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Theorem imo72b2 44162
Description: IMO 1972 B2. (14th International Mathematical Olympiad in Poland, problem B2). (Contributed by Stanislas Polu, 9-Mar-2020.)
Hypotheses
Ref Expression
imo72b2.1 (𝜑𝐹:ℝ⟶ℝ)
imo72b2.2 (𝜑𝐺:ℝ⟶ℝ)
imo72b2.4 (𝜑𝐵 ∈ ℝ)
imo72b2.5 (𝜑 → ∀𝑢 ∈ ℝ ∀𝑣 ∈ ℝ ((𝐹‘(𝑢 + 𝑣)) + (𝐹‘(𝑢𝑣))) = (2 · ((𝐹𝑢) · (𝐺𝑣))))
imo72b2.6 (𝜑 → ∀𝑦 ∈ ℝ (abs‘(𝐹𝑦)) ≤ 1)
imo72b2.7 (𝜑 → ∃𝑥 ∈ ℝ (𝐹𝑥) ≠ 0)
Assertion
Ref Expression
imo72b2 (𝜑 → (abs‘(𝐺𝐵)) ≤ 1)
Distinct variable groups:   𝑢,𝐵,𝑣   𝑥,𝐵   𝑦,𝐵   𝑢,𝐹,𝑣   𝑥,𝐹   𝑦,𝐹   𝑢,𝐺,𝑣   𝑥,𝐺   𝑦,𝐺   𝜑,𝑢,𝑣   𝜑,𝑥   𝜑,𝑦,𝑢

Proof of Theorem imo72b2
Dummy variables 𝑐 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 imo72b2.2 . . . . 5 (𝜑𝐺:ℝ⟶ℝ)
2 imo72b2.4 . . . . 5 (𝜑𝐵 ∈ ℝ)
31, 2ffvelcdmd 7105 . . . 4 (𝜑 → (𝐺𝐵) ∈ ℝ)
43recnd 11287 . . 3 (𝜑 → (𝐺𝐵) ∈ ℂ)
54abscld 15472 . 2 (𝜑 → (abs‘(𝐺𝐵)) ∈ ℝ)
6 1red 11260 . 2 (𝜑 → 1 ∈ ℝ)
7 simpr 484 . . 3 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → 1 < (abs‘(𝐺𝐵)))
81adantr 480 . . . . . . 7 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → 𝐺:ℝ⟶ℝ)
92adantr 480 . . . . . . 7 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → 𝐵 ∈ ℝ)
108, 9ffvelcdmd 7105 . . . . . 6 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → (𝐺𝐵) ∈ ℝ)
1110recnd 11287 . . . . 5 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → (𝐺𝐵) ∈ ℂ)
1211abscld 15472 . . . 4 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → (abs‘(𝐺𝐵)) ∈ ℝ)
136adantr 480 . . . 4 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → 1 ∈ ℝ)
14 ax-resscn 11210 . . . . . . . . 9 ℝ ⊆ ℂ
15 imaco 6273 . . . . . . . . . . . 12 ((abs ∘ 𝐹) “ ℝ) = (abs “ (𝐹 “ ℝ))
1615eqcomi 2744 . . . . . . . . . . 11 (abs “ (𝐹 “ ℝ)) = ((abs ∘ 𝐹) “ ℝ)
17 imassrn 6091 . . . . . . . . . . . . 13 ((abs ∘ 𝐹) “ ℝ) ⊆ ran (abs ∘ 𝐹)
1817a1i 11 . . . . . . . . . . . 12 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → ((abs ∘ 𝐹) “ ℝ) ⊆ ran (abs ∘ 𝐹))
19 imo72b2.1 . . . . . . . . . . . . . . 15 (𝜑𝐹:ℝ⟶ℝ)
2019adantr 480 . . . . . . . . . . . . . 14 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → 𝐹:ℝ⟶ℝ)
21 absf 15373 . . . . . . . . . . . . . . . 16 abs:ℂ⟶ℝ
2221a1i 11 . . . . . . . . . . . . . . 15 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → abs:ℂ⟶ℝ)
2314a1i 11 . . . . . . . . . . . . . . 15 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → ℝ ⊆ ℂ)
2422, 23fssresd 6776 . . . . . . . . . . . . . 14 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → (abs ↾ ℝ):ℝ⟶ℝ)
2520, 24fco2d 44152 . . . . . . . . . . . . 13 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → (abs ∘ 𝐹):ℝ⟶ℝ)
2625frnd 6745 . . . . . . . . . . . 12 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → ran (abs ∘ 𝐹) ⊆ ℝ)
2718, 26sstrd 4006 . . . . . . . . . . 11 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → ((abs ∘ 𝐹) “ ℝ) ⊆ ℝ)
2816, 27eqsstrid 4044 . . . . . . . . . 10 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → (abs “ (𝐹 “ ℝ)) ⊆ ℝ)
29 0re 11261 . . . . . . . . . . . . . . . 16 0 ∈ ℝ
3029ne0ii 4350 . . . . . . . . . . . . . . 15 ℝ ≠ ∅
3130a1i 11 . . . . . . . . . . . . . 14 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → ℝ ≠ ∅)
3231, 25wnefimgd 44151 . . . . . . . . . . . . 13 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → ((abs ∘ 𝐹) “ ℝ) ≠ ∅)
3332necomd 2994 . . . . . . . . . . . 12 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → ∅ ≠ ((abs ∘ 𝐹) “ ℝ))
3416a1i 11 . . . . . . . . . . . 12 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → (abs “ (𝐹 “ ℝ)) = ((abs ∘ 𝐹) “ ℝ))
3533, 34neeqtrrd 3013 . . . . . . . . . . 11 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → ∅ ≠ (abs “ (𝐹 “ ℝ)))
3635necomd 2994 . . . . . . . . . 10 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → (abs “ (𝐹 “ ℝ)) ≠ ∅)
37 simpr 484 . . . . . . . . . . . . 13 (((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) ∧ 𝑐 = 1) → 𝑐 = 1)
3837breq2d 5160 . . . . . . . . . . . 12 (((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) ∧ 𝑐 = 1) → (𝑡𝑐𝑡 ≤ 1))
3938ralbidv 3176 . . . . . . . . . . 11 (((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) ∧ 𝑐 = 1) → (∀𝑡 ∈ (abs “ (𝐹 “ ℝ))𝑡𝑐 ↔ ∀𝑡 ∈ (abs “ (𝐹 “ ℝ))𝑡 ≤ 1))
40 imo72b2.6 . . . . . . . . . . . . 13 (𝜑 → ∀𝑦 ∈ ℝ (abs‘(𝐹𝑦)) ≤ 1)
4119, 40extoimad 44154 . . . . . . . . . . . 12 (𝜑 → ∀𝑡 ∈ (abs “ (𝐹 “ ℝ))𝑡 ≤ 1)
4241adantr 480 . . . . . . . . . . 11 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → ∀𝑡 ∈ (abs “ (𝐹 “ ℝ))𝑡 ≤ 1)
4313, 39, 42rspcedvd 3624 . . . . . . . . . 10 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → ∃𝑐 ∈ ℝ ∀𝑡 ∈ (abs “ (𝐹 “ ℝ))𝑡𝑐)
4428, 36, 43suprcld 12229 . . . . . . . . 9 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → sup((abs “ (𝐹 “ ℝ)), ℝ, < ) ∈ ℝ)
4514, 44sselid 3993 . . . . . . . 8 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → sup((abs “ (𝐹 “ ℝ)), ℝ, < ) ∈ ℂ)
4614, 12sselid 3993 . . . . . . . 8 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → (abs‘(𝐺𝐵)) ∈ ℂ)
4745, 46mulcomd 11280 . . . . . . 7 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → (sup((abs “ (𝐹 “ ℝ)), ℝ, < ) · (abs‘(𝐺𝐵))) = ((abs‘(𝐺𝐵)) · sup((abs “ (𝐹 “ ℝ)), ℝ, < )))
4829a1i 11 . . . . . . . . . . . 12 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → 0 ∈ ℝ)
49 0lt1 11783 . . . . . . . . . . . . 13 0 < 1
5049a1i 11 . . . . . . . . . . . 12 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → 0 < 1)
5148, 13, 12, 50, 7lttrd 11420 . . . . . . . . . . 11 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → 0 < (abs‘(𝐺𝐵)))
5251gt0ne0d 11825 . . . . . . . . . 10 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → (abs‘(𝐺𝐵)) ≠ 0)
5344, 12, 52redivcld 12093 . . . . . . . . 9 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → (sup((abs “ (𝐹 “ ℝ)), ℝ, < ) / (abs‘(𝐺𝐵))) ∈ ℝ)
5420adantr 480 . . . . . . . . . . . 12 (((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) ∧ 𝑢 ∈ ℝ) → 𝐹:ℝ⟶ℝ)
558adantr 480 . . . . . . . . . . . 12 (((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) ∧ 𝑢 ∈ ℝ) → 𝐺:ℝ⟶ℝ)
56 simpr 484 . . . . . . . . . . . 12 (((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) ∧ 𝑢 ∈ ℝ) → 𝑢 ∈ ℝ)
579adantr 480 . . . . . . . . . . . 12 (((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) ∧ 𝑢 ∈ ℝ) → 𝐵 ∈ ℝ)
58 simpr 484 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑣 = 𝐵) → 𝑣 = 𝐵)
5958oveq2d 7447 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑣 = 𝐵) → (𝑢 + 𝑣) = (𝑢 + 𝐵))
6059fveq2d 6911 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑣 = 𝐵) → (𝐹‘(𝑢 + 𝑣)) = (𝐹‘(𝑢 + 𝐵)))
6158oveq2d 7447 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑣 = 𝐵) → (𝑢𝑣) = (𝑢𝐵))
6261fveq2d 6911 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑣 = 𝐵) → (𝐹‘(𝑢𝑣)) = (𝐹‘(𝑢𝐵)))
6360, 62oveq12d 7449 . . . . . . . . . . . . . . . . 17 ((𝜑𝑣 = 𝐵) → ((𝐹‘(𝑢 + 𝑣)) + (𝐹‘(𝑢𝑣))) = ((𝐹‘(𝑢 + 𝐵)) + (𝐹‘(𝑢𝐵))))
6458fveq2d 6911 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑣 = 𝐵) → (𝐺𝑣) = (𝐺𝐵))
6564oveq2d 7447 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑣 = 𝐵) → ((𝐹𝑢) · (𝐺𝑣)) = ((𝐹𝑢) · (𝐺𝐵)))
6665oveq2d 7447 . . . . . . . . . . . . . . . . 17 ((𝜑𝑣 = 𝐵) → (2 · ((𝐹𝑢) · (𝐺𝑣))) = (2 · ((𝐹𝑢) · (𝐺𝐵))))
6763, 66eqeq12d 2751 . . . . . . . . . . . . . . . 16 ((𝜑𝑣 = 𝐵) → (((𝐹‘(𝑢 + 𝑣)) + (𝐹‘(𝑢𝑣))) = (2 · ((𝐹𝑢) · (𝐺𝑣))) ↔ ((𝐹‘(𝑢 + 𝐵)) + (𝐹‘(𝑢𝐵))) = (2 · ((𝐹𝑢) · (𝐺𝐵)))))
6867ralbidv 3176 . . . . . . . . . . . . . . 15 ((𝜑𝑣 = 𝐵) → (∀𝑢 ∈ ℝ ((𝐹‘(𝑢 + 𝑣)) + (𝐹‘(𝑢𝑣))) = (2 · ((𝐹𝑢) · (𝐺𝑣))) ↔ ∀𝑢 ∈ ℝ ((𝐹‘(𝑢 + 𝐵)) + (𝐹‘(𝑢𝐵))) = (2 · ((𝐹𝑢) · (𝐺𝐵)))))
69 imo72b2.5 . . . . . . . . . . . . . . . 16 (𝜑 → ∀𝑢 ∈ ℝ ∀𝑣 ∈ ℝ ((𝐹‘(𝑢 + 𝑣)) + (𝐹‘(𝑢𝑣))) = (2 · ((𝐹𝑢) · (𝐺𝑣))))
70 ralcom 3287 . . . . . . . . . . . . . . . . . . 19 (∀𝑢 ∈ ℝ ∀𝑣 ∈ ℝ ((𝐹‘(𝑢 + 𝑣)) + (𝐹‘(𝑢𝑣))) = (2 · ((𝐹𝑢) · (𝐺𝑣))) ↔ ∀𝑣 ∈ ℝ ∀𝑢 ∈ ℝ ((𝐹‘(𝑢 + 𝑣)) + (𝐹‘(𝑢𝑣))) = (2 · ((𝐹𝑢) · (𝐺𝑣))))
7170biimpi 216 . . . . . . . . . . . . . . . . . 18 (∀𝑢 ∈ ℝ ∀𝑣 ∈ ℝ ((𝐹‘(𝑢 + 𝑣)) + (𝐹‘(𝑢𝑣))) = (2 · ((𝐹𝑢) · (𝐺𝑣))) → ∀𝑣 ∈ ℝ ∀𝑢 ∈ ℝ ((𝐹‘(𝑢 + 𝑣)) + (𝐹‘(𝑢𝑣))) = (2 · ((𝐹𝑢) · (𝐺𝑣))))
7271a1i 11 . . . . . . . . . . . . . . . . 17 (𝜑 → (∀𝑢 ∈ ℝ ∀𝑣 ∈ ℝ ((𝐹‘(𝑢 + 𝑣)) + (𝐹‘(𝑢𝑣))) = (2 · ((𝐹𝑢) · (𝐺𝑣))) → ∀𝑣 ∈ ℝ ∀𝑢 ∈ ℝ ((𝐹‘(𝑢 + 𝑣)) + (𝐹‘(𝑢𝑣))) = (2 · ((𝐹𝑢) · (𝐺𝑣)))))
7372imp 406 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ∀𝑢 ∈ ℝ ∀𝑣 ∈ ℝ ((𝐹‘(𝑢 + 𝑣)) + (𝐹‘(𝑢𝑣))) = (2 · ((𝐹𝑢) · (𝐺𝑣)))) → ∀𝑣 ∈ ℝ ∀𝑢 ∈ ℝ ((𝐹‘(𝑢 + 𝑣)) + (𝐹‘(𝑢𝑣))) = (2 · ((𝐹𝑢) · (𝐺𝑣))))
7469, 73mpdan 687 . . . . . . . . . . . . . . 15 (𝜑 → ∀𝑣 ∈ ℝ ∀𝑢 ∈ ℝ ((𝐹‘(𝑢 + 𝑣)) + (𝐹‘(𝑢𝑣))) = (2 · ((𝐹𝑢) · (𝐺𝑣))))
7568, 2, 74rspcdv2 3617 . . . . . . . . . . . . . 14 (𝜑 → ∀𝑢 ∈ ℝ ((𝐹‘(𝑢 + 𝐵)) + (𝐹‘(𝑢𝐵))) = (2 · ((𝐹𝑢) · (𝐺𝐵))))
7675r19.21bi 3249 . . . . . . . . . . . . 13 ((𝜑𝑢 ∈ ℝ) → ((𝐹‘(𝑢 + 𝐵)) + (𝐹‘(𝑢𝐵))) = (2 · ((𝐹𝑢) · (𝐺𝐵))))
7776adantlr 715 . . . . . . . . . . . 12 (((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) ∧ 𝑢 ∈ ℝ) → ((𝐹‘(𝑢 + 𝐵)) + (𝐹‘(𝑢𝐵))) = (2 · ((𝐹𝑢) · (𝐺𝐵))))
7840ad2antrr 726 . . . . . . . . . . . 12 (((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) ∧ 𝑢 ∈ ℝ) → ∀𝑦 ∈ ℝ (abs‘(𝐹𝑦)) ≤ 1)
7954, 55, 56, 57, 77, 78imo72b2lem0 44155 . . . . . . . . . . 11 (((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) ∧ 𝑢 ∈ ℝ) → ((abs‘(𝐹𝑢)) · (abs‘(𝐺𝐵))) ≤ sup((abs “ (𝐹 “ ℝ)), ℝ, < ))
80 0xr 11306 . . . . . . . . . . . . 13 0 ∈ ℝ*
8180a1i 11 . . . . . . . . . . . 12 (((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) ∧ 𝑢 ∈ ℝ) → 0 ∈ ℝ*)
82 1xr 11318 . . . . . . . . . . . . 13 1 ∈ ℝ*
8382a1i 11 . . . . . . . . . . . 12 (((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) ∧ 𝑢 ∈ ℝ) → 1 ∈ ℝ*)
8412adantr 480 . . . . . . . . . . . . 13 (((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) ∧ 𝑢 ∈ ℝ) → (abs‘(𝐺𝐵)) ∈ ℝ)
8584rexrd 11309 . . . . . . . . . . . 12 (((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) ∧ 𝑢 ∈ ℝ) → (abs‘(𝐺𝐵)) ∈ ℝ*)
8649a1i 11 . . . . . . . . . . . 12 (((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) ∧ 𝑢 ∈ ℝ) → 0 < 1)
87 simplr 769 . . . . . . . . . . . 12 (((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) ∧ 𝑢 ∈ ℝ) → 1 < (abs‘(𝐺𝐵)))
8881, 83, 85, 86, 87xrlttrd 13198 . . . . . . . . . . 11 (((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) ∧ 𝑢 ∈ ℝ) → 0 < (abs‘(𝐺𝐵)))
8920ffvelcdmda 7104 . . . . . . . . . . . . 13 (((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) ∧ 𝑢 ∈ ℝ) → (𝐹𝑢) ∈ ℝ)
9089recnd 11287 . . . . . . . . . . . 12 (((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) ∧ 𝑢 ∈ ℝ) → (𝐹𝑢) ∈ ℂ)
9190abscld 15472 . . . . . . . . . . 11 (((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) ∧ 𝑢 ∈ ℝ) → (abs‘(𝐹𝑢)) ∈ ℝ)
9244adantr 480 . . . . . . . . . . 11 (((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) ∧ 𝑢 ∈ ℝ) → sup((abs “ (𝐹 “ ℝ)), ℝ, < ) ∈ ℝ)
9379, 88, 84, 91, 92lemuldiv3d 44160 . . . . . . . . . 10 (((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) ∧ 𝑢 ∈ ℝ) → (abs‘(𝐹𝑢)) ≤ (sup((abs “ (𝐹 “ ℝ)), ℝ, < ) / (abs‘(𝐺𝐵))))
9493ralrimiva 3144 . . . . . . . . 9 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → ∀𝑢 ∈ ℝ (abs‘(𝐹𝑢)) ≤ (sup((abs “ (𝐹 “ ℝ)), ℝ, < ) / (abs‘(𝐺𝐵))))
9520, 53, 94imo72b2lem2 44157 . . . . . . . 8 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → sup((abs “ (𝐹 “ ℝ)), ℝ, < ) ≤ (sup((abs “ (𝐹 “ ℝ)), ℝ, < ) / (abs‘(𝐺𝐵))))
9695, 51, 12, 44, 44lemuldiv4d 44161 . . . . . . 7 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → (sup((abs “ (𝐹 “ ℝ)), ℝ, < ) · (abs‘(𝐺𝐵))) ≤ sup((abs “ (𝐹 “ ℝ)), ℝ, < ))
9747, 96eqbrtrrd 5172 . . . . . 6 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → ((abs‘(𝐺𝐵)) · sup((abs “ (𝐹 “ ℝ)), ℝ, < )) ≤ sup((abs “ (𝐹 “ ℝ)), ℝ, < ))
98 imo72b2.7 . . . . . . . 8 (𝜑 → ∃𝑥 ∈ ℝ (𝐹𝑥) ≠ 0)
9998adantr 480 . . . . . . 7 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → ∃𝑥 ∈ ℝ (𝐹𝑥) ≠ 0)
10040adantr 480 . . . . . . 7 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → ∀𝑦 ∈ ℝ (abs‘(𝐹𝑦)) ≤ 1)
10120, 99, 100imo72b2lem1 44159 . . . . . 6 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → 0 < sup((abs “ (𝐹 “ ℝ)), ℝ, < ))
10297, 101, 44, 12, 44lemuldiv3d 44160 . . . . 5 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → (abs‘(𝐺𝐵)) ≤ (sup((abs “ (𝐹 “ ℝ)), ℝ, < ) / sup((abs “ (𝐹 “ ℝ)), ℝ, < )))
10323, 44sseldd 3996 . . . . . . 7 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → sup((abs “ (𝐹 “ ℝ)), ℝ, < ) ∈ ℂ)
104101gt0ne0d 11825 . . . . . . 7 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → sup((abs “ (𝐹 “ ℝ)), ℝ, < ) ≠ 0)
105103, 104dividd 12039 . . . . . 6 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → (sup((abs “ (𝐹 “ ℝ)), ℝ, < ) / sup((abs “ (𝐹 “ ℝ)), ℝ, < )) = 1)
106105eqcomd 2741 . . . . 5 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → 1 = (sup((abs “ (𝐹 “ ℝ)), ℝ, < ) / sup((abs “ (𝐹 “ ℝ)), ℝ, < )))
107102, 106breqtrrd 5176 . . . 4 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → (abs‘(𝐺𝐵)) ≤ 1)
10812, 13, 107lensymd 11410 . . 3 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → ¬ 1 < (abs‘(𝐺𝐵)))
1097, 108pm2.65da 817 . 2 (𝜑 → ¬ 1 < (abs‘(𝐺𝐵)))
1105, 6, 109nltled 11409 1 (𝜑 → (abs‘(𝐺𝐵)) ≤ 1)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wne 2938  wral 3059  wrex 3068  wss 3963  c0 4339   class class class wbr 5148  ran crn 5690  cima 5692  ccom 5693  wf 6559  cfv 6563  (class class class)co 7431  supcsup 9478  cc 11151  cr 11152  0cc0 11153  1c1 11154   + caddc 11156   · cmul 11158  *cxr 11292   < clt 11293  cle 11294  cmin 11490   / cdiv 11918  2c2 12319  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: (None)
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