Theorem lo1bdd2 14873

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 14871	. . 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 12823	. . . . . . . . . . 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 725	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ (𝐶[,)+∞)) ∧ (𝑛 ∈ ℝ ∧ ∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → 𝐵 ≤ 𝑛))) ∧ 𝑛 ≤ 𝑀) → 𝑀 ∈ ℝ)
13		simplrl 776	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ (𝐶[,)+∞)) ∧ (𝑛 ∈ ℝ ∧ ∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → 𝐵 ≤ 𝑛))) ∧ ¬ 𝑛 ≤ 𝑀) → 𝑛 ∈ ℝ)
14	12, 13	ifclda 4459	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝐶[,)+∞)) ∧ (𝑛 ∈ ℝ ∧ ∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → 𝐵 ≤ 𝑛))) → if(𝑛 ≤ 𝑀, 𝑀, 𝑛) ∈ ℝ)
15	2	ad2antrr 725	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝐶[,)+∞)) ∧ 𝑛 ∈ ℝ) → 𝐴 ⊆ ℝ)
16	15	sselda 3915	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ (𝐶[,)+∞)) ∧ 𝑛 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ)
17	9	simpld 498	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐶[,)+∞)) → 𝑦 ∈ ℝ)
18	17	ad2antrr 725	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ (𝐶[,)+∞)) ∧ 𝑛 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ ℝ)
19	16, 18	ltnled 10776	. . . . . . . . . 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 651	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ 𝐶 ≤ 𝑦)) ∧ 𝑥 ∈ 𝐴) → (𝑥 < 𝑦 → 𝐵 ≤ 𝑀))
23	9, 22	syldanl 604	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝐶[,)+∞)) ∧ 𝑥 ∈ 𝐴) → (𝑥 < 𝑦 → 𝐵 ≤ 𝑀))
24	23	adantlr 714	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ (𝐶[,)+∞)) ∧ 𝑛 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → (𝑥 < 𝑦 → 𝐵 ≤ 𝑀))
25		simplr 768	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑦 ∈ (𝐶[,)+∞)) ∧ 𝑛 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → 𝑛 ∈ ℝ)
26	11	ad2antrr 725	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑦 ∈ (𝐶[,)+∞)) ∧ 𝑛 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → 𝑀 ∈ ℝ)
27		max2 12568	. . . . . . . . . . . . 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 751	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑦 ∈ (𝐶[,)+∞)) ∧ 𝑛 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)
30	11	ad5ant12 755	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑦 ∈ (𝐶[,)+∞)) ∧ 𝑛 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) ∧ 𝑛 ≤ 𝑀) → 𝑀 ∈ ℝ)
31		simpllr 775	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑦 ∈ (𝐶[,)+∞)) ∧ 𝑛 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) ∧ ¬ 𝑛 ≤ 𝑀) → 𝑛 ∈ ℝ)
32	30, 31	ifclda 4459	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑦 ∈ (𝐶[,)+∞)) ∧ 𝑛 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → if(𝑛 ≤ 𝑀, 𝑀, 𝑛) ∈ ℝ)
33		letr 10723	. . . . . . . . . . . . 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 1368	. . . . . . . . . . . 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 693	. . . . . . . . . . 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 12566	. . . . . . . . . . 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 10723	. . . . . . . . . . 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 1368	. . . . . . . . . 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 693	. . . . . . . . 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 3144	. . . . . . 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 5090	. . . . . 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 3243	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐶[,)+∞)) → (∃𝑛 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → 𝐵 ≤ 𝑛) → ∃𝑚 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑚))
50	49	rexlimdva 3243	. 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 2111  ∀wral 3106  ∃wrex 3107   ⊆ wss 3881  ifcif 4425   class class class wbr 5030   ↦ cmpt 5110  (class class class)co 7135  ℝcr 10525  +∞cpnf 10661   < clt 10664   ≤ cle 10665  [,)cico 12728  ≤𝑂(1)clo1 14836

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583  ax-pre-lttri 10600  ax-pre-lttrn 10601

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-po 5438  df-so 5439  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-er 8272  df-pm 8392  df-en 8493  df-dom 8494  df-sdom 8495  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-ico 12732  df-lo1 14840

This theorem is referenced by:  lo1bddrp  14874  o1bdd2  14890