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Theorem unbdqndv2lem2 36493
Description: Lemma for unbdqndv2 36494. (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 4537 . . . . . 6 ((𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴))) → if((𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴))), 𝑈, 𝑉) = 𝑈)
43adantl 481 . . . . 5 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → if((𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴))), 𝑈, 𝑉) = 𝑈)
52, 4eqtrd 2775 . . . 4 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝑊 = 𝑈)
6 unbdqndv2lem2.u . . . . . . 7 (𝜑𝑈𝑋)
76adantr 480 . . . . . 6 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝑈𝑋)
8 simplr 769 . . . . . . . . 9 (((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) ∧ 𝑈 = 𝐴) → (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴))))
9 fveq2 6907 . . . . . . . . . . . . . . 15 (𝑈 = 𝐴 → (𝐹𝑈) = (𝐹𝐴))
109eqcomd 2741 . . . . . . . . . . . . . 14 (𝑈 = 𝐴 → (𝐹𝐴) = (𝐹𝑈))
1110oveq2d 7447 . . . . . . . . . . . . 13 (𝑈 = 𝐴 → ((𝐹𝑈) − (𝐹𝐴)) = ((𝐹𝑈) − (𝐹𝑈)))
1211fveq2d 6911 . . . . . . . . . . . 12 (𝑈 = 𝐴 → (abs‘((𝐹𝑈) − (𝐹𝐴))) = (abs‘((𝐹𝑈) − (𝐹𝑈))))
1312adantl 481 . . . . . . . . . . 11 ((𝜑𝑈 = 𝐴) → (abs‘((𝐹𝑈) − (𝐹𝐴))) = (abs‘((𝐹𝑈) − (𝐹𝑈))))
14 unbdqndv2lem2.f . . . . . . . . . . . . . . . 16 (𝜑𝐹:𝑋⟶ℂ)
1514, 6ffvelcdmd 7105 . . . . . . . . . . . . . . 15 (𝜑 → (𝐹𝑈) ∈ ℂ)
1615subidd 11606 . . . . . . . . . . . . . 14 (𝜑 → ((𝐹𝑈) − (𝐹𝑈)) = 0)
1716fveq2d 6911 . . . . . . . . . . . . 13 (𝜑 → (abs‘((𝐹𝑈) − (𝐹𝑈))) = (abs‘0))
1817adantr 480 . . . . . . . . . . . 12 ((𝜑𝑈 = 𝐴) → (abs‘((𝐹𝑈) − (𝐹𝑈))) = (abs‘0))
19 abs0 15321 . . . . . . . . . . . . 13 (abs‘0) = 0
2019a1i 11 . . . . . . . . . . . 12 ((𝜑𝑈 = 𝐴) → (abs‘0) = 0)
2118, 20eqtrd 2775 . . . . . . . . . . 11 ((𝜑𝑈 = 𝐴) → (abs‘((𝐹𝑈) − (𝐹𝑈))) = 0)
2213, 21eqtrd 2775 . . . . . . . . . 10 ((𝜑𝑈 = 𝐴) → (abs‘((𝐹𝑈) − (𝐹𝐴))) = 0)
2322adantlr 715 . . . . . . . . 9 (((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) ∧ 𝑈 = 𝐴) → (abs‘((𝐹𝑈) − (𝐹𝐴))) = 0)
248, 23breqtrd 5174 . . . . . . . 8 (((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) ∧ 𝑈 = 𝐴) → (𝐵 · (𝑉𝑈)) ≤ 0)
25 unbdqndv2lem2.b . . . . . . . . . . . . 13 (𝜑𝐵 ∈ ℝ+)
2625rpred 13075 . . . . . . . . . . . 12 (𝜑𝐵 ∈ ℝ)
27 unbdqndv2lem2.x . . . . . . . . . . . . . 14 (𝜑𝑋 ⊆ ℝ)
28 unbdqndv2lem2.v . . . . . . . . . . . . . 14 (𝜑𝑉𝑋)
2927, 28sseldd 3996 . . . . . . . . . . . . 13 (𝜑𝑉 ∈ ℝ)
3027, 6sseldd 3996 . . . . . . . . . . . . 13 (𝜑𝑈 ∈ ℝ)
3129, 30resubcld 11689 . . . . . . . . . . . 12 (𝜑 → (𝑉𝑈) ∈ ℝ)
3225rpgt0d 13078 . . . . . . . . . . . 12 (𝜑 → 0 < 𝐵)
33 unbdqndv2lem2.a . . . . . . . . . . . . . . . 16 (𝜑𝐴𝑋)
3427, 33sseldd 3996 . . . . . . . . . . . . . . 15 (𝜑𝐴 ∈ ℝ)
35 unbdqndv2lem2.2 . . . . . . . . . . . . . . 15 (𝜑𝑈𝐴)
36 unbdqndv2lem2.3 . . . . . . . . . . . . . . 15 (𝜑𝐴𝑉)
3730, 34, 29, 35, 36letrd 11416 . . . . . . . . . . . . . 14 (𝜑𝑈𝑉)
38 unbdqndv2lem2.1 . . . . . . . . . . . . . . 15 (𝜑𝑈𝑉)
3938necomd 2994 . . . . . . . . . . . . . 14 (𝜑𝑉𝑈)
4030, 29, 37, 39leneltd 11413 . . . . . . . . . . . . 13 (𝜑𝑈 < 𝑉)
4130, 29posdifd 11848 . . . . . . . . . . . . 13 (𝜑 → (𝑈 < 𝑉 ↔ 0 < (𝑉𝑈)))
4240, 41mpbid 232 . . . . . . . . . . . 12 (𝜑 → 0 < (𝑉𝑈))
4326, 31, 32, 42mulgt0d 11414 . . . . . . . . . . 11 (𝜑 → 0 < (𝐵 · (𝑉𝑈)))
44 0red 11262 . . . . . . . . . . . 12 (𝜑 → 0 ∈ ℝ)
4526, 31remulcld 11289 . . . . . . . . . . . 12 (𝜑 → (𝐵 · (𝑉𝑈)) ∈ ℝ)
4644, 45ltnled 11406 . . . . . . . . . . 11 (𝜑 → (0 < (𝐵 · (𝑉𝑈)) ↔ ¬ (𝐵 · (𝑉𝑈)) ≤ 0))
4743, 46mpbid 232 . . . . . . . . . 10 (𝜑 → ¬ (𝐵 · (𝑉𝑈)) ≤ 0)
4847adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → ¬ (𝐵 · (𝑉𝑈)) ≤ 0)
4948adantr 480 . . . . . . . 8 (((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) ∧ 𝑈 = 𝐴) → ¬ (𝐵 · (𝑉𝑈)) ≤ 0)
5024, 49pm2.65da 817 . . . . . . 7 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → ¬ 𝑈 = 𝐴)
5150neqned 2945 . . . . . 6 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝑈𝐴)
527, 51jca 511 . . . . 5 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝑈𝑋𝑈𝐴))
53 eldifsn 4791 . . . . 5 (𝑈 ∈ (𝑋 ∖ {𝐴}) ↔ (𝑈𝑋𝑈𝐴))
5452, 53sylibr 234 . . . 4 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝑈 ∈ (𝑋 ∖ {𝐴}))
555, 54eqeltrd 2839 . . 3 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝑊 ∈ (𝑋 ∖ {𝐴}))
565oveq1d 7446 . . . . . . 7 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝑊𝐴) = (𝑈𝐴))
5756fveq2d 6911 . . . . . 6 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (abs‘(𝑊𝐴)) = (abs‘(𝑈𝐴)))
5830, 34, 35abssuble0d 15468 . . . . . . 7 (𝜑 → (abs‘(𝑈𝐴)) = (𝐴𝑈))
5958adantr 480 . . . . . 6 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (abs‘(𝑈𝐴)) = (𝐴𝑈))
6057, 59eqtrd 2775 . . . . 5 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (abs‘(𝑊𝐴)) = (𝐴𝑈))
6134, 30resubcld 11689 . . . . . . 7 (𝜑 → (𝐴𝑈) ∈ ℝ)
62 unbdqndv2lem2.d . . . . . . . 8 (𝜑𝐷 ∈ ℝ+)
6362rpred 13075 . . . . . . 7 (𝜑𝐷 ∈ ℝ)
6434, 29, 30, 36lesub1dd 11877 . . . . . . 7 (𝜑 → (𝐴𝑈) ≤ (𝑉𝑈))
65 unbdqndv2lem2.4 . . . . . . 7 (𝜑 → (𝑉𝑈) < 𝐷)
6661, 31, 63, 64, 65lelttrd 11417 . . . . . 6 (𝜑 → (𝐴𝑈) < 𝐷)
6766adantr 480 . . . . 5 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝐴𝑈) < 𝐷)
6860, 67eqbrtrd 5170 . . . 4 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (abs‘(𝑊𝐴)) < 𝐷)
6926, 61remulcld 11289 . . . . . . . 8 (𝜑 → (𝐵 · (𝐴𝑈)) ∈ ℝ)
7069adantr 480 . . . . . . 7 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝐵 · (𝐴𝑈)) ∈ ℝ)
7145adantr 480 . . . . . . 7 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝐵 · (𝑉𝑈)) ∈ ℝ)
7214, 33ffvelcdmd 7105 . . . . . . . . . 10 (𝜑 → (𝐹𝐴) ∈ ℂ)
7315, 72subcld 11618 . . . . . . . . 9 (𝜑 → ((𝐹𝑈) − (𝐹𝐴)) ∈ ℂ)
7473abscld 15472 . . . . . . . 8 (𝜑 → (abs‘((𝐹𝑈) − (𝐹𝐴))) ∈ ℝ)
7574adantr 480 . . . . . . 7 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (abs‘((𝐹𝑈) − (𝐹𝐴))) ∈ ℝ)
7644, 26, 32ltled 11407 . . . . . . . . 9 (𝜑 → 0 ≤ 𝐵)
7761, 31, 26, 76, 64lemul2ad 12206 . . . . . . . 8 (𝜑 → (𝐵 · (𝐴𝑈)) ≤ (𝐵 · (𝑉𝑈)))
7877adantr 480 . . . . . . 7 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝐵 · (𝐴𝑈)) ≤ (𝐵 · (𝑉𝑈)))
79 simpr 484 . . . . . . 7 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴))))
8070, 71, 75, 78, 79letrd 11416 . . . . . 6 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝐵 · (𝐴𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴))))
8126adantr 480 . . . . . . 7 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝐵 ∈ ℝ)
8261adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝐴𝑈) ∈ ℝ)
8335adantr 480 . . . . . . . . . . . 12 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝑈𝐴)
8451necomd 2994 . . . . . . . . . . . 12 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝐴𝑈)
8583, 84jca 511 . . . . . . . . . . 11 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝑈𝐴𝐴𝑈))
8630, 34ltlend 11404 . . . . . . . . . . . 12 (𝜑 → (𝑈 < 𝐴 ↔ (𝑈𝐴𝐴𝑈)))
8786adantr 480 . . . . . . . . . . 11 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝑈 < 𝐴 ↔ (𝑈𝐴𝐴𝑈)))
8885, 87mpbird 257 . . . . . . . . . 10 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝑈 < 𝐴)
8930, 34posdifd 11848 . . . . . . . . . . 11 (𝜑 → (𝑈 < 𝐴 ↔ 0 < (𝐴𝑈)))
9089adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝑈 < 𝐴 ↔ 0 < (𝐴𝑈)))
9188, 90mpbid 232 . . . . . . . . 9 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 0 < (𝐴𝑈))
9282, 91jca 511 . . . . . . . 8 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → ((𝐴𝑈) ∈ ℝ ∧ 0 < (𝐴𝑈)))
93 elrp 13034 . . . . . . . 8 ((𝐴𝑈) ∈ ℝ+ ↔ ((𝐴𝑈) ∈ ℝ ∧ 0 < (𝐴𝑈)))
9492, 93sylibr 234 . . . . . . 7 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝐴𝑈) ∈ ℝ+)
9581, 75, 94lemuldivd 13124 . . . . . 6 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → ((𝐵 · (𝐴𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴))) ↔ 𝐵 ≤ ((abs‘((𝐹𝑈) − (𝐹𝐴))) / (𝐴𝑈))))
9680, 95mpbid 232 . . . . 5 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝐵 ≤ ((abs‘((𝐹𝑈) − (𝐹𝐴))) / (𝐴𝑈)))
975fveq2d 6911 . . . . . . . . 9 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝐺𝑊) = (𝐺𝑈))
98 unbdqndv2lem2.g . . . . . . . . . 10 𝐺 = (𝑧 ∈ (𝑋 ∖ {𝐴}) ↦ (((𝐹𝑧) − (𝐹𝐴)) / (𝑧𝐴)))
99 fveq2 6907 . . . . . . . . . . . 12 (𝑧 = 𝑈 → (𝐹𝑧) = (𝐹𝑈))
10099oveq1d 7446 . . . . . . . . . . 11 (𝑧 = 𝑈 → ((𝐹𝑧) − (𝐹𝐴)) = ((𝐹𝑈) − (𝐹𝐴)))
101 oveq1 7438 . . . . . . . . . . 11 (𝑧 = 𝑈 → (𝑧𝐴) = (𝑈𝐴))
102100, 101oveq12d 7449 . . . . . . . . . 10 (𝑧 = 𝑈 → (((𝐹𝑧) − (𝐹𝐴)) / (𝑧𝐴)) = (((𝐹𝑈) − (𝐹𝐴)) / (𝑈𝐴)))
103 ovexd 7466 . . . . . . . . . 10 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (((𝐹𝑈) − (𝐹𝐴)) / (𝑈𝐴)) ∈ V)
10498, 102, 54, 103fvmptd3 7039 . . . . . . . . 9 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝐺𝑈) = (((𝐹𝑈) − (𝐹𝐴)) / (𝑈𝐴)))
10597, 104eqtrd 2775 . . . . . . . 8 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝐺𝑊) = (((𝐹𝑈) − (𝐹𝐴)) / (𝑈𝐴)))
106105fveq2d 6911 . . . . . . 7 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (abs‘(𝐺𝑊)) = (abs‘(((𝐹𝑈) − (𝐹𝐴)) / (𝑈𝐴))))
10773adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → ((𝐹𝑈) − (𝐹𝐴)) ∈ ℂ)
10830recnd 11287 . . . . . . . . . . 11 (𝜑𝑈 ∈ ℂ)
10934recnd 11287 . . . . . . . . . . 11 (𝜑𝐴 ∈ ℂ)
110108, 109subcld 11618 . . . . . . . . . 10 (𝜑 → (𝑈𝐴) ∈ ℂ)
111110adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝑈𝐴) ∈ ℂ)
112108adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝑈 ∈ ℂ)
113109adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝐴 ∈ ℂ)
114112, 113, 51subne0d 11627 . . . . . . . . 9 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝑈𝐴) ≠ 0)
115107, 111, 114absdivd 15491 . . . . . . . 8 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (abs‘(((𝐹𝑈) − (𝐹𝐴)) / (𝑈𝐴))) = ((abs‘((𝐹𝑈) − (𝐹𝐴))) / (abs‘(𝑈𝐴))))
11659oveq2d 7447 . . . . . . . 8 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → ((abs‘((𝐹𝑈) − (𝐹𝐴))) / (abs‘(𝑈𝐴))) = ((abs‘((𝐹𝑈) − (𝐹𝐴))) / (𝐴𝑈)))
117115, 116eqtrd 2775 . . . . . . 7 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (abs‘(((𝐹𝑈) − (𝐹𝐴)) / (𝑈𝐴))) = ((abs‘((𝐹𝑈) − (𝐹𝐴))) / (𝐴𝑈)))
118106, 117eqtrd 2775 . . . . . 6 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (abs‘(𝐺𝑊)) = ((abs‘((𝐹𝑈) − (𝐹𝐴))) / (𝐴𝑈)))
119118eqcomd 2741 . . . . 5 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → ((abs‘((𝐹𝑈) − (𝐹𝐴))) / (𝐴𝑈)) = (abs‘(𝐺𝑊)))
12096, 119breqtrd 5174 . . . 4 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝐵 ≤ (abs‘(𝐺𝑊)))
12168, 120jca 511 . . 3 ((𝜑 ∧ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → ((abs‘(𝑊𝐴)) < 𝐷𝐵 ≤ (abs‘(𝐺𝑊))))
12255, 121jca 511 . 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 484 . . . . . 6 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴))))
125124iffalsed 4542 . . . . 5 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → if((𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴))), 𝑈, 𝑉) = 𝑉)
126123, 125eqtrd 2775 . . . 4 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝑊 = 𝑉)
12728adantr 480 . . . . . 6 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝑉𝑋)
12830, 29, 37abssubge0d 15467 . . . . . . . . . . . . . . 15 (𝜑 → (abs‘(𝑉𝑈)) = (𝑉𝑈))
129128oveq2d 7447 . . . . . . . . . . . . . 14 (𝜑 → (𝐵 · (abs‘(𝑉𝑈))) = (𝐵 · (𝑉𝑈)))
130129breq1d 5158 . . . . . . . . . . . . 13 (𝜑 → ((𝐵 · (abs‘(𝑉𝑈))) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴))) ↔ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))))
131130adantr 480 . . . . . . . . . . . 12 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → ((𝐵 · (abs‘(𝑉𝑈))) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴))) ↔ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))))
132124, 131mtbird 325 . . . . . . . . . . 11 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → ¬ (𝐵 · (abs‘(𝑉𝑈))) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴))))
13314, 28ffvelcdmd 7105 . . . . . . . . . . . . 13 (𝜑 → (𝐹𝑉) ∈ ℂ)
13431recnd 11287 . . . . . . . . . . . . 13 (𝜑 → (𝑉𝑈) ∈ ℂ)
13544, 42gtned 11394 . . . . . . . . . . . . 13 (𝜑 → (𝑉𝑈) ≠ 0)
136 unbdqndv2lem2.5 . . . . . . . . . . . . . 14 (𝜑 → (2 · 𝐵) ≤ ((abs‘((𝐹𝑉) − (𝐹𝑈))) / (𝑉𝑈)))
137133, 15subcld 11618 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝐹𝑉) − (𝐹𝑈)) ∈ ℂ)
138137, 134, 135absdivd 15491 . . . . . . . . . . . . . . . 16 (𝜑 → (abs‘(((𝐹𝑉) − (𝐹𝑈)) / (𝑉𝑈))) = ((abs‘((𝐹𝑉) − (𝐹𝑈))) / (abs‘(𝑉𝑈))))
139128oveq2d 7447 . . . . . . . . . . . . . . . 16 (𝜑 → ((abs‘((𝐹𝑉) − (𝐹𝑈))) / (abs‘(𝑉𝑈))) = ((abs‘((𝐹𝑉) − (𝐹𝑈))) / (𝑉𝑈)))
140138, 139eqtrd 2775 . . . . . . . . . . . . . . 15 (𝜑 → (abs‘(((𝐹𝑉) − (𝐹𝑈)) / (𝑉𝑈))) = ((abs‘((𝐹𝑉) − (𝐹𝑈))) / (𝑉𝑈)))
141140eqcomd 2741 . . . . . . . . . . . . . 14 (𝜑 → ((abs‘((𝐹𝑉) − (𝐹𝑈))) / (𝑉𝑈)) = (abs‘(((𝐹𝑉) − (𝐹𝑈)) / (𝑉𝑈))))
142136, 141breqtrd 5174 . . . . . . . . . . . . 13 (𝜑 → (2 · 𝐵) ≤ (abs‘(((𝐹𝑉) − (𝐹𝑈)) / (𝑉𝑈))))
143133, 15, 72, 134, 25, 135, 142unbdqndv2lem1 36492 . . . . . . . . . . . 12 (𝜑 → ((𝐵 · (abs‘(𝑉𝑈))) ≤ (abs‘((𝐹𝑉) − (𝐹𝐴))) ∨ (𝐵 · (abs‘(𝑉𝑈))) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))))
144143adantr 480 . . . . . . . . . . 11 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → ((𝐵 · (abs‘(𝑉𝑈))) ≤ (abs‘((𝐹𝑉) − (𝐹𝐴))) ∨ (𝐵 · (abs‘(𝑉𝑈))) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))))
145 orel2 890 . . . . . . . . . . 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 480 . . . . . . . . 9 (((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) ∧ 𝑉 = 𝐴) → (𝐵 · (abs‘(𝑉𝑈))) ≤ (abs‘((𝐹𝑉) − (𝐹𝐴))))
148 fveq2 6907 . . . . . . . . . . . . . . 15 (𝑉 = 𝐴 → (𝐹𝑉) = (𝐹𝐴))
149148oveq1d 7446 . . . . . . . . . . . . . 14 (𝑉 = 𝐴 → ((𝐹𝑉) − (𝐹𝐴)) = ((𝐹𝐴) − (𝐹𝐴)))
150149adantl 481 . . . . . . . . . . . . 13 ((𝜑𝑉 = 𝐴) → ((𝐹𝑉) − (𝐹𝐴)) = ((𝐹𝐴) − (𝐹𝐴)))
15172subidd 11606 . . . . . . . . . . . . . 14 (𝜑 → ((𝐹𝐴) − (𝐹𝐴)) = 0)
152151adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑉 = 𝐴) → ((𝐹𝐴) − (𝐹𝐴)) = 0)
153150, 152eqtrd 2775 . . . . . . . . . . . 12 ((𝜑𝑉 = 𝐴) → ((𝐹𝑉) − (𝐹𝐴)) = 0)
154153fveq2d 6911 . . . . . . . . . . 11 ((𝜑𝑉 = 𝐴) → (abs‘((𝐹𝑉) − (𝐹𝐴))) = (abs‘0))
15519a1i 11 . . . . . . . . . . 11 ((𝜑𝑉 = 𝐴) → (abs‘0) = 0)
156154, 155eqtrd 2775 . . . . . . . . . 10 ((𝜑𝑉 = 𝐴) → (abs‘((𝐹𝑉) − (𝐹𝐴))) = 0)
157156adantlr 715 . . . . . . . . 9 (((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) ∧ 𝑉 = 𝐴) → (abs‘((𝐹𝑉) − (𝐹𝐴))) = 0)
158147, 157breqtrd 5174 . . . . . . . 8 (((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) ∧ 𝑉 = 𝐴) → (𝐵 · (abs‘(𝑉𝑈))) ≤ 0)
159129breq1d 5158 . . . . . . . . . . 11 (𝜑 → ((𝐵 · (abs‘(𝑉𝑈))) ≤ 0 ↔ (𝐵 · (𝑉𝑈)) ≤ 0))
16047, 159mtbird 325 . . . . . . . . . 10 (𝜑 → ¬ (𝐵 · (abs‘(𝑉𝑈))) ≤ 0)
161160adantr 480 . . . . . . . . 9 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → ¬ (𝐵 · (abs‘(𝑉𝑈))) ≤ 0)
162161adantr 480 . . . . . . . 8 (((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) ∧ 𝑉 = 𝐴) → ¬ (𝐵 · (abs‘(𝑉𝑈))) ≤ 0)
163158, 162pm2.65da 817 . . . . . . 7 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → ¬ 𝑉 = 𝐴)
164163neqned 2945 . . . . . 6 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝑉𝐴)
165127, 164jca 511 . . . . 5 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝑉𝑋𝑉𝐴))
166 eldifsn 4791 . . . . 5 (𝑉 ∈ (𝑋 ∖ {𝐴}) ↔ (𝑉𝑋𝑉𝐴))
167165, 166sylibr 234 . . . 4 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝑉 ∈ (𝑋 ∖ {𝐴}))
168126, 167eqeltrd 2839 . . 3 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝑊 ∈ (𝑋 ∖ {𝐴}))
169126oveq1d 7446 . . . . . . 7 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝑊𝐴) = (𝑉𝐴))
170169fveq2d 6911 . . . . . 6 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (abs‘(𝑊𝐴)) = (abs‘(𝑉𝐴)))
17134, 29, 36abssubge0d 15467 . . . . . . 7 (𝜑 → (abs‘(𝑉𝐴)) = (𝑉𝐴))
172171adantr 480 . . . . . 6 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (abs‘(𝑉𝐴)) = (𝑉𝐴))
173170, 172eqtrd 2775 . . . . 5 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (abs‘(𝑊𝐴)) = (𝑉𝐴))
17429, 34resubcld 11689 . . . . . . 7 (𝜑 → (𝑉𝐴) ∈ ℝ)
17530, 34, 29, 35lesub2dd 11878 . . . . . . 7 (𝜑 → (𝑉𝐴) ≤ (𝑉𝑈))
176174, 31, 63, 175, 65lelttrd 11417 . . . . . 6 (𝜑 → (𝑉𝐴) < 𝐷)
177176adantr 480 . . . . 5 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝑉𝐴) < 𝐷)
178173, 177eqbrtrd 5170 . . . 4 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (abs‘(𝑊𝐴)) < 𝐷)
179171, 174eqeltrd 2839 . . . . . . . . 9 (𝜑 → (abs‘(𝑉𝐴)) ∈ ℝ)
18026, 179remulcld 11289 . . . . . . . 8 (𝜑 → (𝐵 · (abs‘(𝑉𝐴))) ∈ ℝ)
181180adantr 480 . . . . . . 7 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝐵 · (abs‘(𝑉𝐴))) ∈ ℝ)
182129, 45eqeltrd 2839 . . . . . . . 8 (𝜑 → (𝐵 · (abs‘(𝑉𝑈))) ∈ ℝ)
183182adantr 480 . . . . . . 7 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝐵 · (abs‘(𝑉𝑈))) ∈ ℝ)
184133, 72subcld 11618 . . . . . . . . 9 (𝜑 → ((𝐹𝑉) − (𝐹𝐴)) ∈ ℂ)
185184abscld 15472 . . . . . . . 8 (𝜑 → (abs‘((𝐹𝑉) − (𝐹𝐴))) ∈ ℝ)
186185adantr 480 . . . . . . 7 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (abs‘((𝐹𝑉) − (𝐹𝐴))) ∈ ℝ)
187128, 31eqeltrd 2839 . . . . . . . . 9 (𝜑 → (abs‘(𝑉𝑈)) ∈ ℝ)
188175, 171, 1283brtr4d 5180 . . . . . . . . 9 (𝜑 → (abs‘(𝑉𝐴)) ≤ (abs‘(𝑉𝑈)))
189179, 187, 26, 76, 188lemul2ad 12206 . . . . . . . 8 (𝜑 → (𝐵 · (abs‘(𝑉𝐴))) ≤ (𝐵 · (abs‘(𝑉𝑈))))
190189adantr 480 . . . . . . 7 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝐵 · (abs‘(𝑉𝐴))) ≤ (𝐵 · (abs‘(𝑉𝑈))))
191181, 183, 186, 190, 146letrd 11416 . . . . . 6 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝐵 · (abs‘(𝑉𝐴))) ≤ (abs‘((𝐹𝑉) − (𝐹𝐴))))
19226adantr 480 . . . . . . 7 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝐵 ∈ ℝ)
193174recnd 11287 . . . . . . . . 9 (𝜑 → (𝑉𝐴) ∈ ℂ)
194193adantr 480 . . . . . . . 8 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝑉𝐴) ∈ ℂ)
19529recnd 11287 . . . . . . . . . 10 (𝜑𝑉 ∈ ℂ)
196195adantr 480 . . . . . . . . 9 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝑉 ∈ ℂ)
197109adantr 480 . . . . . . . . 9 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝐴 ∈ ℂ)
198196, 197, 164subne0d 11627 . . . . . . . 8 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝑉𝐴) ≠ 0)
199194, 198absrpcld 15484 . . . . . . 7 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (abs‘(𝑉𝐴)) ∈ ℝ+)
200192, 186, 199lemuldivd 13124 . . . . . 6 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → ((𝐵 · (abs‘(𝑉𝐴))) ≤ (abs‘((𝐹𝑉) − (𝐹𝐴))) ↔ 𝐵 ≤ ((abs‘((𝐹𝑉) − (𝐹𝐴))) / (abs‘(𝑉𝐴)))))
201191, 200mpbid 232 . . . . 5 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝐵 ≤ ((abs‘((𝐹𝑉) − (𝐹𝐴))) / (abs‘(𝑉𝐴))))
202126fveq2d 6911 . . . . . . . . 9 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝐺𝑊) = (𝐺𝑉))
203 fveq2 6907 . . . . . . . . . . . 12 (𝑧 = 𝑉 → (𝐹𝑧) = (𝐹𝑉))
204203oveq1d 7446 . . . . . . . . . . 11 (𝑧 = 𝑉 → ((𝐹𝑧) − (𝐹𝐴)) = ((𝐹𝑉) − (𝐹𝐴)))
205 oveq1 7438 . . . . . . . . . . 11 (𝑧 = 𝑉 → (𝑧𝐴) = (𝑉𝐴))
206204, 205oveq12d 7449 . . . . . . . . . 10 (𝑧 = 𝑉 → (((𝐹𝑧) − (𝐹𝐴)) / (𝑧𝐴)) = (((𝐹𝑉) − (𝐹𝐴)) / (𝑉𝐴)))
207 ovexd 7466 . . . . . . . . . 10 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (((𝐹𝑉) − (𝐹𝐴)) / (𝑉𝐴)) ∈ V)
20898, 206, 167, 207fvmptd3 7039 . . . . . . . . 9 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝐺𝑉) = (((𝐹𝑉) − (𝐹𝐴)) / (𝑉𝐴)))
209202, 208eqtrd 2775 . . . . . . . 8 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝐺𝑊) = (((𝐹𝑉) − (𝐹𝐴)) / (𝑉𝐴)))
210209fveq2d 6911 . . . . . . 7 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (abs‘(𝐺𝑊)) = (abs‘(((𝐹𝑉) − (𝐹𝐴)) / (𝑉𝐴))))
211184adantr 480 . . . . . . . 8 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → ((𝐹𝑉) − (𝐹𝐴)) ∈ ℂ)
212211, 194, 198absdivd 15491 . . . . . . 7 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (abs‘(((𝐹𝑉) − (𝐹𝐴)) / (𝑉𝐴))) = ((abs‘((𝐹𝑉) − (𝐹𝐴))) / (abs‘(𝑉𝐴))))
213210, 212eqtrd 2775 . . . . . 6 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (abs‘(𝐺𝑊)) = ((abs‘((𝐹𝑉) − (𝐹𝐴))) / (abs‘(𝑉𝐴))))
214213eqcomd 2741 . . . . 5 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → ((abs‘((𝐹𝑉) − (𝐹𝐴))) / (abs‘(𝑉𝐴))) = (abs‘(𝐺𝑊)))
215201, 214breqtrd 5174 . . . 4 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → 𝐵 ≤ (abs‘(𝐺𝑊)))
216178, 215jca 511 . . 3 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → ((abs‘(𝑊𝐴)) < 𝐷𝐵 ≤ (abs‘(𝐺𝑊))))
217168, 216jca 511 . 2 ((𝜑 ∧ ¬ (𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴)))) → (𝑊 ∈ (𝑋 ∖ {𝐴}) ∧ ((abs‘(𝑊𝐴)) < 𝐷𝐵 ≤ (abs‘(𝐺𝑊)))))
218122, 217pm2.61dan 813 1 (𝜑 → (𝑊 ∈ (𝑋 ∖ {𝐴}) ∧ ((abs‘(𝑊𝐴)) < 𝐷𝐵 ≤ (abs‘(𝐺𝑊)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 847   = wceq 1537  wcel 2106  wne 2938  Vcvv 3478  cdif 3960  wss 3963  ifcif 4531  {csn 4631   class class class wbr 5148  cmpt 5231  wf 6559  cfv 6563  (class class class)co 7431  cc 11151  cr 11152  0cc0 11153   · cmul 11158   < clt 11293  cle 11294  cmin 11490   / cdiv 11918  2c2 12319  +crp 13032  abscabs 15270
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-sup 9480  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-n0 12525  df-z 12612  df-uz 12877  df-rp 13033  df-seq 14040  df-exp 14100  df-cj 15135  df-re 15136  df-im 15137  df-sqrt 15271  df-abs 15272
This theorem is referenced by:  unbdqndv2  36494
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