Theorem lo1resb 15611

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 15606	. 2 ⊢ (𝐹 ∈ ≤𝑂(1) → (𝐹 ↾ (𝐵[,)+∞)) ∈ ≤𝑂(1))
2		lo1resb.1	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ)
3	2	feqmptd 6949	. . . . . 6 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))
4	3	reseq1d 5977	. . . . 5 ⊢ (𝜑 → (𝐹 ↾ (𝐵[,)+∞)) = ((𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) ↾ (𝐵[,)+∞)))
5		resmpt3 6040	. . . . 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 7076	. . . . . 6 ⊢ ((𝐹:𝐴⟶ℝ ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ ℝ)
13	2, 11, 12	syl2an 607	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ (𝐵[,)+∞))) → (𝐹‘𝑥) ∈ ℝ)
14	10, 13	ello1mpt 15568	. . . 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 352	. . . . . . . . 9 ⊢ ((𝑥 ∈ (𝐴 ∩ (𝐵[,)+∞)) → (𝑦 ≤ 𝑥 → (𝐹‘𝑥) ≤ 𝑧)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵[,)+∞)) → (𝑦 ≤ 𝑥 → (𝐹‘𝑥) ≤ 𝑧)))
17		impexp 455	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵[,)+∞)) → (𝑦 ≤ 𝑥 → (𝐹‘𝑥) ≤ 𝑧)) ↔ (𝑥 ∈ 𝐴 → (𝑥 ∈ (𝐵[,)+∞) → (𝑦 ≤ 𝑥 → (𝐹‘𝑥) ≤ 𝑧))))
18	16, 17	bitri 278	. . . . . . . 8 ⊢ ((𝑥 ∈ (𝐴 ∩ (𝐵[,)+∞)) → (𝑦 ≤ 𝑥 → (𝐹‘𝑥) ≤ 𝑧)) ↔ (𝑥 ∈ 𝐴 → (𝑥 ∈ (𝐵[,)+∞) → (𝑦 ≤ 𝑥 → (𝐹‘𝑥) ≤ 𝑧))))
19		impexp 455	. . . . . . . . . 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 738	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ)) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)
22	9	adantr 485	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ)) → 𝐴 ⊆ ℝ)
23	22	sselda 3937	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ)) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ)
24		elicopnf 13467	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ ℝ → (𝑥 ∈ (𝐵[,)+∞) ↔ (𝑥 ∈ ℝ ∧ 𝐵 ≤ 𝑥)))
25	24	baibd 548	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝑥 ∈ (𝐵[,)+∞) ↔ 𝐵 ≤ 𝑥))
26	21, 23, 25	syl2anc 595	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ)) ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ (𝐵[,)+∞) ↔ 𝐵 ≤ 𝑥))
27	26	anbi1d 642	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ)) ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ (𝐵[,)+∞) ∧ 𝑦 ≤ 𝑥) ↔ (𝐵 ≤ 𝑥 ∧ 𝑦 ≤ 𝑥)))
28		simplrl 788	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ)) ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ ℝ)
29		maxle 13212	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (if(𝐵 ≤ 𝑦, 𝑦, 𝐵) ≤ 𝑥 ↔ (𝐵 ≤ 𝑥 ∧ 𝑦 ≤ 𝑥)))
30	21, 28, 23, 29	syl3anc 1398	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ)) ∧ 𝑥 ∈ 𝐴) → (if(𝐵 ≤ 𝑦, 𝑦, 𝐵) ≤ 𝑥 ↔ (𝐵 ≤ 𝑥 ∧ 𝑦 ≤ 𝑥)))
31	27, 30	bitr4d 285	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ)) ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ (𝐵[,)+∞) ∧ 𝑦 ≤ 𝑥) ↔ if(𝐵 ≤ 𝑦, 𝑦, 𝐵) ≤ 𝑥))
32	31	imbi1d 344	. . . . . . . . . 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 288	. . . . . . . . 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 815	. . . . . . . 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 286	. . . . . . 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 3184	. . . . . 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 485	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ)) → 𝐹:𝐴⟶ℝ)
38		simprl 782	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ)) → 𝑦 ∈ ℝ)
39	20	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ)) → 𝐵 ∈ ℝ)
40	38, 39	ifcld 4534	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ)) → if(𝐵 ≤ 𝑦, 𝑦, 𝐵) ∈ ℝ)
41		simprr 784	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ)) → 𝑧 ∈ ℝ)
42		ello12r 15564	. . . . . . . 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 1139	. . . . . . 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 851	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ)) → (∀𝑥 ∈ 𝐴 (if(𝐵 ≤ 𝑦, 𝑦, 𝐵) ≤ 𝑥 → (𝐹‘𝑥) ≤ 𝑧) → 𝐹 ∈ ≤𝑂(1)))
45	36, 44	sylbid 243	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ)) → (∀𝑥 ∈ (𝐴 ∩ (𝐵[,)+∞))(𝑦 ≤ 𝑥 → (𝐹‘𝑥) ≤ 𝑧) → 𝐹 ∈ ≤𝑂(1)))
46	45	rexlimdvva 3222	. . . 4 ⊢ (𝜑 → (∃𝑦 ∈ ℝ ∃𝑧 ∈ ℝ ∀𝑥 ∈ (𝐴 ∩ (𝐵[,)+∞))(𝑦 ≤ 𝑥 → (𝐹‘𝑥) ≤ 𝑧) → 𝐹 ∈ ≤𝑂(1)))
47	14, 46	sylbid 243	. . 3 ⊢ (𝜑 → ((𝑥 ∈ (𝐴 ∩ (𝐵[,)+∞)) ↦ (𝐹‘𝑥)) ∈ ≤𝑂(1) → 𝐹 ∈ ≤𝑂(1)))
48	7, 47	sylbid 243	. 2 ⊢ (𝜑 → ((𝐹 ↾ (𝐵[,)+∞)) ∈ ≤𝑂(1) → 𝐹 ∈ ≤𝑂(1)))
49	1, 48	impbid2 229	1 ⊢ (𝜑 → (𝐹 ∈ ≤𝑂(1) ↔ (𝐹 ↾ (𝐵[,)+∞)) ∈ ≤𝑂(1)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∈ wcel 2143  ∀wral 3079  ∃wrex 3089   ∩ cin 3904   ⊆ wss 3905  ifcif 4487   class class class wbr 5109   ↦ cmpt 5192   ↾ cres 5663  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410  ℝcr 11094  +∞cpnf 11235   ≤ cle 11239  [,)cico 13369  ≤𝑂(1)clo1 15534

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11151  ax-resscn 11152  ax-pre-lttri 11169  ax-pre-lttrn 11170

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-po 5569  df-so 5570  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-er 8690  df-pm 8823  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11240  df-mnf 11241  df-xr 11242  df-ltxr 11243  df-le 11244  df-ico 13373  df-lo1 15538

This theorem is referenced by:  lo1eq  15615