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Theorem imo72b2 40410
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 6850 . . . 4 (𝜑 → (𝐺𝐵) ∈ ℝ)
43recnd 10663 . . 3 (𝜑 → (𝐺𝐵) ∈ ℂ)
54abscld 14791 . 2 (𝜑 → (abs‘(𝐺𝐵)) ∈ ℝ)
6 1red 10636 . 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, 9ffvelrnd 6850 . . . . . 6 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → (𝐺𝐵) ∈ ℝ)
1110recnd 10663 . . . . 5 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → (𝐺𝐵) ∈ ℂ)
1211abscld 14791 . . . 4 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → (abs‘(𝐺𝐵)) ∈ ℝ)
136adantr 481 . . . 4 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → 1 ∈ ℝ)
14 ax-resscn 10588 . . . . . . . . 9 ℝ ⊆ ℂ
15 imaco 6103 . . . . . . . . . . . 12 ((abs ∘ 𝐹) “ ℝ) = (abs “ (𝐹 “ ℝ))
1615eqcomi 2835 . . . . . . . . . . 11 (abs “ (𝐹 “ ℝ)) = ((abs ∘ 𝐹) “ ℝ)
17 imassrn 5939 . . . . . . . . . . . . 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 14692 . . . . . . . . . . . . . . . 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 6544 . . . . . . . . . . . . . 14 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → (abs ↾ ℝ):ℝ⟶ℝ)
2520, 24fco2d 40397 . . . . . . . . . . . . 13 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → (abs ∘ 𝐹):ℝ⟶ℝ)
2625frnd 6520 . . . . . . . . . . . 12 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → ran (abs ∘ 𝐹) ⊆ ℝ)
2718, 26sstrd 3981 . . . . . . . . . . 11 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → ((abs ∘ 𝐹) “ ℝ) ⊆ ℝ)
2816, 27eqsstrid 4019 . . . . . . . . . 10 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → (abs “ (𝐹 “ ℝ)) ⊆ ℝ)
29 0re 10637 . . . . . . . . . . . . . . . 16 0 ∈ ℝ
3029ne0ii 4307 . . . . . . . . . . . . . . 15 ℝ ≠ ∅
3130a1i 11 . . . . . . . . . . . . . 14 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → ℝ ≠ ∅)
3231, 25wnefimgd 40396 . . . . . . . . . . . . 13 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → ((abs ∘ 𝐹) “ ℝ) ≠ ∅)
3332necomd 3076 . . . . . . . . . . . 12 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → ∅ ≠ ((abs ∘ 𝐹) “ ℝ))
3416a1i 11 . . . . . . . . . . . 12 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → (abs “ (𝐹 “ ℝ)) = ((abs ∘ 𝐹) “ ℝ))
3533, 34neeqtrrd 3095 . . . . . . . . . . 11 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → ∅ ≠ (abs “ (𝐹 “ ℝ)))
3635necomd 3076 . . . . . . . . . 10 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → (abs “ (𝐹 “ ℝ)) ≠ ∅)
37 simpr 485 . . . . . . . . . . . . 13 (((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) ∧ 𝑐 = 1) → 𝑐 = 1)
3837breq2d 5075 . . . . . . . . . . . 12 (((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) ∧ 𝑐 = 1) → (𝑡𝑐𝑡 ≤ 1))
3938ralbidv 3202 . . . . . . . . . . 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 40399 . . . . . . . . . . . 12 (𝜑 → ∀𝑡 ∈ (abs “ (𝐹 “ ℝ))𝑡 ≤ 1)
4241adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → ∀𝑡 ∈ (abs “ (𝐹 “ ℝ))𝑡 ≤ 1)
4313, 39, 42rspcedvd 3630 . . . . . . . . . 10 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → ∃𝑐 ∈ ℝ ∀𝑡 ∈ (abs “ (𝐹 “ ℝ))𝑡𝑐)
4428, 36, 43suprcld 11598 . . . . . . . . 9 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → sup((abs “ (𝐹 “ ℝ)), ℝ, < ) ∈ ℝ)
4514, 44sseldi 3969 . . . . . . . 8 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → sup((abs “ (𝐹 “ ℝ)), ℝ, < ) ∈ ℂ)
4614, 12sseldi 3969 . . . . . . . 8 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → (abs‘(𝐺𝐵)) ∈ ℂ)
4745, 46mulcomd 10656 . . . . . . 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 11156 . . . . . . . . . . . . 13 0 < 1
5049a1i 11 . . . . . . . . . . . 12 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → 0 < 1)
5148, 13, 12, 50, 7lttrd 10795 . . . . . . . . . . 11 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → 0 < (abs‘(𝐺𝐵)))
5251gt0ne0d 11198 . . . . . . . . . 10 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → (abs‘(𝐺𝐵)) ≠ 0)
5344, 12, 52redivcld 11462 . . . . . . . . 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 7166 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑣 = 𝐵) → (𝑢 + 𝑣) = (𝑢 + 𝐵))
6059fveq2d 6673 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑣 = 𝐵) → (𝐹‘(𝑢 + 𝑣)) = (𝐹‘(𝑢 + 𝐵)))
6158oveq2d 7166 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑣 = 𝐵) → (𝑢𝑣) = (𝑢𝐵))
6261fveq2d 6673 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑣 = 𝐵) → (𝐹‘(𝑢𝑣)) = (𝐹‘(𝑢𝐵)))
6360, 62oveq12d 7168 . . . . . . . . . . . . . . . . 17 ((𝜑𝑣 = 𝐵) → ((𝐹‘(𝑢 + 𝑣)) + (𝐹‘(𝑢𝑣))) = ((𝐹‘(𝑢 + 𝐵)) + (𝐹‘(𝑢𝐵))))
6458fveq2d 6673 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑣 = 𝐵) → (𝐺𝑣) = (𝐺𝐵))
6564oveq2d 7166 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑣 = 𝐵) → ((𝐹𝑢) · (𝐺𝑣)) = ((𝐹𝑢) · (𝐺𝐵)))
6665oveq2d 7166 . . . . . . . . . . . . . . . . 17 ((𝜑𝑣 = 𝐵) → (2 · ((𝐹𝑢) · (𝐺𝑣))) = (2 · ((𝐹𝑢) · (𝐺𝐵))))
6763, 66eqeq12d 2842 . . . . . . . . . . . . . . . 16 ((𝜑𝑣 = 𝐵) → (((𝐹‘(𝑢 + 𝑣)) + (𝐹‘(𝑢𝑣))) = (2 · ((𝐹𝑢) · (𝐺𝑣))) ↔ ((𝐹‘(𝑢 + 𝐵)) + (𝐹‘(𝑢𝐵))) = (2 · ((𝐹𝑢) · (𝐺𝐵)))))
6867ralbidv 3202 . . . . . . . . . . . . . . 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 ralcom2 3369 . . . . . . . . . . . . . . . . . 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))))
7170a1i 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)))))
7271imp 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))))
7369, 72mpdan 683 . . . . . . . . . . . . . . 15 (𝜑 → ∀𝑣 ∈ ℝ ∀𝑢 ∈ ℝ ((𝐹‘(𝑢 + 𝑣)) + (𝐹‘(𝑢𝑣))) = (2 · ((𝐹𝑢) · (𝐺𝑣))))
7468, 2, 73rspcdvinvd 40409 . . . . . . . . . . . . . 14 (𝜑 → ∀𝑢 ∈ ℝ ((𝐹‘(𝑢 + 𝐵)) + (𝐹‘(𝑢𝐵))) = (2 · ((𝐹𝑢) · (𝐺𝐵))))
7574r19.21bi 3213 . . . . . . . . . . . . 13 ((𝜑𝑢 ∈ ℝ) → ((𝐹‘(𝑢 + 𝐵)) + (𝐹‘(𝑢𝐵))) = (2 · ((𝐹𝑢) · (𝐺𝐵))))
7675adantlr 711 . . . . . . . . . . . 12 (((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) ∧ 𝑢 ∈ ℝ) → ((𝐹‘(𝑢 + 𝐵)) + (𝐹‘(𝑢𝐵))) = (2 · ((𝐹𝑢) · (𝐺𝐵))))
7740ad2antrr 722 . . . . . . . . . . . 12 (((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) ∧ 𝑢 ∈ ℝ) → ∀𝑦 ∈ ℝ (abs‘(𝐹𝑦)) ≤ 1)
7854, 55, 56, 57, 76, 77imo72b2lem0 40400 . . . . . . . . . . 11 (((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) ∧ 𝑢 ∈ ℝ) → ((abs‘(𝐹𝑢)) · (abs‘(𝐺𝐵))) ≤ sup((abs “ (𝐹 “ ℝ)), ℝ, < ))
79 0xr 10682 . . . . . . . . . . . . 13 0 ∈ ℝ*
8079a1i 11 . . . . . . . . . . . 12 (((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) ∧ 𝑢 ∈ ℝ) → 0 ∈ ℝ*)
81 1xr 10694 . . . . . . . . . . . . 13 1 ∈ ℝ*
8281a1i 11 . . . . . . . . . . . 12 (((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) ∧ 𝑢 ∈ ℝ) → 1 ∈ ℝ*)
8312adantr 481 . . . . . . . . . . . . 13 (((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) ∧ 𝑢 ∈ ℝ) → (abs‘(𝐺𝐵)) ∈ ℝ)
8483rexrd 10685 . . . . . . . . . . . 12 (((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) ∧ 𝑢 ∈ ℝ) → (abs‘(𝐺𝐵)) ∈ ℝ*)
8549a1i 11 . . . . . . . . . . . 12 (((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) ∧ 𝑢 ∈ ℝ) → 0 < 1)
86 simplr 765 . . . . . . . . . . . 12 (((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) ∧ 𝑢 ∈ ℝ) → 1 < (abs‘(𝐺𝐵)))
8780, 82, 84, 85, 86xrlttrd 12547 . . . . . . . . . . 11 (((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) ∧ 𝑢 ∈ ℝ) → 0 < (abs‘(𝐺𝐵)))
8820ffvelrnda 6849 . . . . . . . . . . . . 13 (((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) ∧ 𝑢 ∈ ℝ) → (𝐹𝑢) ∈ ℝ)
8988recnd 10663 . . . . . . . . . . . 12 (((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) ∧ 𝑢 ∈ ℝ) → (𝐹𝑢) ∈ ℂ)
9089abscld 14791 . . . . . . . . . . 11 (((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) ∧ 𝑢 ∈ ℝ) → (abs‘(𝐹𝑢)) ∈ ℝ)
9144adantr 481 . . . . . . . . . . 11 (((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) ∧ 𝑢 ∈ ℝ) → sup((abs “ (𝐹 “ ℝ)), ℝ, < ) ∈ ℝ)
9278, 87, 83, 90, 91lemuldiv3d 40407 . . . . . . . . . 10 (((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) ∧ 𝑢 ∈ ℝ) → (abs‘(𝐹𝑢)) ≤ (sup((abs “ (𝐹 “ ℝ)), ℝ, < ) / (abs‘(𝐺𝐵))))
9392ralrimiva 3187 . . . . . . . . 9 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → ∀𝑢 ∈ ℝ (abs‘(𝐹𝑢)) ≤ (sup((abs “ (𝐹 “ ℝ)), ℝ, < ) / (abs‘(𝐺𝐵))))
9420, 53, 93imo72b2lem2 40402 . . . . . . . 8 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → sup((abs “ (𝐹 “ ℝ)), ℝ, < ) ≤ (sup((abs “ (𝐹 “ ℝ)), ℝ, < ) / (abs‘(𝐺𝐵))))
9594, 51, 12, 44, 44lemuldiv4d 40408 . . . . . . 7 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → (sup((abs “ (𝐹 “ ℝ)), ℝ, < ) · (abs‘(𝐺𝐵))) ≤ sup((abs “ (𝐹 “ ℝ)), ℝ, < ))
9647, 95eqbrtrrd 5087 . . . . . 6 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → ((abs‘(𝐺𝐵)) · sup((abs “ (𝐹 “ ℝ)), ℝ, < )) ≤ sup((abs “ (𝐹 “ ℝ)), ℝ, < ))
97 imo72b2.7 . . . . . . . 8 (𝜑 → ∃𝑥 ∈ ℝ (𝐹𝑥) ≠ 0)
9897adantr 481 . . . . . . 7 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → ∃𝑥 ∈ ℝ (𝐹𝑥) ≠ 0)
9940adantr 481 . . . . . . 7 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → ∀𝑦 ∈ ℝ (abs‘(𝐹𝑦)) ≤ 1)
10020, 98, 99imo72b2lem1 40406 . . . . . 6 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → 0 < sup((abs “ (𝐹 “ ℝ)), ℝ, < ))
10196, 100, 44, 12, 44lemuldiv3d 40407 . . . . 5 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → (abs‘(𝐺𝐵)) ≤ (sup((abs “ (𝐹 “ ℝ)), ℝ, < ) / sup((abs “ (𝐹 “ ℝ)), ℝ, < )))
10223, 44sseldd 3972 . . . . . . 7 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → sup((abs “ (𝐹 “ ℝ)), ℝ, < ) ∈ ℂ)
103100gt0ne0d 11198 . . . . . . 7 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → sup((abs “ (𝐹 “ ℝ)), ℝ, < ) ≠ 0)
104102, 103dividd 11408 . . . . . 6 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → (sup((abs “ (𝐹 “ ℝ)), ℝ, < ) / sup((abs “ (𝐹 “ ℝ)), ℝ, < )) = 1)
105104eqcomd 2832 . . . . 5 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → 1 = (sup((abs “ (𝐹 “ ℝ)), ℝ, < ) / sup((abs “ (𝐹 “ ℝ)), ℝ, < )))
106101, 105breqtrrd 5091 . . . 4 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → (abs‘(𝐺𝐵)) ≤ 1)
10712, 13, 106lensymd 10785 . . 3 ((𝜑 ∧ 1 < (abs‘(𝐺𝐵))) → ¬ 1 < (abs‘(𝐺𝐵)))
1087, 107pm2.65da 813 . 2 (𝜑 → ¬ 1 < (abs‘(𝐺𝐵)))
1095, 6, 108nltled 10784 1 (𝜑 → (abs‘(𝐺𝐵)) ≤ 1)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1530  wcel 2107  wne 3021  wral 3143  wrex 3144  wss 3940  c0 4295   class class class wbr 5063  ran crn 5555  cima 5557  ccom 5558  wf 6350  cfv 6354  (class class class)co 7150  supcsup 8898  cc 10529  cr 10530  0cc0 10531  1c1 10532   + caddc 10534   · cmul 10536  *cxr 10668   < clt 10669  cle 10670  cmin 10864   / cdiv 11291  2c2 11686  abscabs 14588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-13 2385  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608  ax-pre-sup 10609
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7574  df-2nd 7686  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-er 8284  df-en 8504  df-dom 8505  df-sdom 8506  df-sup 8900  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-div 11292  df-nn 11633  df-2 11694  df-3 11695  df-n0 11892  df-z 11976  df-uz 12238  df-rp 12385  df-seq 13365  df-exp 13425  df-cj 14453  df-re 14454  df-im 14455  df-sqrt 14589  df-abs 14590
This theorem is referenced by: (None)
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