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Theorem unbdqndv2lem2 34786
Description: Lemma for unbdqndv2 34787. (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 4479 . . . . . 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 2776 . . . 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 766 . . . . . . . . 9 (((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) ∧ 𝑈 = 𝐴) → (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴))))
9 fveq2 6825 . . . . . . . . . . . . . . 15 (𝑈 = 𝐴 → (𝐹𝑈) = (𝐹𝐴))
109eqcomd 2742 . . . . . . . . . . . . . 14 (𝑈 = 𝐴 → (𝐹𝐴) = (𝐹𝑈))
1110oveq2d 7353 . . . . . . . . . . . . 13 (𝑈 = 𝐴 → ((𝐹𝑈) − (𝐹𝐴)) = ((𝐹𝑈) − (𝐹𝑈)))
1211fveq2d 6829 . . . . . . . . . . . 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, 6ffvelcdmd 7018 . . . . . . . . . . . . . . 15 (𝜑 → (𝐹𝑈) ∈ ℂ)
1615subidd 11421 . . . . . . . . . . . . . 14 (𝜑 → ((𝐹𝑈) − (𝐹𝑈)) = 0)
1716fveq2d 6829 . . . . . . . . . . . . 13 (𝜑 → (abs‘((𝐹𝑈) − (𝐹𝑈))) = (abs‘0))
1817adantr 481 . . . . . . . . . . . 12 ((𝜑𝑈 = 𝐴) → (abs‘((𝐹𝑈) − (𝐹𝑈))) = (abs‘0))
19 abs0 15096 . . . . . . . . . . . . 13 (abs‘0) = 0
2019a1i 11 . . . . . . . . . . . 12 ((𝜑𝑈 = 𝐴) → (abs‘0) = 0)
2118, 20eqtrd 2776 . . . . . . . . . . 11 ((𝜑𝑈 = 𝐴) → (abs‘((𝐹𝑈) − (𝐹𝑈))) = 0)
2213, 21eqtrd 2776 . . . . . . . . . 10 ((𝜑𝑈 = 𝐴) → (abs‘((𝐹𝑈) − (𝐹𝐴))) = 0)
2322adantlr 712 . . . . . . . . 9 (((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) ∧ 𝑈 = 𝐴) → (abs‘((𝐹𝑈) − (𝐹𝐴))) = 0)
248, 23breqtrd 5118 . . . . . . . 8 (((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) ∧ 𝑈 = 𝐴) → (𝐵 · (𝑉𝑈)) ≤ 0)
25 unbdqndv2lem2.b . . . . . . . . . . . . 13 (𝜑𝐵 ∈ ℝ+)
2625rpred 12873 . . . . . . . . . . . 12 (𝜑𝐵 ∈ ℝ)
27 unbdqndv2lem2.x . . . . . . . . . . . . . 14 (𝜑𝑋 ⊆ ℝ)
28 unbdqndv2lem2.v . . . . . . . . . . . . . 14 (𝜑𝑉𝑋)
2927, 28sseldd 3933 . . . . . . . . . . . . 13 (𝜑𝑉 ∈ ℝ)
3027, 6sseldd 3933 . . . . . . . . . . . . 13 (𝜑𝑈 ∈ ℝ)
3129, 30resubcld 11504 . . . . . . . . . . . 12 (𝜑 → (𝑉𝑈) ∈ ℝ)
3225rpgt0d 12876 . . . . . . . . . . . 12 (𝜑 → 0 < 𝐵)
33 unbdqndv2lem2.a . . . . . . . . . . . . . . . 16 (𝜑𝐴𝑋)
3427, 33sseldd 3933 . . . . . . . . . . . . . . 15 (𝜑𝐴 ∈ ℝ)
35 unbdqndv2lem2.2 . . . . . . . . . . . . . . 15 (𝜑𝑈𝐴)
36 unbdqndv2lem2.3 . . . . . . . . . . . . . . 15 (𝜑𝐴𝑉)
3730, 34, 29, 35, 36letrd 11233 . . . . . . . . . . . . . 14 (𝜑𝑈𝑉)
38 unbdqndv2lem2.1 . . . . . . . . . . . . . . 15 (𝜑𝑈𝑉)
3938necomd 2996 . . . . . . . . . . . . . 14 (𝜑𝑉𝑈)
4030, 29, 37, 39leneltd 11230 . . . . . . . . . . . . 13 (𝜑𝑈 < 𝑉)
4130, 29posdifd 11663 . . . . . . . . . . . . 13 (𝜑 → (𝑈 < 𝑉 ↔ 0 < (𝑉𝑈)))
4240, 41mpbid 231 . . . . . . . . . . . 12 (𝜑 → 0 < (𝑉𝑈))
4326, 31, 32, 42mulgt0d 11231 . . . . . . . . . . 11 (𝜑 → 0 < (𝐵 · (𝑉𝑈)))
44 0red 11079 . . . . . . . . . . . 12 (𝜑 → 0 ∈ ℝ)
4526, 31remulcld 11106 . . . . . . . . . . . 12 (𝜑 → (𝐵 · (𝑉𝑈)) ∈ ℝ)
4644, 45ltnled 11223 . . . . . . . . . . 11 (𝜑 → (0 < (𝐵 · (𝑉𝑈)) ↔ ¬ (𝐵 · (𝑉𝑈)) ≤ 0))
4743, 46mpbid 231 . . . . . . . . . 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 814 . . . . . . 7 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → ¬ 𝑈 = 𝐴)
5150neqned 2947 . . . . . 6 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝑈𝐴)
527, 51jca 512 . . . . 5 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝑈𝑋𝑈𝐴))
53 eldifsn 4734 . . . . 5 (𝑈 ∈ (𝑋 ∖ {𝐴}) ↔ (𝑈𝑋𝑈𝐴))
5452, 53sylibr 233 . . . 4 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝑈 ∈ (𝑋 ∖ {𝐴}))
555, 54eqeltrd 2837 . . 3 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝑊 ∈ (𝑋 ∖ {𝐴}))
565oveq1d 7352 . . . . . . 7 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝑊𝐴) = (𝑈𝐴))
5756fveq2d 6829 . . . . . 6 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (abs‘(𝑊𝐴)) = (abs‘(𝑈𝐴)))
5830, 34, 35abssuble0d 15243 . . . . . . 7 (𝜑 → (abs‘(𝑈𝐴)) = (𝐴𝑈))
5958adantr 481 . . . . . 6 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (abs‘(𝑈𝐴)) = (𝐴𝑈))
6057, 59eqtrd 2776 . . . . 5 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (abs‘(𝑊𝐴)) = (𝐴𝑈))
6134, 30resubcld 11504 . . . . . . 7 (𝜑 → (𝐴𝑈) ∈ ℝ)
62 unbdqndv2lem2.d . . . . . . . 8 (𝜑𝐷 ∈ ℝ+)
6362rpred 12873 . . . . . . 7 (𝜑𝐷 ∈ ℝ)
6434, 29, 30, 36lesub1dd 11692 . . . . . . 7 (𝜑 → (𝐴𝑈) ≤ (𝑉𝑈))
65 unbdqndv2lem2.4 . . . . . . 7 (𝜑 → (𝑉𝑈) < 𝐷)
6661, 31, 63, 64, 65lelttrd 11234 . . . . . 6 (𝜑 → (𝐴𝑈) < 𝐷)
6766adantr 481 . . . . 5 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝐴𝑈) < 𝐷)
6860, 67eqbrtrd 5114 . . . 4 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (abs‘(𝑊𝐴)) < 𝐷)
6926, 61remulcld 11106 . . . . . . . 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, 33ffvelcdmd 7018 . . . . . . . . . 10 (𝜑 → (𝐹𝐴) ∈ ℂ)
7315, 72subcld 11433 . . . . . . . . 9 (𝜑 → ((𝐹𝑈) − (𝐹𝐴)) ∈ ℂ)
7473abscld 15247 . . . . . . . 8 (𝜑 → (abs‘((𝐹𝑈) − (𝐹𝐴))) ∈ ℝ)
7574adantr 481 . . . . . . 7 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (abs‘((𝐹𝑈) − (𝐹𝐴))) ∈ ℝ)
7644, 26, 32ltled 11224 . . . . . . . . 9 (𝜑 → 0 ≤ 𝐵)
7761, 31, 26, 76, 64lemul2ad 12016 . . . . . . . 8 (𝜑 → (𝐵 · (𝐴𝑈)) ≤ (𝐵 · (𝑉𝑈)))
7877adantr 481 . . . . . . 7 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝐵 · (𝐴𝑈)) ≤ (𝐵 · (𝑉𝑈)))
79 simpr 485 . . . . . . 7 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴))))
8070, 71, 75, 78, 79letrd 11233 . . . . . 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 2996 . . . . . . . . . . . 12 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝐴𝑈)
8583, 84jca 512 . . . . . . . . . . 11 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝑈𝐴𝐴𝑈))
8630, 34ltlend 11221 . . . . . . . . . . . 12 (𝜑 → (𝑈 < 𝐴 ↔ (𝑈𝐴𝐴𝑈)))
8786adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝑈 < 𝐴 ↔ (𝑈𝐴𝐴𝑈)))
8885, 87mpbird 256 . . . . . . . . . 10 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝑈 < 𝐴)
8930, 34posdifd 11663 . . . . . . . . . . 11 (𝜑 → (𝑈 < 𝐴 ↔ 0 < (𝐴𝑈)))
9089adantr 481 . . . . . . . . . 10 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝑈 < 𝐴 ↔ 0 < (𝐴𝑈)))
9188, 90mpbid 231 . . . . . . . . 9 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 0 < (𝐴𝑈))
9282, 91jca 512 . . . . . . . 8 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → ((𝐴𝑈) ∈ ℝ ∧ 0 < (𝐴𝑈)))
93 elrp 12833 . . . . . . . 8 ((𝐴𝑈) ∈ ℝ+ ↔ ((𝐴𝑈) ∈ ℝ ∧ 0 < (𝐴𝑈)))
9492, 93sylibr 233 . . . . . . 7 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝐴𝑈) ∈ ℝ+)
9581, 75, 94lemuldivd 12922 . . . . . 6 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → ((𝐵 · (𝐴𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴))) ↔ 𝐵 ≤ ((abs‘((𝐹𝑈) − (𝐹𝐴))) / (𝐴𝑈))))
9680, 95mpbid 231 . . . . 5 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝐵 ≤ ((abs‘((𝐹𝑈) − (𝐹𝐴))) / (𝐴𝑈)))
975fveq2d 6829 . . . . . . . . 9 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝐺𝑊) = (𝐺𝑈))
98 unbdqndv2lem2.g . . . . . . . . . 10 𝐺 = (𝑧 ∈ (𝑋 ∖ {𝐴}) ↦ (((𝐹𝑧) − (𝐹𝐴)) / (𝑧𝐴)))
99 fveq2 6825 . . . . . . . . . . . 12 (𝑧 = 𝑈 → (𝐹𝑧) = (𝐹𝑈))
10099oveq1d 7352 . . . . . . . . . . 11 (𝑧 = 𝑈 → ((𝐹𝑧) − (𝐹𝐴)) = ((𝐹𝑈) − (𝐹𝐴)))
101 oveq1 7344 . . . . . . . . . . 11 (𝑧 = 𝑈 → (𝑧𝐴) = (𝑈𝐴))
102100, 101oveq12d 7355 . . . . . . . . . 10 (𝑧 = 𝑈 → (((𝐹𝑧) − (𝐹𝐴)) / (𝑧𝐴)) = (((𝐹𝑈) − (𝐹𝐴)) / (𝑈𝐴)))
103 ovexd 7372 . . . . . . . . . 10 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (((𝐹𝑈) − (𝐹𝐴)) / (𝑈𝐴)) ∈ V)
10498, 102, 54, 103fvmptd3 6954 . . . . . . . . 9 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝐺𝑈) = (((𝐹𝑈) − (𝐹𝐴)) / (𝑈𝐴)))
10597, 104eqtrd 2776 . . . . . . . 8 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝐺𝑊) = (((𝐹𝑈) − (𝐹𝐴)) / (𝑈𝐴)))
106105fveq2d 6829 . . . . . . 7 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (abs‘(𝐺𝑊)) = (abs‘(((𝐹𝑈) − (𝐹𝐴)) / (𝑈𝐴))))
10773adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → ((𝐹𝑈) − (𝐹𝐴)) ∈ ℂ)
10830recnd 11104 . . . . . . . . . . 11 (𝜑𝑈 ∈ ℂ)
10934recnd 11104 . . . . . . . . . . 11 (𝜑𝐴 ∈ ℂ)
110108, 109subcld 11433 . . . . . . . . . 10 (𝜑 → (𝑈𝐴) ∈ ℂ)
111110adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝑈𝐴) ∈ ℂ)
112108adantr 481 . . . . . . . . . 10 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝑈 ∈ ℂ)
113109adantr 481 . . . . . . . . . 10 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝐴 ∈ ℂ)
114112, 113, 51subne0d 11442 . . . . . . . . 9 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝑈𝐴) ≠ 0)
115107, 111, 114absdivd 15266 . . . . . . . 8 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (abs‘(((𝐹𝑈) − (𝐹𝐴)) / (𝑈𝐴))) = ((abs‘((𝐹𝑈) − (𝐹𝐴))) / (abs‘(𝑈𝐴))))
11659oveq2d 7353 . . . . . . . 8 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → ((abs‘((𝐹𝑈) − (𝐹𝐴))) / (abs‘(𝑈𝐴))) = ((abs‘((𝐹𝑈) − (𝐹𝐴))) / (𝐴𝑈)))
117115, 116eqtrd 2776 . . . . . . 7 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (abs‘(((𝐹𝑈) − (𝐹𝐴)) / (𝑈𝐴))) = ((abs‘((𝐹𝑈) − (𝐹𝐴))) / (𝐴𝑈)))
118106, 117eqtrd 2776 . . . . . 6 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (abs‘(𝐺𝑊)) = ((abs‘((𝐹𝑈) − (𝐹𝐴))) / (𝐴𝑈)))
119118eqcomd 2742 . . . . 5 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → ((abs‘((𝐹𝑈) − (𝐹𝐴))) / (𝐴𝑈)) = (abs‘(𝐺𝑊)))
12096, 119breqtrd 5118 . . . 4 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝐵 ≤ (abs‘(𝐺𝑊)))
12168, 120jca 512 . . 3 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → ((abs‘(𝑊𝐴)) < 𝐷𝐵 ≤ (abs‘(𝐺𝑊))))
12255, 121jca 512 . 2 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝑊 ∈ (𝑋 ∖ {𝐴}) ∧ ((abs‘(𝑊𝐴)) < 𝐷𝐵 ≤ (abs‘(𝐺𝑊)))))
1231a1i 11 . . . . 5 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝑊 = if((𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴))), 𝑈, 𝑉))
124 simpr 485 . . . . . 6 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴))))
125124iffalsed 4484 . . . . 5 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → if((𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴))), 𝑈, 𝑉) = 𝑉)
126123, 125eqtrd 2776 . . . 4 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝑊 = 𝑉)
12728adantr 481 . . . . . 6 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝑉𝑋)
12830, 29, 37abssubge0d 15242 . . . . . . . . . . . . . . 15 (𝜑 → (abs‘(𝑉𝑈)) = (𝑉𝑈))
129128oveq2d 7353 . . . . . . . . . . . . . 14 (𝜑 → (𝐵 · (abs‘(𝑉𝑈))) = (𝐵 · (𝑉𝑈)))
130129breq1d 5102 . . . . . . . . . . . . 13 (𝜑 → ((𝐵 · (abs‘(𝑉𝑈))) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴))) ↔ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))))
131130adantr 481 . . . . . . . . . . . 12 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → ((𝐵 · (abs‘(𝑉𝑈))) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴))) ↔ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))))
132124, 131mtbird 324 . . . . . . . . . . 11 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → ¬ (𝐵 · (abs‘(𝑉𝑈))) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴))))
13314, 28ffvelcdmd 7018 . . . . . . . . . . . . 13 (𝜑 → (𝐹𝑉) ∈ ℂ)
13431recnd 11104 . . . . . . . . . . . . 13 (𝜑 → (𝑉𝑈) ∈ ℂ)
13544, 42gtned 11211 . . . . . . . . . . . . 13 (𝜑 → (𝑉𝑈) ≠ 0)
136 unbdqndv2lem2.5 . . . . . . . . . . . . . 14 (𝜑 → (2 · 𝐵) ≤ ((abs‘((𝐹𝑉) − (𝐹𝑈))) / (𝑉𝑈)))
137133, 15subcld 11433 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝐹𝑉) − (𝐹𝑈)) ∈ ℂ)
138137, 134, 135absdivd 15266 . . . . . . . . . . . . . . . 16 (𝜑 → (abs‘(((𝐹𝑉) − (𝐹𝑈)) / (𝑉𝑈))) = ((abs‘((𝐹𝑉) − (𝐹𝑈))) / (abs‘(𝑉𝑈))))
139128oveq2d 7353 . . . . . . . . . . . . . . . 16 (𝜑 → ((abs‘((𝐹𝑉) − (𝐹𝑈))) / (abs‘(𝑉𝑈))) = ((abs‘((𝐹𝑉) − (𝐹𝑈))) / (𝑉𝑈)))
140138, 139eqtrd 2776 . . . . . . . . . . . . . . 15 (𝜑 → (abs‘(((𝐹𝑉) − (𝐹𝑈)) / (𝑉𝑈))) = ((abs‘((𝐹𝑉) − (𝐹𝑈))) / (𝑉𝑈)))
141140eqcomd 2742 . . . . . . . . . . . . . 14 (𝜑 → ((abs‘((𝐹𝑉) − (𝐹𝑈))) / (𝑉𝑈)) = (abs‘(((𝐹𝑉) − (𝐹𝑈)) / (𝑉𝑈))))
142136, 141breqtrd 5118 . . . . . . . . . . . . 13 (𝜑 → (2 · 𝐵) ≤ (abs‘(((𝐹𝑉) − (𝐹𝑈)) / (𝑉𝑈))))
143133, 15, 72, 134, 25, 135, 142unbdqndv2lem1 34785 . . . . . . . . . . . 12 (𝜑 → ((𝐵 · (abs‘(𝑉𝑈))) ≤ (abs‘((𝐹𝑉) − (𝐹𝐴))) ∨ (𝐵 · (abs‘(𝑉𝑈))) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))))
144143adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → ((𝐵 · (abs‘(𝑉𝑈))) ≤ (abs‘((𝐹𝑉) − (𝐹𝐴))) ∨ (𝐵 · (abs‘(𝑉𝑈))) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))))
145 orel2 888 . . . . . . . . . . 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)))))
146132, 144, 145sylc 65 . . . . . . . . . 10 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝐵 · (abs‘(𝑉𝑈))) ≤ (abs‘((𝐹𝑉) − (𝐹𝐴))))
147146adantr 481 . . . . . . . . 9 (((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) ∧ 𝑉 = 𝐴) → (𝐵 · (abs‘(𝑉𝑈))) ≤ (abs‘((𝐹𝑉) − (𝐹𝐴))))
148 fveq2 6825 . . . . . . . . . . . . . . 15 (𝑉 = 𝐴 → (𝐹𝑉) = (𝐹𝐴))
149148oveq1d 7352 . . . . . . . . . . . . . 14 (𝑉 = 𝐴 → ((𝐹𝑉) − (𝐹𝐴)) = ((𝐹𝐴) − (𝐹𝐴)))
150149adantl 482 . . . . . . . . . . . . 13 ((𝜑𝑉 = 𝐴) → ((𝐹𝑉) − (𝐹𝐴)) = ((𝐹𝐴) − (𝐹𝐴)))
15172subidd 11421 . . . . . . . . . . . . . 14 (𝜑 → ((𝐹𝐴) − (𝐹𝐴)) = 0)
152151adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑉 = 𝐴) → ((𝐹𝐴) − (𝐹𝐴)) = 0)
153150, 152eqtrd 2776 . . . . . . . . . . . 12 ((𝜑𝑉 = 𝐴) → ((𝐹𝑉) − (𝐹𝐴)) = 0)
154153fveq2d 6829 . . . . . . . . . . 11 ((𝜑𝑉 = 𝐴) → (abs‘((𝐹𝑉) − (𝐹𝐴))) = (abs‘0))
15519a1i 11 . . . . . . . . . . 11 ((𝜑𝑉 = 𝐴) → (abs‘0) = 0)
156154, 155eqtrd 2776 . . . . . . . . . 10 ((𝜑𝑉 = 𝐴) → (abs‘((𝐹𝑉) − (𝐹𝐴))) = 0)
157156adantlr 712 . . . . . . . . 9 (((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) ∧ 𝑉 = 𝐴) → (abs‘((𝐹𝑉) − (𝐹𝐴))) = 0)
158147, 157breqtrd 5118 . . . . . . . 8 (((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) ∧ 𝑉 = 𝐴) → (𝐵 · (abs‘(𝑉𝑈))) ≤ 0)
159129breq1d 5102 . . . . . . . . . . 11 (𝜑 → ((𝐵 · (abs‘(𝑉𝑈))) ≤ 0 ↔ (𝐵 · (𝑉𝑈)) ≤ 0))
16047, 159mtbird 324 . . . . . . . . . 10 (𝜑 → ¬ (𝐵 · (abs‘(𝑉𝑈))) ≤ 0)
161160adantr 481 . . . . . . . . 9 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → ¬ (𝐵 · (abs‘(𝑉𝑈))) ≤ 0)
162161adantr 481 . . . . . . . 8 (((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) ∧ 𝑉 = 𝐴) → ¬ (𝐵 · (abs‘(𝑉𝑈))) ≤ 0)
163158, 162pm2.65da 814 . . . . . . 7 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → ¬ 𝑉 = 𝐴)
164163neqned 2947 . . . . . 6 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝑉𝐴)
165127, 164jca 512 . . . . 5 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝑉𝑋𝑉𝐴))
166 eldifsn 4734 . . . . 5 (𝑉 ∈ (𝑋 ∖ {𝐴}) ↔ (𝑉𝑋𝑉𝐴))
167165, 166sylibr 233 . . . 4 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝑉 ∈ (𝑋 ∖ {𝐴}))
168126, 167eqeltrd 2837 . . 3 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝑊 ∈ (𝑋 ∖ {𝐴}))
169126oveq1d 7352 . . . . . . 7 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝑊𝐴) = (𝑉𝐴))
170169fveq2d 6829 . . . . . 6 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (abs‘(𝑊𝐴)) = (abs‘(𝑉𝐴)))
17134, 29, 36abssubge0d 15242 . . . . . . 7 (𝜑 → (abs‘(𝑉𝐴)) = (𝑉𝐴))
172171adantr 481 . . . . . 6 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (abs‘(𝑉𝐴)) = (𝑉𝐴))
173170, 172eqtrd 2776 . . . . 5 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (abs‘(𝑊𝐴)) = (𝑉𝐴))
17429, 34resubcld 11504 . . . . . . 7 (𝜑 → (𝑉𝐴) ∈ ℝ)
17530, 34, 29, 35lesub2dd 11693 . . . . . . 7 (𝜑 → (𝑉𝐴) ≤ (𝑉𝑈))
176174, 31, 63, 175, 65lelttrd 11234 . . . . . 6 (𝜑 → (𝑉𝐴) < 𝐷)
177176adantr 481 . . . . 5 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝑉𝐴) < 𝐷)
178173, 177eqbrtrd 5114 . . . 4 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (abs‘(𝑊𝐴)) < 𝐷)
179171, 174eqeltrd 2837 . . . . . . . . 9 (𝜑 → (abs‘(𝑉𝐴)) ∈ ℝ)
18026, 179remulcld 11106 . . . . . . . 8 (𝜑 → (𝐵 · (abs‘(𝑉𝐴))) ∈ ℝ)
181180adantr 481 . . . . . . 7 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝐵 · (abs‘(𝑉𝐴))) ∈ ℝ)
182129, 45eqeltrd 2837 . . . . . . . 8 (𝜑 → (𝐵 · (abs‘(𝑉𝑈))) ∈ ℝ)
183182adantr 481 . . . . . . 7 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝐵 · (abs‘(𝑉𝑈))) ∈ ℝ)
184133, 72subcld 11433 . . . . . . . . 9 (𝜑 → ((𝐹𝑉) − (𝐹𝐴)) ∈ ℂ)
185184abscld 15247 . . . . . . . 8 (𝜑 → (abs‘((𝐹𝑉) − (𝐹𝐴))) ∈ ℝ)
186185adantr 481 . . . . . . 7 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (abs‘((𝐹𝑉) − (𝐹𝐴))) ∈ ℝ)
187128, 31eqeltrd 2837 . . . . . . . . 9 (𝜑 → (abs‘(𝑉𝑈)) ∈ ℝ)
188175, 171, 1283brtr4d 5124 . . . . . . . . 9 (𝜑 → (abs‘(𝑉𝐴)) ≤ (abs‘(𝑉𝑈)))
189179, 187, 26, 76, 188lemul2ad 12016 . . . . . . . 8 (𝜑 → (𝐵 · (abs‘(𝑉𝐴))) ≤ (𝐵 · (abs‘(𝑉𝑈))))
190189adantr 481 . . . . . . 7 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝐵 · (abs‘(𝑉𝐴))) ≤ (𝐵 · (abs‘(𝑉𝑈))))
191181, 183, 186, 190, 146letrd 11233 . . . . . 6 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝐵 · (abs‘(𝑉𝐴))) ≤ (abs‘((𝐹𝑉) − (𝐹𝐴))))
19226adantr 481 . . . . . . 7 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝐵 ∈ ℝ)
193174recnd 11104 . . . . . . . . 9 (𝜑 → (𝑉𝐴) ∈ ℂ)
194193adantr 481 . . . . . . . 8 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝑉𝐴) ∈ ℂ)
19529recnd 11104 . . . . . . . . . 10 (𝜑𝑉 ∈ ℂ)
196195adantr 481 . . . . . . . . 9 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝑉 ∈ ℂ)
197109adantr 481 . . . . . . . . 9 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝐴 ∈ ℂ)
198196, 197, 164subne0d 11442 . . . . . . . 8 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝑉𝐴) ≠ 0)
199194, 198absrpcld 15259 . . . . . . 7 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (abs‘(𝑉𝐴)) ∈ ℝ+)
200192, 186, 199lemuldivd 12922 . . . . . 6 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → ((𝐵 · (abs‘(𝑉𝐴))) ≤ (abs‘((𝐹𝑉) − (𝐹𝐴))) ↔ 𝐵 ≤ ((abs‘((𝐹𝑉) − (𝐹𝐴))) / (abs‘(𝑉𝐴)))))
201191, 200mpbid 231 . . . . 5 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝐵 ≤ ((abs‘((𝐹𝑉) − (𝐹𝐴))) / (abs‘(𝑉𝐴))))
202126fveq2d 6829 . . . . . . . . 9 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝐺𝑊) = (𝐺𝑉))
203 fveq2 6825 . . . . . . . . . . . 12 (𝑧 = 𝑉 → (𝐹𝑧) = (𝐹𝑉))
204203oveq1d 7352 . . . . . . . . . . 11 (𝑧 = 𝑉 → ((𝐹𝑧) − (𝐹𝐴)) = ((𝐹𝑉) − (𝐹𝐴)))
205 oveq1 7344 . . . . . . . . . . 11 (𝑧 = 𝑉 → (𝑧𝐴) = (𝑉𝐴))
206204, 205oveq12d 7355 . . . . . . . . . 10 (𝑧 = 𝑉 → (((𝐹𝑧) − (𝐹𝐴)) / (𝑧𝐴)) = (((𝐹𝑉) − (𝐹𝐴)) / (𝑉𝐴)))
207 ovexd 7372 . . . . . . . . . 10 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (((𝐹𝑉) − (𝐹𝐴)) / (𝑉𝐴)) ∈ V)
20898, 206, 167, 207fvmptd3 6954 . . . . . . . . 9 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝐺𝑉) = (((𝐹𝑉) − (𝐹𝐴)) / (𝑉𝐴)))
209202, 208eqtrd 2776 . . . . . . . 8 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝐺𝑊) = (((𝐹𝑉) − (𝐹𝐴)) / (𝑉𝐴)))
210209fveq2d 6829 . . . . . . 7 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (abs‘(𝐺𝑊)) = (abs‘(((𝐹𝑉) − (𝐹𝐴)) / (𝑉𝐴))))
211184adantr 481 . . . . . . . 8 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → ((𝐹𝑉) − (𝐹𝐴)) ∈ ℂ)
212211, 194, 198absdivd 15266 . . . . . . 7 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (abs‘(((𝐹𝑉) − (𝐹𝐴)) / (𝑉𝐴))) = ((abs‘((𝐹𝑉) − (𝐹𝐴))) / (abs‘(𝑉𝐴))))
213210, 212eqtrd 2776 . . . . . 6 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (abs‘(𝐺𝑊)) = ((abs‘((𝐹𝑉) − (𝐹𝐴))) / (abs‘(𝑉𝐴))))
214213eqcomd 2742 . . . . 5 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → ((abs‘((𝐹𝑉) − (𝐹𝐴))) / (abs‘(𝑉𝐴))) = (abs‘(𝐺𝑊)))
215201, 214breqtrd 5118 . . . 4 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝐵 ≤ (abs‘(𝐺𝑊)))
216178, 215jca 512 . . 3 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → ((abs‘(𝑊𝐴)) < 𝐷𝐵 ≤ (abs‘(𝐺𝑊))))
217168, 216jca 512 . 2 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝑊 ∈ (𝑋 ∖ {𝐴}) ∧ ((abs‘(𝑊𝐴)) < 𝐷𝐵 ≤ (abs‘(𝐺𝑊)))))
218122, 217pm2.61dan 810 1 (𝜑 → (𝑊 ∈ (𝑋 ∖ {𝐴}) ∧ ((abs‘(𝑊𝐴)) < 𝐷𝐵 ≤ (abs‘(𝐺𝑊)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  wo 844   = wceq 1540  wcel 2105  wne 2940  Vcvv 3441  cdif 3895  wss 3898  ifcif 4473  {csn 4573   class class class wbr 5092  cmpt 5175  wf 6475  cfv 6479  (class class class)co 7337  cc 10970  cr 10971  0cc0 10972   · cmul 10977   < clt 11110  cle 11111  cmin 11306   / cdiv 11733  2c2 12129  +crp 12831  abscabs 15044
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5243  ax-nul 5250  ax-pow 5308  ax-pr 5372  ax-un 7650  ax-cnex 11028  ax-resscn 11029  ax-1cn 11030  ax-icn 11031  ax-addcl 11032  ax-addrcl 11033  ax-mulcl 11034  ax-mulrcl 11035  ax-mulcom 11036  ax-addass 11037  ax-mulass 11038  ax-distr 11039  ax-i2m1 11040  ax-1ne0 11041  ax-1rid 11042  ax-rnegex 11043  ax-rrecex 11044  ax-cnre 11045  ax-pre-lttri 11046  ax-pre-lttrn 11047  ax-pre-ltadd 11048  ax-pre-mulgt0 11049  ax-pre-sup 11050
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3349  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3917  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5176  df-tr 5210  df-id 5518  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5575  df-we 5577  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6238  df-ord 6305  df-on 6306  df-lim 6307  df-suc 6308  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-f1 6484  df-fo 6485  df-f1o 6486  df-fv 6487  df-riota 7293  df-ov 7340  df-oprab 7341  df-mpo 7342  df-om 7781  df-2nd 7900  df-frecs 8167  df-wrecs 8198  df-recs 8272  df-rdg 8311  df-er 8569  df-en 8805  df-dom 8806  df-sdom 8807  df-sup 9299  df-pnf 11112  df-mnf 11113  df-xr 11114  df-ltxr 11115  df-le 11116  df-sub 11308  df-neg 11309  df-div 11734  df-nn 12075  df-2 12137  df-3 12138  df-n0 12335  df-z 12421  df-uz 12684  df-rp 12832  df-seq 13823  df-exp 13884  df-cj 14909  df-re 14910  df-im 14911  df-sqrt 15045  df-abs 15046
This theorem is referenced by:  unbdqndv2  34787
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