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Theorem imo72b2 42435
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 7036 . . . 4 (𝜑 → (𝐺𝐵) ∈ ℝ)
43recnd 11183 . . 3 (𝜑 → (𝐺𝐵) ∈ ℂ)
54abscld 15321 . 2 (𝜑 → (abs‘(𝐺𝐵)) ∈ ℝ)
6 1red 11156 . 2 (𝜑 → 1 ∈ ℝ)
7 simpr 485 . . 3 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → 1 < (abs‘(𝐺𝐵)))
81adantr 481 . . . . . . 7 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → 𝐺:ℝ⟶ℝ)
92adantr 481 . . . . . . 7 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → 𝐵 ∈ ℝ)
108, 9ffvelcdmd 7036 . . . . . 6 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → (𝐺𝐵) ∈ ℝ)
1110recnd 11183 . . . . 5 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → (𝐺𝐵) ∈ ℂ)
1211abscld 15321 . . . 4 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → (abs‘(𝐺𝐵)) ∈ ℝ)
136adantr 481 . . . 4 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → 1 ∈ ℝ)
14 ax-resscn 11108 . . . . . . . . 9 ℝ ⊆ ℂ
15 imaco 6203 . . . . . . . . . . . 12 ((abs ∘ 𝐹) “ ℝ) = (abs “ (𝐹 “ ℝ))
1615eqcomi 2745 . . . . . . . . . . 11 (abs “ (𝐹 “ ℝ)) = ((abs ∘ 𝐹) “ ℝ)
17 imassrn 6024 . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . 14 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → 𝐹:ℝ⟶ℝ)
21 absf 15222 . . . . . . . . . . . . . . . 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 6709 . . . . . . . . . . . . . 14 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → (abs ↾ ℝ):ℝ⟶ℝ)
2520, 24fco2d 42425 . . . . . . . . . . . . 13 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → (abs ∘ 𝐹):ℝ⟶ℝ)
2625frnd 6676 . . . . . . . . . . . 12 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → ran (abs ∘ 𝐹) ⊆ ℝ)
2718, 26sstrd 3954 . . . . . . . . . . 11 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → ((abs ∘ 𝐹) “ ℝ) ⊆ ℝ)
2816, 27eqsstrid 3992 . . . . . . . . . 10 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → (abs “ (𝐹 “ ℝ)) ⊆ ℝ)
29 0re 11157 . . . . . . . . . . . . . . . 16 0 ∈ ℝ
3029ne0ii 4297 . . . . . . . . . . . . . . 15 ℝ ≠ ∅
3130a1i 11 . . . . . . . . . . . . . 14 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → ℝ ≠ ∅)
3231, 25wnefimgd 42424 . . . . . . . . . . . . 13 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → ((abs ∘ 𝐹) “ ℝ) ≠ ∅)
3332necomd 2999 . . . . . . . . . . . 12 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → ∅ ≠ ((abs ∘ 𝐹) “ ℝ))
3416a1i 11 . . . . . . . . . . . 12 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → (abs “ (𝐹 “ ℝ)) = ((abs ∘ 𝐹) “ ℝ))
3533, 34neeqtrrd 3018 . . . . . . . . . . 11 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → ∅ ≠ (abs “ (𝐹 “ ℝ)))
3635necomd 2999 . . . . . . . . . 10 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → (abs “ (𝐹 “ ℝ)) ≠ ∅)
37 simpr 485 . . . . . . . . . . . . 13 (((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) ∧ 𝑐 = 1) → 𝑐 = 1)
3837breq2d 5117 . . . . . . . . . . . 12 (((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) ∧ 𝑐 = 1) → (𝑡𝑐𝑡 ≤ 1))
3938ralbidv 3174 . . . . . . . . . . 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 42427 . . . . . . . . . . . 12 (𝜑 → ∀𝑡 ∈ (abs “ (𝐹 “ ℝ))𝑡 ≤ 1)
4241adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → ∀𝑡 ∈ (abs “ (𝐹 “ ℝ))𝑡 ≤ 1)
4313, 39, 42rspcedvd 3583 . . . . . . . . . 10 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → ∃𝑐 ∈ ℝ ∀𝑡 ∈ (abs “ (𝐹 “ ℝ))𝑡𝑐)
4428, 36, 43suprcld 12118 . . . . . . . . 9 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → sup((abs “ (𝐹 “ ℝ)), ℝ, < ) ∈ ℝ)
4514, 44sselid 3942 . . . . . . . 8 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → sup((abs “ (𝐹 “ ℝ)), ℝ, < ) ∈ ℂ)
4614, 12sselid 3942 . . . . . . . 8 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → (abs‘(𝐺𝐵)) ∈ ℂ)
4745, 46mulcomd 11176 . . . . . . 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 11677 . . . . . . . . . . . . 13 0 < 1
5049a1i 11 . . . . . . . . . . . 12 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → 0 < 1)
5148, 13, 12, 50, 7lttrd 11316 . . . . . . . . . . 11 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → 0 < (abs‘(𝐺𝐵)))
5251gt0ne0d 11719 . . . . . . . . . 10 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → (abs‘(𝐺𝐵)) ≠ 0)
5344, 12, 52redivcld 11983 . . . . . . . . 9 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → (sup((abs “ (𝐹 “ ℝ)), ℝ, < ) / (abs‘(𝐺𝐵))) ∈ ℝ)
5420adantr 481 . . . . . . . . . . . 12 (((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) ∧ 𝑢 ∈ ℝ) → 𝐹:ℝ⟶ℝ)
558adantr 481 . . . . . . . . . . . 12 (((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) ∧ 𝑢 ∈ ℝ) → 𝐺:ℝ⟶ℝ)
56 simpr 485 . . . . . . . . . . . 12 (((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) ∧ 𝑢 ∈ ℝ) → 𝑢 ∈ ℝ)
579adantr 481 . . . . . . . . . . . 12 (((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) ∧ 𝑢 ∈ ℝ) → 𝐵 ∈ ℝ)
58 simpr 485 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑣 = 𝐵) → 𝑣 = 𝐵)
5958oveq2d 7373 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑣 = 𝐵) → (𝑢 + 𝑣) = (𝑢 + 𝐵))
6059fveq2d 6846 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑣 = 𝐵) → (𝐹‘(𝑢 + 𝑣)) = (𝐹‘(𝑢 + 𝐵)))
6158oveq2d 7373 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑣 = 𝐵) → (𝑢𝑣) = (𝑢𝐵))
6261fveq2d 6846 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑣 = 𝐵) → (𝐹‘(𝑢𝑣)) = (𝐹‘(𝑢𝐵)))
6360, 62oveq12d 7375 . . . . . . . . . . . . . . . . 17 ((𝜑𝑣 = 𝐵) → ((𝐹‘(𝑢 + 𝑣)) + (𝐹‘(𝑢𝑣))) = ((𝐹‘(𝑢 + 𝐵)) + (𝐹‘(𝑢𝐵))))
6458fveq2d 6846 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑣 = 𝐵) → (𝐺𝑣) = (𝐺𝐵))
6564oveq2d 7373 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑣 = 𝐵) → ((𝐹𝑢) · (𝐺𝑣)) = ((𝐹𝑢) · (𝐺𝐵)))
6665oveq2d 7373 . . . . . . . . . . . . . . . . 17 ((𝜑𝑣 = 𝐵) → (2 · ((𝐹𝑢) · (𝐺𝑣))) = (2 · ((𝐹𝑢) · (𝐺𝐵))))
6763, 66eqeq12d 2752 . . . . . . . . . . . . . . . 16 ((𝜑𝑣 = 𝐵) → (((𝐹‘(𝑢 + 𝑣)) + (𝐹‘(𝑢𝑣))) = (2 · ((𝐹𝑢) · (𝐺𝑣))) ↔ ((𝐹‘(𝑢 + 𝐵)) + (𝐹‘(𝑢𝐵))) = (2 · ((𝐹𝑢) · (𝐺𝐵)))))
6867ralbidv 3174 . . . . . . . . . . . . . . 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 3272 . . . . . . . . . . . . . . . . . . 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 215 . . . . . . . . . . . . . . . . . 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 407 . . . . . . . . . . . . . . . 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 685 . . . . . . . . . . . . . . 15 (𝜑 → ∀𝑣 ∈ ℝ ∀𝑢 ∈ ℝ ((𝐹‘(𝑢 + 𝑣)) + (𝐹‘(𝑢𝑣))) = (2 · ((𝐹𝑢) · (𝐺𝑣))))
7568, 2, 74rspcdv2 3576 . . . . . . . . . . . . . 14 (𝜑 → ∀𝑢 ∈ ℝ ((𝐹‘(𝑢 + 𝐵)) + (𝐹‘(𝑢𝐵))) = (2 · ((𝐹𝑢) · (𝐺𝐵))))
7675r19.21bi 3234 . . . . . . . . . . . . 13 ((𝜑𝑢 ∈ ℝ) → ((𝐹‘(𝑢 + 𝐵)) + (𝐹‘(𝑢𝐵))) = (2 · ((𝐹𝑢) · (𝐺𝐵))))
7776adantlr 713 . . . . . . . . . . . 12 (((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) ∧ 𝑢 ∈ ℝ) → ((𝐹‘(𝑢 + 𝐵)) + (𝐹‘(𝑢𝐵))) = (2 · ((𝐹𝑢) · (𝐺𝐵))))
7840ad2antrr 724 . . . . . . . . . . . 12 (((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) ∧ 𝑢 ∈ ℝ) → ∀𝑦 ∈ ℝ (abs‘(𝐹𝑦)) ≤ 1)
7954, 55, 56, 57, 77, 78imo72b2lem0 42428 . . . . . . . . . . 11 (((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) ∧ 𝑢 ∈ ℝ) → ((abs‘(𝐹𝑢)) · (abs‘(𝐺𝐵))) ≤ sup((abs “ (𝐹 “ ℝ)), ℝ, < ))
80 0xr 11202 . . . . . . . . . . . . 13 0 ∈ ℝ*
8180a1i 11 . . . . . . . . . . . 12 (((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) ∧ 𝑢 ∈ ℝ) → 0 ∈ ℝ*)
82 1xr 11214 . . . . . . . . . . . . 13 1 ∈ ℝ*
8382a1i 11 . . . . . . . . . . . 12 (((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) ∧ 𝑢 ∈ ℝ) → 1 ∈ ℝ*)
8412adantr 481 . . . . . . . . . . . . 13 (((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) ∧ 𝑢 ∈ ℝ) → (abs‘(𝐺𝐵)) ∈ ℝ)
8584rexrd 11205 . . . . . . . . . . . 12 (((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) ∧ 𝑢 ∈ ℝ) → (abs‘(𝐺𝐵)) ∈ ℝ*)
8649a1i 11 . . . . . . . . . . . 12 (((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) ∧ 𝑢 ∈ ℝ) → 0 < 1)
87 simplr 767 . . . . . . . . . . . 12 (((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) ∧ 𝑢 ∈ ℝ) → 1 < (abs‘(𝐺𝐵)))
8881, 83, 85, 86, 87xrlttrd 13078 . . . . . . . . . . 11 (((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) ∧ 𝑢 ∈ ℝ) → 0 < (abs‘(𝐺𝐵)))
8920ffvelcdmda 7035 . . . . . . . . . . . . 13 (((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) ∧ 𝑢 ∈ ℝ) → (𝐹𝑢) ∈ ℝ)
9089recnd 11183 . . . . . . . . . . . 12 (((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) ∧ 𝑢 ∈ ℝ) → (𝐹𝑢) ∈ ℂ)
9190abscld 15321 . . . . . . . . . . 11 (((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) ∧ 𝑢 ∈ ℝ) → (abs‘(𝐹𝑢)) ∈ ℝ)
9244adantr 481 . . . . . . . . . . 11 (((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) ∧ 𝑢 ∈ ℝ) → sup((abs “ (𝐹 “ ℝ)), ℝ, < ) ∈ ℝ)
9379, 88, 84, 91, 92lemuldiv3d 42433 . . . . . . . . . 10 (((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) ∧ 𝑢 ∈ ℝ) → (abs‘(𝐹𝑢)) ≤ (sup((abs “ (𝐹 “ ℝ)), ℝ, < ) / (abs‘(𝐺𝐵))))
9493ralrimiva 3143 . . . . . . . . 9 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → ∀𝑢 ∈ ℝ (abs‘(𝐹𝑢)) ≤ (sup((abs “ (𝐹 “ ℝ)), ℝ, < ) / (abs‘(𝐺𝐵))))
9520, 53, 94imo72b2lem2 42430 . . . . . . . 8 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → sup((abs “ (𝐹 “ ℝ)), ℝ, < ) ≤ (sup((abs “ (𝐹 “ ℝ)), ℝ, < ) / (abs‘(𝐺𝐵))))
9695, 51, 12, 44, 44lemuldiv4d 42434 . . . . . . 7 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → (sup((abs “ (𝐹 “ ℝ)), ℝ, < ) · (abs‘(𝐺𝐵))) ≤ sup((abs “ (𝐹 “ ℝ)), ℝ, < ))
9747, 96eqbrtrrd 5129 . . . . . 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 481 . . . . . . 7 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → ∃𝑥 ∈ ℝ (𝐹𝑥) ≠ 0)
10040adantr 481 . . . . . . 7 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → ∀𝑦 ∈ ℝ (abs‘(𝐹𝑦)) ≤ 1)
10120, 99, 100imo72b2lem1 42432 . . . . . 6 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → 0 < sup((abs “ (𝐹 “ ℝ)), ℝ, < ))
10297, 101, 44, 12, 44lemuldiv3d 42433 . . . . 5 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → (abs‘(𝐺𝐵)) ≤ (sup((abs “ (𝐹 “ ℝ)), ℝ, < ) / sup((abs “ (𝐹 “ ℝ)), ℝ, < )))
10323, 44sseldd 3945 . . . . . . 7 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → sup((abs “ (𝐹 “ ℝ)), ℝ, < ) ∈ ℂ)
104101gt0ne0d 11719 . . . . . . 7 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → sup((abs “ (𝐹 “ ℝ)), ℝ, < ) ≠ 0)
105103, 104dividd 11929 . . . . . 6 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → (sup((abs “ (𝐹 “ ℝ)), ℝ, < ) / sup((abs “ (𝐹 “ ℝ)), ℝ, < )) = 1)
106105eqcomd 2742 . . . . 5 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → 1 = (sup((abs “ (𝐹 “ ℝ)), ℝ, < ) / sup((abs “ (𝐹 “ ℝ)), ℝ, < )))
107102, 106breqtrrd 5133 . . . 4 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → (abs‘(𝐺𝐵)) ≤ 1)
10812, 13, 107lensymd 11306 . . 3 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → ¬ 1 < (abs‘(𝐺𝐵)))
1097, 108pm2.65da 815 . 2 (𝜑 → ¬ 1 < (abs‘(𝐺𝐵)))
1105, 6, 109nltled 11305 1 (𝜑 → (abs‘(𝐺𝐵)) ≤ 1)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  wne 2943  wral 3064  wrex 3073  wss 3910  c0 4282   class class class wbr 5105  ran crn 5634  cima 5636  ccom 5637  wf 6492  cfv 6496  (class class class)co 7357  supcsup 9376  cc 11049  cr 11050  0cc0 11051  1c1 11052   + caddc 11054   · cmul 11056  *cxr 11188   < clt 11189  cle 11190  cmin 11385   / cdiv 11812  2c2 12208  abscabs 15119
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-sup 9378  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-z 12500  df-uz 12764  df-rp 12916  df-seq 13907  df-exp 13968  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121
This theorem is referenced by: (None)
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