Theorem lo1resb 15592

Description: The restriction of a function to an unbounded-above interval is eventually upper bounded iff the original is eventually upper bounded. (Contributed by Mario Carneiro, 26-May-2016.)

Hypotheses

Ref	Expression
lo1resb.1	⊢ (𝜑 → 𝐹:𝐴⟶ℝ)
lo1resb.2	⊢ (𝜑 → 𝐴 ⊆ ℝ)
lo1resb.3	⊢ (𝜑 → 𝐵 ∈ ℝ)

Assertion

Ref	Expression
lo1resb	⊢ (𝜑 → (𝐹 ∈ ≤𝑂(1) ↔ (𝐹 ↾ (𝐵[,)+∞)) ∈ ≤𝑂(1)))

Proof of Theorem lo1resb

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lo1res 15587	. 2 ⊢ (𝐹 ∈ ≤𝑂(1) → (𝐹 ↾ (𝐵[,)+∞)) ∈ ≤𝑂(1))
2		lo1resb.1	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ)
3	2	feqmptd 6936	. . . . . 6 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))
4	3	reseq1d 5965	. . . . 5 ⊢ (𝜑 → (𝐹 ↾ (𝐵[,)+∞)) = ((𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) ↾ (𝐵[,)+∞)))
5		resmpt3 6028	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) ↾ (𝐵[,)+∞)) = (𝑥 ∈ (𝐴 ∩ (𝐵[,)+∞)) ↦ (𝐹‘𝑥))
6	4, 5	eqtrdi 2814	. . . 4 ⊢ (𝜑 → (𝐹 ↾ (𝐵[,)+∞)) = (𝑥 ∈ (𝐴 ∩ (𝐵[,)+∞)) ↦ (𝐹‘𝑥)))
7	6	eleq1d 2848	. . 3 ⊢ (𝜑 → ((𝐹 ↾ (𝐵[,)+∞)) ∈ ≤𝑂(1) ↔ (𝑥 ∈ (𝐴 ∩ (𝐵[,)+∞)) ↦ (𝐹‘𝑥)) ∈ ≤𝑂(1)))
8		inss1 4189	. . . . . 6 ⊢ (𝐴 ∩ (𝐵[,)+∞)) ⊆ 𝐴
9		lo1resb.2	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
10	8, 9	sstrid 3948	. . . . 5 ⊢ (𝜑 → (𝐴 ∩ (𝐵[,)+∞)) ⊆ ℝ)
11		elinel1 4154	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∩ (𝐵[,)+∞)) → 𝑥 ∈ 𝐴)
12		ffvelcdm 7063	. . . . . 6 ⊢ ((𝐹:𝐴⟶ℝ ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ ℝ)
13	2, 11, 12	syl2an 605	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ (𝐵[,)+∞))) → (𝐹‘𝑥) ∈ ℝ)
14	10, 13	ello1mpt 15549	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ (𝐴 ∩ (𝐵[,)+∞)) ↦ (𝐹‘𝑥)) ∈ ≤𝑂(1) ↔ ∃𝑦 ∈ ℝ ∃𝑧 ∈ ℝ ∀𝑥 ∈ (𝐴 ∩ (𝐵[,)+∞))(𝑦 ≤ 𝑥 → (𝐹‘𝑥) ≤ 𝑧)))
15		elin 3921	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐴 ∩ (𝐵[,)+∞)) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵[,)+∞)))
16	15	imbi1i 351	. . . . . . . . 9 ⊢ ((𝑥 ∈ (𝐴 ∩ (𝐵[,)+∞)) → (𝑦 ≤ 𝑥 → (𝐹‘𝑥) ≤ 𝑧)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵[,)+∞)) → (𝑦 ≤ 𝑥 → (𝐹‘𝑥) ≤ 𝑧)))
17		impexp 454	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵[,)+∞)) → (𝑦 ≤ 𝑥 → (𝐹‘𝑥) ≤ 𝑧)) ↔ (𝑥 ∈ 𝐴 → (𝑥 ∈ (𝐵[,)+∞) → (𝑦 ≤ 𝑥 → (𝐹‘𝑥) ≤ 𝑧))))
18	16, 17	bitri 277	. . . . . . . 8 ⊢ ((𝑥 ∈ (𝐴 ∩ (𝐵[,)+∞)) → (𝑦 ≤ 𝑥 → (𝐹‘𝑥) ≤ 𝑧)) ↔ (𝑥 ∈ 𝐴 → (𝑥 ∈ (𝐵[,)+∞) → (𝑦 ≤ 𝑥 → (𝐹‘𝑥) ≤ 𝑧))))
19		impexp 454	. . . . . . . . . 10 ⊢ (((𝑥 ∈ (𝐵[,)+∞) ∧ 𝑦 ≤ 𝑥) → (𝐹‘𝑥) ≤ 𝑧) ↔ (𝑥 ∈ (𝐵[,)+∞) → (𝑦 ≤ 𝑥 → (𝐹‘𝑥) ≤ 𝑧)))
20		lo1resb.3	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐵 ∈ ℝ)
21	20	ad2antrr 736	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ)) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)
22	9	adantr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ)) → 𝐴 ⊆ ℝ)
23	22	sselda 3937	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ)) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ)
24		elicopnf 13450	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ ℝ → (𝑥 ∈ (𝐵[,)+∞) ↔ (𝑥 ∈ ℝ ∧ 𝐵 ≤ 𝑥)))
25	24	baibd 547	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝑥 ∈ (𝐵[,)+∞) ↔ 𝐵 ≤ 𝑥))
26	21, 23, 25	syl2anc 593	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ)) ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ (𝐵[,)+∞) ↔ 𝐵 ≤ 𝑥))
27	26	anbi1d 640	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ)) ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ (𝐵[,)+∞) ∧ 𝑦 ≤ 𝑥) ↔ (𝐵 ≤ 𝑥 ∧ 𝑦 ≤ 𝑥)))
28		simplrl 786	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ)) ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ ℝ)
29		maxle 13195	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (if(𝐵 ≤ 𝑦, 𝑦, 𝐵) ≤ 𝑥 ↔ (𝐵 ≤ 𝑥 ∧ 𝑦 ≤ 𝑥)))
30	21, 28, 23, 29	syl3anc 1391	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ)) ∧ 𝑥 ∈ 𝐴) → (if(𝐵 ≤ 𝑦, 𝑦, 𝐵) ≤ 𝑥 ↔ (𝐵 ≤ 𝑥 ∧ 𝑦 ≤ 𝑥)))
31	27, 30	bitr4d 284	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ)) ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ (𝐵[,)+∞) ∧ 𝑦 ≤ 𝑥) ↔ if(𝐵 ≤ 𝑦, 𝑦, 𝐵) ≤ 𝑥))
32	31	imbi1d 343	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ)) ∧ 𝑥 ∈ 𝐴) → (((𝑥 ∈ (𝐵[,)+∞) ∧ 𝑦 ≤ 𝑥) → (𝐹‘𝑥) ≤ 𝑧) ↔ (if(𝐵 ≤ 𝑦, 𝑦, 𝐵) ≤ 𝑥 → (𝐹‘𝑥) ≤ 𝑧)))
33	19, 32	bitr3id 287	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ)) ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ (𝐵[,)+∞) → (𝑦 ≤ 𝑥 → (𝐹‘𝑥) ≤ 𝑧)) ↔ (if(𝐵 ≤ 𝑦, 𝑦, 𝐵) ≤ 𝑥 → (𝐹‘𝑥) ≤ 𝑧)))
34	33	pm5.74da 813	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ)) → ((𝑥 ∈ 𝐴 → (𝑥 ∈ (𝐵[,)+∞) → (𝑦 ≤ 𝑥 → (𝐹‘𝑥) ≤ 𝑧))) ↔ (𝑥 ∈ 𝐴 → (if(𝐵 ≤ 𝑦, 𝑦, 𝐵) ≤ 𝑥 → (𝐹‘𝑥) ≤ 𝑧))))
35	18, 34	bitrid 285	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ)) → ((𝑥 ∈ (𝐴 ∩ (𝐵[,)+∞)) → (𝑦 ≤ 𝑥 → (𝐹‘𝑥) ≤ 𝑧)) ↔ (𝑥 ∈ 𝐴 → (if(𝐵 ≤ 𝑦, 𝑦, 𝐵) ≤ 𝑥 → (𝐹‘𝑥) ≤ 𝑧))))
36	35	ralbidv2 3182	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ)) → (∀𝑥 ∈ (𝐴 ∩ (𝐵[,)+∞))(𝑦 ≤ 𝑥 → (𝐹‘𝑥) ≤ 𝑧) ↔ ∀𝑥 ∈ 𝐴 (if(𝐵 ≤ 𝑦, 𝑦, 𝐵) ≤ 𝑥 → (𝐹‘𝑥) ≤ 𝑧)))
37	2	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ)) → 𝐹:𝐴⟶ℝ)
38		simprl 780	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ)) → 𝑦 ∈ ℝ)
39	20	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ)) → 𝐵 ∈ ℝ)
40	38, 39	ifcld 4528	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ)) → if(𝐵 ≤ 𝑦, 𝑦, 𝐵) ∈ ℝ)
41		simprr 782	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ)) → 𝑧 ∈ ℝ)
42		ello12r 15545	. . . . . . . 8 ⊢ (((𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) ∧ (if(𝐵 ≤ 𝑦, 𝑦, 𝐵) ∈ ℝ ∧ 𝑧 ∈ ℝ) ∧ ∀𝑥 ∈ 𝐴 (if(𝐵 ≤ 𝑦, 𝑦, 𝐵) ≤ 𝑥 → (𝐹‘𝑥) ≤ 𝑧)) → 𝐹 ∈ ≤𝑂(1))
43	42	3expia 1135	. . . . . . 7 ⊢ (((𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) ∧ (if(𝐵 ≤ 𝑦, 𝑦, 𝐵) ∈ ℝ ∧ 𝑧 ∈ ℝ)) → (∀𝑥 ∈ 𝐴 (if(𝐵 ≤ 𝑦, 𝑦, 𝐵) ≤ 𝑥 → (𝐹‘𝑥) ≤ 𝑧) → 𝐹 ∈ ≤𝑂(1)))
44	37, 22, 40, 41, 43	syl22anc 849	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ)) → (∀𝑥 ∈ 𝐴 (if(𝐵 ≤ 𝑦, 𝑦, 𝐵) ≤ 𝑥 → (𝐹‘𝑥) ≤ 𝑧) → 𝐹 ∈ ≤𝑂(1)))
45	36, 44	sylbid 242	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ)) → (∀𝑥 ∈ (𝐴 ∩ (𝐵[,)+∞))(𝑦 ≤ 𝑥 → (𝐹‘𝑥) ≤ 𝑧) → 𝐹 ∈ ≤𝑂(1)))
46	45	rexlimdvva 3220	. . . 4 ⊢ (𝜑 → (∃𝑦 ∈ ℝ ∃𝑧 ∈ ℝ ∀𝑥 ∈ (𝐴 ∩ (𝐵[,)+∞))(𝑦 ≤ 𝑥 → (𝐹‘𝑥) ≤ 𝑧) → 𝐹 ∈ ≤𝑂(1)))
47	14, 46	sylbid 242	. . 3 ⊢ (𝜑 → ((𝑥 ∈ (𝐴 ∩ (𝐵[,)+∞)) ↦ (𝐹‘𝑥)) ∈ ≤𝑂(1) → 𝐹 ∈ ≤𝑂(1)))
48	7, 47	sylbid 242	. 2 ⊢ (𝜑 → ((𝐹 ↾ (𝐵[,)+∞)) ∈ ≤𝑂(1) → 𝐹 ∈ ≤𝑂(1)))
49	1, 48	impbid2 228	1 ⊢ (𝜑 → (𝐹 ∈ ≤𝑂(1) ↔ (𝐹 ↾ (𝐵[,)+∞)) ∈ ≤𝑂(1)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∈ wcel 2143  ∀wral 3077  ∃wrex 3087   ∩ cin 3904   ⊆ wss 3905  ifcif 4481   class class class wbr 5101   ↦ cmpt 5182   ↾ cres 5650  ⟶wf 6518  ‘cfv 6522  (class class class)co 7397  ℝcr 11073  +∞cpnf 11214   ≤ cle 11218  [,)cico 13352  ≤𝑂(1)clo1 15515

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5247  ax-nul 5257  ax-pow 5323  ax-pr 5391  ax-un 7719  ax-cnex 11130  ax-resscn 11131  ax-pre-lttri 11148  ax-pre-lttrn 11149

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3063  df-ral 3078  df-rex 3088  df-rab 3416  df-v 3457  df-sbc 3746  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5102  df-opab 5164  df-mpt 5183  df-id 5543  df-po 5556  df-so 5557  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-iota 6478  df-fun 6524  df-fn 6525  df-f 6526  df-f1 6527  df-fo 6528  df-f1o 6529  df-fv 6530  df-ov 7400  df-oprab 7401  df-mpo 7402  df-er 8679  df-pm 8812  df-en 8929  df-dom 8930  df-sdom 8931  df-pnf 11219  df-mnf 11220  df-xr 11221  df-ltxr 11222  df-le 11223  df-ico 13356  df-lo1 15519

This theorem is referenced by:  lo1eq  15596