imo72b2 - Mathbox for Stanislas Polu

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Theorem imo72b2 44161

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	ffvelcdmd 7057	. . . 4 ⊢ (𝜑 → (𝐺‘𝐵) ∈ ℝ)
4	3	recnd 11202	. . 3 ⊢ (𝜑 → (𝐺‘𝐵) ∈ ℂ)
5	4	abscld 15405	. 2 ⊢ (𝜑 → (abs‘(𝐺‘𝐵)) ∈ ℝ)
6		1red 11175	. 2 ⊢ (𝜑 → 1 ∈ ℝ)
7		simpr 484	. . 3 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → 1 < (abs‘(𝐺‘𝐵)))
8	1	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → 𝐺:ℝ⟶ℝ)
9	2	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → 𝐵 ∈ ℝ)
10	8, 9	ffvelcdmd 7057	. . . . . 6 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → (𝐺‘𝐵) ∈ ℝ)
11	10	recnd 11202	. . . . 5 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → (𝐺‘𝐵) ∈ ℂ)
12	11	abscld 15405	. . . 4 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → (abs‘(𝐺‘𝐵)) ∈ ℝ)
13	6	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → 1 ∈ ℝ)
14		ax-resscn 11125	. . . . . . . . 9 ⊢ ℝ ⊆ ℂ
15		imaco 6224	. . . . . . . . . . . 12 ⊢ ((abs ∘ 𝐹) “ ℝ) = (abs “ (𝐹 “ ℝ))
16	15	eqcomi 2738	. . . . . . . . . . 11 ⊢ (abs “ (𝐹 “ ℝ)) = ((abs ∘ 𝐹) “ ℝ)
17		imassrn 6042	. . . . . . . . . . . . 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 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → 𝐹:ℝ⟶ℝ)
21		absf 15304	. . . . . . . . . . . . . . . 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 6727	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → (abs ↾ ℝ):ℝ⟶ℝ)
25	20, 24	fco2d 44151	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → (abs ∘ 𝐹):ℝ⟶ℝ)
26	25	frnd 6696	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → ran (abs ∘ 𝐹) ⊆ ℝ)
27	18, 26	sstrd 3957	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → ((abs ∘ 𝐹) “ ℝ) ⊆ ℝ)
28	16, 27	eqsstrid 3985	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → (abs “ (𝐹 “ ℝ)) ⊆ ℝ)
29		0re 11176	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℝ
30	29	ne0ii 4307	. . . . . . . . . . . . . . 15 ⊢ ℝ ≠ ∅
31	30	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → ℝ ≠ ∅)
32	31, 25	wnefimgd 44150	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → ((abs ∘ 𝐹) “ ℝ) ≠ ∅)
33	32	necomd 2980	. . . . . . . . . . . 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 2999	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → ∅ ≠ (abs “ (𝐹 “ ℝ)))
36	35	necomd 2980	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → (abs “ (𝐹 “ ℝ)) ≠ ∅)
37		simpr 484	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑐 = 1) → 𝑐 = 1)
38	37	breq2d 5119	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑐 = 1) → (𝑡 ≤ 𝑐 ↔ 𝑡 ≤ 1))
39	38	ralbidv 3156	. . . . . . . . . . 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 44153	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑡 ∈ (abs “ (𝐹 “ ℝ))𝑡 ≤ 1)
42	41	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → ∀𝑡 ∈ (abs “ (𝐹 “ ℝ))𝑡 ≤ 1)
43	13, 39, 42	rspcedvd 3590	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → ∃𝑐 ∈ ℝ ∀𝑡 ∈ (abs “ (𝐹 “ ℝ))𝑡 ≤ 𝑐)
44	28, 36, 43	suprcld 12146	. . . . . . . . 9 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → sup((abs “ (𝐹 “ ℝ)), ℝ, < ) ∈ ℝ)
45	14, 44	sselid 3944	. . . . . . . 8 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → sup((abs “ (𝐹 “ ℝ)), ℝ, < ) ∈ ℂ)
46	14, 12	sselid 3944	. . . . . . . 8 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → (abs‘(𝐺‘𝐵)) ∈ ℂ)
47	45, 46	mulcomd 11195	. . . . . . 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 11700	. . . . . . . . . . . . 13 ⊢ 0 < 1
50	49	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → 0 < 1)
51	48, 13, 12, 50, 7	lttrd 11335	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → 0 < (abs‘(𝐺‘𝐵)))
52	51	gt0ne0d 11742	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → (abs‘(𝐺‘𝐵)) ≠ 0)
53	44, 12, 52	redivcld 12010	. . . . . . . . 9 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → (sup((abs “ (𝐹 “ ℝ)), ℝ, < ) / (abs‘(𝐺‘𝐵))) ∈ ℝ)
54	20	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → 𝐹:ℝ⟶ℝ)
55	8	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → 𝐺:ℝ⟶ℝ)
56		simpr 484	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → 𝑢 ∈ ℝ)
57	9	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → 𝐵 ∈ ℝ)
58		simpr 484	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑣 = 𝐵) → 𝑣 = 𝐵)
59	58	oveq2d 7403	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑣 = 𝐵) → (𝑢 + 𝑣) = (𝑢 + 𝐵))
60	59	fveq2d 6862	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑣 = 𝐵) → (𝐹‘(𝑢 + 𝑣)) = (𝐹‘(𝑢 + 𝐵)))
61	58	oveq2d 7403	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑣 = 𝐵) → (𝑢 − 𝑣) = (𝑢 − 𝐵))
62	61	fveq2d 6862	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑣 = 𝐵) → (𝐹‘(𝑢 − 𝑣)) = (𝐹‘(𝑢 − 𝐵)))
63	60, 62	oveq12d 7405	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑣 = 𝐵) → ((𝐹‘(𝑢 + 𝑣)) + (𝐹‘(𝑢 − 𝑣))) = ((𝐹‘(𝑢 + 𝐵)) + (𝐹‘(𝑢 − 𝐵))))
64	58	fveq2d 6862	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑣 = 𝐵) → (𝐺‘𝑣) = (𝐺‘𝐵))
65	64	oveq2d 7403	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑣 = 𝐵) → ((𝐹‘𝑢) · (𝐺‘𝑣)) = ((𝐹‘𝑢) · (𝐺‘𝐵)))
66	65	oveq2d 7403	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑣 = 𝐵) → (2 · ((𝐹‘𝑢) · (𝐺‘𝑣))) = (2 · ((𝐹‘𝑢) · (𝐺‘𝐵))))
67	63, 66	eqeq12d 2745	. . . . . . . . . . . . . . . 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 3156	. . . . . . . . . . . . . . 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 3265	. . . . . . . . . . . . . . . . . . 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 216	. . . . . . . . . . . . . . . . . 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 406	. . . . . . . . . . . . . . . 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 687	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∀𝑣 ∈ ℝ ∀𝑢 ∈ ℝ ((𝐹‘(𝑢 + 𝑣)) + (𝐹‘(𝑢 − 𝑣))) = (2 · ((𝐹‘𝑢) · (𝐺‘𝑣))))
75	68, 2, 74	rspcdv2 3583	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑢 ∈ ℝ ((𝐹‘(𝑢 + 𝐵)) + (𝐹‘(𝑢 − 𝐵))) = (2 · ((𝐹‘𝑢) · (𝐺‘𝐵))))
76	75	r19.21bi 3229	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑢 ∈ ℝ) → ((𝐹‘(𝑢 + 𝐵)) + (𝐹‘(𝑢 − 𝐵))) = (2 · ((𝐹‘𝑢) · (𝐺‘𝐵))))
77	76	adantlr 715	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → ((𝐹‘(𝑢 + 𝐵)) + (𝐹‘(𝑢 − 𝐵))) = (2 · ((𝐹‘𝑢) · (𝐺‘𝐵))))
78	40	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → ∀𝑦 ∈ ℝ (abs‘(𝐹‘𝑦)) ≤ 1)
79	54, 55, 56, 57, 77, 78	imo72b2lem0 44154	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → ((abs‘(𝐹‘𝑢)) · (abs‘(𝐺‘𝐵))) ≤ sup((abs “ (𝐹 “ ℝ)), ℝ, < ))
80		0xr 11221	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℝ^*
81	80	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → 0 ∈ ℝ^*)
82		1xr 11233	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℝ^*
83	82	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → 1 ∈ ℝ^*)
84	12	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → (abs‘(𝐺‘𝐵)) ∈ ℝ)
85	84	rexrd 11224	. . . . . . . . . . . 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 768	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → 1 < (abs‘(𝐺‘𝐵)))
88	81, 83, 85, 86, 87	xrlttrd 13119	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → 0 < (abs‘(𝐺‘𝐵)))
89	20	ffvelcdmda 7056	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → (𝐹‘𝑢) ∈ ℝ)
90	89	recnd 11202	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → (𝐹‘𝑢) ∈ ℂ)
91	90	abscld 15405	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → (abs‘(𝐹‘𝑢)) ∈ ℝ)
92	44	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → sup((abs “ (𝐹 “ ℝ)), ℝ, < ) ∈ ℝ)
93	79, 88, 84, 91, 92	lemuldiv3d 44159	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → (abs‘(𝐹‘𝑢)) ≤ (sup((abs “ (𝐹 “ ℝ)), ℝ, < ) / (abs‘(𝐺‘𝐵))))
94	93	ralrimiva 3125	. . . . . . . . 9 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → ∀𝑢 ∈ ℝ (abs‘(𝐹‘𝑢)) ≤ (sup((abs “ (𝐹 “ ℝ)), ℝ, < ) / (abs‘(𝐺‘𝐵))))
95	20, 53, 94	imo72b2lem2 44156	. . . . . . . 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 44160	. . . . . . 7 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → (sup((abs “ (𝐹 “ ℝ)), ℝ, < ) · (abs‘(𝐺‘𝐵))) ≤ sup((abs “ (𝐹 “ ℝ)), ℝ, < ))
97	47, 96	eqbrtrrd 5131	. . . . . 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 480	. . . . . . 7 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → ∃𝑥 ∈ ℝ (𝐹‘𝑥) ≠ 0)
100	40	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → ∀𝑦 ∈ ℝ (abs‘(𝐹‘𝑦)) ≤ 1)
101	20, 99, 100	imo72b2lem1 44158	. . . . . 6 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → 0 < sup((abs “ (𝐹 “ ℝ)), ℝ, < ))
102	97, 101, 44, 12, 44	lemuldiv3d 44159	. . . . 5 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → (abs‘(𝐺‘𝐵)) ≤ (sup((abs “ (𝐹 “ ℝ)), ℝ, < ) / sup((abs “ (𝐹 “ ℝ)), ℝ, < )))
103	23, 44	sseldd 3947	. . . . . . 7 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → sup((abs “ (𝐹 “ ℝ)), ℝ, < ) ∈ ℂ)
104	101	gt0ne0d 11742	. . . . . . 7 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → sup((abs “ (𝐹 “ ℝ)), ℝ, < ) ≠ 0)
105	103, 104	dividd 11956	. . . . . 6 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → (sup((abs “ (𝐹 “ ℝ)), ℝ, < ) / sup((abs “ (𝐹 “ ℝ)), ℝ, < )) = 1)
106	105	eqcomd 2735	. . . . 5 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → 1 = (sup((abs “ (𝐹 “ ℝ)), ℝ, < ) / sup((abs “ (𝐹 “ ℝ)), ℝ, < )))
107	102, 106	breqtrrd 5135	. . . 4 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → (abs‘(𝐺‘𝐵)) ≤ 1)
108	12, 13, 107	lensymd 11325	. . 3 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → ¬ 1 < (abs‘(𝐺‘𝐵)))
109	7, 108	pm2.65da 816	. 2 ⊢ (𝜑 → ¬ 1 < (abs‘(𝐺‘𝐵)))
110	5, 6, 109	nltled 11324	1 ⊢ (𝜑 → (abs‘(𝐺‘𝐵)) ≤ 1)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ∀wral 3044 ∃wrex 3053 ⊆ wss 3914 ∅c0 4296 class class class wbr 5107 ran crn 5639 “ cima 5641 ∘ ccom 5642 ⟶wf 6507 ‘cfv 6511 (class class class)co 7387 supcsup 9391 ℂcc 11066 ℝcr 11067 0cc0 11068 1c1 11069 + caddc 11071 · cmul 11073 ℝ^*cxr 11207 < clt 11208 ≤ cle 11209 − cmin 11405 / cdiv 11835 2c2 12241 abscabs 15200

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-er 8671 df-en 8919 df-dom 8920 df-sdom 8921 df-sup 9393 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-n0 12443 df-z 12530 df-uz 12794 df-rp 12952 df-seq 13967 df-exp 14027 df-cj 15065 df-re 15066 df-im 15067 df-sqrt 15201 df-abs 15202

This theorem is referenced by: (None)

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