imo72b2 - Mathbox for Stanislas Polu

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Theorem imo72b2 38475

Description: IMO 1972 B2. (14th International Mathemahics 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 6360	. . . 4 ⊢ (𝜑 → (𝐺‘𝐵) ∈ ℝ)
4	3	recnd 10068	. . 3 ⊢ (𝜑 → (𝐺‘𝐵) ∈ ℂ)
5	4	abscld 14175	. 2 ⊢ (𝜑 → (abs‘(𝐺‘𝐵)) ∈ ℝ)
6		1red 10055	. 2 ⊢ (𝜑 → 1 ∈ ℝ)
7		simpr 477	. . 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 6360	. . . . . 6 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → (𝐺‘𝐵) ∈ ℝ)
11	10	recnd 10068	. . . . 5 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → (𝐺‘𝐵) ∈ ℂ)
12	11	abscld 14175	. . . 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 9993	. . . . . . . . 9 ⊢ ℝ ⊆ ℂ
15		imaco 5640	. . . . . . . . . . . 12 ⊢ ((abs ∘ 𝐹) “ ℝ) = (abs “ (𝐹 “ ℝ))
16	15	eqcomi 2631	. . . . . . . . . . 11 ⊢ (abs “ (𝐹 “ ℝ)) = ((abs ∘ 𝐹) “ ℝ)
17		imassrn 5477	. . . . . . . . . . . . 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 14077	. . . . . . . . . . . . . . . 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 6071	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → (abs ↾ ℝ):ℝ⟶ℝ)
25	20, 24	fco2d 38461	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → (abs ∘ 𝐹):ℝ⟶ℝ)
26		frn 6053	. . . . . . . . . . . . 13 ⊢ ((abs ∘ 𝐹):ℝ⟶ℝ → ran (abs ∘ 𝐹) ⊆ ℝ)
27	25, 26	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → ran (abs ∘ 𝐹) ⊆ ℝ)
28	18, 27	sstrd 3613	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → ((abs ∘ 𝐹) “ ℝ) ⊆ ℝ)
29	16, 28	syl5eqss 3649	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → (abs “ (𝐹 “ ℝ)) ⊆ ℝ)
30		0re 10040	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℝ
31	30	ne0ii 3923	. . . . . . . . . . . . . . 15 ⊢ ℝ ≠ ∅
32	31	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → ℝ ≠ ∅)
33	32, 25	wnefimgd 38460	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → ((abs ∘ 𝐹) “ ℝ) ≠ ∅)
34	33	necomd 2849	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → ∅ ≠ ((abs ∘ 𝐹) “ ℝ))
35	16	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → (abs “ (𝐹 “ ℝ)) = ((abs ∘ 𝐹) “ ℝ))
36	34, 35	neeqtrrd 2868	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → ∅ ≠ (abs “ (𝐹 “ ℝ)))
37	36	necomd 2849	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → (abs “ (𝐹 “ ℝ)) ≠ ∅)
38		simpr 477	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑐 = 1) → 𝑐 = 1)
39	38	breq2d 4665	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑐 = 1) → (𝑡 ≤ 𝑐 ↔ 𝑡 ≤ 1))
40	39	ralbidv 2986	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑐 = 1) → (∀𝑡 ∈ (abs “ (𝐹 “ ℝ))𝑡 ≤ 𝑐 ↔ ∀𝑡 ∈ (abs “ (𝐹 “ ℝ))𝑡 ≤ 1))
41		imo72b2.6	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑦 ∈ ℝ (abs‘(𝐹‘𝑦)) ≤ 1)
42	19, 41	extoimad 38464	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑡 ∈ (abs “ (𝐹 “ ℝ))𝑡 ≤ 1)
43	42	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → ∀𝑡 ∈ (abs “ (𝐹 “ ℝ))𝑡 ≤ 1)
44	13, 40, 43	rspcedvd 3317	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → ∃𝑐 ∈ ℝ ∀𝑡 ∈ (abs “ (𝐹 “ ℝ))𝑡 ≤ 𝑐)
45	29, 37, 44	suprcld 10986	. . . . . . . . 9 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → sup((abs “ (𝐹 “ ℝ)), ℝ, < ) ∈ ℝ)
46	14, 45	sseldi 3601	. . . . . . . 8 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → sup((abs “ (𝐹 “ ℝ)), ℝ, < ) ∈ ℂ)
47	14, 12	sseldi 3601	. . . . . . . 8 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → (abs‘(𝐺‘𝐵)) ∈ ℂ)
48	46, 47	mulcomd 10061	. . . . . . 7 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → (sup((abs “ (𝐹 “ ℝ)), ℝ, < ) · (abs‘(𝐺‘𝐵))) = ((abs‘(𝐺‘𝐵)) · sup((abs “ (𝐹 “ ℝ)), ℝ, < )))
49	30	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → 0 ∈ ℝ)
50		0lt1 10550	. . . . . . . . . . . . 13 ⊢ 0 < 1
51	50	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → 0 < 1)
52	49, 13, 12, 51, 7	lttrd 10198	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → 0 < (abs‘(𝐺‘𝐵)))
53	52	gt0ne0d 10592	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → (abs‘(𝐺‘𝐵)) ≠ 0)
54	45, 12, 53	redivcld 10853	. . . . . . . . 9 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → (sup((abs “ (𝐹 “ ℝ)), ℝ, < ) / (abs‘(𝐺‘𝐵))) ∈ ℝ)
55	20	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → 𝐹:ℝ⟶ℝ)
56	8	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → 𝐺:ℝ⟶ℝ)
57		simpr 477	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → 𝑢 ∈ ℝ)
58	9	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → 𝐵 ∈ ℝ)
59		simpr 477	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑣 = 𝐵) → 𝑣 = 𝐵)
60	59	oveq2d 6666	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑣 = 𝐵) → (𝑢 + 𝑣) = (𝑢 + 𝐵))
61	60	fveq2d 6195	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑣 = 𝐵) → (𝐹‘(𝑢 + 𝑣)) = (𝐹‘(𝑢 + 𝐵)))
62	59	oveq2d 6666	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑣 = 𝐵) → (𝑢 − 𝑣) = (𝑢 − 𝐵))
63	62	fveq2d 6195	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑣 = 𝐵) → (𝐹‘(𝑢 − 𝑣)) = (𝐹‘(𝑢 − 𝐵)))
64	61, 63	oveq12d 6668	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑣 = 𝐵) → ((𝐹‘(𝑢 + 𝑣)) + (𝐹‘(𝑢 − 𝑣))) = ((𝐹‘(𝑢 + 𝐵)) + (𝐹‘(𝑢 − 𝐵))))
65	59	fveq2d 6195	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑣 = 𝐵) → (𝐺‘𝑣) = (𝐺‘𝐵))
66	65	oveq2d 6666	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑣 = 𝐵) → ((𝐹‘𝑢) · (𝐺‘𝑣)) = ((𝐹‘𝑢) · (𝐺‘𝐵)))
67	66	oveq2d 6666	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑣 = 𝐵) → (2 · ((𝐹‘𝑢) · (𝐺‘𝑣))) = (2 · ((𝐹‘𝑢) · (𝐺‘𝐵))))
68	64, 67	eqeq12d 2637	. . . . . . . . . . . . . . . 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)))))
69	68	ralbidv 2986	. . . . . . . . . . . . . . 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)))))
70		imo72b2.5	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ∀𝑢 ∈ ℝ ∀𝑣 ∈ ℝ ((𝐹‘(𝑢 + 𝑣)) + (𝐹‘(𝑢 − 𝑣))) = (2 · ((𝐹‘𝑢) · (𝐺‘𝑣))))
71		ralcom2 3104	. . . . . . . . . . . . . . . . . 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 445	. . . . . . . . . . . . . . . 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	70, 73	mpdan 702	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∀𝑣 ∈ ℝ ∀𝑢 ∈ ℝ ((𝐹‘(𝑢 + 𝑣)) + (𝐹‘(𝑢 − 𝑣))) = (2 · ((𝐹‘𝑢) · (𝐺‘𝑣))))
75	69, 2, 74	rspcdvinvd 38474	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑢 ∈ ℝ ((𝐹‘(𝑢 + 𝐵)) + (𝐹‘(𝑢 − 𝐵))) = (2 · ((𝐹‘𝑢) · (𝐺‘𝐵))))
76	75	r19.21bi 2932	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑢 ∈ ℝ) → ((𝐹‘(𝑢 + 𝐵)) + (𝐹‘(𝑢 − 𝐵))) = (2 · ((𝐹‘𝑢) · (𝐺‘𝐵))))
77	76	adantlr 751	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → ((𝐹‘(𝑢 + 𝐵)) + (𝐹‘(𝑢 − 𝐵))) = (2 · ((𝐹‘𝑢) · (𝐺‘𝐵))))
78	41	ad2antrr 762	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → ∀𝑦 ∈ ℝ (abs‘(𝐹‘𝑦)) ≤ 1)
79	55, 56, 57, 58, 77, 78	imo72b2lem0 38465	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → ((abs‘(𝐹‘𝑢)) · (abs‘(𝐺‘𝐵))) ≤ sup((abs “ (𝐹 “ ℝ)), ℝ, < ))
80		0xr 10086	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℝ^*
81	80	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → 0 ∈ ℝ^*)
82		1re 10039	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℝ
83	82	rexri 10097	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℝ^*
84	83	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → 1 ∈ ℝ^*)
85	12	adantr 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → (abs‘(𝐺‘𝐵)) ∈ ℝ)
86	85	rexrd 10089	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → (abs‘(𝐺‘𝐵)) ∈ ℝ^*)
87	50	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → 0 < 1)
88		simplr 792	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → 1 < (abs‘(𝐺‘𝐵)))
89	81, 84, 86, 87, 88	xrlttrd 11990	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → 0 < (abs‘(𝐺‘𝐵)))
90	20	ffvelrnda 6359	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → (𝐹‘𝑢) ∈ ℝ)
91	90	recnd 10068	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → (𝐹‘𝑢) ∈ ℂ)
92	91	abscld 14175	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → (abs‘(𝐹‘𝑢)) ∈ ℝ)
93	45	adantr 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → sup((abs “ (𝐹 “ ℝ)), ℝ, < ) ∈ ℝ)
94	79, 89, 85, 92, 93	lemuldiv3d 38472	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → (abs‘(𝐹‘𝑢)) ≤ (sup((abs “ (𝐹 “ ℝ)), ℝ, < ) / (abs‘(𝐺‘𝐵))))
95	94	ralrimiva 2966	. . . . . . . . 9 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → ∀𝑢 ∈ ℝ (abs‘(𝐹‘𝑢)) ≤ (sup((abs “ (𝐹 “ ℝ)), ℝ, < ) / (abs‘(𝐺‘𝐵))))
96	20, 54, 95	imo72b2lem2 38467	. . . . . . . 8 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → sup((abs “ (𝐹 “ ℝ)), ℝ, < ) ≤ (sup((abs “ (𝐹 “ ℝ)), ℝ, < ) / (abs‘(𝐺‘𝐵))))
97	96, 52, 12, 45, 45	lemuldiv4d 38473	. . . . . . 7 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → (sup((abs “ (𝐹 “ ℝ)), ℝ, < ) · (abs‘(𝐺‘𝐵))) ≤ sup((abs “ (𝐹 “ ℝ)), ℝ, < ))
98	48, 97	eqbrtrrd 4677	. . . . . 6 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → ((abs‘(𝐺‘𝐵)) · sup((abs “ (𝐹 “ ℝ)), ℝ, < )) ≤ sup((abs “ (𝐹 “ ℝ)), ℝ, < ))
99		imo72b2.7	. . . . . . . 8 ⊢ (𝜑 → ∃𝑥 ∈ ℝ (𝐹‘𝑥) ≠ 0)
100	99	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → ∃𝑥 ∈ ℝ (𝐹‘𝑥) ≠ 0)
101	41	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → ∀𝑦 ∈ ℝ (abs‘(𝐹‘𝑦)) ≤ 1)
102	20, 100, 101	imo72b2lem1 38471	. . . . . 6 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → 0 < sup((abs “ (𝐹 “ ℝ)), ℝ, < ))
103	98, 102, 45, 12, 45	lemuldiv3d 38472	. . . . 5 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → (abs‘(𝐺‘𝐵)) ≤ (sup((abs “ (𝐹 “ ℝ)), ℝ, < ) / sup((abs “ (𝐹 “ ℝ)), ℝ, < )))
104	23, 45	sseldd 3604	. . . . . . 7 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → sup((abs “ (𝐹 “ ℝ)), ℝ, < ) ∈ ℂ)
105	102	gt0ne0d 10592	. . . . . . 7 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → sup((abs “ (𝐹 “ ℝ)), ℝ, < ) ≠ 0)
106	104, 105	dividd 10799	. . . . . 6 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → (sup((abs “ (𝐹 “ ℝ)), ℝ, < ) / sup((abs “ (𝐹 “ ℝ)), ℝ, < )) = 1)
107	106	eqcomd 2628	. . . . 5 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → 1 = (sup((abs “ (𝐹 “ ℝ)), ℝ, < ) / sup((abs “ (𝐹 “ ℝ)), ℝ, < )))
108	103, 107	breqtrrd 4681	. . . 4 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → (abs‘(𝐺‘𝐵)) ≤ 1)
109	12, 13, 108	lensymd 10188	. . 3 ⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → ¬ 1 < (abs‘(𝐺‘𝐵)))
110	7, 109	pm2.65da 600	. 2 ⊢ (𝜑 → ¬ 1 < (abs‘(𝐺‘𝐵)))
111	5, 6, 110	nltled 10187	1 ⊢ (𝜑 → (abs‘(𝐺‘𝐵)) ≤ 1)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 ∃wrex 2913 ⊆ wss 3574 ∅c0 3915 class class class wbr 4653 ran crn 5115 “ cima 5117 ∘ ccom 5118 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 supcsup 8346 ℂcc 9934 ℝcr 9935 0cc0 9936 1c1 9937 + caddc 9939 · cmul 9941 ℝ^*cxr 10073 < clt 10074 ≤ cle 10075 − cmin 10266 / cdiv 10684 2c2 11070 abscabs 13974

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976

This theorem is referenced by: (None)

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