pntrmax - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > pntrmax		Structured version Visualization version GIF version

Theorem pntrmax 27067

Description: There is a bound on the residual valid for all 𝑥. (Contributed by Mario Carneiro, 9-Apr-2016.)

**Hypothesis**
Ref	Expression
pntrval.r	⊢ 𝑅 = (𝑎 ∈ ℝ⁺ ↦ ((ψ‘𝑎) − 𝑎))

**Assertion**
Ref	Expression
pntrmax	⊢ ∃𝑐 ∈ ℝ⁺ ∀𝑥 ∈ ℝ⁺ (abs‘((𝑅‘𝑥) / 𝑥)) ≤ 𝑐

Distinct variable groups: 𝑥,𝑎 𝑥,𝑐,𝑅 Allowed substitution hint: 𝑅(𝑎)

Proof of Theorem pntrmax Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rpssre 12981	. . . 4 ⊢ ℝ⁺ ⊆ ℝ
2	1	a1i 11	. . 3 ⊢ (⊤ → ℝ⁺ ⊆ ℝ)
3		1red 11215	. . 3 ⊢ (⊤ → 1 ∈ ℝ)
4		pntrval.r	. . . . . . . 8 ⊢ 𝑅 = (𝑎 ∈ ℝ⁺ ↦ ((ψ‘𝑎) − 𝑎))
5	4	pntrval 27065	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (𝑅‘𝑥) = ((ψ‘𝑥) − 𝑥))
6		rpre 12982	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
7		chpcl 26628	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → (ψ‘𝑥) ∈ ℝ)
8	6, 7	syl 17	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → (ψ‘𝑥) ∈ ℝ)
9	8, 6	resubcld 11642	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → ((ψ‘𝑥) − 𝑥) ∈ ℝ)
10	5, 9	eqeltrd 2834	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → (𝑅‘𝑥) ∈ ℝ)
11		rerpdivcl 13004	. . . . . 6 ⊢ (((𝑅‘𝑥) ∈ ℝ ∧ 𝑥 ∈ ℝ⁺) → ((𝑅‘𝑥) / 𝑥) ∈ ℝ)
12	10, 11	mpancom 687	. . . . 5 ⊢ (𝑥 ∈ ℝ⁺ → ((𝑅‘𝑥) / 𝑥) ∈ ℝ)
13	12	recnd 11242	. . . 4 ⊢ (𝑥 ∈ ℝ⁺ → ((𝑅‘𝑥) / 𝑥) ∈ ℂ)
14	13	adantl 483	. . 3 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((𝑅‘𝑥) / 𝑥) ∈ ℂ)
15	5	oveq1d 7424	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → ((𝑅‘𝑥) / 𝑥) = (((ψ‘𝑥) − 𝑥) / 𝑥))
16	8	recnd 11242	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (ψ‘𝑥) ∈ ℂ)
17		rpcn 12984	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℂ)
18		rpne0 12990	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ≠ 0)
19	16, 17, 17, 18	divsubdird 12029	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → (((ψ‘𝑥) − 𝑥) / 𝑥) = (((ψ‘𝑥) / 𝑥) − (𝑥 / 𝑥)))
20	17, 18	dividd 11988	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 / 𝑥) = 1)
21	20	oveq2d 7425	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → (((ψ‘𝑥) / 𝑥) − (𝑥 / 𝑥)) = (((ψ‘𝑥) / 𝑥) − 1))
22	15, 19, 21	3eqtrd 2777	. . . . 5 ⊢ (𝑥 ∈ ℝ⁺ → ((𝑅‘𝑥) / 𝑥) = (((ψ‘𝑥) / 𝑥) − 1))
23	22	mpteq2ia 5252	. . . 4 ⊢ (𝑥 ∈ ℝ⁺ ↦ ((𝑅‘𝑥) / 𝑥)) = (𝑥 ∈ ℝ⁺ ↦ (((ψ‘𝑥) / 𝑥) − 1))
24		rerpdivcl 13004	. . . . . . 7 ⊢ (((ψ‘𝑥) ∈ ℝ ∧ 𝑥 ∈ ℝ⁺) → ((ψ‘𝑥) / 𝑥) ∈ ℝ)
25	8, 24	mpancom 687	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → ((ψ‘𝑥) / 𝑥) ∈ ℝ)
26	25	adantl 483	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((ψ‘𝑥) / 𝑥) ∈ ℝ)
27		1red 11215	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → 1 ∈ ℝ)
28		chpo1ub 26983	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)) ∈ 𝑂(1)
29	28	a1i 11	. . . . 5 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)) ∈ 𝑂(1))
30		ax-1cn 11168	. . . . . . 7 ⊢ 1 ∈ ℂ
31		o1const 15564	. . . . . . 7 ⊢ ((ℝ⁺ ⊆ ℝ ∧ 1 ∈ ℂ) → (𝑥 ∈ ℝ⁺ ↦ 1) ∈ 𝑂(1))
32	1, 30, 31	mp2an 691	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ ↦ 1) ∈ 𝑂(1)
33	32	a1i 11	. . . . 5 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ 1) ∈ 𝑂(1))
34	26, 27, 29, 33	o1sub2 15570	. . . 4 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (((ψ‘𝑥) / 𝑥) − 1)) ∈ 𝑂(1))
35	23, 34	eqeltrid 2838	. . 3 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((𝑅‘𝑥) / 𝑥)) ∈ 𝑂(1))
36		chpcl 26628	. . . . 5 ⊢ (𝑦 ∈ ℝ → (ψ‘𝑦) ∈ ℝ)
37		peano2re 11387	. . . . 5 ⊢ ((ψ‘𝑦) ∈ ℝ → ((ψ‘𝑦) + 1) ∈ ℝ)
38	36, 37	syl 17	. . . 4 ⊢ (𝑦 ∈ ℝ → ((ψ‘𝑦) + 1) ∈ ℝ)
39	38	ad2antrl 727	. . 3 ⊢ ((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) → ((ψ‘𝑦) + 1) ∈ ℝ)
40	22	3ad2ant1 1134	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → ((𝑅‘𝑥) / 𝑥) = (((ψ‘𝑥) / 𝑥) − 1))
41	40	fveq2d 6896	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → (abs‘((𝑅‘𝑥) / 𝑥)) = (abs‘(((ψ‘𝑥) / 𝑥) − 1)))
42		1re 11214	. . . . . . . . . 10 ⊢ 1 ∈ ℝ
43	38	3ad2ant2 1135	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → ((ψ‘𝑦) + 1) ∈ ℝ)
44		resubcl 11524	. . . . . . . . . 10 ⊢ ((1 ∈ ℝ ∧ ((ψ‘𝑦) + 1) ∈ ℝ) → (1 − ((ψ‘𝑦) + 1)) ∈ ℝ)
45	42, 43, 44	sylancr 588	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → (1 − ((ψ‘𝑦) + 1)) ∈ ℝ)
46		0red 11217	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → 0 ∈ ℝ)
47	25	3ad2ant1 1134	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → ((ψ‘𝑥) / 𝑥) ∈ ℝ)
48		chpge0 26630	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℝ → 0 ≤ (ψ‘𝑦))
49	48	3ad2ant2 1135	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → 0 ≤ (ψ‘𝑦))
50	36	3ad2ant2 1135	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → (ψ‘𝑦) ∈ ℝ)
51		addge02 11725	. . . . . . . . . . . 12 ⊢ ((1 ∈ ℝ ∧ (ψ‘𝑦) ∈ ℝ) → (0 ≤ (ψ‘𝑦) ↔ 1 ≤ ((ψ‘𝑦) + 1)))
52	42, 50, 51	sylancr 588	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → (0 ≤ (ψ‘𝑦) ↔ 1 ≤ ((ψ‘𝑦) + 1)))
53	49, 52	mpbid 231	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → 1 ≤ ((ψ‘𝑦) + 1))
54		suble0 11728	. . . . . . . . . . 11 ⊢ ((1 ∈ ℝ ∧ ((ψ‘𝑦) + 1) ∈ ℝ) → ((1 − ((ψ‘𝑦) + 1)) ≤ 0 ↔ 1 ≤ ((ψ‘𝑦) + 1)))
55	42, 43, 54	sylancr 588	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → ((1 − ((ψ‘𝑦) + 1)) ≤ 0 ↔ 1 ≤ ((ψ‘𝑦) + 1)))
56	53, 55	mpbird 257	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → (1 − ((ψ‘𝑦) + 1)) ≤ 0)
57	8	3ad2ant1 1134	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → (ψ‘𝑥) ∈ ℝ)
58	6	3ad2ant1 1134	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → 𝑥 ∈ ℝ)
59		chpge0 26630	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ → 0 ≤ (ψ‘𝑥))
60	58, 59	syl 17	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → 0 ≤ (ψ‘𝑥))
61		rpregt0 12988	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℝ ∧ 0 < 𝑥))
62	61	3ad2ant1 1134	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → (𝑥 ∈ ℝ ∧ 0 < 𝑥))
63		divge0 12083	. . . . . . . . . 10 ⊢ ((((ψ‘𝑥) ∈ ℝ ∧ 0 ≤ (ψ‘𝑥)) ∧ (𝑥 ∈ ℝ ∧ 0 < 𝑥)) → 0 ≤ ((ψ‘𝑥) / 𝑥))
64	57, 60, 62, 63	syl21anc 837	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → 0 ≤ ((ψ‘𝑥) / 𝑥))
65	45, 46, 47, 56, 64	letrd 11371	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → (1 − ((ψ‘𝑦) + 1)) ≤ ((ψ‘𝑥) / 𝑥))
66		2re 12286	. . . . . . . . . . 11 ⊢ 2 ∈ ℝ
67		readdcl 11193	. . . . . . . . . . 11 ⊢ (((ψ‘𝑦) ∈ ℝ ∧ 2 ∈ ℝ) → ((ψ‘𝑦) + 2) ∈ ℝ)
68	50, 66, 67	sylancl 587	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → ((ψ‘𝑦) + 2) ∈ ℝ)
69		1red 11215	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → 1 ∈ ℝ)
70	58	adantr 482	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) ∧ 𝑥 ≤ 1) → 𝑥 ∈ ℝ)
71		1red 11215	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) ∧ 𝑥 ≤ 1) → 1 ∈ ℝ)
72	66	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) ∧ 𝑥 ≤ 1) → 2 ∈ ℝ)
73		simpr 486	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) ∧ 𝑥 ≤ 1) → 𝑥 ≤ 1)
74		1lt2 12383	. . . . . . . . . . . . . . . 16 ⊢ 1 < 2
75	74	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) ∧ 𝑥 ≤ 1) → 1 < 2)
76	70, 71, 72, 73, 75	lelttrd 11372	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) ∧ 𝑥 ≤ 1) → 𝑥 < 2)
77		chpeq0 26711	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℝ → ((ψ‘𝑥) = 0 ↔ 𝑥 < 2))
78	70, 77	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) ∧ 𝑥 ≤ 1) → ((ψ‘𝑥) = 0 ↔ 𝑥 < 2))
79	76, 78	mpbird 257	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) ∧ 𝑥 ≤ 1) → (ψ‘𝑥) = 0)
80	79	oveq1d 7424	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) ∧ 𝑥 ≤ 1) → ((ψ‘𝑥) / 𝑥) = (0 / 𝑥))
81		simp1 1137	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → 𝑥 ∈ ℝ⁺)
82	81	rpcnne0d 13025	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
83		div0 11902	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) → (0 / 𝑥) = 0)
84	82, 83	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → (0 / 𝑥) = 0)
85	84, 49	eqbrtrd 5171	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → (0 / 𝑥) ≤ (ψ‘𝑦))
86	85	adantr 482	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) ∧ 𝑥 ≤ 1) → (0 / 𝑥) ≤ (ψ‘𝑦))
87	80, 86	eqbrtrd 5171	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) ∧ 𝑥 ≤ 1) → ((ψ‘𝑥) / 𝑥) ≤ (ψ‘𝑦))
88	47	adantr 482	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) ∧ 1 ≤ 𝑥) → ((ψ‘𝑥) / 𝑥) ∈ ℝ)
89	57	adantr 482	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) ∧ 1 ≤ 𝑥) → (ψ‘𝑥) ∈ ℝ)
90	50	adantr 482	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) ∧ 1 ≤ 𝑥) → (ψ‘𝑦) ∈ ℝ)
91		0lt1 11736	. . . . . . . . . . . . . . . 16 ⊢ 0 < 1
92	91	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → 0 < 1)
93		lediv2a 12108	. . . . . . . . . . . . . . . 16 ⊢ ((((1 ∈ ℝ ∧ 0 < 1) ∧ (𝑥 ∈ ℝ ∧ 0 < 𝑥) ∧ ((ψ‘𝑥) ∈ ℝ ∧ 0 ≤ (ψ‘𝑥))) ∧ 1 ≤ 𝑥) → ((ψ‘𝑥) / 𝑥) ≤ ((ψ‘𝑥) / 1))
94	93	ex 414	. . . . . . . . . . . . . . 15 ⊢ (((1 ∈ ℝ ∧ 0 < 1) ∧ (𝑥 ∈ ℝ ∧ 0 < 𝑥) ∧ ((ψ‘𝑥) ∈ ℝ ∧ 0 ≤ (ψ‘𝑥))) → (1 ≤ 𝑥 → ((ψ‘𝑥) / 𝑥) ≤ ((ψ‘𝑥) / 1)))
95	69, 92, 62, 57, 60, 94	syl212anc 1381	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → (1 ≤ 𝑥 → ((ψ‘𝑥) / 𝑥) ≤ ((ψ‘𝑥) / 1)))
96	95	imp 408	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) ∧ 1 ≤ 𝑥) → ((ψ‘𝑥) / 𝑥) ≤ ((ψ‘𝑥) / 1))
97	89	recnd 11242	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) ∧ 1 ≤ 𝑥) → (ψ‘𝑥) ∈ ℂ)
98	97	div1d 11982	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) ∧ 1 ≤ 𝑥) → ((ψ‘𝑥) / 1) = (ψ‘𝑥))
99	96, 98	breqtrd 5175	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) ∧ 1 ≤ 𝑥) → ((ψ‘𝑥) / 𝑥) ≤ (ψ‘𝑥))
100		simp2 1138	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → 𝑦 ∈ ℝ)
101		ltle 11302	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 < 𝑦 → 𝑥 ≤ 𝑦))
102	6, 101	sylan 581	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ) → (𝑥 < 𝑦 → 𝑥 ≤ 𝑦))
103	102	3impia 1118	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → 𝑥 ≤ 𝑦)
104		chpwordi 26661	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 ≤ 𝑦) → (ψ‘𝑥) ≤ (ψ‘𝑦))
105	58, 100, 103, 104	syl3anc 1372	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → (ψ‘𝑥) ≤ (ψ‘𝑦))
106	105	adantr 482	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) ∧ 1 ≤ 𝑥) → (ψ‘𝑥) ≤ (ψ‘𝑦))
107	88, 89, 90, 99, 106	letrd 11371	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) ∧ 1 ≤ 𝑥) → ((ψ‘𝑥) / 𝑥) ≤ (ψ‘𝑦))
108	58, 69, 87, 107	lecasei 11320	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → ((ψ‘𝑥) / 𝑥) ≤ (ψ‘𝑦))
109		2nn0 12489	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ₀
110		nn0addge1 12518	. . . . . . . . . . 11 ⊢ (((ψ‘𝑦) ∈ ℝ ∧ 2 ∈ ℕ₀) → (ψ‘𝑦) ≤ ((ψ‘𝑦) + 2))
111	50, 109, 110	sylancl 587	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → (ψ‘𝑦) ≤ ((ψ‘𝑦) + 2))
112	47, 50, 68, 108, 111	letrd 11371	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → ((ψ‘𝑥) / 𝑥) ≤ ((ψ‘𝑦) + 2))
113		df-2 12275	. . . . . . . . . . 11 ⊢ 2 = (1 + 1)
114	113	oveq2i 7420	. . . . . . . . . 10 ⊢ ((ψ‘𝑦) + 2) = ((ψ‘𝑦) + (1 + 1))
115	50	recnd 11242	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → (ψ‘𝑦) ∈ ℂ)
116	30	a1i 11	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → 1 ∈ ℂ)
117	115, 116, 116	add12d 11440	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → ((ψ‘𝑦) + (1 + 1)) = (1 + ((ψ‘𝑦) + 1)))
118	114, 117	eqtrid 2785	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → ((ψ‘𝑦) + 2) = (1 + ((ψ‘𝑦) + 1)))
119	112, 118	breqtrd 5175	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → ((ψ‘𝑥) / 𝑥) ≤ (1 + ((ψ‘𝑦) + 1)))
120	47, 69, 43	absdifled 15381	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → ((abs‘(((ψ‘𝑥) / 𝑥) − 1)) ≤ ((ψ‘𝑦) + 1) ↔ ((1 − ((ψ‘𝑦) + 1)) ≤ ((ψ‘𝑥) / 𝑥) ∧ ((ψ‘𝑥) / 𝑥) ≤ (1 + ((ψ‘𝑦) + 1)))))
121	65, 119, 120	mpbir2and 712	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → (abs‘(((ψ‘𝑥) / 𝑥) − 1)) ≤ ((ψ‘𝑦) + 1))
122	41, 121	eqbrtrd 5171	. . . . . 6 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → (abs‘((𝑅‘𝑥) / 𝑥)) ≤ ((ψ‘𝑦) + 1))
123	122	3expb 1121	. . . . 5 ⊢ ((𝑥 ∈ ℝ⁺ ∧ (𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)) → (abs‘((𝑅‘𝑥) / 𝑥)) ≤ ((ψ‘𝑦) + 1))
124	123	adantrlr 722	. . . 4 ⊢ ((𝑥 ∈ ℝ⁺ ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (abs‘((𝑅‘𝑥) / 𝑥)) ≤ ((ψ‘𝑦) + 1))
125	124	adantll 713	. . 3 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (abs‘((𝑅‘𝑥) / 𝑥)) ≤ ((ψ‘𝑦) + 1))
126	2, 3, 14, 35, 39, 125	o1bddrp 15486	. 2 ⊢ (⊤ → ∃𝑐 ∈ ℝ⁺ ∀𝑥 ∈ ℝ⁺ (abs‘((𝑅‘𝑥) / 𝑥)) ≤ 𝑐)
127	126	mptru 1549	1 ⊢ ∃𝑐 ∈ ℝ⁺ ∀𝑥 ∈ ℝ⁺ (abs‘((𝑅‘𝑥) / 𝑥)) ≤ 𝑐

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ⊤wtru 1543 ∈ wcel 2107 ≠ wne 2941 ∀wral 3062 ∃wrex 3071 ⊆ wss 3949 class class class wbr 5149 ↦ cmpt 5232 ‘cfv 6544 (class class class)co 7409 ℂcc 11108 ℝcr 11109 0cc0 11110 1c1 11111 + caddc 11113 < clt 11248 ≤ cle 11249 − cmin 11444 / cdiv 11871 2c2 12267 ℕ₀cn0 12472 ℝ⁺crp 12974 abscabs 15181 𝑂(1)co1 15430 ψcchp 26597

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188 ax-addf 11189 ax-mulf 11190

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-of 7670 df-om 7856 df-1st 7975 df-2nd 7976 df-supp 8147 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-2o 8467 df-oadd 8470 df-er 8703 df-map 8822 df-pm 8823 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9362 df-fi 9406 df-sup 9437 df-inf 9438 df-oi 9505 df-dju 9896 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12473 df-xnn0 12545 df-z 12559 df-dec 12678 df-uz 12823 df-q 12933 df-rp 12975 df-xneg 13092 df-xadd 13093 df-xmul 13094 df-ioo 13328 df-ioc 13329 df-ico 13330 df-icc 13331 df-fz 13485 df-fzo 13628 df-fl 13757 df-mod 13835 df-seq 13967 df-exp 14028 df-fac 14234 df-bc 14263 df-hash 14291 df-shft 15014 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-limsup 15415 df-clim 15432 df-rlim 15433 df-o1 15434 df-lo1 15435 df-sum 15633 df-ef 16011 df-e 16012 df-sin 16013 df-cos 16014 df-pi 16016 df-dvds 16198 df-gcd 16436 df-prm 16609 df-pc 16770 df-struct 17080 df-sets 17097 df-slot 17115 df-ndx 17127 df-base 17145 df-ress 17174 df-plusg 17210 df-mulr 17211 df-starv 17212 df-sca 17213 df-vsca 17214 df-ip 17215 df-tset 17216 df-ple 17217 df-ds 17219 df-unif 17220 df-hom 17221 df-cco 17222 df-rest 17368 df-topn 17369 df-0g 17387 df-gsum 17388 df-topgen 17389 df-pt 17390 df-prds 17393 df-xrs 17448 df-qtop 17453 df-imas 17454 df-xps 17456 df-mre 17530 df-mrc 17531 df-acs 17533 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-submnd 18672 df-mulg 18951 df-cntz 19181 df-cmn 19650 df-psmet 20936 df-xmet 20937 df-met 20938 df-bl 20939 df-mopn 20940 df-fbas 20941 df-fg 20942 df-cnfld 20945 df-top 22396 df-topon 22413 df-topsp 22435 df-bases 22449 df-cld 22523 df-ntr 22524 df-cls 22525 df-nei 22602 df-lp 22640 df-perf 22641 df-cn 22731 df-cnp 22732 df-haus 22819 df-tx 23066 df-hmeo 23259 df-fil 23350 df-fm 23442 df-flim 23443 df-flf 23444 df-xms 23826 df-ms 23827 df-tms 23828 df-cncf 24394 df-limc 25383 df-dv 25384 df-log 26065 df-cxp 26066 df-cht 26601 df-vma 26602 df-chp 26603 df-ppi 26604

This theorem is referenced by: pntrlog2bnd 27087 pntibnd 27096 pnt3 27115

W3C validator