imo72b2 - Mathbox for Stanislas Polu

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Theorem imo72b2 41783

Description: IMO 1972 B2. (14th International Mathematical Olympiad in Poland, problem B2). (Contributed by Stanislas Polu, 9-Mar-2020.)

**Hypotheses**
Ref	Expression
imo72b2.1	⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
imo72b2.2	⊢ (𝜑 → 𝐺:ℝ⟶ℝ)
imo72b2.4	⊢ (𝜑 → 𝐵 ∈ ℝ)
imo72b2.5	⊢ (𝜑 → ∀𝑢 ∈ ℝ ∀𝑣 ∈ ℝ ((𝐹‘(𝑢 + 𝑣)) + (𝐹‘(𝑢 − 𝑣))) = (2 · ((𝐹‘𝑢) · (𝐺‘𝑣))))
imo72b2.6	⊢ (𝜑 → ∀𝑦 ∈ ℝ (abs‘(𝐹‘𝑦)) ≤ 1)
imo72b2.7	⊢ (𝜑 → ∃𝑥 ∈ ℝ (𝐹‘𝑥) ≠ 0)

**Assertion**
Ref	Expression
imo72b2	⊢ (𝜑 → (abs‘(𝐺‘𝐵)) ≤ 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.

Step	Hyp	Ref	Expression
1		imo72b2.2	. . . . 5 ⊢ (𝜑 → 𝐺:ℝ⟶ℝ)
2		imo72b2.4	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ)
3	1, 2	ffvelrnd 6962	. . . 4 ⊢ (𝜑 → (𝐺‘𝐵) ∈ ℝ)
4	3	recnd 11003	. . 3 ⊢ (𝜑 → (𝐺‘𝐵) ∈ ℂ)
5	4	abscld 15148	. 2 ⊢ (𝜑 → (abs‘(𝐺‘𝐵)) ∈ ℝ)
6		1red 10976	. 2 ⊢ (𝜑 → 1 ∈ ℝ)
7		simpr 485	. . 3 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → 1 < (abs‘(𝐺‘𝐵)))
8	1	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → 𝐺:ℝ⟶ℝ)
9	2	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → 𝐵 ∈ ℝ)
10	8, 9	ffvelrnd 6962	. . . . . 6 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → (𝐺‘𝐵) ∈ ℝ)
11	10	recnd 11003	. . . . 5 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → (𝐺‘𝐵) ∈ ℂ)
12	11	abscld 15148	. . . 4 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → (abs‘(𝐺‘𝐵)) ∈ ℝ)
13	6	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → 1 ∈ ℝ)
14		ax-resscn 10928	. . . . . . . . 9 ⊢ ℝ ⊆ ℂ
15		imaco 6155	. . . . . . . . . . . 12 ⊢ ((abs ∘ 𝐹) “ ℝ) = (abs “ (𝐹 “ ℝ))
16	15	eqcomi 2747	. . . . . . . . . . 11 ⊢ (abs “ (𝐹 “ ℝ)) = ((abs ∘ 𝐹) “ ℝ)
17		imassrn 5980	. . . . . . . . . . . . 13 ⊢ ((abs ∘ 𝐹) “ ℝ) ⊆ ran (abs ∘ 𝐹)
18	17	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → ((abs ∘ 𝐹) “ ℝ) ⊆ ran (abs ∘ 𝐹))
19		imo72b2.1	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
20	19	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → 𝐹:ℝ⟶ℝ)
21		absf 15049	. . . . . . . . . . . . . . . 16 ⊢ abs:ℂ⟶ℝ
22	21	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → abs:ℂ⟶ℝ)
23	14	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → ℝ ⊆ ℂ)
24	22, 23	fssresd 6641	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → (abs ↾ ℝ):ℝ⟶ℝ)
25	20, 24	fco2d 41773	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → (abs ∘ 𝐹):ℝ⟶ℝ)
26	25	frnd 6608	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → ran (abs ∘ 𝐹) ⊆ ℝ)
27	18, 26	sstrd 3931	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → ((abs ∘ 𝐹) “ ℝ) ⊆ ℝ)
28	16, 27	eqsstrid 3969	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → (abs “ (𝐹 “ ℝ)) ⊆ ℝ)
29		0re 10977	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℝ
30	29	ne0ii 4271	. . . . . . . . . . . . . . 15 ⊢ ℝ ≠ ∅
31	30	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → ℝ ≠ ∅)
32	31, 25	wnefimgd 41772	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → ((abs ∘ 𝐹) “ ℝ) ≠ ∅)
33	32	necomd 2999	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → ∅ ≠ ((abs ∘ 𝐹) “ ℝ))
34	16	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → (abs “ (𝐹 “ ℝ)) = ((abs ∘ 𝐹) “ ℝ))
35	33, 34	neeqtrrd 3018	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → ∅ ≠ (abs “ (𝐹 “ ℝ)))
36	35	necomd 2999	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → (abs “ (𝐹 “ ℝ)) ≠ ∅)
37		simpr 485	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑐 = 1) → 𝑐 = 1)
38	37	breq2d 5086	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑐 = 1) → (𝑡 ≤ 𝑐 ↔ 𝑡 ≤ 1))
39	38	ralbidv 3112	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑐 = 1) → (∀𝑡 ∈ (abs “ (𝐹 “ ℝ))𝑡 ≤ 𝑐 ↔ ∀𝑡 ∈ (abs “ (𝐹 “ ℝ))𝑡 ≤ 1))
40		imo72b2.6	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑦 ∈ ℝ (abs‘(𝐹‘𝑦)) ≤ 1)
41	19, 40	extoimad 41775	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑡 ∈ (abs “ (𝐹 “ ℝ))𝑡 ≤ 1)
42	41	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → ∀𝑡 ∈ (abs “ (𝐹 “ ℝ))𝑡 ≤ 1)
43	13, 39, 42	rspcedvd 3563	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → ∃𝑐 ∈ ℝ ∀𝑡 ∈ (abs “ (𝐹 “ ℝ))𝑡 ≤ 𝑐)
44	28, 36, 43	suprcld 11938	. . . . . . . . 9 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → sup((abs “ (𝐹 “ ℝ)), ℝ, < ) ∈ ℝ)
45	14, 44	sselid 3919	. . . . . . . 8 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → sup((abs “ (𝐹 “ ℝ)), ℝ, < ) ∈ ℂ)
46	14, 12	sselid 3919	. . . . . . . 8 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → (abs‘(𝐺‘𝐵)) ∈ ℂ)
47	45, 46	mulcomd 10996	. . . . . . 7 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → (sup((abs “ (𝐹 “ ℝ)), ℝ, < ) · (abs‘(𝐺‘𝐵))) = ((abs‘(𝐺‘𝐵)) · sup((abs “ (𝐹 “ ℝ)), ℝ, < )))
48	29	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → 0 ∈ ℝ)
49		0lt1 11497	. . . . . . . . . . . . 13 ⊢ 0 < 1
50	49	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → 0 < 1)
51	48, 13, 12, 50, 7	lttrd 11136	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → 0 < (abs‘(𝐺‘𝐵)))
52	51	gt0ne0d 11539	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → (abs‘(𝐺‘𝐵)) ≠ 0)
53	44, 12, 52	redivcld 11803	. . . . . . . . 9 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → (sup((abs “ (𝐹 “ ℝ)), ℝ, < ) / (abs‘(𝐺‘𝐵))) ∈ ℝ)
54	20	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → 𝐹:ℝ⟶ℝ)
55	8	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → 𝐺:ℝ⟶ℝ)
56		simpr 485	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → 𝑢 ∈ ℝ)
57	9	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → 𝐵 ∈ ℝ)
58		simpr 485	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑣 = 𝐵) → 𝑣 = 𝐵)
59	58	oveq2d 7291	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑣 = 𝐵) → (𝑢 + 𝑣) = (𝑢 + 𝐵))
60	59	fveq2d 6778	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑣 = 𝐵) → (𝐹‘(𝑢 + 𝑣)) = (𝐹‘(𝑢 + 𝐵)))
61	58	oveq2d 7291	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑣 = 𝐵) → (𝑢 − 𝑣) = (𝑢 − 𝐵))
62	61	fveq2d 6778	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑣 = 𝐵) → (𝐹‘(𝑢 − 𝑣)) = (𝐹‘(𝑢 − 𝐵)))
63	60, 62	oveq12d 7293	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑣 = 𝐵) → ((𝐹‘(𝑢 + 𝑣)) + (𝐹‘(𝑢 − 𝑣))) = ((𝐹‘(𝑢 + 𝐵)) + (𝐹‘(𝑢 − 𝐵))))
64	58	fveq2d 6778	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑣 = 𝐵) → (𝐺‘𝑣) = (𝐺‘𝐵))
65	64	oveq2d 7291	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑣 = 𝐵) → ((𝐹‘𝑢) · (𝐺‘𝑣)) = ((𝐹‘𝑢) · (𝐺‘𝐵)))
66	65	oveq2d 7291	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑣 = 𝐵) → (2 · ((𝐹‘𝑢) · (𝐺‘𝑣))) = (2 · ((𝐹‘𝑢) · (𝐺‘𝐵))))
67	63, 66	eqeq12d 2754	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑣 = 𝐵) → (((𝐹‘(𝑢 + 𝑣)) + (𝐹‘(𝑢 − 𝑣))) = (2 · ((𝐹‘𝑢) · (𝐺‘𝑣))) ↔ ((𝐹‘(𝑢 + 𝐵)) + (𝐹‘(𝑢 − 𝐵))) = (2 · ((𝐹‘𝑢) · (𝐺‘𝐵)))))
68	67	ralbidv 3112	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑣 = 𝐵) → (∀𝑢 ∈ ℝ ((𝐹‘(𝑢 + 𝑣)) + (𝐹‘(𝑢 − 𝑣))) = (2 · ((𝐹‘𝑢) · (𝐺‘𝑣))) ↔ ∀𝑢 ∈ ℝ ((𝐹‘(𝑢 + 𝐵)) + (𝐹‘(𝑢 − 𝐵))) = (2 · ((𝐹‘𝑢) · (𝐺‘𝐵)))))
69		imo72b2.5	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ∀𝑢 ∈ ℝ ∀𝑣 ∈ ℝ ((𝐹‘(𝑢 + 𝑣)) + (𝐹‘(𝑢 − 𝑣))) = (2 · ((𝐹‘𝑢) · (𝐺‘𝑣))))
70		ralcom 3166	. . . . . . . . . . . . . . . . . . 19 ⊢ (∀𝑢 ∈ ℝ ∀𝑣 ∈ ℝ ((𝐹‘(𝑢 + 𝑣)) + (𝐹‘(𝑢 − 𝑣))) = (2 · ((𝐹‘𝑢) · (𝐺‘𝑣))) ↔ ∀𝑣 ∈ ℝ ∀𝑢 ∈ ℝ ((𝐹‘(𝑢 + 𝑣)) + (𝐹‘(𝑢 − 𝑣))) = (2 · ((𝐹‘𝑢) · (𝐺‘𝑣))))
71	70	biimpi 215	. . . . . . . . . . . . . . . . . 18 ⊢ (∀𝑢 ∈ ℝ ∀𝑣 ∈ ℝ ((𝐹‘(𝑢 + 𝑣)) + (𝐹‘(𝑢 − 𝑣))) = (2 · ((𝐹‘𝑢) · (𝐺‘𝑣))) → ∀𝑣 ∈ ℝ ∀𝑢 ∈ ℝ ((𝐹‘(𝑢 + 𝑣)) + (𝐹‘(𝑢 − 𝑣))) = (2 · ((𝐹‘𝑢) · (𝐺‘𝑣))))
72	71	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (∀𝑢 ∈ ℝ ∀𝑣 ∈ ℝ ((𝐹‘(𝑢 + 𝑣)) + (𝐹‘(𝑢 − 𝑣))) = (2 · ((𝐹‘𝑢) · (𝐺‘𝑣))) → ∀𝑣 ∈ ℝ ∀𝑢 ∈ ℝ ((𝐹‘(𝑢 + 𝑣)) + (𝐹‘(𝑢 − 𝑣))) = (2 · ((𝐹‘𝑢) · (𝐺‘𝑣)))))
73	72	imp 407	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ∀𝑢 ∈ ℝ ∀𝑣 ∈ ℝ ((𝐹‘(𝑢 + 𝑣)) + (𝐹‘(𝑢 − 𝑣))) = (2 · ((𝐹‘𝑢) · (𝐺‘𝑣)))) → ∀𝑣 ∈ ℝ ∀𝑢 ∈ ℝ ((𝐹‘(𝑢 + 𝑣)) + (𝐹‘(𝑢 − 𝑣))) = (2 · ((𝐹‘𝑢) · (𝐺‘𝑣))))
74	69, 73	mpdan 684	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∀𝑣 ∈ ℝ ∀𝑢 ∈ ℝ ((𝐹‘(𝑢 + 𝑣)) + (𝐹‘(𝑢 − 𝑣))) = (2 · ((𝐹‘𝑢) · (𝐺‘𝑣))))
75	68, 2, 74	rspcdv2 3556	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑢 ∈ ℝ ((𝐹‘(𝑢 + 𝐵)) + (𝐹‘(𝑢 − 𝐵))) = (2 · ((𝐹‘𝑢) · (𝐺‘𝐵))))
76	75	r19.21bi 3134	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑢 ∈ ℝ) → ((𝐹‘(𝑢 + 𝐵)) + (𝐹‘(𝑢 − 𝐵))) = (2 · ((𝐹‘𝑢) · (𝐺‘𝐵))))
77	76	adantlr 712	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → ((𝐹‘(𝑢 + 𝐵)) + (𝐹‘(𝑢 − 𝐵))) = (2 · ((𝐹‘𝑢) · (𝐺‘𝐵))))
78	40	ad2antrr 723	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → ∀𝑦 ∈ ℝ (abs‘(𝐹‘𝑦)) ≤ 1)
79	54, 55, 56, 57, 77, 78	imo72b2lem0 41776	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → ((abs‘(𝐹‘𝑢)) · (abs‘(𝐺‘𝐵))) ≤ sup((abs “ (𝐹 “ ℝ)), ℝ, < ))
80		0xr 11022	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℝ^*
81	80	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → 0 ∈ ℝ^*)
82		1xr 11034	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℝ^*
83	82	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → 1 ∈ ℝ^*)
84	12	adantr 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → (abs‘(𝐺‘𝐵)) ∈ ℝ)
85	84	rexrd 11025	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → (abs‘(𝐺‘𝐵)) ∈ ℝ^*)
86	49	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → 0 < 1)
87		simplr 766	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → 1 < (abs‘(𝐺‘𝐵)))
88	81, 83, 85, 86, 87	xrlttrd 12893	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → 0 < (abs‘(𝐺‘𝐵)))
89	20	ffvelrnda 6961	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → (𝐹‘𝑢) ∈ ℝ)
90	89	recnd 11003	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → (𝐹‘𝑢) ∈ ℂ)
91	90	abscld 15148	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → (abs‘(𝐹‘𝑢)) ∈ ℝ)
92	44	adantr 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → sup((abs “ (𝐹 “ ℝ)), ℝ, < ) ∈ ℝ)
93	79, 88, 84, 91, 92	lemuldiv3d 41781	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → (abs‘(𝐹‘𝑢)) ≤ (sup((abs “ (𝐹 “ ℝ)), ℝ, < ) / (abs‘(𝐺‘𝐵))))
94	93	ralrimiva 3103	. . . . . . . . 9 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → ∀𝑢 ∈ ℝ (abs‘(𝐹‘𝑢)) ≤ (sup((abs “ (𝐹 “ ℝ)), ℝ, < ) / (abs‘(𝐺‘𝐵))))
95	20, 53, 94	imo72b2lem2 41778	. . . . . . . 8 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → sup((abs “ (𝐹 “ ℝ)), ℝ, < ) ≤ (sup((abs “ (𝐹 “ ℝ)), ℝ, < ) / (abs‘(𝐺‘𝐵))))
96	95, 51, 12, 44, 44	lemuldiv4d 41782	. . . . . . 7 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → (sup((abs “ (𝐹 “ ℝ)), ℝ, < ) · (abs‘(𝐺‘𝐵))) ≤ sup((abs “ (𝐹 “ ℝ)), ℝ, < ))
97	47, 96	eqbrtrrd 5098	. . . . . 6 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → ((abs‘(𝐺‘𝐵)) · sup((abs “ (𝐹 “ ℝ)), ℝ, < )) ≤ sup((abs “ (𝐹 “ ℝ)), ℝ, < ))
98		imo72b2.7	. . . . . . . 8 ⊢ (𝜑 → ∃𝑥 ∈ ℝ (𝐹‘𝑥) ≠ 0)
99	98	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → ∃𝑥 ∈ ℝ (𝐹‘𝑥) ≠ 0)
100	40	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → ∀𝑦 ∈ ℝ (abs‘(𝐹‘𝑦)) ≤ 1)
101	20, 99, 100	imo72b2lem1 41780	. . . . . 6 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → 0 < sup((abs “ (𝐹 “ ℝ)), ℝ, < ))
102	97, 101, 44, 12, 44	lemuldiv3d 41781	. . . . 5 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → (abs‘(𝐺‘𝐵)) ≤ (sup((abs “ (𝐹 “ ℝ)), ℝ, < ) / sup((abs “ (𝐹 “ ℝ)), ℝ, < )))
103	23, 44	sseldd 3922	. . . . . . 7 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → sup((abs “ (𝐹 “ ℝ)), ℝ, < ) ∈ ℂ)
104	101	gt0ne0d 11539	. . . . . . 7 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → sup((abs “ (𝐹 “ ℝ)), ℝ, < ) ≠ 0)
105	103, 104	dividd 11749	. . . . . 6 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → (sup((abs “ (𝐹 “ ℝ)), ℝ, < ) / sup((abs “ (𝐹 “ ℝ)), ℝ, < )) = 1)
106	105	eqcomd 2744	. . . . 5 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → 1 = (sup((abs “ (𝐹 “ ℝ)), ℝ, < ) / sup((abs “ (𝐹 “ ℝ)), ℝ, < )))
107	102, 106	breqtrrd 5102	. . . 4 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → (abs‘(𝐺‘𝐵)) ≤ 1)
108	12, 13, 107	lensymd 11126	. . 3 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → ¬ 1 < (abs‘(𝐺‘𝐵)))
109	7, 108	pm2.65da 814	. 2 ⊢ (𝜑 → ¬ 1 < (abs‘(𝐺‘𝐵)))
110	5, 6, 109	nltled 11125	1 ⊢ (𝜑 → (abs‘(𝐺‘𝐵)) ≤ 1)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 ∃wrex 3065 ⊆ wss 3887 ∅c0 4256 class class class wbr 5074 ran crn 5590 “ cima 5592 ∘ ccom 5593 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 supcsup 9199 ℂcc 10869 ℝcr 10870 0cc0 10871 1c1 10872 + caddc 10874 · cmul 10876 ℝ^*cxr 11008 < clt 11009 ≤ cle 11010 − cmin 11205 / cdiv 11632 2c2 12028 abscabs 14945

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-sup 9201 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-n0 12234 df-z 12320 df-uz 12583 df-rp 12731 df-seq 13722 df-exp 13783 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947

This theorem is referenced by: (None)

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