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Theorem imo72b2 41330
Description: IMO 1972 B2. (14th International Mathemahics 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, 2ffvelrnd 6862 . . . 4 (𝜑 → (𝐺𝐵) ∈ ℝ)
43recnd 10747 . . 3 (𝜑 → (𝐺𝐵) ∈ ℂ)
54abscld 14886 . 2 (𝜑 → (abs‘(𝐺𝐵)) ∈ ℝ)
6 1red 10720 . 2 (𝜑 → 1 ∈ ℝ)
7 simpr 488 . . 3 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → 1 < (abs‘(𝐺𝐵)))
81adantr 484 . . . . . . 7 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → 𝐺:ℝ⟶ℝ)
92adantr 484 . . . . . . 7 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → 𝐵 ∈ ℝ)
108, 9ffvelrnd 6862 . . . . . 6 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → (𝐺𝐵) ∈ ℝ)
1110recnd 10747 . . . . 5 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → (𝐺𝐵) ∈ ℂ)
1211abscld 14886 . . . 4 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → (abs‘(𝐺𝐵)) ∈ ℝ)
136adantr 484 . . . 4 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → 1 ∈ ℝ)
14 ax-resscn 10672 . . . . . . . . 9 ℝ ⊆ ℂ
15 imaco 6084 . . . . . . . . . . . 12 ((abs ∘ 𝐹) “ ℝ) = (abs “ (𝐹 “ ℝ))
1615eqcomi 2747 . . . . . . . . . . 11 (abs “ (𝐹 “ ℝ)) = ((abs ∘ 𝐹) “ ℝ)
17 imassrn 5914 . . . . . . . . . . . . 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 484 . . . . . . . . . . . . . 14 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → 𝐹:ℝ⟶ℝ)
21 absf 14787 . . . . . . . . . . . . . . . 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 6545 . . . . . . . . . . . . . 14 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → (abs ↾ ℝ):ℝ⟶ℝ)
2520, 24fco2d 41319 . . . . . . . . . . . . 13 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → (abs ∘ 𝐹):ℝ⟶ℝ)
2625frnd 6512 . . . . . . . . . . . 12 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → ran (abs ∘ 𝐹) ⊆ ℝ)
2718, 26sstrd 3887 . . . . . . . . . . 11 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → ((abs ∘ 𝐹) “ ℝ) ⊆ ℝ)
2816, 27eqsstrid 3925 . . . . . . . . . 10 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → (abs “ (𝐹 “ ℝ)) ⊆ ℝ)
29 0re 10721 . . . . . . . . . . . . . . . 16 0 ∈ ℝ
3029ne0ii 4226 . . . . . . . . . . . . . . 15 ℝ ≠ ∅
3130a1i 11 . . . . . . . . . . . . . 14 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → ℝ ≠ ∅)
3231, 25wnefimgd 41318 . . . . . . . . . . . . 13 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → ((abs ∘ 𝐹) “ ℝ) ≠ ∅)
3332necomd 2989 . . . . . . . . . . . 12 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → ∅ ≠ ((abs ∘ 𝐹) “ ℝ))
3416a1i 11 . . . . . . . . . . . 12 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → (abs “ (𝐹 “ ℝ)) = ((abs ∘ 𝐹) “ ℝ))
3533, 34neeqtrrd 3008 . . . . . . . . . . 11 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → ∅ ≠ (abs “ (𝐹 “ ℝ)))
3635necomd 2989 . . . . . . . . . 10 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → (abs “ (𝐹 “ ℝ)) ≠ ∅)
37 simpr 488 . . . . . . . . . . . . 13 (((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) ∧ 𝑐 = 1) → 𝑐 = 1)
3837breq2d 5042 . . . . . . . . . . . 12 (((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) ∧ 𝑐 = 1) → (𝑡𝑐𝑡 ≤ 1))
3938ralbidv 3109 . . . . . . . . . . 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 41321 . . . . . . . . . . . 12 (𝜑 → ∀𝑡 ∈ (abs “ (𝐹 “ ℝ))𝑡 ≤ 1)
4241adantr 484 . . . . . . . . . . 11 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → ∀𝑡 ∈ (abs “ (𝐹 “ ℝ))𝑡 ≤ 1)
4313, 39, 42rspcedvd 3529 . . . . . . . . . 10 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → ∃𝑐 ∈ ℝ ∀𝑡 ∈ (abs “ (𝐹 “ ℝ))𝑡𝑐)
4428, 36, 43suprcld 11681 . . . . . . . . 9 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → sup((abs “ (𝐹 “ ℝ)), ℝ, < ) ∈ ℝ)
4514, 44sseldi 3875 . . . . . . . 8 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → sup((abs “ (𝐹 “ ℝ)), ℝ, < ) ∈ ℂ)
4614, 12sseldi 3875 . . . . . . . 8 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → (abs‘(𝐺𝐵)) ∈ ℂ)
4745, 46mulcomd 10740 . . . . . . 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 11240 . . . . . . . . . . . . 13 0 < 1
5049a1i 11 . . . . . . . . . . . 12 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → 0 < 1)
5148, 13, 12, 50, 7lttrd 10879 . . . . . . . . . . 11 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → 0 < (abs‘(𝐺𝐵)))
5251gt0ne0d 11282 . . . . . . . . . 10 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → (abs‘(𝐺𝐵)) ≠ 0)
5344, 12, 52redivcld 11546 . . . . . . . . 9 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → (sup((abs “ (𝐹 “ ℝ)), ℝ, < ) / (abs‘(𝐺𝐵))) ∈ ℝ)
5420adantr 484 . . . . . . . . . . . 12 (((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) ∧ 𝑢 ∈ ℝ) → 𝐹:ℝ⟶ℝ)
558adantr 484 . . . . . . . . . . . 12 (((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) ∧ 𝑢 ∈ ℝ) → 𝐺:ℝ⟶ℝ)
56 simpr 488 . . . . . . . . . . . 12 (((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) ∧ 𝑢 ∈ ℝ) → 𝑢 ∈ ℝ)
579adantr 484 . . . . . . . . . . . 12 (((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) ∧ 𝑢 ∈ ℝ) → 𝐵 ∈ ℝ)
58 simpr 488 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑣 = 𝐵) → 𝑣 = 𝐵)
5958oveq2d 7186 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑣 = 𝐵) → (𝑢 + 𝑣) = (𝑢 + 𝐵))
6059fveq2d 6678 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑣 = 𝐵) → (𝐹‘(𝑢 + 𝑣)) = (𝐹‘(𝑢 + 𝐵)))
6158oveq2d 7186 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑣 = 𝐵) → (𝑢𝑣) = (𝑢𝐵))
6261fveq2d 6678 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑣 = 𝐵) → (𝐹‘(𝑢𝑣)) = (𝐹‘(𝑢𝐵)))
6360, 62oveq12d 7188 . . . . . . . . . . . . . . . . 17 ((𝜑𝑣 = 𝐵) → ((𝐹‘(𝑢 + 𝑣)) + (𝐹‘(𝑢𝑣))) = ((𝐹‘(𝑢 + 𝐵)) + (𝐹‘(𝑢𝐵))))
6458fveq2d 6678 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑣 = 𝐵) → (𝐺𝑣) = (𝐺𝐵))
6564oveq2d 7186 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑣 = 𝐵) → ((𝐹𝑢) · (𝐺𝑣)) = ((𝐹𝑢) · (𝐺𝐵)))
6665oveq2d 7186 . . . . . . . . . . . . . . . . 17 ((𝜑𝑣 = 𝐵) → (2 · ((𝐹𝑢) · (𝐺𝑣))) = (2 · ((𝐹𝑢) · (𝐺𝐵))))
6763, 66eqeq12d 2754 . . . . . . . . . . . . . . . 16 ((𝜑𝑣 = 𝐵) → (((𝐹‘(𝑢 + 𝑣)) + (𝐹‘(𝑢𝑣))) = (2 · ((𝐹𝑢) · (𝐺𝑣))) ↔ ((𝐹‘(𝑢 + 𝐵)) + (𝐹‘(𝑢𝐵))) = (2 · ((𝐹𝑢) · (𝐺𝐵)))))
6867ralbidv 3109 . . . . . . . . . . . . . . 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 3258 . . . . . . . . . . . . . . . . . . 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 219 . . . . . . . . . . . . . . . . . 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 410 . . . . . . . . . . . . . . . 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, 74rspcdvinvd 41329 . . . . . . . . . . . . . 14 (𝜑 → ∀𝑢 ∈ ℝ ((𝐹‘(𝑢 + 𝐵)) + (𝐹‘(𝑢𝐵))) = (2 · ((𝐹𝑢) · (𝐺𝐵))))
7675r19.21bi 3121 . . . . . . . . . . . . 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 41322 . . . . . . . . . . 11 (((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) ∧ 𝑢 ∈ ℝ) → ((abs‘(𝐹𝑢)) · (abs‘(𝐺𝐵))) ≤ sup((abs “ (𝐹 “ ℝ)), ℝ, < ))
80 0xr 10766 . . . . . . . . . . . . 13 0 ∈ ℝ*
8180a1i 11 . . . . . . . . . . . 12 (((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) ∧ 𝑢 ∈ ℝ) → 0 ∈ ℝ*)
82 1xr 10778 . . . . . . . . . . . . 13 1 ∈ ℝ*
8382a1i 11 . . . . . . . . . . . 12 (((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) ∧ 𝑢 ∈ ℝ) → 1 ∈ ℝ*)
8412adantr 484 . . . . . . . . . . . . 13 (((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) ∧ 𝑢 ∈ ℝ) → (abs‘(𝐺𝐵)) ∈ ℝ)
8584rexrd 10769 . . . . . . . . . . . 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 12635 . . . . . . . . . . 11 (((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) ∧ 𝑢 ∈ ℝ) → 0 < (abs‘(𝐺𝐵)))
8920ffvelrnda 6861 . . . . . . . . . . . . 13 (((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) ∧ 𝑢 ∈ ℝ) → (𝐹𝑢) ∈ ℝ)
9089recnd 10747 . . . . . . . . . . . 12 (((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) ∧ 𝑢 ∈ ℝ) → (𝐹𝑢) ∈ ℂ)
9190abscld 14886 . . . . . . . . . . 11 (((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) ∧ 𝑢 ∈ ℝ) → (abs‘(𝐹𝑢)) ∈ ℝ)
9244adantr 484 . . . . . . . . . . 11 (((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) ∧ 𝑢 ∈ ℝ) → sup((abs “ (𝐹 “ ℝ)), ℝ, < ) ∈ ℝ)
9379, 88, 84, 91, 92lemuldiv3d 41327 . . . . . . . . . 10 (((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) ∧ 𝑢 ∈ ℝ) → (abs‘(𝐹𝑢)) ≤ (sup((abs “ (𝐹 “ ℝ)), ℝ, < ) / (abs‘(𝐺𝐵))))
9493ralrimiva 3096 . . . . . . . . 9 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → ∀𝑢 ∈ ℝ (abs‘(𝐹𝑢)) ≤ (sup((abs “ (𝐹 “ ℝ)), ℝ, < ) / (abs‘(𝐺𝐵))))
9520, 53, 94imo72b2lem2 41324 . . . . . . . 8 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → sup((abs “ (𝐹 “ ℝ)), ℝ, < ) ≤ (sup((abs “ (𝐹 “ ℝ)), ℝ, < ) / (abs‘(𝐺𝐵))))
9695, 51, 12, 44, 44lemuldiv4d 41328 . . . . . . 7 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → (sup((abs “ (𝐹 “ ℝ)), ℝ, < ) · (abs‘(𝐺𝐵))) ≤ sup((abs “ (𝐹 “ ℝ)), ℝ, < ))
9747, 96eqbrtrrd 5054 . . . . . 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 484 . . . . . . 7 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → ∃𝑥 ∈ ℝ (𝐹𝑥) ≠ 0)
10040adantr 484 . . . . . . 7 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → ∀𝑦 ∈ ℝ (abs‘(𝐹𝑦)) ≤ 1)
10120, 99, 100imo72b2lem1 41326 . . . . . 6 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → 0 < sup((abs “ (𝐹 “ ℝ)), ℝ, < ))
10297, 101, 44, 12, 44lemuldiv3d 41327 . . . . 5 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → (abs‘(𝐺𝐵)) ≤ (sup((abs “ (𝐹 “ ℝ)), ℝ, < ) / sup((abs “ (𝐹 “ ℝ)), ℝ, < )))
10323, 44sseldd 3878 . . . . . . 7 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → sup((abs “ (𝐹 “ ℝ)), ℝ, < ) ∈ ℂ)
104101gt0ne0d 11282 . . . . . . 7 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → sup((abs “ (𝐹 “ ℝ)), ℝ, < ) ≠ 0)
105103, 104dividd 11492 . . . . . 6 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → (sup((abs “ (𝐹 “ ℝ)), ℝ, < ) / sup((abs “ (𝐹 “ ℝ)), ℝ, < )) = 1)
106105eqcomd 2744 . . . . 5 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → 1 = (sup((abs “ (𝐹 “ ℝ)), ℝ, < ) / sup((abs “ (𝐹 “ ℝ)), ℝ, < )))
107102, 106breqtrrd 5058 . . . 4 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → (abs‘(𝐺𝐵)) ≤ 1)
10812, 13, 107lensymd 10869 . . 3 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → ¬ 1 < (abs‘(𝐺𝐵)))
1097, 108pm2.65da 817 . 2 (𝜑 → ¬ 1 < (abs‘(𝐺𝐵)))
1105, 6, 109nltled 10868 1 (𝜑 → (abs‘(𝐺𝐵)) ≤ 1)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1542  wcel 2114  wne 2934  wral 3053  wrex 3054  wss 3843  c0 4211   class class class wbr 5030  ran crn 5526  cima 5528  ccom 5529  wf 6335  cfv 6339  (class class class)co 7170  supcsup 8977  cc 10613  cr 10614  0cc0 10615  1c1 10616   + caddc 10618   · cmul 10620  *cxr 10752   < clt 10753  cle 10754  cmin 10948   / cdiv 11375  2c2 11771  abscabs 14683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479  ax-cnex 10671  ax-resscn 10672  ax-1cn 10673  ax-icn 10674  ax-addcl 10675  ax-addrcl 10676  ax-mulcl 10677  ax-mulrcl 10678  ax-mulcom 10679  ax-addass 10680  ax-mulass 10681  ax-distr 10682  ax-i2m1 10683  ax-1ne0 10684  ax-1rid 10685  ax-rnegex 10686  ax-rrecex 10687  ax-cnre 10688  ax-pre-lttri 10689  ax-pre-lttrn 10690  ax-pre-ltadd 10691  ax-pre-mulgt0 10692  ax-pre-sup 10693
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-nel 3039  df-ral 3058  df-rex 3059  df-reu 3060  df-rmo 3061  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-riota 7127  df-ov 7173  df-oprab 7174  df-mpo 7175  df-om 7600  df-2nd 7715  df-wrecs 7976  df-recs 8037  df-rdg 8075  df-er 8320  df-en 8556  df-dom 8557  df-sdom 8558  df-sup 8979  df-pnf 10755  df-mnf 10756  df-xr 10757  df-ltxr 10758  df-le 10759  df-sub 10950  df-neg 10951  df-div 11376  df-nn 11717  df-2 11779  df-3 11780  df-n0 11977  df-z 12063  df-uz 12325  df-rp 12473  df-seq 13461  df-exp 13522  df-cj 14548  df-re 14549  df-im 14550  df-sqrt 14684  df-abs 14685
This theorem is referenced by: (None)
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