imo72b2 - Mathbox for Stanislas Polu

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Theorem imo72b2 44712

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 7062	. . . 4 ⊢ (𝜑 → (𝐺‘𝐵) ∈ ℝ)
4	3	recnd 11207	. . 3 ⊢ (𝜑 → (𝐺‘𝐵) ∈ ℂ)
5	4	abscld 15449	. 2 ⊢ (𝜑 → (abs‘(𝐺‘𝐵)) ∈ ℝ)
6		1red 11179	. 2 ⊢ (𝜑 → 1 ∈ ℝ)
7		simpr 488	. . 3 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → 1 < (abs‘(𝐺‘𝐵)))
8	1	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → 𝐺:ℝ⟶ℝ)
9	2	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → 𝐵 ∈ ℝ)
10	8, 9	ffvelcdmd 7062	. . . . . 6 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → (𝐺‘𝐵) ∈ ℝ)
11	10	recnd 11207	. . . . 5 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → (𝐺‘𝐵) ∈ ℂ)
12	11	abscld 15449	. . . 4 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → (abs‘(𝐺‘𝐵)) ∈ ℝ)
13	6	adantr 484	. . . 4 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → 1 ∈ ℝ)
14		ax-resscn 11127	. . . . . . . . 9 ⊢ ℝ ⊆ ℂ
15		imaco 6234	. . . . . . . . . . . 12 ⊢ ((abs ∘ 𝐹) “ ℝ) = (abs “ (𝐹 “ ℝ))
16	15	eqcomi 2770	. . . . . . . . . . 11 ⊢ (abs “ (𝐹 “ ℝ)) = ((abs ∘ 𝐹) “ ℝ)
17		imassrn 6057	. . . . . . . . . . . . 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 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → 𝐹:ℝ⟶ℝ)
21		absf 15348	. . . . . . . . . . . . . . . 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 44702	. . . . . . . . . . . . 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 3946	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → ((abs ∘ 𝐹) “ ℝ) ⊆ ℝ)
28	16, 27	eqsstrid 3974	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → (abs “ (𝐹 “ ℝ)) ⊆ ℝ)
29		0re 11180	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℝ
30	29	ne0ii 4296	. . . . . . . . . . . . . . 15 ⊢ ℝ ≠ ∅
31	30	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → ℝ ≠ ∅)
32	31, 25	wnefimgd 44701	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → ((abs ∘ 𝐹) “ ℝ) ≠ ∅)
33	32	necomd 3011	. . . . . . . . . . . 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 3030	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → ∅ ≠ (abs “ (𝐹 “ ℝ)))
36	35	necomd 3011	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → (abs “ (𝐹 “ ℝ)) ≠ ∅)
37		simpr 488	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑐 = 1) → 𝑐 = 1)
38	37	breq2d 5111	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑐 = 1) → (𝑡 ≤ 𝑐 ↔ 𝑡 ≤ 1))
39	38	ralbidv 3184	. . . . . . . . . . 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 44704	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑡 ∈ (abs “ (𝐹 “ ℝ))𝑡 ≤ 1)
42	41	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → ∀𝑡 ∈ (abs “ (𝐹 “ ℝ))𝑡 ≤ 1)
43	13, 39, 42	rspcedvd 3583	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → ∃𝑐 ∈ ℝ ∀𝑡 ∈ (abs “ (𝐹 “ ℝ))𝑡 ≤ 𝑐)
44	28, 36, 43	suprcld 12152	. . . . . . . . 9 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → sup((abs “ (𝐹 “ ℝ)), ℝ, < ) ∈ ℝ)
45	14, 44	sselid 3934	. . . . . . . 8 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → sup((abs “ (𝐹 “ ℝ)), ℝ, < ) ∈ ℂ)
46	14, 12	sselid 3934	. . . . . . . 8 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → (abs‘(𝐺‘𝐵)) ∈ ℂ)
47	45, 46	mulcomd 11200	. . . . . . 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 11706	. . . . . . . . . . . . 13 ⊢ 0 < 1
50	49	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → 0 < 1)
51	48, 13, 12, 50, 7	lttrd 11341	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → 0 < (abs‘(𝐺‘𝐵)))
52	51	gt0ne0d 11748	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → (abs‘(𝐺‘𝐵)) ≠ 0)
53	44, 12, 52	redivcld 12016	. . . . . . . . 9 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → (sup((abs “ (𝐹 “ ℝ)), ℝ, < ) / (abs‘(𝐺‘𝐵))) ∈ ℝ)
54	20	adantr 484	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → 𝐹:ℝ⟶ℝ)
55	8	adantr 484	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → 𝐺:ℝ⟶ℝ)
56		simpr 488	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → 𝑢 ∈ ℝ)
57	9	adantr 484	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → 𝐵 ∈ ℝ)
58		simpr 488	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑣 = 𝐵) → 𝑣 = 𝐵)
59	58	oveq2d 7408	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑣 = 𝐵) → (𝑢 + 𝑣) = (𝑢 + 𝐵))
60	59	fveq2d 6867	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑣 = 𝐵) → (𝐹‘(𝑢 + 𝑣)) = (𝐹‘(𝑢 + 𝐵)))
61	58	oveq2d 7408	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑣 = 𝐵) → (𝑢 − 𝑣) = (𝑢 − 𝐵))
62	61	fveq2d 6867	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑣 = 𝐵) → (𝐹‘(𝑢 − 𝑣)) = (𝐹‘(𝑢 − 𝐵)))
63	60, 62	oveq12d 7410	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑣 = 𝐵) → ((𝐹‘(𝑢 + 𝑣)) + (𝐹‘(𝑢 − 𝑣))) = ((𝐹‘(𝑢 + 𝐵)) + (𝐹‘(𝑢 − 𝐵))))
64	58	fveq2d 6867	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑣 = 𝐵) → (𝐺‘𝑣) = (𝐺‘𝐵))
65	64	oveq2d 7408	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑣 = 𝐵) → ((𝐹‘𝑢) · (𝐺‘𝑣)) = ((𝐹‘𝑢) · (𝐺‘𝐵)))
66	65	oveq2d 7408	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑣 = 𝐵) → (2 · ((𝐹‘𝑢) · (𝐺‘𝑣))) = (2 · ((𝐹‘𝑢) · (𝐺‘𝐵))))
67	63, 66	eqeq12d 2777	. . . . . . . . . . . . . . . 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 3184	. . . . . . . . . . . . . . 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 3289	. . . . . . . . . . . . . . . . 17 ⊢ (∀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	bilani 508	. . . . . . . . . . . . . . . 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))))
72	69, 71	mpdan 697	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∀𝑣 ∈ ℝ ∀𝑢 ∈ ℝ ((𝐹‘(𝑢 + 𝑣)) + (𝐹‘(𝑢 − 𝑣))) = (2 · ((𝐹‘𝑢) · (𝐺‘𝑣))))
73	68, 2, 72	rspcdv2 3576	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑢 ∈ ℝ ((𝐹‘(𝑢 + 𝐵)) + (𝐹‘(𝑢 − 𝐵))) = (2 · ((𝐹‘𝑢) · (𝐺‘𝐵))))
74	73	r19.21bi 3253	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑢 ∈ ℝ) → ((𝐹‘(𝑢 + 𝐵)) + (𝐹‘(𝑢 − 𝐵))) = (2 · ((𝐹‘𝑢) · (𝐺‘𝐵))))
75	74	adantlr 725	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → ((𝐹‘(𝑢 + 𝐵)) + (𝐹‘(𝑢 − 𝐵))) = (2 · ((𝐹‘𝑢) · (𝐺‘𝐵))))
76	40	ad2antrr 736	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → ∀𝑦 ∈ ℝ (abs‘(𝐹‘𝑦)) ≤ 1)
77	54, 55, 56, 57, 75, 76	imo72b2lem0 44705	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → ((abs‘(𝐹‘𝑢)) · (abs‘(𝐺‘𝐵))) ≤ sup((abs “ (𝐹 “ ℝ)), ℝ, < ))
78		0xr 11226	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℝ^*
79	78	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → 0 ∈ ℝ^*)
80		1xr 11238	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℝ^*
81	80	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → 1 ∈ ℝ^*)
82	12	adantr 484	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → (abs‘(𝐺‘𝐵)) ∈ ℝ)
83	82	rexrd 11229	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → (abs‘(𝐺‘𝐵)) ∈ ℝ^*)
84	49	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → 0 < 1)
85		simplr 778	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → 1 < (abs‘(𝐺‘𝐵)))
86	79, 81, 83, 84, 85	xrlttrd 13158	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → 0 < (abs‘(𝐺‘𝐵)))
87	20	ffvelcdmda 7061	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → (𝐹‘𝑢) ∈ ℝ)
88	87	recnd 11207	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → (𝐹‘𝑢) ∈ ℂ)
89	88	abscld 15449	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → (abs‘(𝐹‘𝑢)) ∈ ℝ)
90	44	adantr 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → sup((abs “ (𝐹 “ ℝ)), ℝ, < ) ∈ ℝ)
91	77, 86, 82, 89, 90	lemuldiv3d 44710	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → (abs‘(𝐹‘𝑢)) ≤ (sup((abs “ (𝐹 “ ℝ)), ℝ, < ) / (abs‘(𝐺‘𝐵))))
92	91	ralrimiva 3153	. . . . . . . . 9 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → ∀𝑢 ∈ ℝ (abs‘(𝐹‘𝑢)) ≤ (sup((abs “ (𝐹 “ ℝ)), ℝ, < ) / (abs‘(𝐺‘𝐵))))
93	20, 53, 92	imo72b2lem2 44707	. . . . . . . 8 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → sup((abs “ (𝐹 “ ℝ)), ℝ, < ) ≤ (sup((abs “ (𝐹 “ ℝ)), ℝ, < ) / (abs‘(𝐺‘𝐵))))
94	93, 51, 12, 44, 44	lemuldiv4d 44711	. . . . . . 7 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → (sup((abs “ (𝐹 “ ℝ)), ℝ, < ) · (abs‘(𝐺‘𝐵))) ≤ sup((abs “ (𝐹 “ ℝ)), ℝ, < ))
95	47, 94	eqbrtrrd 5123	. . . . . 6 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → ((abs‘(𝐺‘𝐵)) · sup((abs “ (𝐹 “ ℝ)), ℝ, < )) ≤ sup((abs “ (𝐹 “ ℝ)), ℝ, < ))
96		imo72b2.7	. . . . . . . 8 ⊢ (𝜑 → ∃𝑥 ∈ ℝ (𝐹‘𝑥) ≠ 0)
97	96	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → ∃𝑥 ∈ ℝ (𝐹‘𝑥) ≠ 0)
98	40	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → ∀𝑦 ∈ ℝ (abs‘(𝐹‘𝑦)) ≤ 1)
99	20, 97, 98	imo72b2lem1 44709	. . . . . 6 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → 0 < sup((abs “ (𝐹 “ ℝ)), ℝ, < ))
100	95, 99, 44, 12, 44	lemuldiv3d 44710	. . . . 5 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → (abs‘(𝐺‘𝐵)) ≤ (sup((abs “ (𝐹 “ ℝ)), ℝ, < ) / sup((abs “ (𝐹 “ ℝ)), ℝ, < )))
101	23, 44	sseldd 3937	. . . . . . 7 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → sup((abs “ (𝐹 “ ℝ)), ℝ, < ) ∈ ℂ)
102	99	gt0ne0d 11748	. . . . . . 7 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → sup((abs “ (𝐹 “ ℝ)), ℝ, < ) ≠ 0)
103	101, 102	dividd 11962	. . . . . 6 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → (sup((abs “ (𝐹 “ ℝ)), ℝ, < ) / sup((abs “ (𝐹 “ ℝ)), ℝ, < )) = 1)
104	103	eqcomd 2767	. . . . 5 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → 1 = (sup((abs “ (𝐹 “ ℝ)), ℝ, < ) / sup((abs “ (𝐹 “ ℝ)), ℝ, < )))
105	100, 104	breqtrrd 5127	. . . 4 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → (abs‘(𝐺‘𝐵)) ≤ 1)
106	12, 13, 105	lensymd 11331	. . 3 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → ¬ 1 < (abs‘(𝐺‘𝐵)))
107	7, 106	pm2.65da 826	. 2 ⊢ (𝜑 → ¬ 1 < (abs‘(𝐺‘𝐵)))
108	5, 6, 107	nltled 11330	1 ⊢ (𝜑 → (abs‘(𝐺‘𝐵)) ≤ 1)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 = wceq 1559 ∈ wcel 2141 ≠ wne 2956 ∀wral 3075 ∃wrex 3085 ⊆ wss 3904 ∅c0 4285 class class class wbr 5099 ran crn 5646 “ cima 5648 ∘ ccom 5649 ⟶wf 6513 ‘cfv 6517 (class class class)co 7392 supcsup 9383 ℂcc 11068 ℝcr 11069 0cc0 11070 1c1 11071 + caddc 11073 · cmul 11075 ℝ^*cxr 11212 < clt 11213 ≤ cle 11214 − cmin 11411 / cdiv 11841 2c2 12269 abscabs 15244

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-sep 5245 ax-nul 5255 ax-pow 5321 ax-pr 5389 ax-un 7714 ax-cnex 11126 ax-resscn 11127 ax-1cn 11128 ax-icn 11129 ax-addcl 11130 ax-addrcl 11131 ax-mulcl 11132 ax-mulrcl 11133 ax-mulcom 11134 ax-addass 11135 ax-mulass 11136 ax-distr 11137 ax-i2m1 11138 ax-1ne0 11139 ax-1rid 11140 ax-rnegex 11141 ax-rrecex 11142 ax-cnre 11143 ax-pre-lttri 11144 ax-pre-lttrn 11145 ax-pre-ltadd 11146 ax-pre-mulgt0 11147 ax-pre-sup 11148

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4582 df-pr 4584 df-op 4588 df-uni 4865 df-iun 4950 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5540 df-eprel 5545 df-po 5553 df-so 5554 df-fr 5598 df-we 5600 df-xp 5651 df-rel 5652 df-cnv 5653 df-co 5654 df-dm 5655 df-rn 5656 df-res 5657 df-ima 5658 df-pred 6284 df-ord 6345 df-on 6346 df-lim 6347 df-suc 6348 df-iota 6473 df-fun 6519 df-fn 6520 df-f 6521 df-f1 6522 df-fo 6523 df-f1o 6524 df-fv 6525 df-riota 7349 df-ov 7395 df-oprab 7396 df-mpo 7397 df-om 7843 df-2nd 7967 df-frecs 8257 df-wrecs 8288 df-recs 8337 df-rdg 8376 df-er 8673 df-en 8924 df-dom 8925 df-sdom 8926 df-sup 9385 df-pnf 11215 df-mnf 11216 df-xr 11217 df-ltxr 11218 df-le 11219 df-sub 11413 df-neg 11414 df-div 11842 df-nn 12208 df-2 12277 df-3 12278 df-n0 12479 df-z 12566 df-uz 12837 df-rp 12991 df-seq 14012 df-exp 14072 df-cj 15109 df-re 15110 df-im 15111 df-sqrt 15245 df-abs 15246

This theorem is referenced by: (None)

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