Theorem lo1bdd2 15448

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 15446	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ ≤𝑂(1) ↔ ∃𝑦 ∈ (𝐶[,)+∞)∃𝑛 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → 𝐵 ≤ 𝑛)))
6	1, 5	mpbid 232	. 2 ⊢ (𝜑 → ∃𝑦 ∈ (𝐶[,)+∞)∃𝑛 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → 𝐵 ≤ 𝑛))
7		elicopnf 13362	. . . . . . . . . . 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 476	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐶[,)+∞)) → (𝑦 ∈ ℝ ∧ 𝐶 ≤ 𝑦))
10		lo1bdd2.5	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ ∧ 𝐶 ≤ 𝑦)) → 𝑀 ∈ ℝ)
11	9, 10	syldan 592	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐶[,)+∞)) → 𝑀 ∈ ℝ)
12	11	ad2antrr 727	. . . . . . 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 4503	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝐶[,)+∞)) ∧ (𝑛 ∈ ℝ ∧ ∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → 𝐵 ≤ 𝑛))) → if(𝑛 ≤ 𝑀, 𝑀, 𝑛) ∈ ℝ)
15	2	ad2antrr 727	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝐶[,)+∞)) ∧ 𝑛 ∈ ℝ) → 𝐴 ⊆ ℝ)
16	15	sselda 3922	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ (𝐶[,)+∞)) ∧ 𝑛 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ)
17	9	simpld 494	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐶[,)+∞)) → 𝑦 ∈ ℝ)
18	17	ad2antrr 727	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ (𝐶[,)+∞)) ∧ 𝑛 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ ℝ)
19	16, 18	ltnled 11281	. . . . . . . . . 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 456	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ (𝑦 ∈ ℝ ∧ 𝐶 ≤ 𝑦)) → (𝑥 < 𝑦 → 𝐵 ≤ 𝑀))
22	21	an32s 653	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ 𝐶 ≤ 𝑦)) ∧ 𝑥 ∈ 𝐴) → (𝑥 < 𝑦 → 𝐵 ≤ 𝑀))
23	9, 22	syldanl 603	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝐶[,)+∞)) ∧ 𝑥 ∈ 𝐴) → (𝑥 < 𝑦 → 𝐵 ≤ 𝑀))
24	23	adantlr 716	. . . . . . . . . . 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 727	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑦 ∈ (𝐶[,)+∞)) ∧ 𝑛 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → 𝑀 ∈ ℝ)
27		max2 13103	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℝ ∧ 𝑀 ∈ ℝ) → 𝑀 ≤ if(𝑛 ≤ 𝑀, 𝑀, 𝑛))
28	25, 26, 27	syl2anc 585	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ (𝐶[,)+∞)) ∧ 𝑛 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → 𝑀 ≤ if(𝑛 ≤ 𝑀, 𝑀, 𝑛))
29	3	ad4ant14 753	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑦 ∈ (𝐶[,)+∞)) ∧ 𝑛 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)
30	11	ad3antrrr 731	. . . . . . . . . . . . . 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 4503	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑦 ∈ (𝐶[,)+∞)) ∧ 𝑛 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → if(𝑛 ≤ 𝑀, 𝑀, 𝑛) ∈ ℝ)
33		letr 11228	. . . . . . . . . . . . 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 1374	. . . . . . . . . . . 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 695	. . . . . . . . . . 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 260	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ (𝐶[,)+∞)) ∧ 𝑛 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → (¬ 𝑦 ≤ 𝑥 → 𝐵 ≤ if(𝑛 ≤ 𝑀, 𝑀, 𝑛)))
38		max1 13101	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℝ ∧ 𝑀 ∈ ℝ) → 𝑛 ≤ if(𝑛 ≤ 𝑀, 𝑀, 𝑛))
39	25, 26, 38	syl2anc 585	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ (𝐶[,)+∞)) ∧ 𝑛 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → 𝑛 ≤ if(𝑛 ≤ 𝑀, 𝑀, 𝑛))
40		letr 11228	. . . . . . . . . . 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 1374	. . . . . . . . . 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 695	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ (𝐶[,)+∞)) ∧ 𝑛 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → (𝐵 ≤ 𝑛 → 𝐵 ≤ if(𝑛 ≤ 𝑀, 𝑀, 𝑛)))
43	37, 42	jad 187	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ (𝐶[,)+∞)) ∧ 𝑛 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → ((𝑦 ≤ 𝑥 → 𝐵 ≤ 𝑛) → 𝐵 ≤ if(𝑛 ≤ 𝑀, 𝑀, 𝑛)))
44	43	ralimdva 3150	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝐶[,)+∞)) ∧ 𝑛 ∈ ℝ) → (∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → 𝐵 ≤ 𝑛) → ∀𝑥 ∈ 𝐴 𝐵 ≤ if(𝑛 ≤ 𝑀, 𝑀, 𝑛)))
45	44	impr 454	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝐶[,)+∞)) ∧ (𝑛 ∈ ℝ ∧ ∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → 𝐵 ≤ 𝑛))) → ∀𝑥 ∈ 𝐴 𝐵 ≤ if(𝑛 ≤ 𝑀, 𝑀, 𝑛))
46		brralrspcev 5146	. . . . . 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 585	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝐶[,)+∞)) ∧ (𝑛 ∈ ℝ ∧ ∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → 𝐵 ≤ 𝑛))) → ∃𝑚 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑚)
48	47	expr 456	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝐶[,)+∞)) ∧ 𝑛 ∈ ℝ) → (∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → 𝐵 ≤ 𝑛) → ∃𝑚 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑚))
49	48	rexlimdva 3139	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐶[,)+∞)) → (∃𝑛 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → 𝐵 ≤ 𝑛) → ∃𝑚 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑚))
50	49	rexlimdva 3139	. 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 206   ∧ wa 395   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062   ⊆ wss 3890  ifcif 4467   class class class wbr 5086   ↦ cmpt 5167  (class class class)co 7358  ℝcr 11026  +∞cpnf 11164   < clt 11167   ≤ cle 11168  [,)cico 13264  ≤𝑂(1)clo1 15411

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pow 5300  ax-pr 5368  ax-un 7680  ax-cnex 11083  ax-resscn 11084  ax-pre-lttri 11101  ax-pre-lttrn 11102

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5517  df-po 5530  df-so 5531  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-ov 7361  df-oprab 7362  df-mpo 7363  df-er 8634  df-pm 8767  df-en 8885  df-dom 8886  df-sdom 8887  df-pnf 11169  df-mnf 11170  df-xr 11171  df-ltxr 11172  df-le 11173  df-ico 13268  df-lo1 15415

This theorem is referenced by:  lo1bddrp  15449  o1bdd2  15465