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Theorem rlimno1 15626

Description: A function whose inverse converges to zero is unbounded. (Contributed by Mario Carneiro, 30-May-2016.)

**Hypotheses**
Ref	Expression
rlimno1.1	⊢ (𝜑 → sup(𝐴, ℝ^*, < ) = +∞)
rlimno1.2	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (1 / 𝐵)) ⇝_𝑟 0)
rlimno1.3	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ)
rlimno1.4	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ≠ 0)

**Assertion**
Ref	Expression
rlimno1	⊢ (𝜑 → ¬ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝑂(1))

Distinct variable groups: 𝑥,𝐴 𝜑,𝑥 Allowed substitution hint: 𝐵(𝑥)

Proof of Theorem rlimno1 Dummy variables 𝑐 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fal 1548	. . . 4 ⊢ ¬ ⊥
2		rlimno1.3	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ)
3		rlimno1.4	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ≠ 0)
4	2, 3	reccld 12007	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (1 / 𝐵) ∈ ℂ)
5	4	ralrimiva 3142	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (1 / 𝐵) ∈ ℂ)
6	5	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → ∀𝑥 ∈ 𝐴 (1 / 𝐵) ∈ ℂ)
7		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ)
8		1re 11238	. . . . . . . . 9 ⊢ 1 ∈ ℝ
9		ifcl 4569	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℝ ∧ 1 ∈ ℝ) → if(1 ≤ 𝑦, 𝑦, 1) ∈ ℝ)
10	7, 8, 9	sylancl 585	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → if(1 ≤ 𝑦, 𝑦, 1) ∈ ℝ)
11		1rp 13004	. . . . . . . . 9 ⊢ 1 ∈ ℝ⁺
12	11	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 1 ∈ ℝ⁺)
13		max1 13190	. . . . . . . . 9 ⊢ ((1 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 1 ≤ if(1 ≤ 𝑦, 𝑦, 1))
14	8, 7, 13	sylancr 586	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 1 ≤ if(1 ≤ 𝑦, 𝑦, 1))
15	10, 12, 14	rpgecld 13081	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → if(1 ≤ 𝑦, 𝑦, 1) ∈ ℝ⁺)
16	15	rpreccld 13052	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (1 / if(1 ≤ 𝑦, 𝑦, 1)) ∈ ℝ⁺)
17		rlimno1.2	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (1 / 𝐵)) ⇝_𝑟 0)
18	17	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝑥 ∈ 𝐴 ↦ (1 / 𝐵)) ⇝_𝑟 0)
19	6, 16, 18	rlimi 15483	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → ∃𝑐 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑐 ≤ 𝑥 → (abs‘((1 / 𝐵) − 0)) < (1 / if(1 ≤ 𝑦, 𝑦, 1))))
20		dmmptg 6240	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ 𝐴 (1 / 𝐵) ∈ ℂ → dom (𝑥 ∈ 𝐴 ↦ (1 / 𝐵)) = 𝐴)
21	5, 20	syl 17	. . . . . . . . 9 ⊢ (𝜑 → dom (𝑥 ∈ 𝐴 ↦ (1 / 𝐵)) = 𝐴)
22		rlimss 15472	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ↦ (1 / 𝐵)) ⇝_𝑟 0 → dom (𝑥 ∈ 𝐴 ↦ (1 / 𝐵)) ⊆ ℝ)
23	17, 22	syl 17	. . . . . . . . 9 ⊢ (𝜑 → dom (𝑥 ∈ 𝐴 ↦ (1 / 𝐵)) ⊆ ℝ)
24	21, 23	eqsstrrd 4017	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
25	24	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝐴 ⊆ ℝ)
26		rexanre 15319	. . . . . . 7 ⊢ (𝐴 ⊆ ℝ → (∃𝑐 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑐 ≤ 𝑥 → ((abs‘((1 / 𝐵) − 0)) < (1 / if(1 ≤ 𝑦, 𝑦, 1)) ∧ (abs‘𝐵) ≤ 𝑦)) ↔ (∃𝑐 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑐 ≤ 𝑥 → (abs‘((1 / 𝐵) − 0)) < (1 / if(1 ≤ 𝑦, 𝑦, 1))) ∧ ∃𝑐 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑐 ≤ 𝑥 → (abs‘𝐵) ≤ 𝑦))))
27	25, 26	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (∃𝑐 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑐 ≤ 𝑥 → ((abs‘((1 / 𝐵) − 0)) < (1 / if(1 ≤ 𝑦, 𝑦, 1)) ∧ (abs‘𝐵) ≤ 𝑦)) ↔ (∃𝑐 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑐 ≤ 𝑥 → (abs‘((1 / 𝐵) − 0)) < (1 / if(1 ≤ 𝑦, 𝑦, 1))) ∧ ∃𝑐 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑐 ≤ 𝑥 → (abs‘𝐵) ≤ 𝑦))))
28		rlimno1.1	. . . . . . . . 9 ⊢ (𝜑 → sup(𝐴, ℝ^*, < ) = +∞)
29		ressxr 11282	. . . . . . . . . . 11 ⊢ ℝ ⊆ ℝ^*
30	24, 29	sstrdi 3990	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ⊆ ℝ^*)
31		supxrunb1 13324	. . . . . . . . . 10 ⊢ (𝐴 ⊆ ℝ^* → (∀𝑐 ∈ ℝ ∃𝑥 ∈ 𝐴 𝑐 ≤ 𝑥 ↔ sup(𝐴, ℝ^*, < ) = +∞))
32	30, 31	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (∀𝑐 ∈ ℝ ∃𝑥 ∈ 𝐴 𝑐 ≤ 𝑥 ↔ sup(𝐴, ℝ^*, < ) = +∞))
33	28, 32	mpbird 257	. . . . . . . 8 ⊢ (𝜑 → ∀𝑐 ∈ ℝ ∃𝑥 ∈ 𝐴 𝑐 ≤ 𝑥)
34	33	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → ∀𝑐 ∈ ℝ ∃𝑥 ∈ 𝐴 𝑐 ≤ 𝑥)
35		r19.29 3110	. . . . . . . 8 ⊢ ((∀𝑐 ∈ ℝ ∃𝑥 ∈ 𝐴 𝑐 ≤ 𝑥 ∧ ∃𝑐 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑐 ≤ 𝑥 → ((abs‘((1 / 𝐵) − 0)) < (1 / if(1 ≤ 𝑦, 𝑦, 1)) ∧ (abs‘𝐵) ≤ 𝑦))) → ∃𝑐 ∈ ℝ (∃𝑥 ∈ 𝐴 𝑐 ≤ 𝑥 ∧ ∀𝑥 ∈ 𝐴 (𝑐 ≤ 𝑥 → ((abs‘((1 / 𝐵) − 0)) < (1 / if(1 ≤ 𝑦, 𝑦, 1)) ∧ (abs‘𝐵) ≤ 𝑦))))
36		r19.29r 3112	. . . . . . . . . 10 ⊢ ((∃𝑥 ∈ 𝐴 𝑐 ≤ 𝑥 ∧ ∀𝑥 ∈ 𝐴 (𝑐 ≤ 𝑥 → ((abs‘((1 / 𝐵) − 0)) < (1 / if(1 ≤ 𝑦, 𝑦, 1)) ∧ (abs‘𝐵) ≤ 𝑦))) → ∃𝑥 ∈ 𝐴 (𝑐 ≤ 𝑥 ∧ (𝑐 ≤ 𝑥 → ((abs‘((1 / 𝐵) − 0)) < (1 / if(1 ≤ 𝑦, 𝑦, 1)) ∧ (abs‘𝐵) ≤ 𝑦))))
37	2	adantlr 714	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ)
38	37	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) ∧ (abs‘𝐵) ≤ 𝑦) → 𝐵 ∈ ℂ)
39	3	adantlr 714	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → 𝐵 ≠ 0)
40	39	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) ∧ (abs‘𝐵) ≤ 𝑦) → 𝐵 ≠ 0)
41	38, 40	reccld 12007	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) ∧ (abs‘𝐵) ≤ 𝑦) → (1 / 𝐵) ∈ ℂ)
42	41	subid1d 11584	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) ∧ (abs‘𝐵) ≤ 𝑦) → ((1 / 𝐵) − 0) = (1 / 𝐵))
43	42	fveq2d 6895	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) ∧ (abs‘𝐵) ≤ 𝑦) → (abs‘((1 / 𝐵) − 0)) = (abs‘(1 / 𝐵)))
44		1cnd 11233	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) ∧ (abs‘𝐵) ≤ 𝑦) → 1 ∈ ℂ)
45	44, 38, 40	absdivd 15428	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) ∧ (abs‘𝐵) ≤ 𝑦) → (abs‘(1 / 𝐵)) = ((abs‘1) / (abs‘𝐵)))
46	8	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) ∧ (abs‘𝐵) ≤ 𝑦) → 1 ∈ ℝ)
47		0le1 11761	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 0 ≤ 1
48	47	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) ∧ (abs‘𝐵) ≤ 𝑦) → 0 ≤ 1)
49	46, 48	absidd 15395	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) ∧ (abs‘𝐵) ≤ 𝑦) → (abs‘1) = 1)
50	49	oveq1d 7429	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) ∧ (abs‘𝐵) ≤ 𝑦) → ((abs‘1) / (abs‘𝐵)) = (1 / (abs‘𝐵)))
51	43, 45, 50	3eqtrd 2772	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) ∧ (abs‘𝐵) ≤ 𝑦) → (abs‘((1 / 𝐵) − 0)) = (1 / (abs‘𝐵)))
52	15	ad2antrr 725	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) ∧ (abs‘𝐵) ≤ 𝑦) → if(1 ≤ 𝑦, 𝑦, 1) ∈ ℝ⁺)
53	52	rprecred 13053	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) ∧ (abs‘𝐵) ≤ 𝑦) → (1 / if(1 ≤ 𝑦, 𝑦, 1)) ∈ ℝ)
54	37, 39	absrpcld 15421	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → (abs‘𝐵) ∈ ℝ⁺)
55	54	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) ∧ (abs‘𝐵) ≤ 𝑦) → (abs‘𝐵) ∈ ℝ⁺)
56	55	rprecred 13053	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) ∧ (abs‘𝐵) ≤ 𝑦) → (1 / (abs‘𝐵)) ∈ ℝ)
57	55	rpred 13042	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) ∧ (abs‘𝐵) ≤ 𝑦) → (abs‘𝐵) ∈ ℝ)
58	7	ad2antrr 725	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) ∧ (abs‘𝐵) ≤ 𝑦) → 𝑦 ∈ ℝ)
59	10	ad2antrr 725	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) ∧ (abs‘𝐵) ≤ 𝑦) → if(1 ≤ 𝑦, 𝑦, 1) ∈ ℝ)
60		simpr 484	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) ∧ (abs‘𝐵) ≤ 𝑦) → (abs‘𝐵) ≤ 𝑦)
61		max2 13192	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((1 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑦 ≤ if(1 ≤ 𝑦, 𝑦, 1))
62	8, 58, 61	sylancr 586	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) ∧ (abs‘𝐵) ≤ 𝑦) → 𝑦 ≤ if(1 ≤ 𝑦, 𝑦, 1))
63	57, 58, 59, 60, 62	letrd 11395	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) ∧ (abs‘𝐵) ≤ 𝑦) → (abs‘𝐵) ≤ if(1 ≤ 𝑦, 𝑦, 1))
64	55, 52, 46, 48, 63	lediv2ad 13064	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) ∧ (abs‘𝐵) ≤ 𝑦) → (1 / if(1 ≤ 𝑦, 𝑦, 1)) ≤ (1 / (abs‘𝐵)))
65	53, 56, 64	lensymd 11389	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) ∧ (abs‘𝐵) ≤ 𝑦) → ¬ (1 / (abs‘𝐵)) < (1 / if(1 ≤ 𝑦, 𝑦, 1)))
66	51, 65	eqnbrtrd 5160	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) ∧ (abs‘𝐵) ≤ 𝑦) → ¬ (abs‘((1 / 𝐵) − 0)) < (1 / if(1 ≤ 𝑦, 𝑦, 1)))
67	66	pm2.21d 121	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) ∧ (abs‘𝐵) ≤ 𝑦) → ((abs‘((1 / 𝐵) − 0)) < (1 / if(1 ≤ 𝑦, 𝑦, 1)) → ⊥))
68	67	expimpd 453	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → (((abs‘𝐵) ≤ 𝑦 ∧ (abs‘((1 / 𝐵) − 0)) < (1 / if(1 ≤ 𝑦, 𝑦, 1))) → ⊥))
69	68	ancomsd 465	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → (((abs‘((1 / 𝐵) − 0)) < (1 / if(1 ≤ 𝑦, 𝑦, 1)) ∧ (abs‘𝐵) ≤ 𝑦) → ⊥))
70	69	imim2d 57	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → ((𝑐 ≤ 𝑥 → ((abs‘((1 / 𝐵) − 0)) < (1 / if(1 ≤ 𝑦, 𝑦, 1)) ∧ (abs‘𝐵) ≤ 𝑦)) → (𝑐 ≤ 𝑥 → ⊥)))
71	70	impcomd 411	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → ((𝑐 ≤ 𝑥 ∧ (𝑐 ≤ 𝑥 → ((abs‘((1 / 𝐵) − 0)) < (1 / if(1 ≤ 𝑦, 𝑦, 1)) ∧ (abs‘𝐵) ≤ 𝑦))) → ⊥))
72	71	rexlimdva 3151	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (∃𝑥 ∈ 𝐴 (𝑐 ≤ 𝑥 ∧ (𝑐 ≤ 𝑥 → ((abs‘((1 / 𝐵) − 0)) < (1 / if(1 ≤ 𝑦, 𝑦, 1)) ∧ (abs‘𝐵) ≤ 𝑦))) → ⊥))
73	36, 72	syl5 34	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → ((∃𝑥 ∈ 𝐴 𝑐 ≤ 𝑥 ∧ ∀𝑥 ∈ 𝐴 (𝑐 ≤ 𝑥 → ((abs‘((1 / 𝐵) − 0)) < (1 / if(1 ≤ 𝑦, 𝑦, 1)) ∧ (abs‘𝐵) ≤ 𝑦))) → ⊥))
74	73	rexlimdvw 3156	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (∃𝑐 ∈ ℝ (∃𝑥 ∈ 𝐴 𝑐 ≤ 𝑥 ∧ ∀𝑥 ∈ 𝐴 (𝑐 ≤ 𝑥 → ((abs‘((1 / 𝐵) − 0)) < (1 / if(1 ≤ 𝑦, 𝑦, 1)) ∧ (abs‘𝐵) ≤ 𝑦))) → ⊥))
75	35, 74	syl5 34	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → ((∀𝑐 ∈ ℝ ∃𝑥 ∈ 𝐴 𝑐 ≤ 𝑥 ∧ ∃𝑐 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑐 ≤ 𝑥 → ((abs‘((1 / 𝐵) − 0)) < (1 / if(1 ≤ 𝑦, 𝑦, 1)) ∧ (abs‘𝐵) ≤ 𝑦))) → ⊥))
76	34, 75	mpand 694	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (∃𝑐 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑐 ≤ 𝑥 → ((abs‘((1 / 𝐵) − 0)) < (1 / if(1 ≤ 𝑦, 𝑦, 1)) ∧ (abs‘𝐵) ≤ 𝑦)) → ⊥))
77	27, 76	sylbird 260	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → ((∃𝑐 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑐 ≤ 𝑥 → (abs‘((1 / 𝐵) − 0)) < (1 / if(1 ≤ 𝑦, 𝑦, 1))) ∧ ∃𝑐 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑐 ≤ 𝑥 → (abs‘𝐵) ≤ 𝑦)) → ⊥))
78	19, 77	mpand 694	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (∃𝑐 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑐 ≤ 𝑥 → (abs‘𝐵) ≤ 𝑦) → ⊥))
79	1, 78	mtoi 198	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → ¬ ∃𝑐 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑐 ≤ 𝑥 → (abs‘𝐵) ≤ 𝑦))
80	79	nrexdv 3145	. 2 ⊢ (𝜑 → ¬ ∃𝑦 ∈ ℝ ∃𝑐 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑐 ≤ 𝑥 → (abs‘𝐵) ≤ 𝑦))
81	24, 2	elo1mpt 15504	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝑂(1) ↔ ∃𝑐 ∈ ℝ ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑐 ≤ 𝑥 → (abs‘𝐵) ≤ 𝑦)))
82		rexcom 3283	. . 3 ⊢ (∃𝑐 ∈ ℝ ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑐 ≤ 𝑥 → (abs‘𝐵) ≤ 𝑦) ↔ ∃𝑦 ∈ ℝ ∃𝑐 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑐 ≤ 𝑥 → (abs‘𝐵) ≤ 𝑦))
83	81, 82	bitrdi 287	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝑂(1) ↔ ∃𝑦 ∈ ℝ ∃𝑐 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑐 ≤ 𝑥 → (abs‘𝐵) ≤ 𝑦)))
84	80, 83	mtbird 325	1 ⊢ (𝜑 → ¬ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝑂(1))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ⊥wfal 1546 ∈ wcel 2099 ≠ wne 2936 ∀wral 3057 ∃wrex 3066 ⊆ wss 3945 ifcif 4524 class class class wbr 5142 ↦ cmpt 5225 dom cdm 5672 ‘cfv 6542 (class class class)co 7414 supcsup 9457 ℂcc 11130 ℝcr 11131 0cc0 11132 1c1 11133 +∞cpnf 11269 ℝ^*cxr 11271 < clt 11272 ≤ cle 11273 − cmin 11468 / cdiv 11895 ℝ⁺crp 13000 abscabs 15207 ⇝_𝑟 crli 15455 𝑂(1)co1 15456

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 ax-pre-sup 11210

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-pm 8841 df-en 8958 df-dom 8959 df-sdom 8960 df-sup 9459 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-div 11896 df-nn 12237 df-2 12299 df-3 12300 df-n0 12497 df-z 12583 df-uz 12847 df-rp 13001 df-ico 13356 df-seq 13993 df-exp 14053 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-rlim 15459 df-o1 15460 df-lo1 15461

This theorem is referenced by: logno1 26563

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