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Theorem pntrmax 26445

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 12593	. . . 4 ⊢ ℝ⁺ ⊆ ℝ
2	1	a1i 11	. . 3 ⊢ (⊤ → ℝ⁺ ⊆ ℝ)
3		1red 10834	. . 3 ⊢ (⊤ → 1 ∈ ℝ)
4		pntrval.r	. . . . . . . 8 ⊢ 𝑅 = (𝑎 ∈ ℝ⁺ ↦ ((ψ‘𝑎) − 𝑎))
5	4	pntrval 26443	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (𝑅‘𝑥) = ((ψ‘𝑥) − 𝑥))
6		rpre 12594	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
7		chpcl 26006	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → (ψ‘𝑥) ∈ ℝ)
8	6, 7	syl 17	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → (ψ‘𝑥) ∈ ℝ)
9	8, 6	resubcld 11260	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → ((ψ‘𝑥) − 𝑥) ∈ ℝ)
10	5, 9	eqeltrd 2838	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → (𝑅‘𝑥) ∈ ℝ)
11		rerpdivcl 12616	. . . . . 6 ⊢ (((𝑅‘𝑥) ∈ ℝ ∧ 𝑥 ∈ ℝ⁺) → ((𝑅‘𝑥) / 𝑥) ∈ ℝ)
12	10, 11	mpancom 688	. . . . 5 ⊢ (𝑥 ∈ ℝ⁺ → ((𝑅‘𝑥) / 𝑥) ∈ ℝ)
13	12	recnd 10861	. . . 4 ⊢ (𝑥 ∈ ℝ⁺ → ((𝑅‘𝑥) / 𝑥) ∈ ℂ)
14	13	adantl 485	. . 3 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((𝑅‘𝑥) / 𝑥) ∈ ℂ)
15	5	oveq1d 7228	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → ((𝑅‘𝑥) / 𝑥) = (((ψ‘𝑥) − 𝑥) / 𝑥))
16	8	recnd 10861	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (ψ‘𝑥) ∈ ℂ)
17		rpcn 12596	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℂ)
18		rpne0 12602	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ≠ 0)
19	16, 17, 17, 18	divsubdird 11647	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → (((ψ‘𝑥) − 𝑥) / 𝑥) = (((ψ‘𝑥) / 𝑥) − (𝑥 / 𝑥)))
20	17, 18	dividd 11606	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 / 𝑥) = 1)
21	20	oveq2d 7229	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → (((ψ‘𝑥) / 𝑥) − (𝑥 / 𝑥)) = (((ψ‘𝑥) / 𝑥) − 1))
22	15, 19, 21	3eqtrd 2781	. . . . 5 ⊢ (𝑥 ∈ ℝ⁺ → ((𝑅‘𝑥) / 𝑥) = (((ψ‘𝑥) / 𝑥) − 1))
23	22	mpteq2ia 5146	. . . 4 ⊢ (𝑥 ∈ ℝ⁺ ↦ ((𝑅‘𝑥) / 𝑥)) = (𝑥 ∈ ℝ⁺ ↦ (((ψ‘𝑥) / 𝑥) − 1))
24		rerpdivcl 12616	. . . . . . 7 ⊢ (((ψ‘𝑥) ∈ ℝ ∧ 𝑥 ∈ ℝ⁺) → ((ψ‘𝑥) / 𝑥) ∈ ℝ)
25	8, 24	mpancom 688	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → ((ψ‘𝑥) / 𝑥) ∈ ℝ)
26	25	adantl 485	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((ψ‘𝑥) / 𝑥) ∈ ℝ)
27		1red 10834	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → 1 ∈ ℝ)
28		chpo1ub 26361	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)) ∈ 𝑂(1)
29	28	a1i 11	. . . . 5 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)) ∈ 𝑂(1))
30		ax-1cn 10787	. . . . . . 7 ⊢ 1 ∈ ℂ
31		o1const 15181	. . . . . . 7 ⊢ ((ℝ⁺ ⊆ ℝ ∧ 1 ∈ ℂ) → (𝑥 ∈ ℝ⁺ ↦ 1) ∈ 𝑂(1))
32	1, 30, 31	mp2an 692	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ ↦ 1) ∈ 𝑂(1)
33	32	a1i 11	. . . . 5 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ 1) ∈ 𝑂(1))
34	26, 27, 29, 33	o1sub2 15187	. . . 4 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (((ψ‘𝑥) / 𝑥) − 1)) ∈ 𝑂(1))
35	23, 34	eqeltrid 2842	. . 3 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((𝑅‘𝑥) / 𝑥)) ∈ 𝑂(1))
36		chpcl 26006	. . . . 5 ⊢ (𝑦 ∈ ℝ → (ψ‘𝑦) ∈ ℝ)
37		peano2re 11005	. . . . 5 ⊢ ((ψ‘𝑦) ∈ ℝ → ((ψ‘𝑦) + 1) ∈ ℝ)
38	36, 37	syl 17	. . . 4 ⊢ (𝑦 ∈ ℝ → ((ψ‘𝑦) + 1) ∈ ℝ)
39	38	ad2antrl 728	. . 3 ⊢ ((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) → ((ψ‘𝑦) + 1) ∈ ℝ)
40	22	3ad2ant1 1135	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → ((𝑅‘𝑥) / 𝑥) = (((ψ‘𝑥) / 𝑥) − 1))
41	40	fveq2d 6721	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → (abs‘((𝑅‘𝑥) / 𝑥)) = (abs‘(((ψ‘𝑥) / 𝑥) − 1)))
42		1re 10833	. . . . . . . . . 10 ⊢ 1 ∈ ℝ
43	38	3ad2ant2 1136	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → ((ψ‘𝑦) + 1) ∈ ℝ)
44		resubcl 11142	. . . . . . . . . 10 ⊢ ((1 ∈ ℝ ∧ ((ψ‘𝑦) + 1) ∈ ℝ) → (1 − ((ψ‘𝑦) + 1)) ∈ ℝ)
45	42, 43, 44	sylancr 590	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → (1 − ((ψ‘𝑦) + 1)) ∈ ℝ)
46		0red 10836	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → 0 ∈ ℝ)
47	25	3ad2ant1 1135	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → ((ψ‘𝑥) / 𝑥) ∈ ℝ)
48		chpge0 26008	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℝ → 0 ≤ (ψ‘𝑦))
49	48	3ad2ant2 1136	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → 0 ≤ (ψ‘𝑦))
50	36	3ad2ant2 1136	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → (ψ‘𝑦) ∈ ℝ)
51		addge02 11343	. . . . . . . . . . . 12 ⊢ ((1 ∈ ℝ ∧ (ψ‘𝑦) ∈ ℝ) → (0 ≤ (ψ‘𝑦) ↔ 1 ≤ ((ψ‘𝑦) + 1)))
52	42, 50, 51	sylancr 590	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → (0 ≤ (ψ‘𝑦) ↔ 1 ≤ ((ψ‘𝑦) + 1)))
53	49, 52	mpbid 235	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → 1 ≤ ((ψ‘𝑦) + 1))
54		suble0 11346	. . . . . . . . . . 11 ⊢ ((1 ∈ ℝ ∧ ((ψ‘𝑦) + 1) ∈ ℝ) → ((1 − ((ψ‘𝑦) + 1)) ≤ 0 ↔ 1 ≤ ((ψ‘𝑦) + 1)))
55	42, 43, 54	sylancr 590	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → ((1 − ((ψ‘𝑦) + 1)) ≤ 0 ↔ 1 ≤ ((ψ‘𝑦) + 1)))
56	53, 55	mpbird 260	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → (1 − ((ψ‘𝑦) + 1)) ≤ 0)
57	8	3ad2ant1 1135	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → (ψ‘𝑥) ∈ ℝ)
58	6	3ad2ant1 1135	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → 𝑥 ∈ ℝ)
59		chpge0 26008	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ → 0 ≤ (ψ‘𝑥))
60	58, 59	syl 17	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → 0 ≤ (ψ‘𝑥))
61		rpregt0 12600	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℝ ∧ 0 < 𝑥))
62	61	3ad2ant1 1135	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → (𝑥 ∈ ℝ ∧ 0 < 𝑥))
63		divge0 11701	. . . . . . . . . 10 ⊢ ((((ψ‘𝑥) ∈ ℝ ∧ 0 ≤ (ψ‘𝑥)) ∧ (𝑥 ∈ ℝ ∧ 0 < 𝑥)) → 0 ≤ ((ψ‘𝑥) / 𝑥))
64	57, 60, 62, 63	syl21anc 838	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → 0 ≤ ((ψ‘𝑥) / 𝑥))
65	45, 46, 47, 56, 64	letrd 10989	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → (1 − ((ψ‘𝑦) + 1)) ≤ ((ψ‘𝑥) / 𝑥))
66		2re 11904	. . . . . . . . . . 11 ⊢ 2 ∈ ℝ
67		readdcl 10812	. . . . . . . . . . 11 ⊢ (((ψ‘𝑦) ∈ ℝ ∧ 2 ∈ ℝ) → ((ψ‘𝑦) + 2) ∈ ℝ)
68	50, 66, 67	sylancl 589	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → ((ψ‘𝑦) + 2) ∈ ℝ)
69		1red 10834	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → 1 ∈ ℝ)
70	58	adantr 484	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) ∧ 𝑥 ≤ 1) → 𝑥 ∈ ℝ)
71		1red 10834	. . . . . . . . . . . . . . 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 488	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) ∧ 𝑥 ≤ 1) → 𝑥 ≤ 1)
74		1lt2 12001	. . . . . . . . . . . . . . . 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 10990	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) ∧ 𝑥 ≤ 1) → 𝑥 < 2)
77		chpeq0 26089	. . . . . . . . . . . . . . 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 260	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) ∧ 𝑥 ≤ 1) → (ψ‘𝑥) = 0)
80	79	oveq1d 7228	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) ∧ 𝑥 ≤ 1) → ((ψ‘𝑥) / 𝑥) = (0 / 𝑥))
81		simp1 1138	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → 𝑥 ∈ ℝ⁺)
82	81	rpcnne0d 12637	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
83		div0 11520	. . . . . . . . . . . . . . 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 5075	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → (0 / 𝑥) ≤ (ψ‘𝑦))
86	85	adantr 484	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) ∧ 𝑥 ≤ 1) → (0 / 𝑥) ≤ (ψ‘𝑦))
87	80, 86	eqbrtrd 5075	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) ∧ 𝑥 ≤ 1) → ((ψ‘𝑥) / 𝑥) ≤ (ψ‘𝑦))
88	47	adantr 484	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) ∧ 1 ≤ 𝑥) → ((ψ‘𝑥) / 𝑥) ∈ ℝ)
89	57	adantr 484	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) ∧ 1 ≤ 𝑥) → (ψ‘𝑥) ∈ ℝ)
90	50	adantr 484	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) ∧ 1 ≤ 𝑥) → (ψ‘𝑦) ∈ ℝ)
91		0lt1 11354	. . . . . . . . . . . . . . . 16 ⊢ 0 < 1
92	91	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → 0 < 1)
93		lediv2a 11726	. . . . . . . . . . . . . . . 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 416	. . . . . . . . . . . . . . 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 1382	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → (1 ≤ 𝑥 → ((ψ‘𝑥) / 𝑥) ≤ ((ψ‘𝑥) / 1)))
96	95	imp 410	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) ∧ 1 ≤ 𝑥) → ((ψ‘𝑥) / 𝑥) ≤ ((ψ‘𝑥) / 1))
97	89	recnd 10861	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) ∧ 1 ≤ 𝑥) → (ψ‘𝑥) ∈ ℂ)
98	97	div1d 11600	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) ∧ 1 ≤ 𝑥) → ((ψ‘𝑥) / 1) = (ψ‘𝑥))
99	96, 98	breqtrd 5079	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) ∧ 1 ≤ 𝑥) → ((ψ‘𝑥) / 𝑥) ≤ (ψ‘𝑥))
100		simp2 1139	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → 𝑦 ∈ ℝ)
101		ltle 10921	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 < 𝑦 → 𝑥 ≤ 𝑦))
102	6, 101	sylan 583	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ) → (𝑥 < 𝑦 → 𝑥 ≤ 𝑦))
103	102	3impia 1119	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → 𝑥 ≤ 𝑦)
104		chpwordi 26039	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 ≤ 𝑦) → (ψ‘𝑥) ≤ (ψ‘𝑦))
105	58, 100, 103, 104	syl3anc 1373	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → (ψ‘𝑥) ≤ (ψ‘𝑦))
106	105	adantr 484	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) ∧ 1 ≤ 𝑥) → (ψ‘𝑥) ≤ (ψ‘𝑦))
107	88, 89, 90, 99, 106	letrd 10989	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) ∧ 1 ≤ 𝑥) → ((ψ‘𝑥) / 𝑥) ≤ (ψ‘𝑦))
108	58, 69, 87, 107	lecasei 10938	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → ((ψ‘𝑥) / 𝑥) ≤ (ψ‘𝑦))
109		2nn0 12107	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ₀
110		nn0addge1 12136	. . . . . . . . . . 11 ⊢ (((ψ‘𝑦) ∈ ℝ ∧ 2 ∈ ℕ₀) → (ψ‘𝑦) ≤ ((ψ‘𝑦) + 2))
111	50, 109, 110	sylancl 589	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → (ψ‘𝑦) ≤ ((ψ‘𝑦) + 2))
112	47, 50, 68, 108, 111	letrd 10989	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → ((ψ‘𝑥) / 𝑥) ≤ ((ψ‘𝑦) + 2))
113		df-2 11893	. . . . . . . . . . 11 ⊢ 2 = (1 + 1)
114	113	oveq2i 7224	. . . . . . . . . 10 ⊢ ((ψ‘𝑦) + 2) = ((ψ‘𝑦) + (1 + 1))
115	50	recnd 10861	. . . . . . . . . . 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 11058	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → ((ψ‘𝑦) + (1 + 1)) = (1 + ((ψ‘𝑦) + 1)))
118	114, 117	syl5eq 2790	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → ((ψ‘𝑦) + 2) = (1 + ((ψ‘𝑦) + 1)))
119	112, 118	breqtrd 5079	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → ((ψ‘𝑥) / 𝑥) ≤ (1 + ((ψ‘𝑦) + 1)))
120	47, 69, 43	absdifled 14998	. . . . . . . 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 713	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → (abs‘(((ψ‘𝑥) / 𝑥) − 1)) ≤ ((ψ‘𝑦) + 1))
122	41, 121	eqbrtrd 5075	. . . . . 6 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → (abs‘((𝑅‘𝑥) / 𝑥)) ≤ ((ψ‘𝑦) + 1))
123	122	3expb 1122	. . . . 5 ⊢ ((𝑥 ∈ ℝ⁺ ∧ (𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)) → (abs‘((𝑅‘𝑥) / 𝑥)) ≤ ((ψ‘𝑦) + 1))
124	123	adantrlr 723	. . . 4 ⊢ ((𝑥 ∈ ℝ⁺ ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (abs‘((𝑅‘𝑥) / 𝑥)) ≤ ((ψ‘𝑦) + 1))
125	124	adantll 714	. . 3 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (abs‘((𝑅‘𝑥) / 𝑥)) ≤ ((ψ‘𝑦) + 1))
126	2, 3, 14, 35, 39, 125	o1bddrp 15103	. 2 ⊢ (⊤ → ∃𝑐 ∈ ℝ⁺ ∀𝑥 ∈ ℝ⁺ (abs‘((𝑅‘𝑥) / 𝑥)) ≤ 𝑐)
127	126	mptru 1550	1 ⊢ ∃𝑐 ∈ ℝ⁺ ∀𝑥 ∈ ℝ⁺ (abs‘((𝑅‘𝑥) / 𝑥)) ≤ 𝑐

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ⊤wtru 1544 ∈ wcel 2110 ≠ wne 2940 ∀wral 3061 ∃wrex 3062 ⊆ wss 3866 class class class wbr 5053 ↦ cmpt 5135 ‘cfv 6380 (class class class)co 7213 ℂcc 10727 ℝcr 10728 0cc0 10729 1c1 10730 + caddc 10732 < clt 10867 ≤ cle 10868 − cmin 11062 / cdiv 11489 2c2 11885 ℕ₀cn0 12090 ℝ⁺crp 12586 abscabs 14797 𝑂(1)co1 15047 ψcchp 25975

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-inf2 9256 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-pre-sup 10807 ax-addf 10808 ax-mulf 10809

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-iin 4907 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-se 5510 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-isom 6389 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-of 7469 df-om 7645 df-1st 7761 df-2nd 7762 df-supp 7904 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-2o 8203 df-oadd 8206 df-er 8391 df-map 8510 df-pm 8511 df-ixp 8579 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-fsupp 8986 df-fi 9027 df-sup 9058 df-inf 9059 df-oi 9126 df-dju 9517 df-card 9555 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-nn 11831 df-2 11893 df-3 11894 df-4 11895 df-5 11896 df-6 11897 df-7 11898 df-8 11899 df-9 11900 df-n0 12091 df-xnn0 12163 df-z 12177 df-dec 12294 df-uz 12439 df-q 12545 df-rp 12587 df-xneg 12704 df-xadd 12705 df-xmul 12706 df-ioo 12939 df-ioc 12940 df-ico 12941 df-icc 12942 df-fz 13096 df-fzo 13239 df-fl 13367 df-mod 13443 df-seq 13575 df-exp 13636 df-fac 13840 df-bc 13869 df-hash 13897 df-shft 14630 df-cj 14662 df-re 14663 df-im 14664 df-sqrt 14798 df-abs 14799 df-limsup 15032 df-clim 15049 df-rlim 15050 df-o1 15051 df-lo1 15052 df-sum 15250 df-ef 15629 df-e 15630 df-sin 15631 df-cos 15632 df-pi 15634 df-dvds 15816 df-gcd 16054 df-prm 16229 df-pc 16390 df-struct 16700 df-sets 16717 df-slot 16735 df-ndx 16745 df-base 16761 df-ress 16785 df-plusg 16815 df-mulr 16816 df-starv 16817 df-sca 16818 df-vsca 16819 df-ip 16820 df-tset 16821 df-ple 16822 df-ds 16824 df-unif 16825 df-hom 16826 df-cco 16827 df-rest 16927 df-topn 16928 df-0g 16946 df-gsum 16947 df-topgen 16948 df-pt 16949 df-prds 16952 df-xrs 17007 df-qtop 17012 df-imas 17013 df-xps 17015 df-mre 17089 df-mrc 17090 df-acs 17092 df-mgm 18114 df-sgrp 18163 df-mnd 18174 df-submnd 18219 df-mulg 18489 df-cntz 18711 df-cmn 19172 df-psmet 20355 df-xmet 20356 df-met 20357 df-bl 20358 df-mopn 20359 df-fbas 20360 df-fg 20361 df-cnfld 20364 df-top 21791 df-topon 21808 df-topsp 21830 df-bases 21843 df-cld 21916 df-ntr 21917 df-cls 21918 df-nei 21995 df-lp 22033 df-perf 22034 df-cn 22124 df-cnp 22125 df-haus 22212 df-tx 22459 df-hmeo 22652 df-fil 22743 df-fm 22835 df-flim 22836 df-flf 22837 df-xms 23218 df-ms 23219 df-tms 23220 df-cncf 23775 df-limc 24763 df-dv 24764 df-log 25445 df-cxp 25446 df-cht 25979 df-vma 25980 df-chp 25981 df-ppi 25982

This theorem is referenced by: pntrlog2bnd 26465 pntibnd 26474 pnt3 26493

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