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Theorem unbdqndv2lem2 32807
Description: Lemma for unbdqndv2 32808. (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 4236 . . . . . 6 ((𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴))) → if((𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴))), 𝑈, 𝑉) = 𝑈)
43adantl 473 . . . . 5 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → if((𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴))), 𝑈, 𝑉) = 𝑈)
52, 4eqtrd 2794 . . . 4 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝑊 = 𝑈)
6 unbdqndv2lem2.u . . . . . . 7 (𝜑𝑈𝑋)
76adantr 472 . . . . . 6 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝑈𝑋)
8 simplr 809 . . . . . . . . 9 (((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) ∧ 𝑈 = 𝐴) → (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴))))
9 fveq2 6352 . . . . . . . . . . . . . . 15 (𝑈 = 𝐴 → (𝐹𝑈) = (𝐹𝐴))
109eqcomd 2766 . . . . . . . . . . . . . 14 (𝑈 = 𝐴 → (𝐹𝐴) = (𝐹𝑈))
1110oveq2d 6829 . . . . . . . . . . . . 13 (𝑈 = 𝐴 → ((𝐹𝑈) − (𝐹𝐴)) = ((𝐹𝑈) − (𝐹𝑈)))
1211fveq2d 6356 . . . . . . . . . . . 12 (𝑈 = 𝐴 → (abs‘((𝐹𝑈) − (𝐹𝐴))) = (abs‘((𝐹𝑈) − (𝐹𝑈))))
1312adantl 473 . . . . . . . . . . 11 ((𝜑𝑈 = 𝐴) → (abs‘((𝐹𝑈) − (𝐹𝐴))) = (abs‘((𝐹𝑈) − (𝐹𝑈))))
14 unbdqndv2lem2.f . . . . . . . . . . . . . . . 16 (𝜑𝐹:𝑋⟶ℂ)
1514, 6ffvelrnd 6523 . . . . . . . . . . . . . . 15 (𝜑 → (𝐹𝑈) ∈ ℂ)
1615subidd 10572 . . . . . . . . . . . . . 14 (𝜑 → ((𝐹𝑈) − (𝐹𝑈)) = 0)
1716fveq2d 6356 . . . . . . . . . . . . 13 (𝜑 → (abs‘((𝐹𝑈) − (𝐹𝑈))) = (abs‘0))
1817adantr 472 . . . . . . . . . . . 12 ((𝜑𝑈 = 𝐴) → (abs‘((𝐹𝑈) − (𝐹𝑈))) = (abs‘0))
19 abs0 14224 . . . . . . . . . . . . 13 (abs‘0) = 0
2019a1i 11 . . . . . . . . . . . 12 ((𝜑𝑈 = 𝐴) → (abs‘0) = 0)
2118, 20eqtrd 2794 . . . . . . . . . . 11 ((𝜑𝑈 = 𝐴) → (abs‘((𝐹𝑈) − (𝐹𝑈))) = 0)
2213, 21eqtrd 2794 . . . . . . . . . 10 ((𝜑𝑈 = 𝐴) → (abs‘((𝐹𝑈) − (𝐹𝐴))) = 0)
2322adantlr 753 . . . . . . . . 9 (((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) ∧ 𝑈 = 𝐴) → (abs‘((𝐹𝑈) − (𝐹𝐴))) = 0)
248, 23breqtrd 4830 . . . . . . . 8 (((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) ∧ 𝑈 = 𝐴) → (𝐵 · (𝑉𝑈)) ≤ 0)
25 unbdqndv2lem2.b . . . . . . . . . . . . 13 (𝜑𝐵 ∈ ℝ+)
2625rpred 12065 . . . . . . . . . . . 12 (𝜑𝐵 ∈ ℝ)
27 unbdqndv2lem2.x . . . . . . . . . . . . . 14 (𝜑𝑋 ⊆ ℝ)
28 unbdqndv2lem2.v . . . . . . . . . . . . . 14 (𝜑𝑉𝑋)
2927, 28sseldd 3745 . . . . . . . . . . . . 13 (𝜑𝑉 ∈ ℝ)
3027, 6sseldd 3745 . . . . . . . . . . . . 13 (𝜑𝑈 ∈ ℝ)
3129, 30resubcld 10650 . . . . . . . . . . . 12 (𝜑 → (𝑉𝑈) ∈ ℝ)
3225rpgt0d 12068 . . . . . . . . . . . 12 (𝜑 → 0 < 𝐵)
33 unbdqndv2lem2.a . . . . . . . . . . . . . . . 16 (𝜑𝐴𝑋)
3427, 33sseldd 3745 . . . . . . . . . . . . . . 15 (𝜑𝐴 ∈ ℝ)
35 unbdqndv2lem2.2 . . . . . . . . . . . . . . 15 (𝜑𝑈𝐴)
36 unbdqndv2lem2.3 . . . . . . . . . . . . . . 15 (𝜑𝐴𝑉)
3730, 34, 29, 35, 36letrd 10386 . . . . . . . . . . . . . 14 (𝜑𝑈𝑉)
38 unbdqndv2lem2.1 . . . . . . . . . . . . . . 15 (𝜑𝑈𝑉)
3938necomd 2987 . . . . . . . . . . . . . 14 (𝜑𝑉𝑈)
4030, 29, 37, 39leneltd 10383 . . . . . . . . . . . . 13 (𝜑𝑈 < 𝑉)
4130, 29posdifd 10806 . . . . . . . . . . . . 13 (𝜑 → (𝑈 < 𝑉 ↔ 0 < (𝑉𝑈)))
4240, 41mpbid 222 . . . . . . . . . . . 12 (𝜑 → 0 < (𝑉𝑈))
4326, 31, 32, 42mulgt0d 10384 . . . . . . . . . . 11 (𝜑 → 0 < (𝐵 · (𝑉𝑈)))
44 0red 10233 . . . . . . . . . . . 12 (𝜑 → 0 ∈ ℝ)
4526, 31remulcld 10262 . . . . . . . . . . . 12 (𝜑 → (𝐵 · (𝑉𝑈)) ∈ ℝ)
4644, 45ltnled 10376 . . . . . . . . . . 11 (𝜑 → (0 < (𝐵 · (𝑉𝑈)) ↔ ¬ (𝐵 · (𝑉𝑈)) ≤ 0))
4743, 46mpbid 222 . . . . . . . . . 10 (𝜑 → ¬ (𝐵 · (𝑉𝑈)) ≤ 0)
4847adantr 472 . . . . . . . . 9 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → ¬ (𝐵 · (𝑉𝑈)) ≤ 0)
4948adantr 472 . . . . . . . 8 (((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) ∧ 𝑈 = 𝐴) → ¬ (𝐵 · (𝑉𝑈)) ≤ 0)
5024, 49pm2.65da 601 . . . . . . 7 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → ¬ 𝑈 = 𝐴)
5150neqned 2939 . . . . . 6 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝑈𝐴)
527, 51jca 555 . . . . 5 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝑈𝑋𝑈𝐴))
53 eldifsn 4462 . . . . 5 (𝑈 ∈ (𝑋 ∖ {𝐴}) ↔ (𝑈𝑋𝑈𝐴))
5452, 53sylibr 224 . . . 4 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝑈 ∈ (𝑋 ∖ {𝐴}))
555, 54eqeltrd 2839 . . 3 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝑊 ∈ (𝑋 ∖ {𝐴}))
565oveq1d 6828 . . . . . . 7 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝑊𝐴) = (𝑈𝐴))
5756fveq2d 6356 . . . . . 6 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (abs‘(𝑊𝐴)) = (abs‘(𝑈𝐴)))
5830, 34, 35abssuble0d 14370 . . . . . . 7 (𝜑 → (abs‘(𝑈𝐴)) = (𝐴𝑈))
5958adantr 472 . . . . . 6 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (abs‘(𝑈𝐴)) = (𝐴𝑈))
6057, 59eqtrd 2794 . . . . 5 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (abs‘(𝑊𝐴)) = (𝐴𝑈))
6134, 30resubcld 10650 . . . . . . 7 (𝜑 → (𝐴𝑈) ∈ ℝ)
62 unbdqndv2lem2.d . . . . . . . 8 (𝜑𝐷 ∈ ℝ+)
6362rpred 12065 . . . . . . 7 (𝜑𝐷 ∈ ℝ)
6434, 29, 30, 36lesub1dd 10835 . . . . . . 7 (𝜑 → (𝐴𝑈) ≤ (𝑉𝑈))
65 unbdqndv2lem2.4 . . . . . . 7 (𝜑 → (𝑉𝑈) < 𝐷)
6661, 31, 63, 64, 65lelttrd 10387 . . . . . 6 (𝜑 → (𝐴𝑈) < 𝐷)
6766adantr 472 . . . . 5 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝐴𝑈) < 𝐷)
6860, 67eqbrtrd 4826 . . . 4 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (abs‘(𝑊𝐴)) < 𝐷)
6926, 61remulcld 10262 . . . . . . . 8 (𝜑 → (𝐵 · (𝐴𝑈)) ∈ ℝ)
7069adantr 472 . . . . . . 7 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝐵 · (𝐴𝑈)) ∈ ℝ)
7145adantr 472 . . . . . . 7 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝐵 · (𝑉𝑈)) ∈ ℝ)
7214, 33ffvelrnd 6523 . . . . . . . . . 10 (𝜑 → (𝐹𝐴) ∈ ℂ)
7315, 72subcld 10584 . . . . . . . . 9 (𝜑 → ((𝐹𝑈) − (𝐹𝐴)) ∈ ℂ)
7473abscld 14374 . . . . . . . 8 (𝜑 → (abs‘((𝐹𝑈) − (𝐹𝐴))) ∈ ℝ)
7574adantr 472 . . . . . . 7 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (abs‘((𝐹𝑈) − (𝐹𝐴))) ∈ ℝ)
7644, 26, 32ltled 10377 . . . . . . . . 9 (𝜑 → 0 ≤ 𝐵)
7761, 31, 26, 76, 64lemul2ad 11156 . . . . . . . 8 (𝜑 → (𝐵 · (𝐴𝑈)) ≤ (𝐵 · (𝑉𝑈)))
7877adantr 472 . . . . . . 7 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝐵 · (𝐴𝑈)) ≤ (𝐵 · (𝑉𝑈)))
79 simpr 479 . . . . . . 7 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴))))
8070, 71, 75, 78, 79letrd 10386 . . . . . 6 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝐵 · (𝐴𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴))))
8126adantr 472 . . . . . . 7 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝐵 ∈ ℝ)
8261adantr 472 . . . . . . . . 9 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝐴𝑈) ∈ ℝ)
8335adantr 472 . . . . . . . . . . . 12 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝑈𝐴)
8451necomd 2987 . . . . . . . . . . . 12 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝐴𝑈)
8583, 84jca 555 . . . . . . . . . . 11 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝑈𝐴𝐴𝑈))
8630, 34ltlend 10374 . . . . . . . . . . . 12 (𝜑 → (𝑈 < 𝐴 ↔ (𝑈𝐴𝐴𝑈)))
8786adantr 472 . . . . . . . . . . 11 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝑈 < 𝐴 ↔ (𝑈𝐴𝐴𝑈)))
8885, 87mpbird 247 . . . . . . . . . 10 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝑈 < 𝐴)
8930, 34posdifd 10806 . . . . . . . . . . 11 (𝜑 → (𝑈 < 𝐴 ↔ 0 < (𝐴𝑈)))
9089adantr 472 . . . . . . . . . 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 555 . . . . . . . 8 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → ((𝐴𝑈) ∈ ℝ ∧ 0 < (𝐴𝑈)))
93 elrp 12027 . . . . . . . 8 ((𝐴𝑈) ∈ ℝ+ ↔ ((𝐴𝑈) ∈ ℝ ∧ 0 < (𝐴𝑈)))
9492, 93sylibr 224 . . . . . . 7 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝐴𝑈) ∈ ℝ+)
9581, 75, 94lemuldivd 12114 . . . . . 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 6356 . . . . . . . . 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 6352 . . . . . . . . . . . . 13 (𝑧 = 𝑈 → (𝐹𝑧) = (𝐹𝑈))
101100oveq1d 6828 . . . . . . . . . . . 12 (𝑧 = 𝑈 → ((𝐹𝑧) − (𝐹𝐴)) = ((𝐹𝑈) − (𝐹𝐴)))
102 oveq1 6820 . . . . . . . . . . . 12 (𝑧 = 𝑈 → (𝑧𝐴) = (𝑈𝐴))
103101, 102oveq12d 6831 . . . . . . . . . . 11 (𝑧 = 𝑈 → (((𝐹𝑧) − (𝐹𝐴)) / (𝑧𝐴)) = (((𝐹𝑈) − (𝐹𝐴)) / (𝑈𝐴)))
104103adantl 473 . . . . . . . . . 10 (((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) ∧ 𝑧 = 𝑈) → (((𝐹𝑧) − (𝐹𝐴)) / (𝑧𝐴)) = (((𝐹𝑈) − (𝐹𝐴)) / (𝑈𝐴)))
105 ovexd 6843 . . . . . . . . . 10 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (((𝐹𝑈) − (𝐹𝐴)) / (𝑈𝐴)) ∈ V)
10699, 104, 54, 105fvmptd 6450 . . . . . . . . 9 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝐺𝑈) = (((𝐹𝑈) − (𝐹𝐴)) / (𝑈𝐴)))
10797, 106eqtrd 2794 . . . . . . . 8 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝐺𝑊) = (((𝐹𝑈) − (𝐹𝐴)) / (𝑈𝐴)))
108107fveq2d 6356 . . . . . . 7 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (abs‘(𝐺𝑊)) = (abs‘(((𝐹𝑈) − (𝐹𝐴)) / (𝑈𝐴))))
10973adantr 472 . . . . . . . . 9 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → ((𝐹𝑈) − (𝐹𝐴)) ∈ ℂ)
11030recnd 10260 . . . . . . . . . . 11 (𝜑𝑈 ∈ ℂ)
11134recnd 10260 . . . . . . . . . . 11 (𝜑𝐴 ∈ ℂ)
112110, 111subcld 10584 . . . . . . . . . 10 (𝜑 → (𝑈𝐴) ∈ ℂ)
113112adantr 472 . . . . . . . . 9 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝑈𝐴) ∈ ℂ)
114110adantr 472 . . . . . . . . . 10 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝑈 ∈ ℂ)
115111adantr 472 . . . . . . . . . 10 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝐴 ∈ ℂ)
116114, 115, 51subne0d 10593 . . . . . . . . 9 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝑈𝐴) ≠ 0)
117109, 113, 116absdivd 14393 . . . . . . . 8 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (abs‘(((𝐹𝑈) − (𝐹𝐴)) / (𝑈𝐴))) = ((abs‘((𝐹𝑈) − (𝐹𝐴))) / (abs‘(𝑈𝐴))))
11859oveq2d 6829 . . . . . . . 8 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → ((abs‘((𝐹𝑈) − (𝐹𝐴))) / (abs‘(𝑈𝐴))) = ((abs‘((𝐹𝑈) − (𝐹𝐴))) / (𝐴𝑈)))
119117, 118eqtrd 2794 . . . . . . 7 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (abs‘(((𝐹𝑈) − (𝐹𝐴)) / (𝑈𝐴))) = ((abs‘((𝐹𝑈) − (𝐹𝐴))) / (𝐴𝑈)))
120108, 119eqtrd 2794 . . . . . 6 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (abs‘(𝐺𝑊)) = ((abs‘((𝐹𝑈) − (𝐹𝐴))) / (𝐴𝑈)))
121120eqcomd 2766 . . . . 5 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → ((abs‘((𝐹𝑈) − (𝐹𝐴))) / (𝐴𝑈)) = (abs‘(𝐺𝑊)))
12296, 121breqtrd 4830 . . . 4 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝐵 ≤ (abs‘(𝐺𝑊)))
12368, 122jca 555 . . 3 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → ((abs‘(𝑊𝐴)) < 𝐷𝐵 ≤ (abs‘(𝐺𝑊))))
12455, 123jca 555 . 2 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝑊 ∈ (𝑋 ∖ {𝐴}) ∧ ((abs‘(𝑊𝐴)) < 𝐷𝐵 ≤ (abs‘(𝐺𝑊)))))
1251a1i 11 . . . . 5 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝑊 = if((𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴))), 𝑈, 𝑉))
126 simpr 479 . . . . . 6 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴))))
127126iffalsed 4241 . . . . 5 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → if((𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴))), 𝑈, 𝑉) = 𝑉)
128125, 127eqtrd 2794 . . . 4 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝑊 = 𝑉)
12928adantr 472 . . . . . 6 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝑉𝑋)
13030, 29, 37abssubge0d 14369 . . . . . . . . . . . . . . 15 (𝜑 → (abs‘(𝑉𝑈)) = (𝑉𝑈))
131130oveq2d 6829 . . . . . . . . . . . . . 14 (𝜑 → (𝐵 · (abs‘(𝑉𝑈))) = (𝐵 · (𝑉𝑈)))
132131breq1d 4814 . . . . . . . . . . . . 13 (𝜑 → ((𝐵 · (abs‘(𝑉𝑈))) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴))) ↔ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))))
133132adantr 472 . . . . . . . . . . . 12 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → ((𝐵 · (abs‘(𝑉𝑈))) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴))) ↔ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))))
134126, 133mtbird 314 . . . . . . . . . . 11 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → ¬ (𝐵 · (abs‘(𝑉𝑈))) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴))))
13514, 28ffvelrnd 6523 . . . . . . . . . . . . 13 (𝜑 → (𝐹𝑉) ∈ ℂ)
13631recnd 10260 . . . . . . . . . . . . 13 (𝜑 → (𝑉𝑈) ∈ ℂ)
13744, 42gtned 10364 . . . . . . . . . . . . 13 (𝜑 → (𝑉𝑈) ≠ 0)
138 unbdqndv2lem2.5 . . . . . . . . . . . . . 14 (𝜑 → (2 · 𝐵) ≤ ((abs‘((𝐹𝑉) − (𝐹𝑈))) / (𝑉𝑈)))
139135, 15subcld 10584 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝐹𝑉) − (𝐹𝑈)) ∈ ℂ)
140139, 136, 137absdivd 14393 . . . . . . . . . . . . . . . 16 (𝜑 → (abs‘(((𝐹𝑉) − (𝐹𝑈)) / (𝑉𝑈))) = ((abs‘((𝐹𝑉) − (𝐹𝑈))) / (abs‘(𝑉𝑈))))
141130oveq2d 6829 . . . . . . . . . . . . . . . 16 (𝜑 → ((abs‘((𝐹𝑉) − (𝐹𝑈))) / (abs‘(𝑉𝑈))) = ((abs‘((𝐹𝑉) − (𝐹𝑈))) / (𝑉𝑈)))
142140, 141eqtrd 2794 . . . . . . . . . . . . . . 15 (𝜑 → (abs‘(((𝐹𝑉) − (𝐹𝑈)) / (𝑉𝑈))) = ((abs‘((𝐹𝑉) − (𝐹𝑈))) / (𝑉𝑈)))
143142eqcomd 2766 . . . . . . . . . . . . . 14 (𝜑 → ((abs‘((𝐹𝑉) − (𝐹𝑈))) / (𝑉𝑈)) = (abs‘(((𝐹𝑉) − (𝐹𝑈)) / (𝑉𝑈))))
144138, 143breqtrd 4830 . . . . . . . . . . . . 13 (𝜑 → (2 · 𝐵) ≤ (abs‘(((𝐹𝑉) − (𝐹𝑈)) / (𝑉𝑈))))
145135, 15, 72, 136, 25, 137, 144unbdqndv2lem1 32806 . . . . . . . . . . . 12 (𝜑 → ((𝐵 · (abs‘(𝑉𝑈))) ≤ (abs‘((𝐹𝑉) − (𝐹𝐴))) ∨ (𝐵 · (abs‘(𝑉𝑈))) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))))
146145adantr 472 . . . . . . . . . . 11 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → ((𝐵 · (abs‘(𝑉𝑈))) ≤ (abs‘((𝐹𝑉) − (𝐹𝐴))) ∨ (𝐵 · (abs‘(𝑉𝑈))) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))))
147 orel2 397 . . . . . . . . . . 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)))))
148134, 146, 147sylc 65 . . . . . . . . . 10 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝐵 · (abs‘(𝑉𝑈))) ≤ (abs‘((𝐹𝑉) − (𝐹𝐴))))
149148adantr 472 . . . . . . . . 9 (((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) ∧ 𝑉 = 𝐴) → (𝐵 · (abs‘(𝑉𝑈))) ≤ (abs‘((𝐹𝑉) − (𝐹𝐴))))
150 fveq2 6352 . . . . . . . . . . . . . . 15 (𝑉 = 𝐴 → (𝐹𝑉) = (𝐹𝐴))
151150oveq1d 6828 . . . . . . . . . . . . . 14 (𝑉 = 𝐴 → ((𝐹𝑉) − (𝐹𝐴)) = ((𝐹𝐴) − (𝐹𝐴)))
152151adantl 473 . . . . . . . . . . . . 13 ((𝜑𝑉 = 𝐴) → ((𝐹𝑉) − (𝐹𝐴)) = ((𝐹𝐴) − (𝐹𝐴)))
15372subidd 10572 . . . . . . . . . . . . . 14 (𝜑 → ((𝐹𝐴) − (𝐹𝐴)) = 0)
154153adantr 472 . . . . . . . . . . . . 13 ((𝜑𝑉 = 𝐴) → ((𝐹𝐴) − (𝐹𝐴)) = 0)
155152, 154eqtrd 2794 . . . . . . . . . . . 12 ((𝜑𝑉 = 𝐴) → ((𝐹𝑉) − (𝐹𝐴)) = 0)
156155fveq2d 6356 . . . . . . . . . . 11 ((𝜑𝑉 = 𝐴) → (abs‘((𝐹𝑉) − (𝐹𝐴))) = (abs‘0))
15719a1i 11 . . . . . . . . . . 11 ((𝜑𝑉 = 𝐴) → (abs‘0) = 0)
158156, 157eqtrd 2794 . . . . . . . . . 10 ((𝜑𝑉 = 𝐴) → (abs‘((𝐹𝑉) − (𝐹𝐴))) = 0)
159158adantlr 753 . . . . . . . . 9 (((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) ∧ 𝑉 = 𝐴) → (abs‘((𝐹𝑉) − (𝐹𝐴))) = 0)
160149, 159breqtrd 4830 . . . . . . . 8 (((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) ∧ 𝑉 = 𝐴) → (𝐵 · (abs‘(𝑉𝑈))) ≤ 0)
161131breq1d 4814 . . . . . . . . . . 11 (𝜑 → ((𝐵 · (abs‘(𝑉𝑈))) ≤ 0 ↔ (𝐵 · (𝑉𝑈)) ≤ 0))
16247, 161mtbird 314 . . . . . . . . . 10 (𝜑 → ¬ (𝐵 · (abs‘(𝑉𝑈))) ≤ 0)
163162adantr 472 . . . . . . . . 9 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → ¬ (𝐵 · (abs‘(𝑉𝑈))) ≤ 0)
164163adantr 472 . . . . . . . 8 (((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) ∧ 𝑉 = 𝐴) → ¬ (𝐵 · (abs‘(𝑉𝑈))) ≤ 0)
165160, 164pm2.65da 601 . . . . . . 7 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → ¬ 𝑉 = 𝐴)
166165neqned 2939 . . . . . 6 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝑉𝐴)
167129, 166jca 555 . . . . 5 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝑉𝑋𝑉𝐴))
168 eldifsn 4462 . . . . 5 (𝑉 ∈ (𝑋 ∖ {𝐴}) ↔ (𝑉𝑋𝑉𝐴))
169167, 168sylibr 224 . . . 4 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝑉 ∈ (𝑋 ∖ {𝐴}))
170128, 169eqeltrd 2839 . . 3 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝑊 ∈ (𝑋 ∖ {𝐴}))
171128oveq1d 6828 . . . . . . 7 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝑊𝐴) = (𝑉𝐴))
172171fveq2d 6356 . . . . . 6 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (abs‘(𝑊𝐴)) = (abs‘(𝑉𝐴)))
17334, 29, 36abssubge0d 14369 . . . . . . 7 (𝜑 → (abs‘(𝑉𝐴)) = (𝑉𝐴))
174173adantr 472 . . . . . 6 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (abs‘(𝑉𝐴)) = (𝑉𝐴))
175172, 174eqtrd 2794 . . . . 5 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (abs‘(𝑊𝐴)) = (𝑉𝐴))
17629, 34resubcld 10650 . . . . . . 7 (𝜑 → (𝑉𝐴) ∈ ℝ)
17730, 34, 29, 35lesub2dd 10836 . . . . . . 7 (𝜑 → (𝑉𝐴) ≤ (𝑉𝑈))
178176, 31, 63, 177, 65lelttrd 10387 . . . . . 6 (𝜑 → (𝑉𝐴) < 𝐷)
179178adantr 472 . . . . 5 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝑉𝐴) < 𝐷)
180175, 179eqbrtrd 4826 . . . 4 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (abs‘(𝑊𝐴)) < 𝐷)
181173, 176eqeltrd 2839 . . . . . . . . 9 (𝜑 → (abs‘(𝑉𝐴)) ∈ ℝ)
18226, 181remulcld 10262 . . . . . . . 8 (𝜑 → (𝐵 · (abs‘(𝑉𝐴))) ∈ ℝ)
183182adantr 472 . . . . . . 7 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝐵 · (abs‘(𝑉𝐴))) ∈ ℝ)
184131, 45eqeltrd 2839 . . . . . . . 8 (𝜑 → (𝐵 · (abs‘(𝑉𝑈))) ∈ ℝ)
185184adantr 472 . . . . . . 7 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝐵 · (abs‘(𝑉𝑈))) ∈ ℝ)
186135, 72subcld 10584 . . . . . . . . 9 (𝜑 → ((𝐹𝑉) − (𝐹𝐴)) ∈ ℂ)
187186abscld 14374 . . . . . . . 8 (𝜑 → (abs‘((𝐹𝑉) − (𝐹𝐴))) ∈ ℝ)
188187adantr 472 . . . . . . 7 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (abs‘((𝐹𝑉) − (𝐹𝐴))) ∈ ℝ)
189130, 31eqeltrd 2839 . . . . . . . . 9 (𝜑 → (abs‘(𝑉𝑈)) ∈ ℝ)
190177, 173, 1303brtr4d 4836 . . . . . . . . 9 (𝜑 → (abs‘(𝑉𝐴)) ≤ (abs‘(𝑉𝑈)))
191181, 189, 26, 76, 190lemul2ad 11156 . . . . . . . 8 (𝜑 → (𝐵 · (abs‘(𝑉𝐴))) ≤ (𝐵 · (abs‘(𝑉𝑈))))
192191adantr 472 . . . . . . 7 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝐵 · (abs‘(𝑉𝐴))) ≤ (𝐵 · (abs‘(𝑉𝑈))))
193183, 185, 188, 192, 148letrd 10386 . . . . . 6 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝐵 · (abs‘(𝑉𝐴))) ≤ (abs‘((𝐹𝑉) − (𝐹𝐴))))
19426adantr 472 . . . . . . 7 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝐵 ∈ ℝ)
195176recnd 10260 . . . . . . . . 9 (𝜑 → (𝑉𝐴) ∈ ℂ)
196195adantr 472 . . . . . . . 8 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝑉𝐴) ∈ ℂ)
19729recnd 10260 . . . . . . . . . 10 (𝜑𝑉 ∈ ℂ)
198197adantr 472 . . . . . . . . 9 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝑉 ∈ ℂ)
199111adantr 472 . . . . . . . . 9 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝐴 ∈ ℂ)
200198, 199, 166subne0d 10593 . . . . . . . 8 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝑉𝐴) ≠ 0)
201196, 200absrpcld 14386 . . . . . . 7 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (abs‘(𝑉𝐴)) ∈ ℝ+)
202194, 188, 201lemuldivd 12114 . . . . . 6 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → ((𝐵 · (abs‘(𝑉𝐴))) ≤ (abs‘((𝐹𝑉) − (𝐹𝐴))) ↔ 𝐵 ≤ ((abs‘((𝐹𝑉) − (𝐹𝐴))) / (abs‘(𝑉𝐴)))))
203193, 202mpbid 222 . . . . 5 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝐵 ≤ ((abs‘((𝐹𝑉) − (𝐹𝐴))) / (abs‘(𝑉𝐴))))
204128fveq2d 6356 . . . . . . . . 9 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝐺𝑊) = (𝐺𝑉))
20598a1i 11 . . . . . . . . . 10 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝐺 = (𝑧 ∈ (𝑋 ∖ {𝐴}) ↦ (((𝐹𝑧) − (𝐹𝐴)) / (𝑧𝐴))))
206 fveq2 6352 . . . . . . . . . . . . 13 (𝑧 = 𝑉 → (𝐹𝑧) = (𝐹𝑉))
207206oveq1d 6828 . . . . . . . . . . . 12 (𝑧 = 𝑉 → ((𝐹𝑧) − (𝐹𝐴)) = ((𝐹𝑉) − (𝐹𝐴)))
208 oveq1 6820 . . . . . . . . . . . 12 (𝑧 = 𝑉 → (𝑧𝐴) = (𝑉𝐴))
209207, 208oveq12d 6831 . . . . . . . . . . 11 (𝑧 = 𝑉 → (((𝐹𝑧) − (𝐹𝐴)) / (𝑧𝐴)) = (((𝐹𝑉) − (𝐹𝐴)) / (𝑉𝐴)))
210209adantl 473 . . . . . . . . . 10 (((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) ∧ 𝑧 = 𝑉) → (((𝐹𝑧) − (𝐹𝐴)) / (𝑧𝐴)) = (((𝐹𝑉) − (𝐹𝐴)) / (𝑉𝐴)))
211 ovexd 6843 . . . . . . . . . 10 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (((𝐹𝑉) − (𝐹𝐴)) / (𝑉𝐴)) ∈ V)
212205, 210, 169, 211fvmptd 6450 . . . . . . . . 9 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝐺𝑉) = (((𝐹𝑉) − (𝐹𝐴)) / (𝑉𝐴)))
213204, 212eqtrd 2794 . . . . . . . 8 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝐺𝑊) = (((𝐹𝑉) − (𝐹𝐴)) / (𝑉𝐴)))
214213fveq2d 6356 . . . . . . 7 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (abs‘(𝐺𝑊)) = (abs‘(((𝐹𝑉) − (𝐹𝐴)) / (𝑉𝐴))))
215186adantr 472 . . . . . . . 8 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → ((𝐹𝑉) − (𝐹𝐴)) ∈ ℂ)
216215, 196, 200absdivd 14393 . . . . . . 7 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (abs‘(((𝐹𝑉) − (𝐹𝐴)) / (𝑉𝐴))) = ((abs‘((𝐹𝑉) − (𝐹𝐴))) / (abs‘(𝑉𝐴))))
217214, 216eqtrd 2794 . . . . . 6 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (abs‘(𝐺𝑊)) = ((abs‘((𝐹𝑉) − (𝐹𝐴))) / (abs‘(𝑉𝐴))))
218217eqcomd 2766 . . . . 5 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → ((abs‘((𝐹𝑉) − (𝐹𝐴))) / (abs‘(𝑉𝐴))) = (abs‘(𝐺𝑊)))
219203, 218breqtrd 4830 . . . 4 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝐵 ≤ (abs‘(𝐺𝑊)))
220180, 219jca 555 . . 3 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → ((abs‘(𝑊𝐴)) < 𝐷𝐵 ≤ (abs‘(𝐺𝑊))))
221170, 220jca 555 . 2 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝑊 ∈ (𝑋 ∖ {𝐴}) ∧ ((abs‘(𝑊𝐴)) < 𝐷𝐵 ≤ (abs‘(𝐺𝑊)))))
222124, 221pm2.61dan 867 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 382  wa 383   = wceq 1632  wcel 2139  wne 2932  Vcvv 3340  cdif 3712  wss 3715  ifcif 4230  {csn 4321   class class class wbr 4804  cmpt 4881  wf 6045  cfv 6049  (class class class)co 6813  cc 10126  cr 10127  0cc0 10128   · cmul 10133   < clt 10266  cle 10267  cmin 10458   / cdiv 10876  2c2 11262  +crp 12025  abscabs 14173
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205  ax-pre-sup 10206
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-er 7911  df-en 8122  df-dom 8123  df-sdom 8124  df-sup 8513  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-div 10877  df-nn 11213  df-2 11271  df-3 11272  df-n0 11485  df-z 11570  df-uz 11880  df-rp 12026  df-seq 12996  df-exp 13055  df-cj 14038  df-re 14039  df-im 14040  df-sqrt 14174  df-abs 14175
This theorem is referenced by:  unbdqndv2  32808
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