Theorem lo1bdd2 14972

Description: If an eventually bounded function is bounded on every interval 𝐴 ∩ (-∞, 𝑦) by a function 𝑀(𝑦), then the function is bounded on the whole domain. (Contributed by Mario Carneiro, 9-Apr-2016.)

Hypotheses

Ref	Expression
lo1bdd2.1	⊢ (𝜑 → 𝐴 ⊆ ℝ)
lo1bdd2.2	⊢ (𝜑 → 𝐶 ∈ ℝ)
lo1bdd2.3	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)
lo1bdd2.4	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ ≤𝑂(1))
lo1bdd2.5	⊢ ((𝜑 ∧ (𝑦 ∈ ℝ ∧ 𝐶 ≤ 𝑦)) → 𝑀 ∈ ℝ)
lo1bdd2.6	⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ ((𝑦 ∈ ℝ ∧ 𝐶 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → 𝐵 ≤ 𝑀)

Assertion

Ref	Expression
lo1bdd2	⊢ (𝜑 → ∃𝑚 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑚)

Distinct variable groups:   𝑥,𝑚,𝑦,𝐴   𝐵,𝑚,𝑦   𝑥,𝐶,𝑦   𝜑,𝑥,𝑦   𝑚,𝑀,𝑥

Allowed substitution hints:   𝜑(𝑚)   𝐵(𝑥)   𝐶(𝑚)   𝑀(𝑦)

Proof of Theorem lo1bdd2

Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lo1bdd2.4	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ ≤𝑂(1))
2		lo1bdd2.1	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
3		lo1bdd2.3	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)
4		lo1bdd2.2	. . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ)
5	2, 3, 4	ello1mpt2 14970	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ ≤𝑂(1) ↔ ∃𝑦 ∈ (𝐶[,)+∞)∃𝑛 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → 𝐵 ≤ 𝑛)))
6	1, 5	mpbid 235	. 2 ⊢ (𝜑 → ∃𝑦 ∈ (𝐶[,)+∞)∃𝑛 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → 𝐵 ≤ 𝑛))
7		elicopnf 12920	. . . . . . . . . . 11 ⊢ (𝐶 ∈ ℝ → (𝑦 ∈ (𝐶[,)+∞) ↔ (𝑦 ∈ ℝ ∧ 𝐶 ≤ 𝑦)))
8	4, 7	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝑦 ∈ (𝐶[,)+∞) ↔ (𝑦 ∈ ℝ ∧ 𝐶 ≤ 𝑦)))
9	8	biimpa 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐶[,)+∞)) → (𝑦 ∈ ℝ ∧ 𝐶 ≤ 𝑦))
10		lo1bdd2.5	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ ∧ 𝐶 ≤ 𝑦)) → 𝑀 ∈ ℝ)
11	9, 10	syldan 594	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐶[,)+∞)) → 𝑀 ∈ ℝ)
12	11	ad2antrr 726	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ (𝐶[,)+∞)) ∧ (𝑛 ∈ ℝ ∧ ∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → 𝐵 ≤ 𝑛))) ∧ 𝑛 ≤ 𝑀) → 𝑀 ∈ ℝ)
13		simplrl 777	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ (𝐶[,)+∞)) ∧ (𝑛 ∈ ℝ ∧ ∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → 𝐵 ≤ 𝑛))) ∧ ¬ 𝑛 ≤ 𝑀) → 𝑛 ∈ ℝ)
14	12, 13	ifclda 4450	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝐶[,)+∞)) ∧ (𝑛 ∈ ℝ ∧ ∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → 𝐵 ≤ 𝑛))) → if(𝑛 ≤ 𝑀, 𝑀, 𝑛) ∈ ℝ)
15	2	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝐶[,)+∞)) ∧ 𝑛 ∈ ℝ) → 𝐴 ⊆ ℝ)
16	15	sselda 3878	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ (𝐶[,)+∞)) ∧ 𝑛 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ)
17	9	simpld 498	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐶[,)+∞)) → 𝑦 ∈ ℝ)
18	17	ad2antrr 726	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ (𝐶[,)+∞)) ∧ 𝑛 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ ℝ)
19	16, 18	ltnled 10866	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ (𝐶[,)+∞)) ∧ 𝑛 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → (𝑥 < 𝑦 ↔ ¬ 𝑦 ≤ 𝑥))
20		lo1bdd2.6	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ ((𝑦 ∈ ℝ ∧ 𝐶 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → 𝐵 ≤ 𝑀)
21	20	expr 460	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ (𝑦 ∈ ℝ ∧ 𝐶 ≤ 𝑦)) → (𝑥 < 𝑦 → 𝐵 ≤ 𝑀))
22	21	an32s 652	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ 𝐶 ≤ 𝑦)) ∧ 𝑥 ∈ 𝐴) → (𝑥 < 𝑦 → 𝐵 ≤ 𝑀))
23	9, 22	syldanl 605	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝐶[,)+∞)) ∧ 𝑥 ∈ 𝐴) → (𝑥 < 𝑦 → 𝐵 ≤ 𝑀))
24	23	adantlr 715	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ (𝐶[,)+∞)) ∧ 𝑛 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → (𝑥 < 𝑦 → 𝐵 ≤ 𝑀))
25		simplr 769	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑦 ∈ (𝐶[,)+∞)) ∧ 𝑛 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → 𝑛 ∈ ℝ)
26	11	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑦 ∈ (𝐶[,)+∞)) ∧ 𝑛 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → 𝑀 ∈ ℝ)
27		max2 12664	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℝ ∧ 𝑀 ∈ ℝ) → 𝑀 ≤ if(𝑛 ≤ 𝑀, 𝑀, 𝑛))
28	25, 26, 27	syl2anc 587	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ (𝐶[,)+∞)) ∧ 𝑛 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → 𝑀 ≤ if(𝑛 ≤ 𝑀, 𝑀, 𝑛))
29	3	ad4ant14 752	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑦 ∈ (𝐶[,)+∞)) ∧ 𝑛 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)
30	11	ad5ant12 756	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑦 ∈ (𝐶[,)+∞)) ∧ 𝑛 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) ∧ 𝑛 ≤ 𝑀) → 𝑀 ∈ ℝ)
31		simpllr 776	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑦 ∈ (𝐶[,)+∞)) ∧ 𝑛 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) ∧ ¬ 𝑛 ≤ 𝑀) → 𝑛 ∈ ℝ)
32	30, 31	ifclda 4450	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑦 ∈ (𝐶[,)+∞)) ∧ 𝑛 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → if(𝑛 ≤ 𝑀, 𝑀, 𝑛) ∈ ℝ)
33		letr 10813	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ if(𝑛 ≤ 𝑀, 𝑀, 𝑛) ∈ ℝ) → ((𝐵 ≤ 𝑀 ∧ 𝑀 ≤ if(𝑛 ≤ 𝑀, 𝑀, 𝑛)) → 𝐵 ≤ if(𝑛 ≤ 𝑀, 𝑀, 𝑛)))
34	29, 26, 32, 33	syl3anc 1372	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ (𝐶[,)+∞)) ∧ 𝑛 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → ((𝐵 ≤ 𝑀 ∧ 𝑀 ≤ if(𝑛 ≤ 𝑀, 𝑀, 𝑛)) → 𝐵 ≤ if(𝑛 ≤ 𝑀, 𝑀, 𝑛)))
35	28, 34	mpan2d 694	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ (𝐶[,)+∞)) ∧ 𝑛 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → (𝐵 ≤ 𝑀 → 𝐵 ≤ if(𝑛 ≤ 𝑀, 𝑀, 𝑛)))
36	24, 35	syld 47	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ (𝐶[,)+∞)) ∧ 𝑛 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → (𝑥 < 𝑦 → 𝐵 ≤ if(𝑛 ≤ 𝑀, 𝑀, 𝑛)))
37	19, 36	sylbird 263	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ (𝐶[,)+∞)) ∧ 𝑛 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → (¬ 𝑦 ≤ 𝑥 → 𝐵 ≤ if(𝑛 ≤ 𝑀, 𝑀, 𝑛)))
38		max1 12662	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℝ ∧ 𝑀 ∈ ℝ) → 𝑛 ≤ if(𝑛 ≤ 𝑀, 𝑀, 𝑛))
39	25, 26, 38	syl2anc 587	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ (𝐶[,)+∞)) ∧ 𝑛 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → 𝑛 ≤ if(𝑛 ≤ 𝑀, 𝑀, 𝑛))
40		letr 10813	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ℝ ∧ 𝑛 ∈ ℝ ∧ if(𝑛 ≤ 𝑀, 𝑀, 𝑛) ∈ ℝ) → ((𝐵 ≤ 𝑛 ∧ 𝑛 ≤ if(𝑛 ≤ 𝑀, 𝑀, 𝑛)) → 𝐵 ≤ if(𝑛 ≤ 𝑀, 𝑀, 𝑛)))
41	29, 25, 32, 40	syl3anc 1372	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ (𝐶[,)+∞)) ∧ 𝑛 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → ((𝐵 ≤ 𝑛 ∧ 𝑛 ≤ if(𝑛 ≤ 𝑀, 𝑀, 𝑛)) → 𝐵 ≤ if(𝑛 ≤ 𝑀, 𝑀, 𝑛)))
42	39, 41	mpan2d 694	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ (𝐶[,)+∞)) ∧ 𝑛 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → (𝐵 ≤ 𝑛 → 𝐵 ≤ if(𝑛 ≤ 𝑀, 𝑀, 𝑛)))
43	37, 42	jad 190	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ (𝐶[,)+∞)) ∧ 𝑛 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → ((𝑦 ≤ 𝑥 → 𝐵 ≤ 𝑛) → 𝐵 ≤ if(𝑛 ≤ 𝑀, 𝑀, 𝑛)))
44	43	ralimdva 3091	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝐶[,)+∞)) ∧ 𝑛 ∈ ℝ) → (∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → 𝐵 ≤ 𝑛) → ∀𝑥 ∈ 𝐴 𝐵 ≤ if(𝑛 ≤ 𝑀, 𝑀, 𝑛)))
45	44	impr 458	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝐶[,)+∞)) ∧ (𝑛 ∈ ℝ ∧ ∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → 𝐵 ≤ 𝑛))) → ∀𝑥 ∈ 𝐴 𝐵 ≤ if(𝑛 ≤ 𝑀, 𝑀, 𝑛))
46		brralrspcev 5091	. . . . . 6 ⊢ ((if(𝑛 ≤ 𝑀, 𝑀, 𝑛) ∈ ℝ ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ if(𝑛 ≤ 𝑀, 𝑀, 𝑛)) → ∃𝑚 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑚)
47	14, 45, 46	syl2anc 587	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝐶[,)+∞)) ∧ (𝑛 ∈ ℝ ∧ ∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → 𝐵 ≤ 𝑛))) → ∃𝑚 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑚)
48	47	expr 460	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝐶[,)+∞)) ∧ 𝑛 ∈ ℝ) → (∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → 𝐵 ≤ 𝑛) → ∃𝑚 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑚))
49	48	rexlimdva 3194	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐶[,)+∞)) → (∃𝑛 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → 𝐵 ≤ 𝑛) → ∃𝑚 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑚))
50	49	rexlimdva 3194	. 2 ⊢ (𝜑 → (∃𝑦 ∈ (𝐶[,)+∞)∃𝑛 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → 𝐵 ≤ 𝑛) → ∃𝑚 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑚))
51	6, 50	mpd 15	1 ⊢ (𝜑 → ∃𝑚 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑚)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∈ wcel 2113  ∀wral 3053  ∃wrex 3054   ⊆ wss 3844  ifcif 4415   class class class wbr 5031   ↦ cmpt 5111  (class class class)co 7171  ℝcr 10615  +∞cpnf 10751   < clt 10754   ≤ cle 10755  [,)cico 12824  ≤𝑂(1)clo1 14935

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-sep 5168  ax-nul 5175  ax-pow 5233  ax-pr 5297  ax-un 7480  ax-cnex 10672  ax-resscn 10673  ax-pre-lttri 10690  ax-pre-lttrn 10691

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-nel 3039  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-sbc 3683  df-csb 3792  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-nul 4213  df-if 4416  df-pw 4491  df-sn 4518  df-pr 4520  df-op 4524  df-uni 4798  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5430  df-po 5443  df-so 5444  df-xp 5532  df-rel 5533  df-cnv 5534  df-co 5535  df-dm 5536  df-rn 5537  df-res 5538  df-ima 5539  df-iota 6298  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-ov 7174  df-oprab 7175  df-mpo 7176  df-er 8321  df-pm 8441  df-en 8557  df-dom 8558  df-sdom 8559  df-pnf 10756  df-mnf 10757  df-xr 10758  df-ltxr 10759  df-le 10760  df-ico 12828  df-lo1 14939

This theorem is referenced by:  lo1bddrp  14973  o1bdd2  14989