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Theorem unbdqndv2lem2 32140
Description: Lemma for unbdqndv2 32141. (Contributed by Asger C. Ipsen, 12-May-2021.)
Hypotheses
Ref Expression
unbdqndv2lem2.g 𝐺 = (𝑧 ∈ (𝑋 ∖ {𝐴}) ↦ (((𝐹𝑧) − (𝐹𝐴)) / (𝑧𝐴)))
unbdqndv2lem2.w 𝑊 = if((𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴))), 𝑈, 𝑉)
unbdqndv2lem2.x (𝜑𝑋 ⊆ ℝ)
unbdqndv2lem2.f (𝜑𝐹:𝑋⟶ℂ)
unbdqndv2lem2.a (𝜑𝐴𝑋)
unbdqndv2lem2.b (𝜑𝐵 ∈ ℝ+)
unbdqndv2lem2.d (𝜑𝐷 ∈ ℝ+)
unbdqndv2lem2.u (𝜑𝑈𝑋)
unbdqndv2lem2.v (𝜑𝑉𝑋)
unbdqndv2lem2.1 (𝜑𝑈𝑉)
unbdqndv2lem2.2 (𝜑𝑈𝐴)
unbdqndv2lem2.3 (𝜑𝐴𝑉)
unbdqndv2lem2.4 (𝜑 → (𝑉𝑈) < 𝐷)
unbdqndv2lem2.5 (𝜑 → (2 · 𝐵) ≤ ((abs‘((𝐹𝑉) − (𝐹𝑈))) / (𝑉𝑈)))
Assertion
Ref Expression
unbdqndv2lem2 (𝜑 → (𝑊 ∈ (𝑋 ∖ {𝐴}) ∧ ((abs‘(𝑊𝐴)) < 𝐷𝐵 ≤ (abs‘(𝐺𝑊)))))
Distinct variable groups:   𝑧,𝐴   𝑧,𝐵   𝑧,𝐹   𝑧,𝑈   𝑧,𝑉   𝑧,𝑋   𝜑,𝑧
Allowed substitution hints:   𝐷(𝑧)   𝐺(𝑧)   𝑊(𝑧)

Proof of Theorem unbdqndv2lem2
StepHypRef Expression
1 unbdqndv2lem2.w . . . . . 6 𝑊 = if((𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴))), 𝑈, 𝑉)
21a1i 11 . . . . 5 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝑊 = if((𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴))), 𝑈, 𝑉))
3 iftrue 4064 . . . . . 6 ((𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴))) → if((𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴))), 𝑈, 𝑉) = 𝑈)
43adantl 482 . . . . 5 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → if((𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴))), 𝑈, 𝑉) = 𝑈)
52, 4eqtrd 2655 . . . 4 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝑊 = 𝑈)
6 unbdqndv2lem2.u . . . . . . 7 (𝜑𝑈𝑋)
76adantr 481 . . . . . 6 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝑈𝑋)
8 simplr 791 . . . . . . . . 9 (((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) ∧ 𝑈 = 𝐴) → (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴))))
9 fveq2 6148 . . . . . . . . . . . . . . 15 (𝑈 = 𝐴 → (𝐹𝑈) = (𝐹𝐴))
109eqcomd 2627 . . . . . . . . . . . . . 14 (𝑈 = 𝐴 → (𝐹𝐴) = (𝐹𝑈))
1110oveq2d 6620 . . . . . . . . . . . . 13 (𝑈 = 𝐴 → ((𝐹𝑈) − (𝐹𝐴)) = ((𝐹𝑈) − (𝐹𝑈)))
1211fveq2d 6152 . . . . . . . . . . . 12 (𝑈 = 𝐴 → (abs‘((𝐹𝑈) − (𝐹𝐴))) = (abs‘((𝐹𝑈) − (𝐹𝑈))))
1312adantl 482 . . . . . . . . . . 11 ((𝜑𝑈 = 𝐴) → (abs‘((𝐹𝑈) − (𝐹𝐴))) = (abs‘((𝐹𝑈) − (𝐹𝑈))))
14 unbdqndv2lem2.f . . . . . . . . . . . . . . . 16 (𝜑𝐹:𝑋⟶ℂ)
1514, 6ffvelrnd 6316 . . . . . . . . . . . . . . 15 (𝜑 → (𝐹𝑈) ∈ ℂ)
1615subidd 10324 . . . . . . . . . . . . . 14 (𝜑 → ((𝐹𝑈) − (𝐹𝑈)) = 0)
1716fveq2d 6152 . . . . . . . . . . . . 13 (𝜑 → (abs‘((𝐹𝑈) − (𝐹𝑈))) = (abs‘0))
1817adantr 481 . . . . . . . . . . . 12 ((𝜑𝑈 = 𝐴) → (abs‘((𝐹𝑈) − (𝐹𝑈))) = (abs‘0))
19 abs0 13959 . . . . . . . . . . . . 13 (abs‘0) = 0
2019a1i 11 . . . . . . . . . . . 12 ((𝜑𝑈 = 𝐴) → (abs‘0) = 0)
2118, 20eqtrd 2655 . . . . . . . . . . 11 ((𝜑𝑈 = 𝐴) → (abs‘((𝐹𝑈) − (𝐹𝑈))) = 0)
2213, 21eqtrd 2655 . . . . . . . . . 10 ((𝜑𝑈 = 𝐴) → (abs‘((𝐹𝑈) − (𝐹𝐴))) = 0)
2322adantlr 750 . . . . . . . . 9 (((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) ∧ 𝑈 = 𝐴) → (abs‘((𝐹𝑈) − (𝐹𝐴))) = 0)
248, 23breqtrd 4639 . . . . . . . 8 (((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) ∧ 𝑈 = 𝐴) → (𝐵 · (𝑉𝑈)) ≤ 0)
25 unbdqndv2lem2.b . . . . . . . . . . . . 13 (𝜑𝐵 ∈ ℝ+)
2625rpred 11816 . . . . . . . . . . . 12 (𝜑𝐵 ∈ ℝ)
27 unbdqndv2lem2.x . . . . . . . . . . . . . 14 (𝜑𝑋 ⊆ ℝ)
28 unbdqndv2lem2.v . . . . . . . . . . . . . 14 (𝜑𝑉𝑋)
2927, 28sseldd 3584 . . . . . . . . . . . . 13 (𝜑𝑉 ∈ ℝ)
3027, 6sseldd 3584 . . . . . . . . . . . . 13 (𝜑𝑈 ∈ ℝ)
3129, 30resubcld 10402 . . . . . . . . . . . 12 (𝜑 → (𝑉𝑈) ∈ ℝ)
3225rpgt0d 11819 . . . . . . . . . . . 12 (𝜑 → 0 < 𝐵)
33 unbdqndv2lem2.a . . . . . . . . . . . . . . . 16 (𝜑𝐴𝑋)
3427, 33sseldd 3584 . . . . . . . . . . . . . . 15 (𝜑𝐴 ∈ ℝ)
35 unbdqndv2lem2.2 . . . . . . . . . . . . . . 15 (𝜑𝑈𝐴)
36 unbdqndv2lem2.3 . . . . . . . . . . . . . . 15 (𝜑𝐴𝑉)
3730, 34, 29, 35, 36letrd 10138 . . . . . . . . . . . . . 14 (𝜑𝑈𝑉)
38 unbdqndv2lem2.1 . . . . . . . . . . . . . . 15 (𝜑𝑈𝑉)
3938necomd 2845 . . . . . . . . . . . . . 14 (𝜑𝑉𝑈)
4030, 29, 37, 39leneltd 10135 . . . . . . . . . . . . 13 (𝜑𝑈 < 𝑉)
4130, 29posdifd 10558 . . . . . . . . . . . . 13 (𝜑 → (𝑈 < 𝑉 ↔ 0 < (𝑉𝑈)))
4240, 41mpbid 222 . . . . . . . . . . . 12 (𝜑 → 0 < (𝑉𝑈))
4326, 31, 32, 42mulgt0d 10136 . . . . . . . . . . 11 (𝜑 → 0 < (𝐵 · (𝑉𝑈)))
44 0red 9985 . . . . . . . . . . . 12 (𝜑 → 0 ∈ ℝ)
4526, 31remulcld 10014 . . . . . . . . . . . 12 (𝜑 → (𝐵 · (𝑉𝑈)) ∈ ℝ)
4644, 45ltnled 10128 . . . . . . . . . . 11 (𝜑 → (0 < (𝐵 · (𝑉𝑈)) ↔ ¬ (𝐵 · (𝑉𝑈)) ≤ 0))
4743, 46mpbid 222 . . . . . . . . . 10 (𝜑 → ¬ (𝐵 · (𝑉𝑈)) ≤ 0)
4847adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → ¬ (𝐵 · (𝑉𝑈)) ≤ 0)
4948adantr 481 . . . . . . . 8 (((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) ∧ 𝑈 = 𝐴) → ¬ (𝐵 · (𝑉𝑈)) ≤ 0)
5024, 49pm2.65da 599 . . . . . . 7 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → ¬ 𝑈 = 𝐴)
5150neqned 2797 . . . . . 6 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝑈𝐴)
527, 51jca 554 . . . . 5 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝑈𝑋𝑈𝐴))
53 eldifsn 4287 . . . . 5 (𝑈 ∈ (𝑋 ∖ {𝐴}) ↔ (𝑈𝑋𝑈𝐴))
5452, 53sylibr 224 . . . 4 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝑈 ∈ (𝑋 ∖ {𝐴}))
555, 54eqeltrd 2698 . . 3 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝑊 ∈ (𝑋 ∖ {𝐴}))
565oveq1d 6619 . . . . . . 7 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝑊𝐴) = (𝑈𝐴))
5756fveq2d 6152 . . . . . 6 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (abs‘(𝑊𝐴)) = (abs‘(𝑈𝐴)))
5830, 34, 35abssuble0d 14105 . . . . . . 7 (𝜑 → (abs‘(𝑈𝐴)) = (𝐴𝑈))
5958adantr 481 . . . . . 6 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (abs‘(𝑈𝐴)) = (𝐴𝑈))
6057, 59eqtrd 2655 . . . . 5 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (abs‘(𝑊𝐴)) = (𝐴𝑈))
6134, 30resubcld 10402 . . . . . . 7 (𝜑 → (𝐴𝑈) ∈ ℝ)
62 unbdqndv2lem2.d . . . . . . . 8 (𝜑𝐷 ∈ ℝ+)
6362rpred 11816 . . . . . . 7 (𝜑𝐷 ∈ ℝ)
6434, 29, 30, 36lesub1dd 10587 . . . . . . 7 (𝜑 → (𝐴𝑈) ≤ (𝑉𝑈))
65 unbdqndv2lem2.4 . . . . . . 7 (𝜑 → (𝑉𝑈) < 𝐷)
6661, 31, 63, 64, 65lelttrd 10139 . . . . . 6 (𝜑 → (𝐴𝑈) < 𝐷)
6766adantr 481 . . . . 5 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝐴𝑈) < 𝐷)
6860, 67eqbrtrd 4635 . . . 4 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (abs‘(𝑊𝐴)) < 𝐷)
6926, 61remulcld 10014 . . . . . . . 8 (𝜑 → (𝐵 · (𝐴𝑈)) ∈ ℝ)
7069adantr 481 . . . . . . 7 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝐵 · (𝐴𝑈)) ∈ ℝ)
7145adantr 481 . . . . . . 7 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝐵 · (𝑉𝑈)) ∈ ℝ)
7214, 33ffvelrnd 6316 . . . . . . . . . 10 (𝜑 → (𝐹𝐴) ∈ ℂ)
7315, 72subcld 10336 . . . . . . . . 9 (𝜑 → ((𝐹𝑈) − (𝐹𝐴)) ∈ ℂ)
7473abscld 14109 . . . . . . . 8 (𝜑 → (abs‘((𝐹𝑈) − (𝐹𝐴))) ∈ ℝ)
7574adantr 481 . . . . . . 7 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (abs‘((𝐹𝑈) − (𝐹𝐴))) ∈ ℝ)
7644, 26, 32ltled 10129 . . . . . . . . 9 (𝜑 → 0 ≤ 𝐵)
7761, 31, 26, 76, 64lemul2ad 10908 . . . . . . . 8 (𝜑 → (𝐵 · (𝐴𝑈)) ≤ (𝐵 · (𝑉𝑈)))
7877adantr 481 . . . . . . 7 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝐵 · (𝐴𝑈)) ≤ (𝐵 · (𝑉𝑈)))
79 simpr 477 . . . . . . 7 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴))))
8070, 71, 75, 78, 79letrd 10138 . . . . . 6 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝐵 · (𝐴𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴))))
8126adantr 481 . . . . . . 7 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝐵 ∈ ℝ)
8261adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝐴𝑈) ∈ ℝ)
8335adantr 481 . . . . . . . . . . . 12 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝑈𝐴)
8451necomd 2845 . . . . . . . . . . . 12 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝐴𝑈)
8583, 84jca 554 . . . . . . . . . . 11 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝑈𝐴𝐴𝑈))
8630, 34ltlend 10126 . . . . . . . . . . . 12 (𝜑 → (𝑈 < 𝐴 ↔ (𝑈𝐴𝐴𝑈)))
8786adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝑈 < 𝐴 ↔ (𝑈𝐴𝐴𝑈)))
8885, 87mpbird 247 . . . . . . . . . 10 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝑈 < 𝐴)
8930, 34posdifd 10558 . . . . . . . . . . 11 (𝜑 → (𝑈 < 𝐴 ↔ 0 < (𝐴𝑈)))
9089adantr 481 . . . . . . . . . 10 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝑈 < 𝐴 ↔ 0 < (𝐴𝑈)))
9188, 90mpbid 222 . . . . . . . . 9 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 0 < (𝐴𝑈))
9282, 91jca 554 . . . . . . . 8 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → ((𝐴𝑈) ∈ ℝ ∧ 0 < (𝐴𝑈)))
93 elrp 11778 . . . . . . . 8 ((𝐴𝑈) ∈ ℝ+ ↔ ((𝐴𝑈) ∈ ℝ ∧ 0 < (𝐴𝑈)))
9492, 93sylibr 224 . . . . . . 7 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝐴𝑈) ∈ ℝ+)
9581, 75, 94lemuldivd 11865 . . . . . 6 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → ((𝐵 · (𝐴𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴))) ↔ 𝐵 ≤ ((abs‘((𝐹𝑈) − (𝐹𝐴))) / (𝐴𝑈))))
9680, 95mpbid 222 . . . . 5 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝐵 ≤ ((abs‘((𝐹𝑈) − (𝐹𝐴))) / (𝐴𝑈)))
975fveq2d 6152 . . . . . . . . 9 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝐺𝑊) = (𝐺𝑈))
98 unbdqndv2lem2.g . . . . . . . . . . 11 𝐺 = (𝑧 ∈ (𝑋 ∖ {𝐴}) ↦ (((𝐹𝑧) − (𝐹𝐴)) / (𝑧𝐴)))
9998a1i 11 . . . . . . . . . 10 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝐺 = (𝑧 ∈ (𝑋 ∖ {𝐴}) ↦ (((𝐹𝑧) − (𝐹𝐴)) / (𝑧𝐴))))
100 fveq2 6148 . . . . . . . . . . . . 13 (𝑧 = 𝑈 → (𝐹𝑧) = (𝐹𝑈))
101100oveq1d 6619 . . . . . . . . . . . 12 (𝑧 = 𝑈 → ((𝐹𝑧) − (𝐹𝐴)) = ((𝐹𝑈) − (𝐹𝐴)))
102 oveq1 6611 . . . . . . . . . . . 12 (𝑧 = 𝑈 → (𝑧𝐴) = (𝑈𝐴))
103101, 102oveq12d 6622 . . . . . . . . . . 11 (𝑧 = 𝑈 → (((𝐹𝑧) − (𝐹𝐴)) / (𝑧𝐴)) = (((𝐹𝑈) − (𝐹𝐴)) / (𝑈𝐴)))
104103adantl 482 . . . . . . . . . 10 (((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) ∧ 𝑧 = 𝑈) → (((𝐹𝑧) − (𝐹𝐴)) / (𝑧𝐴)) = (((𝐹𝑈) − (𝐹𝐴)) / (𝑈𝐴)))
105 ovex 6632 . . . . . . . . . . 11 (((𝐹𝑈) − (𝐹𝐴)) / (𝑈𝐴)) ∈ V
106105a1i 11 . . . . . . . . . 10 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (((𝐹𝑈) − (𝐹𝐴)) / (𝑈𝐴)) ∈ V)
10799, 104, 54, 106fvmptd 6245 . . . . . . . . 9 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝐺𝑈) = (((𝐹𝑈) − (𝐹𝐴)) / (𝑈𝐴)))
10897, 107eqtrd 2655 . . . . . . . 8 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝐺𝑊) = (((𝐹𝑈) − (𝐹𝐴)) / (𝑈𝐴)))
109108fveq2d 6152 . . . . . . 7 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (abs‘(𝐺𝑊)) = (abs‘(((𝐹𝑈) − (𝐹𝐴)) / (𝑈𝐴))))
11073adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → ((𝐹𝑈) − (𝐹𝐴)) ∈ ℂ)
11130recnd 10012 . . . . . . . . . . 11 (𝜑𝑈 ∈ ℂ)
11234recnd 10012 . . . . . . . . . . 11 (𝜑𝐴 ∈ ℂ)
113111, 112subcld 10336 . . . . . . . . . 10 (𝜑 → (𝑈𝐴) ∈ ℂ)
114113adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝑈𝐴) ∈ ℂ)
115111adantr 481 . . . . . . . . . 10 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝑈 ∈ ℂ)
116112adantr 481 . . . . . . . . . 10 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝐴 ∈ ℂ)
117115, 116, 51subne0d 10345 . . . . . . . . 9 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝑈𝐴) ≠ 0)
118110, 114, 117absdivd 14128 . . . . . . . 8 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (abs‘(((𝐹𝑈) − (𝐹𝐴)) / (𝑈𝐴))) = ((abs‘((𝐹𝑈) − (𝐹𝐴))) / (abs‘(𝑈𝐴))))
11959oveq2d 6620 . . . . . . . 8 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → ((abs‘((𝐹𝑈) − (𝐹𝐴))) / (abs‘(𝑈𝐴))) = ((abs‘((𝐹𝑈) − (𝐹𝐴))) / (𝐴𝑈)))
120118, 119eqtrd 2655 . . . . . . 7 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (abs‘(((𝐹𝑈) − (𝐹𝐴)) / (𝑈𝐴))) = ((abs‘((𝐹𝑈) − (𝐹𝐴))) / (𝐴𝑈)))
121109, 120eqtrd 2655 . . . . . 6 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (abs‘(𝐺𝑊)) = ((abs‘((𝐹𝑈) − (𝐹𝐴))) / (𝐴𝑈)))
122121eqcomd 2627 . . . . 5 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → ((abs‘((𝐹𝑈) − (𝐹𝐴))) / (𝐴𝑈)) = (abs‘(𝐺𝑊)))
12396, 122breqtrd 4639 . . . 4 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝐵 ≤ (abs‘(𝐺𝑊)))
12468, 123jca 554 . . 3 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → ((abs‘(𝑊𝐴)) < 𝐷𝐵 ≤ (abs‘(𝐺𝑊))))
12555, 124jca 554 . 2 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝑊 ∈ (𝑋 ∖ {𝐴}) ∧ ((abs‘(𝑊𝐴)) < 𝐷𝐵 ≤ (abs‘(𝐺𝑊)))))
1261a1i 11 . . . . 5 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝑊 = if((𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴))), 𝑈, 𝑉))
127 simpr 477 . . . . . 6 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴))))
128127iffalsed 4069 . . . . 5 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → if((𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴))), 𝑈, 𝑉) = 𝑉)
129126, 128eqtrd 2655 . . . 4 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝑊 = 𝑉)
13028adantr 481 . . . . . 6 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝑉𝑋)
13130, 29, 37abssubge0d 14104 . . . . . . . . . . . . . . 15 (𝜑 → (abs‘(𝑉𝑈)) = (𝑉𝑈))
132131oveq2d 6620 . . . . . . . . . . . . . 14 (𝜑 → (𝐵 · (abs‘(𝑉𝑈))) = (𝐵 · (𝑉𝑈)))
133132breq1d 4623 . . . . . . . . . . . . 13 (𝜑 → ((𝐵 · (abs‘(𝑉𝑈))) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴))) ↔ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))))
134133adantr 481 . . . . . . . . . . . 12 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → ((𝐵 · (abs‘(𝑉𝑈))) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴))) ↔ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))))
135127, 134mtbird 315 . . . . . . . . . . 11 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → ¬ (𝐵 · (abs‘(𝑉𝑈))) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴))))
13614, 28ffvelrnd 6316 . . . . . . . . . . . . 13 (𝜑 → (𝐹𝑉) ∈ ℂ)
13731recnd 10012 . . . . . . . . . . . . 13 (𝜑 → (𝑉𝑈) ∈ ℂ)
13844, 42gtned 10116 . . . . . . . . . . . . 13 (𝜑 → (𝑉𝑈) ≠ 0)
139 unbdqndv2lem2.5 . . . . . . . . . . . . . 14 (𝜑 → (2 · 𝐵) ≤ ((abs‘((𝐹𝑉) − (𝐹𝑈))) / (𝑉𝑈)))
140136, 15subcld 10336 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝐹𝑉) − (𝐹𝑈)) ∈ ℂ)
141140, 137, 138absdivd 14128 . . . . . . . . . . . . . . . 16 (𝜑 → (abs‘(((𝐹𝑉) − (𝐹𝑈)) / (𝑉𝑈))) = ((abs‘((𝐹𝑉) − (𝐹𝑈))) / (abs‘(𝑉𝑈))))
142131oveq2d 6620 . . . . . . . . . . . . . . . 16 (𝜑 → ((abs‘((𝐹𝑉) − (𝐹𝑈))) / (abs‘(𝑉𝑈))) = ((abs‘((𝐹𝑉) − (𝐹𝑈))) / (𝑉𝑈)))
143141, 142eqtrd 2655 . . . . . . . . . . . . . . 15 (𝜑 → (abs‘(((𝐹𝑉) − (𝐹𝑈)) / (𝑉𝑈))) = ((abs‘((𝐹𝑉) − (𝐹𝑈))) / (𝑉𝑈)))
144143eqcomd 2627 . . . . . . . . . . . . . 14 (𝜑 → ((abs‘((𝐹𝑉) − (𝐹𝑈))) / (𝑉𝑈)) = (abs‘(((𝐹𝑉) − (𝐹𝑈)) / (𝑉𝑈))))
145139, 144breqtrd 4639 . . . . . . . . . . . . 13 (𝜑 → (2 · 𝐵) ≤ (abs‘(((𝐹𝑉) − (𝐹𝑈)) / (𝑉𝑈))))
146136, 15, 72, 137, 25, 138, 145unbdqndv2lem1 32139 . . . . . . . . . . . 12 (𝜑 → ((𝐵 · (abs‘(𝑉𝑈))) ≤ (abs‘((𝐹𝑉) − (𝐹𝐴))) ∨ (𝐵 · (abs‘(𝑉𝑈))) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))))
147146adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → ((𝐵 · (abs‘(𝑉𝑈))) ≤ (abs‘((𝐹𝑉) − (𝐹𝐴))) ∨ (𝐵 · (abs‘(𝑉𝑈))) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))))
148 orel2 398 . . . . . . . . . . 11 (¬ (𝐵 · (abs‘(𝑉𝑈))) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴))) → (((𝐵 · (abs‘(𝑉𝑈))) ≤ (abs‘((𝐹𝑉) − (𝐹𝐴))) ∨ (𝐵 · (abs‘(𝑉𝑈))) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝐵 · (abs‘(𝑉𝑈))) ≤ (abs‘((𝐹𝑉) − (𝐹𝐴)))))
149135, 147, 148sylc 65 . . . . . . . . . 10 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝐵 · (abs‘(𝑉𝑈))) ≤ (abs‘((𝐹𝑉) − (𝐹𝐴))))
150149adantr 481 . . . . . . . . 9 (((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) ∧ 𝑉 = 𝐴) → (𝐵 · (abs‘(𝑉𝑈))) ≤ (abs‘((𝐹𝑉) − (𝐹𝐴))))
151 fveq2 6148 . . . . . . . . . . . . . . 15 (𝑉 = 𝐴 → (𝐹𝑉) = (𝐹𝐴))
152151oveq1d 6619 . . . . . . . . . . . . . 14 (𝑉 = 𝐴 → ((𝐹𝑉) − (𝐹𝐴)) = ((𝐹𝐴) − (𝐹𝐴)))
153152adantl 482 . . . . . . . . . . . . 13 ((𝜑𝑉 = 𝐴) → ((𝐹𝑉) − (𝐹𝐴)) = ((𝐹𝐴) − (𝐹𝐴)))
15472subidd 10324 . . . . . . . . . . . . . 14 (𝜑 → ((𝐹𝐴) − (𝐹𝐴)) = 0)
155154adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑉 = 𝐴) → ((𝐹𝐴) − (𝐹𝐴)) = 0)
156153, 155eqtrd 2655 . . . . . . . . . . . 12 ((𝜑𝑉 = 𝐴) → ((𝐹𝑉) − (𝐹𝐴)) = 0)
157156fveq2d 6152 . . . . . . . . . . 11 ((𝜑𝑉 = 𝐴) → (abs‘((𝐹𝑉) − (𝐹𝐴))) = (abs‘0))
15819a1i 11 . . . . . . . . . . 11 ((𝜑𝑉 = 𝐴) → (abs‘0) = 0)
159157, 158eqtrd 2655 . . . . . . . . . 10 ((𝜑𝑉 = 𝐴) → (abs‘((𝐹𝑉) − (𝐹𝐴))) = 0)
160159adantlr 750 . . . . . . . . 9 (((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) ∧ 𝑉 = 𝐴) → (abs‘((𝐹𝑉) − (𝐹𝐴))) = 0)
161150, 160breqtrd 4639 . . . . . . . 8 (((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) ∧ 𝑉 = 𝐴) → (𝐵 · (abs‘(𝑉𝑈))) ≤ 0)
162132breq1d 4623 . . . . . . . . . . 11 (𝜑 → ((𝐵 · (abs‘(𝑉𝑈))) ≤ 0 ↔ (𝐵 · (𝑉𝑈)) ≤ 0))
16347, 162mtbird 315 . . . . . . . . . 10 (𝜑 → ¬ (𝐵 · (abs‘(𝑉𝑈))) ≤ 0)
164163adantr 481 . . . . . . . . 9 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → ¬ (𝐵 · (abs‘(𝑉𝑈))) ≤ 0)
165164adantr 481 . . . . . . . 8 (((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) ∧ 𝑉 = 𝐴) → ¬ (𝐵 · (abs‘(𝑉𝑈))) ≤ 0)
166161, 165pm2.65da 599 . . . . . . 7 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → ¬ 𝑉 = 𝐴)
167166neqned 2797 . . . . . 6 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝑉𝐴)
168130, 167jca 554 . . . . 5 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝑉𝑋𝑉𝐴))
169 eldifsn 4287 . . . . 5 (𝑉 ∈ (𝑋 ∖ {𝐴}) ↔ (𝑉𝑋𝑉𝐴))
170168, 169sylibr 224 . . . 4 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝑉 ∈ (𝑋 ∖ {𝐴}))
171129, 170eqeltrd 2698 . . 3 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝑊 ∈ (𝑋 ∖ {𝐴}))
172129oveq1d 6619 . . . . . . 7 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝑊𝐴) = (𝑉𝐴))
173172fveq2d 6152 . . . . . 6 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (abs‘(𝑊𝐴)) = (abs‘(𝑉𝐴)))
17434, 29, 36abssubge0d 14104 . . . . . . 7 (𝜑 → (abs‘(𝑉𝐴)) = (𝑉𝐴))
175174adantr 481 . . . . . 6 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (abs‘(𝑉𝐴)) = (𝑉𝐴))
176173, 175eqtrd 2655 . . . . 5 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (abs‘(𝑊𝐴)) = (𝑉𝐴))
17729, 34resubcld 10402 . . . . . . 7 (𝜑 → (𝑉𝐴) ∈ ℝ)
17830, 34, 29, 35lesub2dd 10588 . . . . . . 7 (𝜑 → (𝑉𝐴) ≤ (𝑉𝑈))
179177, 31, 63, 178, 65lelttrd 10139 . . . . . 6 (𝜑 → (𝑉𝐴) < 𝐷)
180179adantr 481 . . . . 5 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝑉𝐴) < 𝐷)
181176, 180eqbrtrd 4635 . . . 4 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (abs‘(𝑊𝐴)) < 𝐷)
182174, 177eqeltrd 2698 . . . . . . . . 9 (𝜑 → (abs‘(𝑉𝐴)) ∈ ℝ)
18326, 182remulcld 10014 . . . . . . . 8 (𝜑 → (𝐵 · (abs‘(𝑉𝐴))) ∈ ℝ)
184183adantr 481 . . . . . . 7 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝐵 · (abs‘(𝑉𝐴))) ∈ ℝ)
185132, 45eqeltrd 2698 . . . . . . . 8 (𝜑 → (𝐵 · (abs‘(𝑉𝑈))) ∈ ℝ)
186185adantr 481 . . . . . . 7 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝐵 · (abs‘(𝑉𝑈))) ∈ ℝ)
187136, 72subcld 10336 . . . . . . . . 9 (𝜑 → ((𝐹𝑉) − (𝐹𝐴)) ∈ ℂ)
188187abscld 14109 . . . . . . . 8 (𝜑 → (abs‘((𝐹𝑉) − (𝐹𝐴))) ∈ ℝ)
189188adantr 481 . . . . . . 7 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (abs‘((𝐹𝑉) − (𝐹𝐴))) ∈ ℝ)
190131, 31eqeltrd 2698 . . . . . . . . 9 (𝜑 → (abs‘(𝑉𝑈)) ∈ ℝ)
191178, 174, 1313brtr4d 4645 . . . . . . . . 9 (𝜑 → (abs‘(𝑉𝐴)) ≤ (abs‘(𝑉𝑈)))
192182, 190, 26, 76, 191lemul2ad 10908 . . . . . . . 8 (𝜑 → (𝐵 · (abs‘(𝑉𝐴))) ≤ (𝐵 · (abs‘(𝑉𝑈))))
193192adantr 481 . . . . . . 7 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝐵 · (abs‘(𝑉𝐴))) ≤ (𝐵 · (abs‘(𝑉𝑈))))
194184, 186, 189, 193, 149letrd 10138 . . . . . 6 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝐵 · (abs‘(𝑉𝐴))) ≤ (abs‘((𝐹𝑉) − (𝐹𝐴))))
19526adantr 481 . . . . . . 7 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝐵 ∈ ℝ)
196177recnd 10012 . . . . . . . . 9 (𝜑 → (𝑉𝐴) ∈ ℂ)
197196adantr 481 . . . . . . . 8 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝑉𝐴) ∈ ℂ)
19829recnd 10012 . . . . . . . . . 10 (𝜑𝑉 ∈ ℂ)
199198adantr 481 . . . . . . . . 9 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝑉 ∈ ℂ)
200112adantr 481 . . . . . . . . 9 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝐴 ∈ ℂ)
201199, 200, 167subne0d 10345 . . . . . . . 8 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝑉𝐴) ≠ 0)
202197, 201absrpcld 14121 . . . . . . 7 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (abs‘(𝑉𝐴)) ∈ ℝ+)
203195, 189, 202lemuldivd 11865 . . . . . 6 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → ((𝐵 · (abs‘(𝑉𝐴))) ≤ (abs‘((𝐹𝑉) − (𝐹𝐴))) ↔ 𝐵 ≤ ((abs‘((𝐹𝑉) − (𝐹𝐴))) / (abs‘(𝑉𝐴)))))
204194, 203mpbid 222 . . . . 5 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝐵 ≤ ((abs‘((𝐹𝑉) − (𝐹𝐴))) / (abs‘(𝑉𝐴))))
205129fveq2d 6152 . . . . . . . . 9 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝐺𝑊) = (𝐺𝑉))
20698a1i 11 . . . . . . . . . 10 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝐺 = (𝑧 ∈ (𝑋 ∖ {𝐴}) ↦ (((𝐹𝑧) − (𝐹𝐴)) / (𝑧𝐴))))
207 fveq2 6148 . . . . . . . . . . . . 13 (𝑧 = 𝑉 → (𝐹𝑧) = (𝐹𝑉))
208207oveq1d 6619 . . . . . . . . . . . 12 (𝑧 = 𝑉 → ((𝐹𝑧) − (𝐹𝐴)) = ((𝐹𝑉) − (𝐹𝐴)))
209 oveq1 6611 . . . . . . . . . . . 12 (𝑧 = 𝑉 → (𝑧𝐴) = (𝑉𝐴))
210208, 209oveq12d 6622 . . . . . . . . . . 11 (𝑧 = 𝑉 → (((𝐹𝑧) − (𝐹𝐴)) / (𝑧𝐴)) = (((𝐹𝑉) − (𝐹𝐴)) / (𝑉𝐴)))
211210adantl 482 . . . . . . . . . 10 (((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) ∧ 𝑧 = 𝑉) → (((𝐹𝑧) − (𝐹𝐴)) / (𝑧𝐴)) = (((𝐹𝑉) − (𝐹𝐴)) / (𝑉𝐴)))
212 ovex 6632 . . . . . . . . . . 11 (((𝐹𝑉) − (𝐹𝐴)) / (𝑉𝐴)) ∈ V
213212a1i 11 . . . . . . . . . 10 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (((𝐹𝑉) − (𝐹𝐴)) / (𝑉𝐴)) ∈ V)
214206, 211, 170, 213fvmptd 6245 . . . . . . . . 9 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝐺𝑉) = (((𝐹𝑉) − (𝐹𝐴)) / (𝑉𝐴)))
215205, 214eqtrd 2655 . . . . . . . 8 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝐺𝑊) = (((𝐹𝑉) − (𝐹𝐴)) / (𝑉𝐴)))
216215fveq2d 6152 . . . . . . 7 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (abs‘(𝐺𝑊)) = (abs‘(((𝐹𝑉) − (𝐹𝐴)) / (𝑉𝐴))))
217187adantr 481 . . . . . . . 8 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → ((𝐹𝑉) − (𝐹𝐴)) ∈ ℂ)
218217, 197, 201absdivd 14128 . . . . . . 7 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (abs‘(((𝐹𝑉) − (𝐹𝐴)) / (𝑉𝐴))) = ((abs‘((𝐹𝑉) − (𝐹𝐴))) / (abs‘(𝑉𝐴))))
219216, 218eqtrd 2655 . . . . . 6 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (abs‘(𝐺𝑊)) = ((abs‘((𝐹𝑉) − (𝐹𝐴))) / (abs‘(𝑉𝐴))))
220219eqcomd 2627 . . . . 5 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → ((abs‘((𝐹𝑉) − (𝐹𝐴))) / (abs‘(𝑉𝐴))) = (abs‘(𝐺𝑊)))
221204, 220breqtrd 4639 . . . 4 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝐵 ≤ (abs‘(𝐺𝑊)))
222181, 221jca 554 . . 3 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → ((abs‘(𝑊𝐴)) < 𝐷𝐵 ≤ (abs‘(𝐺𝑊))))
223171, 222jca 554 . 2 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝑊 ∈ (𝑋 ∖ {𝐴}) ∧ ((abs‘(𝑊𝐴)) < 𝐷𝐵 ≤ (abs‘(𝐺𝑊)))))
224125, 223pm2.61dan 831 1 (𝜑 → (𝑊 ∈ (𝑋 ∖ {𝐴}) ∧ ((abs‘(𝑊𝐴)) < 𝐷𝐵 ≤ (abs‘(𝐺𝑊)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384   = wceq 1480  wcel 1987  wne 2790  Vcvv 3186  cdif 3552  wss 3555  ifcif 4058  {csn 4148   class class class wbr 4613  cmpt 4673  wf 5843  cfv 5847  (class class class)co 6604  cc 9878  cr 9879  0cc0 9880   · cmul 9885   < clt 10018  cle 10019  cmin 10210   / cdiv 10628  2c2 11014  +crp 11776  abscabs 13908
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-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957  ax-pre-sup 9958
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 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902  df-sup 8292  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-div 10629  df-nn 10965  df-2 11023  df-3 11024  df-n0 11237  df-z 11322  df-uz 11632  df-rp 11777  df-seq 12742  df-exp 12801  df-cj 13773  df-re 13774  df-im 13775  df-sqrt 13909  df-abs 13910
This theorem is referenced by:  unbdqndv2  32141
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