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Theorem pntrmax 27517

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 13021	. . . 4 ⊢ ℝ⁺ ⊆ ℝ
2	1	a1i 11	. . 3 ⊢ (⊤ → ℝ⁺ ⊆ ℝ)
3		1red 11253	. . 3 ⊢ (⊤ → 1 ∈ ℝ)
4		pntrval.r	. . . . . . . 8 ⊢ 𝑅 = (𝑎 ∈ ℝ⁺ ↦ ((ψ‘𝑎) − 𝑎))
5	4	pntrval 27515	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (𝑅‘𝑥) = ((ψ‘𝑥) − 𝑥))
6		rpre 13022	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
7		chpcl 27076	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → (ψ‘𝑥) ∈ ℝ)
8	6, 7	syl 17	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → (ψ‘𝑥) ∈ ℝ)
9	8, 6	resubcld 11680	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → ((ψ‘𝑥) − 𝑥) ∈ ℝ)
10	5, 9	eqeltrd 2829	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → (𝑅‘𝑥) ∈ ℝ)
11		rerpdivcl 13044	. . . . . 6 ⊢ (((𝑅‘𝑥) ∈ ℝ ∧ 𝑥 ∈ ℝ⁺) → ((𝑅‘𝑥) / 𝑥) ∈ ℝ)
12	10, 11	mpancom 686	. . . . 5 ⊢ (𝑥 ∈ ℝ⁺ → ((𝑅‘𝑥) / 𝑥) ∈ ℝ)
13	12	recnd 11280	. . . 4 ⊢ (𝑥 ∈ ℝ⁺ → ((𝑅‘𝑥) / 𝑥) ∈ ℂ)
14	13	adantl 480	. . 3 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((𝑅‘𝑥) / 𝑥) ∈ ℂ)
15	5	oveq1d 7441	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → ((𝑅‘𝑥) / 𝑥) = (((ψ‘𝑥) − 𝑥) / 𝑥))
16	8	recnd 11280	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (ψ‘𝑥) ∈ ℂ)
17		rpcn 13024	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℂ)
18		rpne0 13030	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ≠ 0)
19	16, 17, 17, 18	divsubdird 12067	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → (((ψ‘𝑥) − 𝑥) / 𝑥) = (((ψ‘𝑥) / 𝑥) − (𝑥 / 𝑥)))
20	17, 18	dividd 12026	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 / 𝑥) = 1)
21	20	oveq2d 7442	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → (((ψ‘𝑥) / 𝑥) − (𝑥 / 𝑥)) = (((ψ‘𝑥) / 𝑥) − 1))
22	15, 19, 21	3eqtrd 2772	. . . . 5 ⊢ (𝑥 ∈ ℝ⁺ → ((𝑅‘𝑥) / 𝑥) = (((ψ‘𝑥) / 𝑥) − 1))
23	22	mpteq2ia 5255	. . . 4 ⊢ (𝑥 ∈ ℝ⁺ ↦ ((𝑅‘𝑥) / 𝑥)) = (𝑥 ∈ ℝ⁺ ↦ (((ψ‘𝑥) / 𝑥) − 1))
24		rerpdivcl 13044	. . . . . . 7 ⊢ (((ψ‘𝑥) ∈ ℝ ∧ 𝑥 ∈ ℝ⁺) → ((ψ‘𝑥) / 𝑥) ∈ ℝ)
25	8, 24	mpancom 686	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → ((ψ‘𝑥) / 𝑥) ∈ ℝ)
26	25	adantl 480	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((ψ‘𝑥) / 𝑥) ∈ ℝ)
27		1red 11253	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → 1 ∈ ℝ)
28		chpo1ub 27433	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)) ∈ 𝑂(1)
29	28	a1i 11	. . . . 5 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)) ∈ 𝑂(1))
30		ax-1cn 11204	. . . . . . 7 ⊢ 1 ∈ ℂ
31		o1const 15604	. . . . . . 7 ⊢ ((ℝ⁺ ⊆ ℝ ∧ 1 ∈ ℂ) → (𝑥 ∈ ℝ⁺ ↦ 1) ∈ 𝑂(1))
32	1, 30, 31	mp2an 690	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ ↦ 1) ∈ 𝑂(1)
33	32	a1i 11	. . . . 5 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ 1) ∈ 𝑂(1))
34	26, 27, 29, 33	o1sub2 15610	. . . 4 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (((ψ‘𝑥) / 𝑥) − 1)) ∈ 𝑂(1))
35	23, 34	eqeltrid 2833	. . 3 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((𝑅‘𝑥) / 𝑥)) ∈ 𝑂(1))
36		chpcl 27076	. . . . 5 ⊢ (𝑦 ∈ ℝ → (ψ‘𝑦) ∈ ℝ)
37		peano2re 11425	. . . . 5 ⊢ ((ψ‘𝑦) ∈ ℝ → ((ψ‘𝑦) + 1) ∈ ℝ)
38	36, 37	syl 17	. . . 4 ⊢ (𝑦 ∈ ℝ → ((ψ‘𝑦) + 1) ∈ ℝ)
39	38	ad2antrl 726	. . 3 ⊢ ((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) → ((ψ‘𝑦) + 1) ∈ ℝ)
40	22	3ad2ant1 1130	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → ((𝑅‘𝑥) / 𝑥) = (((ψ‘𝑥) / 𝑥) − 1))
41	40	fveq2d 6906	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → (abs‘((𝑅‘𝑥) / 𝑥)) = (abs‘(((ψ‘𝑥) / 𝑥) − 1)))
42		1re 11252	. . . . . . . . . 10 ⊢ 1 ∈ ℝ
43	38	3ad2ant2 1131	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → ((ψ‘𝑦) + 1) ∈ ℝ)
44		resubcl 11562	. . . . . . . . . 10 ⊢ ((1 ∈ ℝ ∧ ((ψ‘𝑦) + 1) ∈ ℝ) → (1 − ((ψ‘𝑦) + 1)) ∈ ℝ)
45	42, 43, 44	sylancr 585	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → (1 − ((ψ‘𝑦) + 1)) ∈ ℝ)
46		0red 11255	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → 0 ∈ ℝ)
47	25	3ad2ant1 1130	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → ((ψ‘𝑥) / 𝑥) ∈ ℝ)
48		chpge0 27078	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℝ → 0 ≤ (ψ‘𝑦))
49	48	3ad2ant2 1131	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → 0 ≤ (ψ‘𝑦))
50	36	3ad2ant2 1131	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → (ψ‘𝑦) ∈ ℝ)
51		addge02 11763	. . . . . . . . . . . 12 ⊢ ((1 ∈ ℝ ∧ (ψ‘𝑦) ∈ ℝ) → (0 ≤ (ψ‘𝑦) ↔ 1 ≤ ((ψ‘𝑦) + 1)))
52	42, 50, 51	sylancr 585	. . . . . . . . . . 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 11766	. . . . . . . . . . 11 ⊢ ((1 ∈ ℝ ∧ ((ψ‘𝑦) + 1) ∈ ℝ) → ((1 − ((ψ‘𝑦) + 1)) ≤ 0 ↔ 1 ≤ ((ψ‘𝑦) + 1)))
55	42, 43, 54	sylancr 585	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → ((1 − ((ψ‘𝑦) + 1)) ≤ 0 ↔ 1 ≤ ((ψ‘𝑦) + 1)))
56	53, 55	mpbird 256	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → (1 − ((ψ‘𝑦) + 1)) ≤ 0)
57	8	3ad2ant1 1130	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → (ψ‘𝑥) ∈ ℝ)
58	6	3ad2ant1 1130	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → 𝑥 ∈ ℝ)
59		chpge0 27078	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ → 0 ≤ (ψ‘𝑥))
60	58, 59	syl 17	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → 0 ≤ (ψ‘𝑥))
61		rpregt0 13028	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℝ ∧ 0 < 𝑥))
62	61	3ad2ant1 1130	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → (𝑥 ∈ ℝ ∧ 0 < 𝑥))
63		divge0 12121	. . . . . . . . . 10 ⊢ ((((ψ‘𝑥) ∈ ℝ ∧ 0 ≤ (ψ‘𝑥)) ∧ (𝑥 ∈ ℝ ∧ 0 < 𝑥)) → 0 ≤ ((ψ‘𝑥) / 𝑥))
64	57, 60, 62, 63	syl21anc 836	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → 0 ≤ ((ψ‘𝑥) / 𝑥))
65	45, 46, 47, 56, 64	letrd 11409	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → (1 − ((ψ‘𝑦) + 1)) ≤ ((ψ‘𝑥) / 𝑥))
66		2re 12324	. . . . . . . . . . 11 ⊢ 2 ∈ ℝ
67		readdcl 11229	. . . . . . . . . . 11 ⊢ (((ψ‘𝑦) ∈ ℝ ∧ 2 ∈ ℝ) → ((ψ‘𝑦) + 2) ∈ ℝ)
68	50, 66, 67	sylancl 584	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → ((ψ‘𝑦) + 2) ∈ ℝ)
69		1red 11253	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → 1 ∈ ℝ)
70	58	adantr 479	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) ∧ 𝑥 ≤ 1) → 𝑥 ∈ ℝ)
71		1red 11253	. . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) ∧ 𝑥 ≤ 1) → 𝑥 ≤ 1)
74		1lt2 12421	. . . . . . . . . . . . . . . 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 11410	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) ∧ 𝑥 ≤ 1) → 𝑥 < 2)
77		chpeq0 27161	. . . . . . . . . . . . . . 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 256	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) ∧ 𝑥 ≤ 1) → (ψ‘𝑥) = 0)
80	79	oveq1d 7441	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) ∧ 𝑥 ≤ 1) → ((ψ‘𝑥) / 𝑥) = (0 / 𝑥))
81		simp1 1133	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → 𝑥 ∈ ℝ⁺)
82	81	rpcnne0d 13065	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
83		div0 11940	. . . . . . . . . . . . . . 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 5174	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → (0 / 𝑥) ≤ (ψ‘𝑦))
86	85	adantr 479	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) ∧ 𝑥 ≤ 1) → (0 / 𝑥) ≤ (ψ‘𝑦))
87	80, 86	eqbrtrd 5174	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) ∧ 𝑥 ≤ 1) → ((ψ‘𝑥) / 𝑥) ≤ (ψ‘𝑦))
88	47	adantr 479	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) ∧ 1 ≤ 𝑥) → ((ψ‘𝑥) / 𝑥) ∈ ℝ)
89	57	adantr 479	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) ∧ 1 ≤ 𝑥) → (ψ‘𝑥) ∈ ℝ)
90	50	adantr 479	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) ∧ 1 ≤ 𝑥) → (ψ‘𝑦) ∈ ℝ)
91		0lt1 11774	. . . . . . . . . . . . . . . 16 ⊢ 0 < 1
92	91	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → 0 < 1)
93		lediv2a 12146	. . . . . . . . . . . . . . . 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 411	. . . . . . . . . . . . . . 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 1377	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → (1 ≤ 𝑥 → ((ψ‘𝑥) / 𝑥) ≤ ((ψ‘𝑥) / 1)))
96	95	imp 405	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) ∧ 1 ≤ 𝑥) → ((ψ‘𝑥) / 𝑥) ≤ ((ψ‘𝑥) / 1))
97	89	recnd 11280	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) ∧ 1 ≤ 𝑥) → (ψ‘𝑥) ∈ ℂ)
98	97	div1d 12020	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) ∧ 1 ≤ 𝑥) → ((ψ‘𝑥) / 1) = (ψ‘𝑥))
99	96, 98	breqtrd 5178	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) ∧ 1 ≤ 𝑥) → ((ψ‘𝑥) / 𝑥) ≤ (ψ‘𝑥))
100		simp2 1134	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → 𝑦 ∈ ℝ)
101		ltle 11340	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 < 𝑦 → 𝑥 ≤ 𝑦))
102	6, 101	sylan 578	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ) → (𝑥 < 𝑦 → 𝑥 ≤ 𝑦))
103	102	3impia 1114	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → 𝑥 ≤ 𝑦)
104		chpwordi 27109	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 ≤ 𝑦) → (ψ‘𝑥) ≤ (ψ‘𝑦))
105	58, 100, 103, 104	syl3anc 1368	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → (ψ‘𝑥) ≤ (ψ‘𝑦))
106	105	adantr 479	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) ∧ 1 ≤ 𝑥) → (ψ‘𝑥) ≤ (ψ‘𝑦))
107	88, 89, 90, 99, 106	letrd 11409	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) ∧ 1 ≤ 𝑥) → ((ψ‘𝑥) / 𝑥) ≤ (ψ‘𝑦))
108	58, 69, 87, 107	lecasei 11358	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → ((ψ‘𝑥) / 𝑥) ≤ (ψ‘𝑦))
109		2nn0 12527	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ₀
110		nn0addge1 12556	. . . . . . . . . . 11 ⊢ (((ψ‘𝑦) ∈ ℝ ∧ 2 ∈ ℕ₀) → (ψ‘𝑦) ≤ ((ψ‘𝑦) + 2))
111	50, 109, 110	sylancl 584	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → (ψ‘𝑦) ≤ ((ψ‘𝑦) + 2))
112	47, 50, 68, 108, 111	letrd 11409	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → ((ψ‘𝑥) / 𝑥) ≤ ((ψ‘𝑦) + 2))
113		df-2 12313	. . . . . . . . . . 11 ⊢ 2 = (1 + 1)
114	113	oveq2i 7437	. . . . . . . . . 10 ⊢ ((ψ‘𝑦) + 2) = ((ψ‘𝑦) + (1 + 1))
115	50	recnd 11280	. . . . . . . . . . 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 11478	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → ((ψ‘𝑦) + (1 + 1)) = (1 + ((ψ‘𝑦) + 1)))
118	114, 117	eqtrid 2780	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → ((ψ‘𝑦) + 2) = (1 + ((ψ‘𝑦) + 1)))
119	112, 118	breqtrd 5178	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → ((ψ‘𝑥) / 𝑥) ≤ (1 + ((ψ‘𝑦) + 1)))
120	47, 69, 43	absdifled 15421	. . . . . . . 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 711	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → (abs‘(((ψ‘𝑥) / 𝑥) − 1)) ≤ ((ψ‘𝑦) + 1))
122	41, 121	eqbrtrd 5174	. . . . . 6 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → (abs‘((𝑅‘𝑥) / 𝑥)) ≤ ((ψ‘𝑦) + 1))
123	122	3expb 1117	. . . . 5 ⊢ ((𝑥 ∈ ℝ⁺ ∧ (𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)) → (abs‘((𝑅‘𝑥) / 𝑥)) ≤ ((ψ‘𝑦) + 1))
124	123	adantrlr 721	. . . 4 ⊢ ((𝑥 ∈ ℝ⁺ ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (abs‘((𝑅‘𝑥) / 𝑥)) ≤ ((ψ‘𝑦) + 1))
125	124	adantll 712	. . 3 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (abs‘((𝑅‘𝑥) / 𝑥)) ≤ ((ψ‘𝑦) + 1))
126	2, 3, 14, 35, 39, 125	o1bddrp 15526	. 2 ⊢ (⊤ → ∃𝑐 ∈ ℝ⁺ ∀𝑥 ∈ ℝ⁺ (abs‘((𝑅‘𝑥) / 𝑥)) ≤ 𝑐)
127	126	mptru 1540	1 ⊢ ∃𝑐 ∈ ℝ⁺ ∀𝑥 ∈ ℝ⁺ (abs‘((𝑅‘𝑥) / 𝑥)) ≤ 𝑐

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ⊤wtru 1534 ∈ wcel 2098 ≠ wne 2937 ∀wral 3058 ∃wrex 3067 ⊆ wss 3949 class class class wbr 5152 ↦ cmpt 5235 ‘cfv 6553 (class class class)co 7426 ℂcc 11144 ℝcr 11145 0cc0 11146 1c1 11147 + caddc 11149 < clt 11286 ≤ cle 11287 − cmin 11482 / cdiv 11909 2c2 12305 ℕ₀cn0 12510 ℝ⁺crp 13014 abscabs 15221 𝑂(1)co1 15470 ψcchp 27045

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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9672 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 ax-addf 11225

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7691 df-om 7877 df-1st 7999 df-2nd 8000 df-supp 8172 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-2o 8494 df-oadd 8497 df-er 8731 df-map 8853 df-pm 8854 df-ixp 8923 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-fsupp 9394 df-fi 9442 df-sup 9473 df-inf 9474 df-oi 9541 df-dju 9932 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-xnn0 12583 df-z 12597 df-dec 12716 df-uz 12861 df-q 12971 df-rp 13015 df-xneg 13132 df-xadd 13133 df-xmul 13134 df-ioo 13368 df-ioc 13369 df-ico 13370 df-icc 13371 df-fz 13525 df-fzo 13668 df-fl 13797 df-mod 13875 df-seq 14007 df-exp 14067 df-fac 14273 df-bc 14302 df-hash 14330 df-shft 15054 df-cj 15086 df-re 15087 df-im 15088 df-sqrt 15222 df-abs 15223 df-limsup 15455 df-clim 15472 df-rlim 15473 df-o1 15474 df-lo1 15475 df-sum 15673 df-ef 16051 df-e 16052 df-sin 16053 df-cos 16054 df-pi 16056 df-dvds 16239 df-gcd 16477 df-prm 16650 df-pc 16813 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-mulr 17254 df-starv 17255 df-sca 17256 df-vsca 17257 df-ip 17258 df-tset 17259 df-ple 17260 df-ds 17262 df-unif 17263 df-hom 17264 df-cco 17265 df-rest 17411 df-topn 17412 df-0g 17430 df-gsum 17431 df-topgen 17432 df-pt 17433 df-prds 17436 df-xrs 17491 df-qtop 17496 df-imas 17497 df-xps 17499 df-mre 17573 df-mrc 17574 df-acs 17576 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-submnd 18748 df-mulg 19031 df-cntz 19275 df-cmn 19744 df-psmet 21278 df-xmet 21279 df-met 21280 df-bl 21281 df-mopn 21282 df-fbas 21283 df-fg 21284 df-cnfld 21287 df-top 22816 df-topon 22833 df-topsp 22855 df-bases 22869 df-cld 22943 df-ntr 22944 df-cls 22945 df-nei 23022 df-lp 23060 df-perf 23061 df-cn 23151 df-cnp 23152 df-haus 23239 df-tx 23486 df-hmeo 23679 df-fil 23770 df-fm 23862 df-flim 23863 df-flf 23864 df-xms 24246 df-ms 24247 df-tms 24248 df-cncf 24818 df-limc 25815 df-dv 25816 df-log 26510 df-cxp 26511 df-cht 27049 df-vma 27050 df-chp 27051 df-ppi 27052

This theorem is referenced by: pntrlog2bnd 27537 pntibnd 27546 pnt3 27565

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