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Theorem pntrmax 27897

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 13132	. . . 4 ⊢ ℝ⁺ ⊆ ℝ
2	1	a1i 11	. . 3 ⊢ (⊤ → ℝ⁺ ⊆ ℝ)
3		1red 11313	. . 3 ⊢ (⊤ → 1 ∈ ℝ)
4		pntrval.r	. . . . . . . 8 ⊢ 𝑅 = (𝑎 ∈ ℝ⁺ ↦ ((ψ‘𝑎) − 𝑎))
5	4	pntrval 27895	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (𝑅‘𝑥) = ((ψ‘𝑥) − 𝑥))
6		rpre 13133	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
7		chpcl 27457	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → (ψ‘𝑥) ∈ ℝ)
8	6, 7	syl 18	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → (ψ‘𝑥) ∈ ℝ)
9	8, 6	resubcld 11748	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → ((ψ‘𝑥) − 𝑥) ∈ ℝ)
10	5, 9	eqeltrd 2860	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → (𝑅‘𝑥) ∈ ℝ)
11		rerpdivcl 13156	. . . . . 6 ⊢ (((𝑅‘𝑥) ∈ ℝ ∧ 𝑥 ∈ ℝ⁺) → ((𝑅‘𝑥) / 𝑥) ∈ ℝ)
12	10, 11	mpancom 701	. . . . 5 ⊢ (𝑥 ∈ ℝ⁺ → ((𝑅‘𝑥) / 𝑥) ∈ ℝ)
13	12	recnd 11341	. . . 4 ⊢ (𝑥 ∈ ℝ⁺ → ((𝑅‘𝑥) / 𝑥) ∈ ℂ)
14	13	adantl 487	. . 3 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((𝑅‘𝑥) / 𝑥) ∈ ℂ)
15	5	oveq1d 7436	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → ((𝑅‘𝑥) / 𝑥) = (((ψ‘𝑥) − 𝑥) / 𝑥))
16	8	recnd 11341	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (ψ‘𝑥) ∈ ℂ)
17		rpcn 13135	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℂ)
18		rpne0 13141	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ≠ 0)
19	16, 17, 17, 18	divsubdird 12136	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → (((ψ‘𝑥) − 𝑥) / 𝑥) = (((ψ‘𝑥) / 𝑥) − (𝑥 / 𝑥)))
20	17, 18	dividd 12095	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 / 𝑥) = 1)
21	20	oveq2d 7437	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → (((ψ‘𝑥) / 𝑥) − (𝑥 / 𝑥)) = (((ψ‘𝑥) / 𝑥) − 1))
22	15, 19, 21	3eqtrd 2799	. . . . 5 ⊢ (𝑥 ∈ ℝ⁺ → ((𝑅‘𝑥) / 𝑥) = (((ψ‘𝑥) / 𝑥) − 1))
23	22	mpteq2ia 5199	. . . 4 ⊢ (𝑥 ∈ ℝ⁺ ↦ ((𝑅‘𝑥) / 𝑥)) = (𝑥 ∈ ℝ⁺ ↦ (((ψ‘𝑥) / 𝑥) − 1))
24		rerpdivcl 13156	. . . . . . 7 ⊢ (((ψ‘𝑥) ∈ ℝ ∧ 𝑥 ∈ ℝ⁺) → ((ψ‘𝑥) / 𝑥) ∈ ℝ)
25	8, 24	mpancom 701	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → ((ψ‘𝑥) / 𝑥) ∈ ℝ)
26	25	adantl 487	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((ψ‘𝑥) / 𝑥) ∈ ℝ)
27		1red 11313	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → 1 ∈ ℝ)
28		chpo1ub 27813	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)) ∈ 𝑂(1)
29	28	a1i 11	. . . . 5 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)) ∈ 𝑂(1))
30		ax-1cn 11262	. . . . . . 7 ⊢ 1 ∈ ℂ
31		o1const 15793	. . . . . . 7 ⊢ ((ℝ⁺ ⊆ ℝ ∧ 1 ∈ ℂ) → (𝑥 ∈ ℝ⁺ ↦ 1) ∈ 𝑂(1))
32	1, 30, 31	mp2an 705	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ ↦ 1) ∈ 𝑂(1)
33	32	a1i 11	. . . . 5 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ 1) ∈ 𝑂(1))
34	26, 27, 29, 33	o1sub2 15799	. . . 4 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (((ψ‘𝑥) / 𝑥) − 1)) ∈ 𝑂(1))
35	23, 34	eqeltrid 2864	. . 3 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((𝑅‘𝑥) / 𝑥)) ∈ 𝑂(1))
36		chpcl 27457	. . . . 5 ⊢ (𝑦 ∈ ℝ → (ψ‘𝑦) ∈ ℝ)
37		peano2re 11487	. . . . 5 ⊢ ((ψ‘𝑦) ∈ ℝ → ((ψ‘𝑦) + 1) ∈ ℝ)
38	36, 37	syl 18	. . . 4 ⊢ (𝑦 ∈ ℝ → ((ψ‘𝑦) + 1) ∈ ℝ)
39	38	ad2antrl 741	. . 3 ⊢ ((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) → ((ψ‘𝑦) + 1) ∈ ℝ)
40	22	3ad2ant1 1151	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → ((𝑅‘𝑥) / 𝑥) = (((ψ‘𝑥) / 𝑥) − 1))
41	40	fveq2d 6890	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → (abs‘((𝑅‘𝑥) / 𝑥)) = (abs‘(((ψ‘𝑥) / 𝑥) − 1)))
42		1re 11312	. . . . . . . . . 10 ⊢ 1 ∈ ℝ
43	38	3ad2ant2 1152	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → ((ψ‘𝑦) + 1) ∈ ℝ)
44		resubcl 11626	. . . . . . . . . 10 ⊢ ((1 ∈ ℝ ∧ ((ψ‘𝑦) + 1) ∈ ℝ) → (1 − ((ψ‘𝑦) + 1)) ∈ ℝ)
45	42, 43, 44	sylancr 599	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → (1 − ((ψ‘𝑦) + 1)) ∈ ℝ)
46		0red 11315	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → 0 ∈ ℝ)
47	25	3ad2ant1 1151	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → ((ψ‘𝑥) / 𝑥) ∈ ℝ)
48		chpge0 27459	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℝ → 0 ≤ (ψ‘𝑦))
49	48	3ad2ant2 1152	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → 0 ≤ (ψ‘𝑦))
50	36	3ad2ant2 1152	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → (ψ‘𝑦) ∈ ℝ)
51		addge02 11831	. . . . . . . . . . . 12 ⊢ ((1 ∈ ℝ ∧ (ψ‘𝑦) ∈ ℝ) → (0 ≤ (ψ‘𝑦) ↔ 1 ≤ ((ψ‘𝑦) + 1)))
52	42, 50, 51	sylancr 599	. . . . . . . . . . 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 11834	. . . . . . . . . . 11 ⊢ ((1 ∈ ℝ ∧ ((ψ‘𝑦) + 1) ∈ ℝ) → ((1 − ((ψ‘𝑦) + 1)) ≤ 0 ↔ 1 ≤ ((ψ‘𝑦) + 1)))
55	42, 43, 54	sylancr 599	. . . . . . . . . 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 1151	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → (ψ‘𝑥) ∈ ℝ)
58	6	3ad2ant1 1151	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → 𝑥 ∈ ℝ)
59		chpge0 27459	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ → 0 ≤ (ψ‘𝑥))
60	58, 59	syl 18	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → 0 ≤ (ψ‘𝑥))
61		rpregt0 13139	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℝ ∧ 0 < 𝑥))
62	61	3ad2ant1 1151	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → (𝑥 ∈ ℝ ∧ 0 < 𝑥))
63		divge0 12190	. . . . . . . . . 10 ⊢ ((((ψ‘𝑥) ∈ ℝ ∧ 0 ≤ (ψ‘𝑥)) ∧ (𝑥 ∈ ℝ ∧ 0 < 𝑥)) → 0 ≤ ((ψ‘𝑥) / 𝑥))
64	57, 60, 62, 63	syl21anc 851	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → 0 ≤ ((ψ‘𝑥) / 𝑥))
65	45, 46, 47, 56, 64	letrd 11471	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → (1 − ((ψ‘𝑦) + 1)) ≤ ((ψ‘𝑥) / 𝑥))
66		2re 12421	. . . . . . . . . . 11 ⊢ 2 ∈ ℝ
67		readdcl 11287	. . . . . . . . . . 11 ⊢ (((ψ‘𝑦) ∈ ℝ ∧ 2 ∈ ℝ) → ((ψ‘𝑦) + 2) ∈ ℝ)
68	50, 66, 67	sylancl 598	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → ((ψ‘𝑦) + 2) ∈ ℝ)
69		1red 11313	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → 1 ∈ ℝ)
70	58	adantr 486	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) ∧ 𝑥 ≤ 1) → 𝑥 ∈ ℝ)
71		1red 11313	. . . . . . . . . . . . . . 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 490	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) ∧ 𝑥 ≤ 1) → 𝑥 ≤ 1)
74		1lt2 12519	. . . . . . . . . . . . . . . 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 11472	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) ∧ 𝑥 ≤ 1) → 𝑥 < 2)
77		chpeq0 27541	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℝ → ((ψ‘𝑥) = 0 ↔ 𝑥 < 2))
78	70, 77	syl 18	. . . . . . . . . . . . . 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 7436	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) ∧ 𝑥 ≤ 1) → ((ψ‘𝑥) / 𝑥) = (0 / 𝑥))
81		simp1 1154	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → 𝑥 ∈ ℝ⁺)
82	81	rpcnne0d 13177	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
83		div0 12009	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) → (0 / 𝑥) = 0)
84	82, 83	syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → (0 / 𝑥) = 0)
85	84, 49	eqbrtrd 5126	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → (0 / 𝑥) ≤ (ψ‘𝑦))
86	85	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) ∧ 𝑥 ≤ 1) → (0 / 𝑥) ≤ (ψ‘𝑦))
87	80, 86	eqbrtrd 5126	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) ∧ 𝑥 ≤ 1) → ((ψ‘𝑥) / 𝑥) ≤ (ψ‘𝑦))
88	47	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) ∧ 1 ≤ 𝑥) → ((ψ‘𝑥) / 𝑥) ∈ ℝ)
89	57	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) ∧ 1 ≤ 𝑥) → (ψ‘𝑥) ∈ ℝ)
90	50	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) ∧ 1 ≤ 𝑥) → (ψ‘𝑦) ∈ ℝ)
91		0lt1 11842	. . . . . . . . . . . . . . . 16 ⊢ 0 < 1
92	91	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → 0 < 1)
93		lediv2a 12215	. . . . . . . . . . . . . . . 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 418	. . . . . . . . . . . . . . 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 1407	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → (1 ≤ 𝑥 → ((ψ‘𝑥) / 𝑥) ≤ ((ψ‘𝑥) / 1)))
96	95	imp 412	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) ∧ 1 ≤ 𝑥) → ((ψ‘𝑥) / 𝑥) ≤ ((ψ‘𝑥) / 1))
97	89	recnd 11341	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) ∧ 1 ≤ 𝑥) → (ψ‘𝑥) ∈ ℂ)
98	97	div1d 12089	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) ∧ 1 ≤ 𝑥) → ((ψ‘𝑥) / 1) = (ψ‘𝑥))
99	96, 98	breqtrd 5130	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) ∧ 1 ≤ 𝑥) → ((ψ‘𝑥) / 𝑥) ≤ (ψ‘𝑥))
100		simp2 1155	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → 𝑦 ∈ ℝ)
101		ltle 11402	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 < 𝑦 → 𝑥 ≤ 𝑦))
102	6, 101	sylan 592	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ) → (𝑥 < 𝑦 → 𝑥 ≤ 𝑦))
103	102	3impia 1135	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → 𝑥 ≤ 𝑦)
104		chpwordi 27490	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 ≤ 𝑦) → (ψ‘𝑥) ≤ (ψ‘𝑦))
105	58, 100, 103, 104	syl3anc 1398	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → (ψ‘𝑥) ≤ (ψ‘𝑦))
106	105	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) ∧ 1 ≤ 𝑥) → (ψ‘𝑥) ≤ (ψ‘𝑦))
107	88, 89, 90, 99, 106	letrd 11471	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) ∧ 1 ≤ 𝑥) → ((ψ‘𝑥) / 𝑥) ≤ (ψ‘𝑦))
108	58, 69, 87, 107	lecasei 11420	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → ((ψ‘𝑥) / 𝑥) ≤ (ψ‘𝑦))
109		2nn0 12627	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ₀
110		nn0addge1 12656	. . . . . . . . . . 11 ⊢ (((ψ‘𝑦) ∈ ℝ ∧ 2 ∈ ℕ₀) → (ψ‘𝑦) ≤ ((ψ‘𝑦) + 2))
111	50, 109, 110	sylancl 598	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → (ψ‘𝑦) ≤ ((ψ‘𝑦) + 2))
112	47, 50, 68, 108, 111	letrd 11471	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → ((ψ‘𝑥) / 𝑥) ≤ ((ψ‘𝑦) + 2))
113		df-2 12409	. . . . . . . . . . 11 ⊢ 2 = (1 + 1)
114	113	oveq2i 7432	. . . . . . . . . 10 ⊢ ((ψ‘𝑦) + 2) = ((ψ‘𝑦) + (1 + 1))
115	50	recnd 11341	. . . . . . . . . . 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 11541	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → ((ψ‘𝑦) + (1 + 1)) = (1 + ((ψ‘𝑦) + 1)))
118	114, 117	eqtrid 2807	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → ((ψ‘𝑦) + 2) = (1 + ((ψ‘𝑦) + 1)))
119	112, 118	breqtrd 5130	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → ((ψ‘𝑥) / 𝑥) ≤ (1 + ((ψ‘𝑦) + 1)))
120	47, 69, 43	absdifled 15610	. . . . . . . 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 726	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → (abs‘(((ψ‘𝑥) / 𝑥) − 1)) ≤ ((ψ‘𝑦) + 1))
122	41, 121	eqbrtrd 5126	. . . . . 6 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → (abs‘((𝑅‘𝑥) / 𝑥)) ≤ ((ψ‘𝑦) + 1))
123	122	3expb 1138	. . . . 5 ⊢ ((𝑥 ∈ ℝ⁺ ∧ (𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)) → (abs‘((𝑅‘𝑥) / 𝑥)) ≤ ((ψ‘𝑦) + 1))
124	123	adantrlr 736	. . . 4 ⊢ ((𝑥 ∈ ℝ⁺ ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (abs‘((𝑅‘𝑥) / 𝑥)) ≤ ((ψ‘𝑦) + 1))
125	124	adantll 727	. . 3 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (abs‘((𝑅‘𝑥) / 𝑥)) ≤ ((ψ‘𝑦) + 1))
126	2, 3, 14, 35, 39, 125	o1bddrp 15715	. 2 ⊢ (⊤ → ∃𝑐 ∈ ℝ⁺ ∀𝑥 ∈ ℝ⁺ (abs‘((𝑅‘𝑥) / 𝑥)) ≤ 𝑐)
127	126	mptru 1576	1 ⊢ ∃𝑐 ∈ ℝ⁺ ∀𝑥 ∈ ℝ⁺ (abs‘((𝑅‘𝑥) / 𝑥)) ≤ 𝑐

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1569 ⊤wtru 1570 ∈ wcel 2144 ≠ wne 2955 ∀wral 3076 ∃wrex 3086 ⊆ wss 3895 class class class wbr 5102 ↦ cmpt 5185 ‘cfv 6539 (class class class)co 7421 ℂcc 11202 ℝcr 11203 0cc0 11204 1c1 11205 + caddc 11207 < clt 11347 ≤ cle 11348 − cmin 11545 / cdiv 11977 2c2 12401 ℕ₀cn0 12610 ℝ⁺crp 13124 abscabs 15407 𝑂(1)co1 15659 ψcchp 27426

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7752 ax-inf2 9646 ax-cnex 11260 ax-resscn 11261 ax-1cn 11262 ax-icn 11263 ax-addcl 11264 ax-addrcl 11265 ax-mulcl 11266 ax-mulrcl 11267 ax-mulcom 11268 ax-addass 11269 ax-mulass 11270 ax-distr 11271 ax-i2m1 11272 ax-1ne0 11273 ax-1rid 11274 ax-rnegex 11275 ax-rrecex 11276 ax-cnre 11277 ax-pre-lttri 11278 ax-pre-lttrn 11279 ax-pre-ltadd 11280 ax-pre-mulgt0 11281 ax-pre-sup 11282 ax-addf 11283

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4835 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-se 5604 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-isom 6548 df-riota 7378 df-ov 7424 df-oprab 7425 df-mpo 7426 df-of 7694 df-om 7879 df-1st 8002 df-2nd 8003 df-supp 8181 df-frecs 8303 df-wrecs 8334 df-recs 8383 df-rdg 8422 df-1o 8480 df-2o 8481 df-oadd 8484 df-er 8721 df-map 8853 df-pm 8854 df-ixp 8930 df-en 8978 df-dom 8979 df-sdom 8980 df-fin 8981 df-fsupp 9358 df-fi 9407 df-sup 9438 df-inf 9439 df-oi 9508 df-dju 9986 df-card 10024 df-pnf 11349 df-mnf 11350 df-xr 11351 df-ltxr 11352 df-le 11353 df-sub 11547 df-neg 11548 df-div 11978 df-nn 12340 df-2 12409 df-3 12410 df-4 12411 df-5 12412 df-6 12413 df-7 12414 df-8 12415 df-9 12416 df-n0 12611 df-xnn0 12684 df-z 12698 df-dec 12819 df-uz 12970 df-q 13080 df-rp 13125 df-xneg 13245 df-xadd 13246 df-xmul 13247 df-ioo 13484 df-ioc 13485 df-ico 13486 df-icc 13487 df-fz 13644 df-fzo 13793 df-fl 13936 df-mod 14014 df-seq 14149 df-exp 14209 df-fac 14422 df-bc 14451 df-hash 14479 df-shft 15226 df-cj 15272 df-re 15273 df-im 15274 df-sqrt 15408 df-abs 15409 df-limsup 15644 df-clim 15661 df-rlim 15662 df-o1 15663 df-lo1 15664 df-sum 15860 df-ef 16239 df-e 16240 df-sin 16241 df-cos 16242 df-pi 16244 df-dvds 16429 df-gcd 16671 df-prm 16853 df-pc 17021 df-struct 17331 df-sets 17348 df-slot 17366 df-ndx 17378 df-base 17394 df-ress 17415 df-plusg 17447 df-mulr 17448 df-starv 17449 df-sca 17450 df-vsca 17451 df-ip 17452 df-tset 17453 df-ple 17454 df-ds 17456 df-unif 17457 df-hom 17458 df-cco 17459 df-rest 17599 df-topn 17600 df-0g 17618 df-gsum 17619 df-topgen 17620 df-pt 17621 df-prds 17624 df-xrs 17680 df-qtop 17685 df-imas 17686 df-xps 17688 df-mre 17762 df-mrc 17763 df-acs 17765 df-mgm 18822 df-sgrp 18914 df-mnd 18930 df-submnd 18985 df-mulg 19284 df-cntz 19537 df-cmn 20002 df-psmet 21676 df-xmet 21677 df-met 21678 df-bl 21679 df-mopn 21680 df-fbas 21681 df-fg 21682 df-cnfld 21685 df-top 23218 df-topon 23235 df-topsp 23257 df-bases 23270 df-cld 23343 df-ntr 23344 df-cls 23345 df-nei 23422 df-lp 23460 df-perf 23461 df-cn 23551 df-cnp 23552 df-haus 23639 df-tx 23887 df-hmeo 24080 df-fil 24171 df-fm 24263 df-flim 24264 df-flf 24265 df-xms 24645 df-ms 24646 df-tms 24647 df-cncf 25205 df-limc 26192 df-dv 26193 df-log 26890 df-cxp 26891 df-cht 27430 df-vma 27431 df-chp 27432 df-ppi 27433

This theorem is used by: pntrlog2bnd 27917 pntibnd 27926 pnt3 27945

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