imo72b2lem0 - Mathbox for Stanislas Polu

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Theorem imo72b2lem0 43498

Description: Lemma for imo72b2 43505. (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 7081	. . . . . . 7 ⊢ (𝜑 → (𝐹‘𝐴) ∈ ℝ)
4	3	recnd 11246	. . . . . 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 7081	. . . . . . 7 ⊢ (𝜑 → (𝐺‘𝐵) ∈ ℝ)
9	8	recnd 11246	. . . . . 6 ⊢ (𝜑 → (𝐺‘𝐵) ∈ ℂ)
10	9	idi 1	. . . . 5 ⊢ (𝜑 → (𝐺‘𝐵) ∈ ℂ)
11	5, 10	mulcld 11238	. . . 4 ⊢ (𝜑 → ((𝐹‘𝐴) · (𝐺‘𝐵)) ∈ ℂ)
12	11	abscld 15389	. . 3 ⊢ (𝜑 → (abs‘((𝐹‘𝐴) · (𝐺‘𝐵))) ∈ ℝ)
13		imaco 6244	. . . . . 6 ⊢ ((abs ∘ 𝐹) “ ℝ) = (abs “ (𝐹 “ ℝ))
14	13	eqcomi 2735	. . . . 5 ⊢ (abs “ (𝐹 “ ℝ)) = ((abs ∘ 𝐹) “ ℝ)
15		imassrn 6064	. . . . . . 7 ⊢ ((abs ∘ 𝐹) “ ℝ) ⊆ ran (abs ∘ 𝐹)
16	15	a1i 11	. . . . . 6 ⊢ (𝜑 → ((abs ∘ 𝐹) “ ℝ) ⊆ ran (abs ∘ 𝐹))
17		absf 15290	. . . . . . . . . 10 ⊢ abs:ℂ⟶ℝ
18	17	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → abs:ℂ⟶ℝ)
19		ax-resscn 11169	. . . . . . . . . 10 ⊢ ℝ ⊆ ℂ
20	19	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ℝ ⊆ ℂ)
21	18, 20	fssresd 6752	. . . . . . . 8 ⊢ (𝜑 → (abs ↾ ℝ):ℝ⟶ℝ)
22	1, 21	fco2d 43495	. . . . . . 7 ⊢ (𝜑 → (abs ∘ 𝐹):ℝ⟶ℝ)
23	22	frnd 6719	. . . . . 6 ⊢ (𝜑 → ran (abs ∘ 𝐹) ⊆ ℝ)
24	16, 23	sstrd 3987	. . . . 5 ⊢ (𝜑 → ((abs ∘ 𝐹) “ ℝ) ⊆ ℝ)
25	14, 24	eqsstrid 4025	. . . 4 ⊢ (𝜑 → (abs “ (𝐹 “ ℝ)) ⊆ ℝ)
26		0re 11220	. . . . . . . . . 10 ⊢ 0 ∈ ℝ
27	26	ne0ii 4332	. . . . . . . . 9 ⊢ ℝ ≠ ∅
28	27	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ℝ ≠ ∅)
29	28, 22	wnefimgd 43494	. . . . . . 7 ⊢ (𝜑 → ((abs ∘ 𝐹) “ ℝ) ≠ ∅)
30	29	necomd 2990	. . . . . 6 ⊢ (𝜑 → ∅ ≠ ((abs ∘ 𝐹) “ ℝ))
31	14	a1i 11	. . . . . 6 ⊢ (𝜑 → (abs “ (𝐹 “ ℝ)) = ((abs ∘ 𝐹) “ ℝ))
32	30, 31	neeqtrrd 3009	. . . . 5 ⊢ (𝜑 → ∅ ≠ (abs “ (𝐹 “ ℝ)))
33	32	necomd 2990	. . . 4 ⊢ (𝜑 → (abs “ (𝐹 “ ℝ)) ≠ ∅)
34		1red 11219	. . . . 5 ⊢ (𝜑 → 1 ∈ ℝ)
35		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 = 1) → 𝑐 = 1)
36	35	breq2d 5153	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 = 1) → (𝑥 ≤ 𝑐 ↔ 𝑥 ≤ 1))
37	36	ralbidv 3171	. . . . 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 43497	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ (abs “ (𝐹 “ ℝ))𝑥 ≤ 1)
40	34, 37, 39	rspcedvd 3608	. . . 4 ⊢ (𝜑 → ∃𝑐 ∈ ℝ ∀𝑥 ∈ (abs “ (𝐹 “ ℝ))𝑥 ≤ 𝑐)
41	25, 33, 40	suprcld 12181	. . 3 ⊢ (𝜑 → sup((abs “ (𝐹 “ ℝ)), ℝ, < ) ∈ ℝ)
42		2re 12290	. . . 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 6889	. . . . . . 7 ⊢ (𝜑 → (abs‘((𝐹‘(𝐴 + 𝐵)) + (𝐹‘(𝐴 − 𝐵)))) = (abs‘(2 · ((𝐹‘𝐴) · (𝐺‘𝐵)))))
47		2cnd 12294	. . . . . . . . 9 ⊢ (𝜑 → 2 ∈ ℂ)
48	47, 11	mulcld 11238	. . . . . . . 8 ⊢ (𝜑 → (2 · ((𝐹‘𝐴) · (𝐺‘𝐵))) ∈ ℂ)
49	48	abscld 15389	. . . . . . 7 ⊢ (𝜑 → (abs‘(2 · ((𝐹‘𝐴) · (𝐺‘𝐵)))) ∈ ℝ)
50	46, 49	eqeltrd 2827	. . . . . 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 11247	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℝ)
55	51, 54	ffvelcdmd 7081	. . . . . . . . 9 ⊢ (𝜑 → (𝐹‘(𝐴 + 𝐵)) ∈ ℝ)
56	55	recnd 11246	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘(𝐴 + 𝐵)) ∈ ℂ)
57	56	abscld 15389	. . . . . . 7 ⊢ (𝜑 → (abs‘(𝐹‘(𝐴 + 𝐵))) ∈ ℝ)
58	52, 53	resubcld 11646	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 − 𝐵) ∈ ℝ)
59	51, 58	ffvelcdmd 7081	. . . . . . . . 9 ⊢ (𝜑 → (𝐹‘(𝐴 − 𝐵)) ∈ ℝ)
60	59	recnd 11246	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘(𝐴 − 𝐵)) ∈ ℂ)
61	60	abscld 15389	. . . . . . 7 ⊢ (𝜑 → (abs‘(𝐹‘(𝐴 − 𝐵))) ∈ ℝ)
62	57, 61	readdcld 11247	. . . . . 6 ⊢ (𝜑 → ((abs‘(𝐹‘(𝐴 + 𝐵))) + (abs‘(𝐹‘(𝐴 − 𝐵)))) ∈ ℝ)
63	43, 41	remulcld 11248	. . . . . 6 ⊢ (𝜑 → (2 · sup((abs “ (𝐹 “ ℝ)), ℝ, < )) ∈ ℝ)
64	56, 60	abstrid 15409	. . . . . 6 ⊢ (𝜑 → (abs‘((𝐹‘(𝐴 + 𝐵)) + (𝐹‘(𝐴 − 𝐵)))) ≤ ((abs‘(𝐹‘(𝐴 + 𝐵))) + (abs‘(𝐹‘(𝐴 − 𝐵)))))
65	1, 54	fvco3d 6985	. . . . . . . . . 10 ⊢ (𝜑 → ((abs ∘ 𝐹)‘(𝐴 + 𝐵)) = (abs‘(𝐹‘(𝐴 + 𝐵))))
66	54, 22	wfximgfd 43496	. . . . . . . . . . 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 2830	. . . . . . . . . 10 ⊢ (𝜑 → ((abs ∘ 𝐹)‘(𝐴 + 𝐵)) ∈ (abs “ (𝐹 “ ℝ)))
69	65, 68	eqeltrrd 2828	. . . . . . . . 9 ⊢ (𝜑 → (abs‘(𝐹‘(𝐴 + 𝐵))) ∈ (abs “ (𝐹 “ ℝ)))
70	25, 33, 40, 69	suprubd 12180	. . . . . . . 8 ⊢ (𝜑 → (abs‘(𝐹‘(𝐴 + 𝐵))) ≤ sup((abs “ (𝐹 “ ℝ)), ℝ, < ))
71	1, 58	fvco3d 6985	. . . . . . . . . 10 ⊢ (𝜑 → ((abs ∘ 𝐹)‘(𝐴 − 𝐵)) = (abs‘(𝐹‘(𝐴 − 𝐵))))
72	58, 22	wfximgfd 43496	. . . . . . . . . . 11 ⊢ (𝜑 → ((abs ∘ 𝐹)‘(𝐴 − 𝐵)) ∈ ((abs ∘ 𝐹) “ ℝ))
73	72, 31	eleqtrrd 2830	. . . . . . . . . 10 ⊢ (𝜑 → ((abs ∘ 𝐹)‘(𝐴 − 𝐵)) ∈ (abs “ (𝐹 “ ℝ)))
74	71, 73	eqeltrrd 2828	. . . . . . . . 9 ⊢ (𝜑 → (abs‘(𝐹‘(𝐴 − 𝐵))) ∈ (abs “ (𝐹 “ ℝ)))
75	25, 33, 40, 74	suprubd 12180	. . . . . . . 8 ⊢ (𝜑 → (abs‘(𝐹‘(𝐴 − 𝐵))) ≤ sup((abs “ (𝐹 “ ℝ)), ℝ, < ))
76	57, 61, 41, 41, 70, 75	le2addd 11837	. . . . . . 7 ⊢ (𝜑 → ((abs‘(𝐹‘(𝐴 + 𝐵))) + (abs‘(𝐹‘(𝐴 − 𝐵)))) ≤ (sup((abs “ (𝐹 “ ℝ)), ℝ, < ) + sup((abs “ (𝐹 “ ℝ)), ℝ, < )))
77	41	recnd 11246	. . . . . . . . 9 ⊢ (𝜑 → sup((abs “ (𝐹 “ ℝ)), ℝ, < ) ∈ ℂ)
78	77	2timesd 12459	. . . . . . . 8 ⊢ (𝜑 → (2 · sup((abs “ (𝐹 “ ℝ)), ℝ, < )) = (sup((abs “ (𝐹 “ ℝ)), ℝ, < ) + sup((abs “ (𝐹 “ ℝ)), ℝ, < )))
79	78	eqcomd 2732	. . . . . . 7 ⊢ (𝜑 → (sup((abs “ (𝐹 “ ℝ)), ℝ, < ) + sup((abs “ (𝐹 “ ℝ)), ℝ, < )) = (2 · sup((abs “ (𝐹 “ ℝ)), ℝ, < )))
80	79, 63	eqeltrd 2827	. . . . . . 7 ⊢ (𝜑 → (sup((abs “ (𝐹 “ ℝ)), ℝ, < ) + sup((abs “ (𝐹 “ ℝ)), ℝ, < )) ∈ ℝ)
81	76, 79, 62, 80	leeq2d 43490	. . . . . 6 ⊢ (𝜑 → ((abs‘(𝐹‘(𝐴 + 𝐵))) + (abs‘(𝐹‘(𝐴 − 𝐵)))) ≤ (2 · sup((abs “ (𝐹 “ ℝ)), ℝ, < )))
82	50, 62, 63, 64, 81	letrd 11375	. . . . 5 ⊢ (𝜑 → (abs‘((𝐹‘(𝐴 + 𝐵)) + (𝐹‘(𝐴 − 𝐵)))) ≤ (2 · sup((abs “ (𝐹 “ ℝ)), ℝ, < )))
83	82, 46, 50, 63	leeq1d 43489	. . . 4 ⊢ (𝜑 → (abs‘(2 · ((𝐹‘𝐴) · (𝐺‘𝐵)))) ≤ (2 · sup((abs “ (𝐹 “ ℝ)), ℝ, < )))
84		0le2 12318	. . . . . 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 11248	. . . . 5 ⊢ (𝜑 → ((𝐹‘𝐴) · (𝐺‘𝐵)) ∈ ℝ)
89	85, 43, 88	absmulrposd 43491	. . . 4 ⊢ (𝜑 → (abs‘(2 · ((𝐹‘𝐴) · (𝐺‘𝐵)))) = (2 · (abs‘((𝐹‘𝐴) · (𝐺‘𝐵)))))
90	83, 89, 49, 63	leeq1d 43489	. . 3 ⊢ (𝜑 → (2 · (abs‘((𝐹‘𝐴) · (𝐺‘𝐵)))) ≤ (2 · sup((abs “ (𝐹 “ ℝ)), ℝ, < )))
91		2pos 12319	. . . 4 ⊢ 0 < 2
92	91	a1i 11	. . 3 ⊢ (𝜑 → 0 < 2)
93	12, 41, 43, 90, 92	wwlemuld 43488	. 2 ⊢ (𝜑 → (abs‘((𝐹‘𝐴) · (𝐺‘𝐵))) ≤ sup((abs “ (𝐹 “ ℝ)), ℝ, < ))
94	4, 9	absmuld 15407	. . 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 43489	1 ⊢ (𝜑 → ((abs‘(𝐹‘𝐴)) · (abs‘(𝐺‘𝐵))) ≤ sup((abs “ (𝐹 “ ℝ)), ℝ, < ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ≠ wne 2934 ∀wral 3055 ⊆ wss 3943 ∅c0 4317 class class class wbr 5141 ran crn 5670 “ cima 5672 ∘ ccom 5673 ⟶wf 6533 ‘cfv 6537 (class class class)co 7405 supcsup 9437 ℂcc 11110 ℝcr 11111 0cc0 11112 1c1 11113 + caddc 11115 · cmul 11117 < clt 11252 ≤ cle 11253 − cmin 11448 2c2 12271 abscabs 15187

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-sup 9439 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-n0 12477 df-z 12563 df-uz 12827 df-rp 12981 df-seq 13973 df-exp 14033 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189

This theorem is referenced by: imo72b2 43505

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