132 2cn 12228 . . . . . . . . . 10 2 ∈ ℂ
13368recnd 11183 . . . . . . . . . 10 (𝜑 → (𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)))) ∈ ℂ)
134 sqmul 14024 . . . . . . . . . 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)))
135132, 133, 134sylancr 587 . . . . . . . . 9 (𝜑 → ((2 · (𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)))))↑2) = ((2↑2) · ((𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))↑2)))
136 sq2 14101 . . . . . . . . . 10 (2↑2) = 4
137136oveq1i 7367 . . . . . . . . 9 ((2↑2) · ((𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))↑2)) = (4 · ((𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))↑2))
138135, 137eqtrdi 2792 . . . . . . . 8 (𝜑 → ((2 · (𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)))))↑2) = (4 · ((𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))↑2)))
139132a1i 11 . . . . . . . . . . . 12 (𝜑 → 2 ∈ ℂ)
1403, 57, 5nvs 29605 . . . . . . . . . . . 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))))))
14129, 139, 66, 140syl3anc 1371 . . . . . . . . . . 11 (𝜑 → (𝑁‘(2( ·𝑠OLD𝑈)(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))) = ((abs‘2) · (𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))))
142 0le2 12255 . . . . . . . . . . . . 13 0 ≤ 2
143 absid 15181 . . . . . . . . . . . . 13 ((2 ∈ ℝ ∧ 0 ≤ 2) → (abs‘2) = 2)
144117, 142, 143mp2an 690 . . . . . . . . . . . 12 (abs‘2) = 2
145144oveq1i 7367 . . . . . . . . . . 11 ((abs‘2) · (𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))) = (2 · (𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)))))
146141, 145eqtrdi 2792 . . . . . . . . . 10 (𝜑 → (𝑁‘(2( ·𝑠OLD𝑈)(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))) = (2 · (𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))))
1473, 4, 57nvmdi 29590 . . . . . . . . . . . . 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)))))
14829, 139, 9, 64, 147syl13anc 1372 . . . . . . . . . . . 12 (𝜑 → (2( ·𝑠OLD𝑈)(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)))) = ((2( ·𝑠OLD𝑈)𝐴)𝑀(2( ·𝑠OLD𝑈)((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)))))
1493, 48, 57nv2 29574 . . . . . . . . . . . . . 14 ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋) → (𝐴( +𝑣𝑈)𝐴) = (2( ·𝑠OLD𝑈)𝐴))
15029, 9, 149syl2anc 584 . . . . . . . . . . . . 13 (𝜑 → (𝐴( +𝑣𝑈)𝐴) = (2( ·𝑠OLD𝑈)𝐴))
151 2ne0 12257 . . . . . . . . . . . . . . . . 17 2 ≠ 0
152132, 151recidi 11886 . . . . . . . . . . . . . . . 16 (2 · (1 / 2)) = 1
153152oveq1i 7367 . . . . . . . . . . . . . . 15 ((2 · (1 / 2))( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)) = (1( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))
1543, 48nvgcl 29562 . . . . . . . . . . . . . . . . 17 ((𝑈 ∈ NrmCVec ∧ 𝐾𝑋𝐿𝑋) → (𝐾( +𝑣𝑈)𝐿) ∈ 𝑋)
15529, 38, 40, 154syl3anc 1371 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐾( +𝑣𝑈)𝐿) ∈ 𝑋)
1563, 57nvsid 29569 . . . . . . . . . . . . . . . 16 ((𝑈 ∈ NrmCVec ∧ (𝐾( +𝑣𝑈)𝐿) ∈ 𝑋) → (1( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)) = (𝐾( +𝑣𝑈)𝐿))
15729, 155, 156syl2anc 584 . . . . . . . . . . . . . . 15 (𝜑 → (1( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)) = (𝐾( +𝑣𝑈)𝐿))
158153, 157eqtrid 2788 . . . . . . . . . . . . . 14 (𝜑 → ((2 · (1 / 2))( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)) = (𝐾( +𝑣𝑈)𝐿))
1593, 57nvsass 29570 . . . . . . . . . . . . . . 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))))
16029, 139, 47, 155, 159syl13anc 1372 . . . . . . . . . . . . . 14 (𝜑 → ((2 · (1 / 2))( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)) = (2( ·𝑠OLD𝑈)((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))
161158, 160eqtr3d 2778 . . . . . . . . . . . . 13 (𝜑 → (𝐾( +𝑣𝑈)𝐿) = (2( ·𝑠OLD𝑈)((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))
162150, 161oveq12d 7375 . . . . . . . . . . . 12 (𝜑 → ((𝐴( +𝑣𝑈)𝐴)𝑀(𝐾( +𝑣𝑈)𝐿)) = ((2( ·𝑠OLD𝑈)𝐴)𝑀(2( ·𝑠OLD𝑈)((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)))))
1633, 48, 4nvaddsub4 29599 . . . . . . . . . . . . 13 ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐴𝑋) ∧ (𝐾𝑋𝐿𝑋)) → ((𝐴( +𝑣𝑈)𝐴)𝑀(𝐾( +𝑣𝑈)𝐿)) = ((𝐴𝑀𝐾)( +𝑣𝑈)(𝐴𝑀𝐿)))
16429, 9, 9, 38, 40, 163syl122anc 1379 . . . . . . . . . . . 12 (𝜑 → ((𝐴( +𝑣𝑈)𝐴)𝑀(𝐾( +𝑣𝑈)𝐿)) = ((𝐴𝑀𝐾)( +𝑣𝑈)(𝐴𝑀𝐿)))
165148, 162, 1643eqtr2d 2782 . . . . . . . . . . 11 (𝜑 → (2( ·𝑠OLD𝑈)(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)))) = ((𝐴𝑀𝐾)( +𝑣𝑈)(𝐴𝑀𝐿)))
166165fveq2d 6846 . . . . . . . . . 10 (𝜑 → (𝑁‘(2( ·𝑠OLD𝑈)(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))) = (𝑁‘((𝐴𝑀𝐾)( +𝑣𝑈)(𝐴𝑀𝐿))))
167146, 166eqtr3d 2778 . . . . . . . . 9 (𝜑 → (2 · (𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))) = (𝑁‘((𝐴𝑀𝐾)( +𝑣𝑈)(𝐴𝑀𝐿))))
168167oveq1d 7372 . . . . . . . 8 (𝜑 → ((2 · (𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿)))))↑2) = ((𝑁‘((𝐴𝑀𝐾)( +𝑣𝑈)(𝐴𝑀𝐿)))↑2))
169138, 168eqtr3d 2778 . . . . . . 7 (𝜑 → (4 · ((𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))↑2)) = ((𝑁‘((𝐴𝑀𝐾)( +𝑣𝑈)(𝐴𝑀𝐿)))↑2))
1703, 4, 5, 10imsdval 29628 . . . . . . . . . 10 ((𝑈 ∈ NrmCVec ∧ 𝐿𝑋𝐾𝑋) → (𝐿𝐷𝐾) = (𝑁‘(𝐿𝑀𝐾)))
17129, 40, 38, 170syl3anc 1371 . . . . . . . . 9 (𝜑 → (𝐿𝐷𝐾) = (𝑁‘(𝐿𝑀𝐾)))
172 metsym 23703 . . . . . . . . . 10 ((𝐷 ∈ (Met‘𝑋) ∧ 𝐾𝑋𝐿𝑋) → (𝐾𝐷𝐿) = (𝐿𝐷𝐾))
17331, 38, 40, 172syl3anc 1371 . . . . . . . . 9 (𝜑 → (𝐾𝐷𝐿) = (𝐿𝐷𝐾))
1743, 4nvnnncan1 29589 . . . . . . . . . . 11 ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐾𝑋𝐿𝑋)) → ((𝐴𝑀𝐾)𝑀(𝐴𝑀𝐿)) = (𝐿𝑀𝐾))
17529, 9, 38, 40, 174syl13anc 1372 . . . . . . . . . 10 (𝜑 → ((𝐴𝑀𝐾)𝑀(𝐴𝑀𝐿)) = (𝐿𝑀𝐾))
176175fveq2d 6846 . . . . . . . . 9 (𝜑 → (𝑁‘((𝐴𝑀𝐾)𝑀(𝐴𝑀𝐿))) = (𝑁‘(𝐿𝑀𝐾)))
177171, 173, 1763eqtr4d 2786 . . . . . . . 8 (𝜑 → (𝐾𝐷𝐿) = (𝑁‘((𝐴𝑀𝐾)𝑀(𝐴𝑀𝐿))))
178177oveq1d 7372 . . . . . . 7 (𝜑 → ((𝐾𝐷𝐿)↑2) = ((𝑁‘((𝐴𝑀𝐾)𝑀(𝐴𝑀𝐿)))↑2))
179169, 178oveq12d 7375 . . . . . 6 (𝜑 → ((4 · ((𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))↑2)) + ((𝐾𝐷𝐿)↑2)) = (((𝑁‘((𝐴𝑀𝐾)( +𝑣𝑈)(𝐴𝑀𝐿)))↑2) + ((𝑁‘((𝐴𝑀𝐾)𝑀(𝐴𝑀𝐿)))↑2)))
1803, 4, 5, 10imsdval 29628 . . . . . . . . . 10 ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐾𝑋) → (𝐴𝐷𝐾) = (𝑁‘(𝐴𝑀𝐾)))
18129, 9, 38, 180syl3anc 1371 . . . . . . . . 9 (𝜑 → (𝐴𝐷𝐾) = (𝑁‘(𝐴𝑀𝐾)))
182181oveq1d 7372 . . . . . . . 8 (𝜑 → ((𝐴𝐷𝐾)↑2) = ((𝑁‘(𝐴𝑀𝐾))↑2))
1833, 4, 5, 10imsdval 29628 . . . . . . . . . 10 ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐿𝑋) → (𝐴𝐷𝐿) = (𝑁‘(𝐴𝑀𝐿)))
18429, 9, 40, 183syl3anc 1371 . . . . . . . . 9 (𝜑 → (𝐴𝐷𝐿) = (𝑁‘(𝐴𝑀𝐿)))
185184oveq1d 7372 . . . . . . . 8 (𝜑 → ((𝐴𝐷𝐿)↑2) = ((𝑁‘(𝐴𝑀𝐿))↑2))
186182, 185oveq12d 7375 . . . . . . 7 (𝜑 → (((𝐴𝐷𝐾)↑2) + ((𝐴𝐷𝐿)↑2)) = (((𝑁‘(𝐴𝑀𝐾))↑2) + ((𝑁‘(𝐴𝑀𝐿))↑2)))
187186oveq2d 7373 . . . . . 6 (𝜑 → (2 · (((𝐴𝐷𝐾)↑2) + ((𝐴𝐷𝐿)↑2))) = (2 · (((𝑁‘(𝐴𝑀𝐾))↑2) + ((𝑁‘(𝐴𝑀𝐿))↑2))))
188131, 179, 1873eqtr4d 2786 . . . . 5 (𝜑 → ((4 · ((𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))↑2)) + ((𝐾𝐷𝐿)↑2)) = (2 · (((𝐴𝐷𝐾)↑2) + ((𝐴𝐷𝐿)↑2))))
189 2t2e4 12317 . . . . . . 7 (2 · 2) = 4
190189oveq1i 7367 . . . . . 6 ((2 · 2) · ((𝑆↑2) + 𝐵)) = (4 · ((𝑆↑2) + 𝐵))
191139, 139, 113mulassd 11178 . . . . . 6 (𝜑 → ((2 · 2) · ((𝑆↑2) + 𝐵)) = (2 · (2 · ((𝑆↑2) + 𝐵))))
192190, 191eqtr3id 2790 . . . . 5 (𝜑 → (4 · ((𝑆↑2) + 𝐵)) = (2 · (2 · ((𝑆↑2) + 𝐵))))
193125, 188, 1923brtr4d 5137 . . . 4 (𝜑 → ((4 · ((𝑁‘(𝐴𝑀((1 / 2)( ·𝑠OLD𝑈)(𝐾( +𝑣𝑈)𝐿))))↑2)) + ((𝐾𝐷𝐿)↑2)) ≤ (4 · ((𝑆↑2) + 𝐵)))
19444, 72, 76, 103, 193letrd 11312 . . 3 (𝜑 → ((4 · (𝑆↑2)) + ((𝐾𝐷𝐿)↑2)) ≤ (4 · ((𝑆↑2) + 𝐵)))
195 4cn 12238 . . . . 5 4 ∈ ℂ
196195a1i 11 . . . 4 (𝜑 → 4 ∈ ℂ)
19725recnd 11183 . . . 4 (𝜑 → (𝑆↑2) ∈ ℂ)
19873recnd 11183 . . . 4 (𝜑𝐵 ∈ ℂ)
199196, 197, 198adddid 11179 . . 3 (𝜑 → (4 · ((𝑆↑2) + 𝐵)) = ((4 · (𝑆↑2)) + (4 · 𝐵)))
200194, 199breqtrd 5131 . 2 (𝜑 → ((4 · (𝑆↑2)) + ((𝐾𝐷𝐿)↑2)) ≤ ((4 · (𝑆↑2)) + (4 · 𝐵)))
201 remulcl 11136 . . . 4 ((4 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (4 · 𝐵) ∈ ℝ)
2021, 73, 201sylancr 587 . . 3 (𝜑 → (4 · 𝐵) ∈ ℝ)
20343, 202, 27leadd2d 11750 . 2 (𝜑 → (((𝐾𝐷𝐿)↑2) ≤ (4 · 𝐵) ↔ ((4 · (𝑆↑2)) + ((𝐾𝐷𝐿)↑2)) ≤ ((4 · (𝑆↑2)) + (4 · 𝐵))))
204200, 203mpbird 256 1 (𝜑 → ((𝐾𝐷𝐿)↑2) ≤ (4 · 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wne 2943  wral 3064  wrex 3073  cin 3909  wss 3910  c0 4282   class class class wbr 5105  cmpt 5188  ran crn 5634  cfv 6496  (class class class)co 7357  infcinf 9377  cc 11049  cr 11050  0cc0 11051  1c1 11052   + caddc 11054   · cmul 11056   < clt 11189  cle 11190   / cdiv 11812  2c2 12208  4c4 12210  cexp 13967  abscabs 15119  Metcmet 20782  MetOpencmopn 20786  NrmCVeccnv 29526   +𝑣 cpv 29527  BaseSetcba 29528   ·𝑠OLD cns 29529  𝑣 cnsb 29531  normCVcnmcv 29532  IndMetcims 29533  SubSpcss 29663  CPreHilOLDccphlo 29754  CBanccbn 29804
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129  ax-addf 11130  ax-mulf 11131
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-sup 9378  df-inf 9379  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-n0 12414  df-z 12500  df-uz 12764  df-rp 12916  df-xadd 13034  df-seq 13907  df-exp 13968  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-xmet 20789  df-met 20790  df-grpo 29435  df-gid 29436  df-ginv 29437  df-gdiv 29438  df-ablo 29487  df-vc 29501  df-nv 29534  df-va 29537  df-ba 29538  df-sm 29539  df-0v 29540  df-vs 29541  df-nmcv 29542  df-ims 29543  df-ssp 29664  df-ph 29755  df-cbn 29805
This theorem is referenced by:  minvecolem3  29818  minvecolem7  29825
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