imo72b2lem0 - Mathbox for Stanislas Polu

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Theorem imo72b2lem0 42907

Description: Lemma for imo72b2 42914. (Contributed by Stanislas Polu, 9-Mar-2020.)

**Hypotheses**
Ref	Expression
imo72b2lem0.1	⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
imo72b2lem0.2	⊢ (𝜑 → 𝐺:ℝ⟶ℝ)
imo72b2lem0.3	⊢ (𝜑 → 𝐴 ∈ ℝ)
imo72b2lem0.4	⊢ (𝜑 → 𝐵 ∈ ℝ)
imo72b2lem0.5	⊢ (𝜑 → ((𝐹‘(𝐴 + 𝐵)) + (𝐹‘(𝐴 − 𝐵))) = (2 · ((𝐹‘𝐴) · (𝐺‘𝐵))))
imo72b2lem0.6	⊢ (𝜑 → ∀𝑦 ∈ ℝ (abs‘(𝐹‘𝑦)) ≤ 1)

**Assertion**
Ref	Expression
imo72b2lem0	⊢ (𝜑 → ((abs‘(𝐹‘𝐴)) · (abs‘(𝐺‘𝐵))) ≤ sup((abs “ (𝐹 “ ℝ)), ℝ, < ))

Distinct variable groups: 𝑦,𝐹 𝜑,𝑦 Allowed substitution hints: 𝐴(𝑦) 𝐵(𝑦) 𝐺(𝑦)

Proof of Theorem imo72b2lem0 Dummy variables 𝑐 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		imo72b2lem0.1	. . . . . . . 8 ⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
2		imo72b2lem0.3	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℝ)
3	1, 2	ffvelcdmd 7087	. . . . . . 7 ⊢ (𝜑 → (𝐹‘𝐴) ∈ ℝ)
4	3	recnd 11241	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝐴) ∈ ℂ)
5	4	idi 1	. . . . 5 ⊢ (𝜑 → (𝐹‘𝐴) ∈ ℂ)
6		imo72b2lem0.2	. . . . . . . 8 ⊢ (𝜑 → 𝐺:ℝ⟶ℝ)
7		imo72b2lem0.4	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℝ)
8	6, 7	ffvelcdmd 7087	. . . . . . 7 ⊢ (𝜑 → (𝐺‘𝐵) ∈ ℝ)
9	8	recnd 11241	. . . . . 6 ⊢ (𝜑 → (𝐺‘𝐵) ∈ ℂ)
10	9	idi 1	. . . . 5 ⊢ (𝜑 → (𝐺‘𝐵) ∈ ℂ)
11	5, 10	mulcld 11233	. . . 4 ⊢ (𝜑 → ((𝐹‘𝐴) · (𝐺‘𝐵)) ∈ ℂ)
12	11	abscld 15382	. . 3 ⊢ (𝜑 → (abs‘((𝐹‘𝐴) · (𝐺‘𝐵))) ∈ ℝ)
13		imaco 6250	. . . . . 6 ⊢ ((abs ∘ 𝐹) “ ℝ) = (abs “ (𝐹 “ ℝ))
14	13	eqcomi 2741	. . . . 5 ⊢ (abs “ (𝐹 “ ℝ)) = ((abs ∘ 𝐹) “ ℝ)
15		imassrn 6070	. . . . . . 7 ⊢ ((abs ∘ 𝐹) “ ℝ) ⊆ ran (abs ∘ 𝐹)
16	15	a1i 11	. . . . . 6 ⊢ (𝜑 → ((abs ∘ 𝐹) “ ℝ) ⊆ ran (abs ∘ 𝐹))
17		absf 15283	. . . . . . . . . 10 ⊢ abs:ℂ⟶ℝ
18	17	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → abs:ℂ⟶ℝ)
19		ax-resscn 11166	. . . . . . . . . 10 ⊢ ℝ ⊆ ℂ
20	19	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ℝ ⊆ ℂ)
21	18, 20	fssresd 6758	. . . . . . . 8 ⊢ (𝜑 → (abs ↾ ℝ):ℝ⟶ℝ)
22	1, 21	fco2d 42904	. . . . . . 7 ⊢ (𝜑 → (abs ∘ 𝐹):ℝ⟶ℝ)
23	22	frnd 6725	. . . . . 6 ⊢ (𝜑 → ran (abs ∘ 𝐹) ⊆ ℝ)
24	16, 23	sstrd 3992	. . . . 5 ⊢ (𝜑 → ((abs ∘ 𝐹) “ ℝ) ⊆ ℝ)
25	14, 24	eqsstrid 4030	. . . 4 ⊢ (𝜑 → (abs “ (𝐹 “ ℝ)) ⊆ ℝ)
26		0re 11215	. . . . . . . . . 10 ⊢ 0 ∈ ℝ
27	26	ne0ii 4337	. . . . . . . . 9 ⊢ ℝ ≠ ∅
28	27	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ℝ ≠ ∅)
29	28, 22	wnefimgd 42903	. . . . . . 7 ⊢ (𝜑 → ((abs ∘ 𝐹) “ ℝ) ≠ ∅)
30	29	necomd 2996	. . . . . 6 ⊢ (𝜑 → ∅ ≠ ((abs ∘ 𝐹) “ ℝ))
31	14	a1i 11	. . . . . 6 ⊢ (𝜑 → (abs “ (𝐹 “ ℝ)) = ((abs ∘ 𝐹) “ ℝ))
32	30, 31	neeqtrrd 3015	. . . . 5 ⊢ (𝜑 → ∅ ≠ (abs “ (𝐹 “ ℝ)))
33	32	necomd 2996	. . . 4 ⊢ (𝜑 → (abs “ (𝐹 “ ℝ)) ≠ ∅)
34		1red 11214	. . . . 5 ⊢ (𝜑 → 1 ∈ ℝ)
35		simpr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 = 1) → 𝑐 = 1)
36	35	breq2d 5160	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 = 1) → (𝑥 ≤ 𝑐 ↔ 𝑥 ≤ 1))
37	36	ralbidv 3177	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 = 1) → (∀𝑥 ∈ (abs “ (𝐹 “ ℝ))𝑥 ≤ 𝑐 ↔ ∀𝑥 ∈ (abs “ (𝐹 “ ℝ))𝑥 ≤ 1))
38		imo72b2lem0.6	. . . . . 6 ⊢ (𝜑 → ∀𝑦 ∈ ℝ (abs‘(𝐹‘𝑦)) ≤ 1)
39	1, 38	extoimad 42906	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ (abs “ (𝐹 “ ℝ))𝑥 ≤ 1)
40	34, 37, 39	rspcedvd 3614	. . . 4 ⊢ (𝜑 → ∃𝑐 ∈ ℝ ∀𝑥 ∈ (abs “ (𝐹 “ ℝ))𝑥 ≤ 𝑐)
41	25, 33, 40	suprcld 12176	. . 3 ⊢ (𝜑 → sup((abs “ (𝐹 “ ℝ)), ℝ, < ) ∈ ℝ)
42		2re 12285	. . . 4 ⊢ 2 ∈ ℝ
43	42	a1i 11	. . 3 ⊢ (𝜑 → 2 ∈ ℝ)
44		imo72b2lem0.5	. . . . . . . . 9 ⊢ (𝜑 → ((𝐹‘(𝐴 + 𝐵)) + (𝐹‘(𝐴 − 𝐵))) = (2 · ((𝐹‘𝐴) · (𝐺‘𝐵))))
45	44	idi 1	. . . . . . . 8 ⊢ (𝜑 → ((𝐹‘(𝐴 + 𝐵)) + (𝐹‘(𝐴 − 𝐵))) = (2 · ((𝐹‘𝐴) · (𝐺‘𝐵))))
46	45	fveq2d 6895	. . . . . . 7 ⊢ (𝜑 → (abs‘((𝐹‘(𝐴 + 𝐵)) + (𝐹‘(𝐴 − 𝐵)))) = (abs‘(2 · ((𝐹‘𝐴) · (𝐺‘𝐵)))))
47		2cnd 12289	. . . . . . . . 9 ⊢ (𝜑 → 2 ∈ ℂ)
48	47, 11	mulcld 11233	. . . . . . . 8 ⊢ (𝜑 → (2 · ((𝐹‘𝐴) · (𝐺‘𝐵))) ∈ ℂ)
49	48	abscld 15382	. . . . . . 7 ⊢ (𝜑 → (abs‘(2 · ((𝐹‘𝐴) · (𝐺‘𝐵)))) ∈ ℝ)
50	46, 49	eqeltrd 2833	. . . . . 6 ⊢ (𝜑 → (abs‘((𝐹‘(𝐴 + 𝐵)) + (𝐹‘(𝐴 − 𝐵)))) ∈ ℝ)
51	1	idi 1	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
52	2	idi 1	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ ℝ)
53	7	idi 1	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ ℝ)
54	52, 53	readdcld 11242	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℝ)
55	51, 54	ffvelcdmd 7087	. . . . . . . . 9 ⊢ (𝜑 → (𝐹‘(𝐴 + 𝐵)) ∈ ℝ)
56	55	recnd 11241	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘(𝐴 + 𝐵)) ∈ ℂ)
57	56	abscld 15382	. . . . . . 7 ⊢ (𝜑 → (abs‘(𝐹‘(𝐴 + 𝐵))) ∈ ℝ)
58	52, 53	resubcld 11641	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 − 𝐵) ∈ ℝ)
59	51, 58	ffvelcdmd 7087	. . . . . . . . 9 ⊢ (𝜑 → (𝐹‘(𝐴 − 𝐵)) ∈ ℝ)
60	59	recnd 11241	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘(𝐴 − 𝐵)) ∈ ℂ)
61	60	abscld 15382	. . . . . . 7 ⊢ (𝜑 → (abs‘(𝐹‘(𝐴 − 𝐵))) ∈ ℝ)
62	57, 61	readdcld 11242	. . . . . 6 ⊢ (𝜑 → ((abs‘(𝐹‘(𝐴 + 𝐵))) + (abs‘(𝐹‘(𝐴 − 𝐵)))) ∈ ℝ)
63	43, 41	remulcld 11243	. . . . . 6 ⊢ (𝜑 → (2 · sup((abs “ (𝐹 “ ℝ)), ℝ, < )) ∈ ℝ)
64	56, 60	abstrid 15402	. . . . . 6 ⊢ (𝜑 → (abs‘((𝐹‘(𝐴 + 𝐵)) + (𝐹‘(𝐴 − 𝐵)))) ≤ ((abs‘(𝐹‘(𝐴 + 𝐵))) + (abs‘(𝐹‘(𝐴 − 𝐵)))))
65	1, 54	fvco3d 6991	. . . . . . . . . 10 ⊢ (𝜑 → ((abs ∘ 𝐹)‘(𝐴 + 𝐵)) = (abs‘(𝐹‘(𝐴 + 𝐵))))
66	54, 22	wfximgfd 42905	. . . . . . . . . . 11 ⊢ (𝜑 → ((abs ∘ 𝐹)‘(𝐴 + 𝐵)) ∈ ((abs ∘ 𝐹) “ ℝ))
67	31	idi 1	. . . . . . . . . . 11 ⊢ (𝜑 → (abs “ (𝐹 “ ℝ)) = ((abs ∘ 𝐹) “ ℝ))
68	66, 67	eleqtrrd 2836	. . . . . . . . . 10 ⊢ (𝜑 → ((abs ∘ 𝐹)‘(𝐴 + 𝐵)) ∈ (abs “ (𝐹 “ ℝ)))
69	65, 68	eqeltrrd 2834	. . . . . . . . 9 ⊢ (𝜑 → (abs‘(𝐹‘(𝐴 + 𝐵))) ∈ (abs “ (𝐹 “ ℝ)))
70	25, 33, 40, 69	suprubd 12175	. . . . . . . 8 ⊢ (𝜑 → (abs‘(𝐹‘(𝐴 + 𝐵))) ≤ sup((abs “ (𝐹 “ ℝ)), ℝ, < ))
71	1, 58	fvco3d 6991	. . . . . . . . . 10 ⊢ (𝜑 → ((abs ∘ 𝐹)‘(𝐴 − 𝐵)) = (abs‘(𝐹‘(𝐴 − 𝐵))))
72	58, 22	wfximgfd 42905	. . . . . . . . . . 11 ⊢ (𝜑 → ((abs ∘ 𝐹)‘(𝐴 − 𝐵)) ∈ ((abs ∘ 𝐹) “ ℝ))
73	72, 31	eleqtrrd 2836	. . . . . . . . . 10 ⊢ (𝜑 → ((abs ∘ 𝐹)‘(𝐴 − 𝐵)) ∈ (abs “ (𝐹 “ ℝ)))
74	71, 73	eqeltrrd 2834	. . . . . . . . 9 ⊢ (𝜑 → (abs‘(𝐹‘(𝐴 − 𝐵))) ∈ (abs “ (𝐹 “ ℝ)))
75	25, 33, 40, 74	suprubd 12175	. . . . . . . 8 ⊢ (𝜑 → (abs‘(𝐹‘(𝐴 − 𝐵))) ≤ sup((abs “ (𝐹 “ ℝ)), ℝ, < ))
76	57, 61, 41, 41, 70, 75	le2addd 11832	. . . . . . 7 ⊢ (𝜑 → ((abs‘(𝐹‘(𝐴 + 𝐵))) + (abs‘(𝐹‘(𝐴 − 𝐵)))) ≤ (sup((abs “ (𝐹 “ ℝ)), ℝ, < ) + sup((abs “ (𝐹 “ ℝ)), ℝ, < )))
77	41	recnd 11241	. . . . . . . . 9 ⊢ (𝜑 → sup((abs “ (𝐹 “ ℝ)), ℝ, < ) ∈ ℂ)
78	77	2timesd 12454	. . . . . . . 8 ⊢ (𝜑 → (2 · sup((abs “ (𝐹 “ ℝ)), ℝ, < )) = (sup((abs “ (𝐹 “ ℝ)), ℝ, < ) + sup((abs “ (𝐹 “ ℝ)), ℝ, < )))
79	78	eqcomd 2738	. . . . . . 7 ⊢ (𝜑 → (sup((abs “ (𝐹 “ ℝ)), ℝ, < ) + sup((abs “ (𝐹 “ ℝ)), ℝ, < )) = (2 · sup((abs “ (𝐹 “ ℝ)), ℝ, < )))
80	79, 63	eqeltrd 2833	. . . . . . 7 ⊢ (𝜑 → (sup((abs “ (𝐹 “ ℝ)), ℝ, < ) + sup((abs “ (𝐹 “ ℝ)), ℝ, < )) ∈ ℝ)
81	76, 79, 62, 80	leeq2d 42899	. . . . . 6 ⊢ (𝜑 → ((abs‘(𝐹‘(𝐴 + 𝐵))) + (abs‘(𝐹‘(𝐴 − 𝐵)))) ≤ (2 · sup((abs “ (𝐹 “ ℝ)), ℝ, < )))
82	50, 62, 63, 64, 81	letrd 11370	. . . . 5 ⊢ (𝜑 → (abs‘((𝐹‘(𝐴 + 𝐵)) + (𝐹‘(𝐴 − 𝐵)))) ≤ (2 · sup((abs “ (𝐹 “ ℝ)), ℝ, < )))
83	82, 46, 50, 63	leeq1d 42898	. . . 4 ⊢ (𝜑 → (abs‘(2 · ((𝐹‘𝐴) · (𝐺‘𝐵)))) ≤ (2 · sup((abs “ (𝐹 “ ℝ)), ℝ, < )))
84		0le2 12313	. . . . . 6 ⊢ 0 ≤ 2
85	84	a1i 11	. . . . 5 ⊢ (𝜑 → 0 ≤ 2)
86	3	idi 1	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝐴) ∈ ℝ)
87	8	idi 1	. . . . . 6 ⊢ (𝜑 → (𝐺‘𝐵) ∈ ℝ)
88	86, 87	remulcld 11243	. . . . 5 ⊢ (𝜑 → ((𝐹‘𝐴) · (𝐺‘𝐵)) ∈ ℝ)
89	85, 43, 88	absmulrposd 42900	. . . 4 ⊢ (𝜑 → (abs‘(2 · ((𝐹‘𝐴) · (𝐺‘𝐵)))) = (2 · (abs‘((𝐹‘𝐴) · (𝐺‘𝐵)))))
90	83, 89, 49, 63	leeq1d 42898	. . 3 ⊢ (𝜑 → (2 · (abs‘((𝐹‘𝐴) · (𝐺‘𝐵)))) ≤ (2 · sup((abs “ (𝐹 “ ℝ)), ℝ, < )))
91		2pos 12314	. . . 4 ⊢ 0 < 2
92	91	a1i 11	. . 3 ⊢ (𝜑 → 0 < 2)
93	12, 41, 43, 90, 92	wwlemuld 42897	. 2 ⊢ (𝜑 → (abs‘((𝐹‘𝐴) · (𝐺‘𝐵))) ≤ sup((abs “ (𝐹 “ ℝ)), ℝ, < ))
94	4, 9	absmuld 15400	. . 3 ⊢ (𝜑 → (abs‘((𝐹‘𝐴) · (𝐺‘𝐵))) = ((abs‘(𝐹‘𝐴)) · (abs‘(𝐺‘𝐵))))
95	94	idi 1	. 2 ⊢ (𝜑 → (abs‘((𝐹‘𝐴) · (𝐺‘𝐵))) = ((abs‘(𝐹‘𝐴)) · (abs‘(𝐺‘𝐵))))
96	93, 95, 12, 41	leeq1d 42898	1 ⊢ (𝜑 → ((abs‘(𝐹‘𝐴)) · (abs‘(𝐺‘𝐵))) ≤ sup((abs “ (𝐹 “ ℝ)), ℝ, < ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∀wral 3061 ⊆ wss 3948 ∅c0 4322 class class class wbr 5148 ran crn 5677 “ cima 5679 ∘ ccom 5680 ⟶wf 6539 ‘cfv 6543 (class class class)co 7408 supcsup 9434 ℂcc 11107 ℝcr 11108 0cc0 11109 1c1 11110 + caddc 11112 · cmul 11114 < clt 11247 ≤ cle 11248 − cmin 11443 2c2 12266 abscabs 15180

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 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 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-sup 9436 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-2 12274 df-3 12275 df-n0 12472 df-z 12558 df-uz 12822 df-rp 12974 df-seq 13966 df-exp 14027 df-cj 15045 df-re 15046 df-im 15047 df-sqrt 15181 df-abs 15182

This theorem is referenced by: imo72b2 42914

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