Theorem o1co 14935

Description: Sufficient condition for transforming the index set of an eventually bounded function. (Contributed by Mario Carneiro, 12-May-2016.)

Hypotheses

Ref	Expression
o1co.1	⊢ (𝜑 → 𝐹:𝐴⟶ℂ)
o1co.2	⊢ (𝜑 → 𝐹 ∈ 𝑂(1))
o1co.3	⊢ (𝜑 → 𝐺:𝐵⟶𝐴)
o1co.4	⊢ (𝜑 → 𝐵 ⊆ ℝ)
o1co.5	⊢ ((𝜑 ∧ 𝑚 ∈ ℝ) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → 𝑚 ≤ (𝐺‘𝑦)))

Assertion

Ref	Expression
o1co	⊢ (𝜑 → (𝐹 ∘ 𝐺) ∈ 𝑂(1))

Distinct variable groups:   𝑥,𝑚,𝑦,𝐴   𝑚,𝐹,𝑥,𝑦   𝑚,𝐺,𝑥,𝑦   𝜑,𝑚,𝑥,𝑦   𝐵,𝑚,𝑥,𝑦

Proof of Theorem o1co

Dummy variables 𝑛 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		o1co.2	. . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝑂(1))
2		o1co.1	. . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶ℂ)
3	2	fdmd 6497	. . . . . 6 ⊢ (𝜑 → dom 𝐹 = 𝐴)
4		o1dm 14879	. . . . . . 7 ⊢ (𝐹 ∈ 𝑂(1) → dom 𝐹 ⊆ ℝ)
5	1, 4	syl 17	. . . . . 6 ⊢ (𝜑 → dom 𝐹 ⊆ ℝ)
6	3, 5	eqsstrrd 3954	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
7		elo12 14876	. . . . 5 ⊢ ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℝ) → (𝐹 ∈ 𝑂(1) ↔ ∃𝑚 ∈ ℝ ∃𝑛 ∈ ℝ ∀𝑧 ∈ 𝐴 (𝑚 ≤ 𝑧 → (abs‘(𝐹‘𝑧)) ≤ 𝑛)))
8	2, 6, 7	syl2anc 587	. . . 4 ⊢ (𝜑 → (𝐹 ∈ 𝑂(1) ↔ ∃𝑚 ∈ ℝ ∃𝑛 ∈ ℝ ∀𝑧 ∈ 𝐴 (𝑚 ≤ 𝑧 → (abs‘(𝐹‘𝑧)) ≤ 𝑛)))
9	1, 8	mpbid 235	. . 3 ⊢ (𝜑 → ∃𝑚 ∈ ℝ ∃𝑛 ∈ ℝ ∀𝑧 ∈ 𝐴 (𝑚 ≤ 𝑧 → (abs‘(𝐹‘𝑧)) ≤ 𝑛))
10		o1co.5	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℝ) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → 𝑚 ≤ (𝐺‘𝑦)))
11		reeanv 3320	. . . . . 6 ⊢ (∃𝑥 ∈ ℝ ∃𝑛 ∈ ℝ (∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → 𝑚 ≤ (𝐺‘𝑦)) ∧ ∀𝑧 ∈ 𝐴 (𝑚 ≤ 𝑧 → (abs‘(𝐹‘𝑧)) ≤ 𝑛)) ↔ (∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → 𝑚 ≤ (𝐺‘𝑦)) ∧ ∃𝑛 ∈ ℝ ∀𝑧 ∈ 𝐴 (𝑚 ≤ 𝑧 → (abs‘(𝐹‘𝑧)) ≤ 𝑛)))
12		o1co.3	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐺:𝐵⟶𝐴)
13	12	ad3antrrr 729	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℝ) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℝ) → 𝐺:𝐵⟶𝐴)
14	13	ffvelrnda 6828	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑚 ∈ ℝ) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℝ) ∧ 𝑦 ∈ 𝐵) → (𝐺‘𝑦) ∈ 𝐴)
15		breq2 5034	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = (𝐺‘𝑦) → (𝑚 ≤ 𝑧 ↔ 𝑚 ≤ (𝐺‘𝑦)))
16		2fveq3 6650	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = (𝐺‘𝑦) → (abs‘(𝐹‘𝑧)) = (abs‘(𝐹‘(𝐺‘𝑦))))
17	16	breq1d 5040	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = (𝐺‘𝑦) → ((abs‘(𝐹‘𝑧)) ≤ 𝑛 ↔ (abs‘(𝐹‘(𝐺‘𝑦))) ≤ 𝑛))
18	15, 17	imbi12d 348	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (𝐺‘𝑦) → ((𝑚 ≤ 𝑧 → (abs‘(𝐹‘𝑧)) ≤ 𝑛) ↔ (𝑚 ≤ (𝐺‘𝑦) → (abs‘(𝐹‘(𝐺‘𝑦))) ≤ 𝑛)))
19	18	rspcva 3569	. . . . . . . . . . . . . . 15 ⊢ (((𝐺‘𝑦) ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 (𝑚 ≤ 𝑧 → (abs‘(𝐹‘𝑧)) ≤ 𝑛)) → (𝑚 ≤ (𝐺‘𝑦) → (abs‘(𝐹‘(𝐺‘𝑦))) ≤ 𝑛))
20	14, 19	sylan 583	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑚 ∈ ℝ) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℝ) ∧ 𝑦 ∈ 𝐵) ∧ ∀𝑧 ∈ 𝐴 (𝑚 ≤ 𝑧 → (abs‘(𝐹‘𝑧)) ≤ 𝑛)) → (𝑚 ≤ (𝐺‘𝑦) → (abs‘(𝐹‘(𝐺‘𝑦))) ≤ 𝑛))
21	20	an32s 651	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑚 ∈ ℝ) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℝ) ∧ ∀𝑧 ∈ 𝐴 (𝑚 ≤ 𝑧 → (abs‘(𝐹‘𝑧)) ≤ 𝑛)) ∧ 𝑦 ∈ 𝐵) → (𝑚 ≤ (𝐺‘𝑦) → (abs‘(𝐹‘(𝐺‘𝑦))) ≤ 𝑛))
22	13	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑚 ∈ ℝ) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℝ) ∧ ∀𝑧 ∈ 𝐴 (𝑚 ≤ 𝑧 → (abs‘(𝐹‘𝑧)) ≤ 𝑛)) → 𝐺:𝐵⟶𝐴)
23		fvco3 6737	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺:𝐵⟶𝐴 ∧ 𝑦 ∈ 𝐵) → ((𝐹 ∘ 𝐺)‘𝑦) = (𝐹‘(𝐺‘𝑦)))
24	22, 23	sylan 583	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑚 ∈ ℝ) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℝ) ∧ ∀𝑧 ∈ 𝐴 (𝑚 ≤ 𝑧 → (abs‘(𝐹‘𝑧)) ≤ 𝑛)) ∧ 𝑦 ∈ 𝐵) → ((𝐹 ∘ 𝐺)‘𝑦) = (𝐹‘(𝐺‘𝑦)))
25	24	fveq2d 6649	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑚 ∈ ℝ) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℝ) ∧ ∀𝑧 ∈ 𝐴 (𝑚 ≤ 𝑧 → (abs‘(𝐹‘𝑧)) ≤ 𝑛)) ∧ 𝑦 ∈ 𝐵) → (abs‘((𝐹 ∘ 𝐺)‘𝑦)) = (abs‘(𝐹‘(𝐺‘𝑦))))
26	25	breq1d 5040	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑚 ∈ ℝ) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℝ) ∧ ∀𝑧 ∈ 𝐴 (𝑚 ≤ 𝑧 → (abs‘(𝐹‘𝑧)) ≤ 𝑛)) ∧ 𝑦 ∈ 𝐵) → ((abs‘((𝐹 ∘ 𝐺)‘𝑦)) ≤ 𝑛 ↔ (abs‘(𝐹‘(𝐺‘𝑦))) ≤ 𝑛))
27	21, 26	sylibrd 262	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑚 ∈ ℝ) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℝ) ∧ ∀𝑧 ∈ 𝐴 (𝑚 ≤ 𝑧 → (abs‘(𝐹‘𝑧)) ≤ 𝑛)) ∧ 𝑦 ∈ 𝐵) → (𝑚 ≤ (𝐺‘𝑦) → (abs‘((𝐹 ∘ 𝐺)‘𝑦)) ≤ 𝑛))
28	27	imim2d 57	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑚 ∈ ℝ) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℝ) ∧ ∀𝑧 ∈ 𝐴 (𝑚 ≤ 𝑧 → (abs‘(𝐹‘𝑧)) ≤ 𝑛)) ∧ 𝑦 ∈ 𝐵) → ((𝑥 ≤ 𝑦 → 𝑚 ≤ (𝐺‘𝑦)) → (𝑥 ≤ 𝑦 → (abs‘((𝐹 ∘ 𝐺)‘𝑦)) ≤ 𝑛)))
29	28	ralimdva 3144	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑚 ∈ ℝ) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℝ) ∧ ∀𝑧 ∈ 𝐴 (𝑚 ≤ 𝑧 → (abs‘(𝐹‘𝑧)) ≤ 𝑛)) → (∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → 𝑚 ≤ (𝐺‘𝑦)) → ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → (abs‘((𝐹 ∘ 𝐺)‘𝑦)) ≤ 𝑛)))
30	29	expimpd 457	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℝ) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℝ) → ((∀𝑧 ∈ 𝐴 (𝑚 ≤ 𝑧 → (abs‘(𝐹‘𝑧)) ≤ 𝑛) ∧ ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → 𝑚 ≤ (𝐺‘𝑦))) → ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → (abs‘((𝐹 ∘ 𝐺)‘𝑦)) ≤ 𝑛)))
31	30	ancomsd 469	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℝ) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℝ) → ((∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → 𝑚 ≤ (𝐺‘𝑦)) ∧ ∀𝑧 ∈ 𝐴 (𝑚 ≤ 𝑧 → (abs‘(𝐹‘𝑧)) ≤ 𝑛)) → ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → (abs‘((𝐹 ∘ 𝐺)‘𝑦)) ≤ 𝑛)))
32	31	reximdva 3233	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℝ) ∧ 𝑥 ∈ ℝ) → (∃𝑛 ∈ ℝ (∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → 𝑚 ≤ (𝐺‘𝑦)) ∧ ∀𝑧 ∈ 𝐴 (𝑚 ≤ 𝑧 → (abs‘(𝐹‘𝑧)) ≤ 𝑛)) → ∃𝑛 ∈ ℝ ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → (abs‘((𝐹 ∘ 𝐺)‘𝑦)) ≤ 𝑛)))
33	32	reximdva 3233	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℝ) → (∃𝑥 ∈ ℝ ∃𝑛 ∈ ℝ (∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → 𝑚 ≤ (𝐺‘𝑦)) ∧ ∀𝑧 ∈ 𝐴 (𝑚 ≤ 𝑧 → (abs‘(𝐹‘𝑧)) ≤ 𝑛)) → ∃𝑥 ∈ ℝ ∃𝑛 ∈ ℝ ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → (abs‘((𝐹 ∘ 𝐺)‘𝑦)) ≤ 𝑛)))
34	11, 33	syl5bir 246	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℝ) → ((∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → 𝑚 ≤ (𝐺‘𝑦)) ∧ ∃𝑛 ∈ ℝ ∀𝑧 ∈ 𝐴 (𝑚 ≤ 𝑧 → (abs‘(𝐹‘𝑧)) ≤ 𝑛)) → ∃𝑥 ∈ ℝ ∃𝑛 ∈ ℝ ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → (abs‘((𝐹 ∘ 𝐺)‘𝑦)) ≤ 𝑛)))
35	10, 34	mpand 694	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℝ) → (∃𝑛 ∈ ℝ ∀𝑧 ∈ 𝐴 (𝑚 ≤ 𝑧 → (abs‘(𝐹‘𝑧)) ≤ 𝑛) → ∃𝑥 ∈ ℝ ∃𝑛 ∈ ℝ ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → (abs‘((𝐹 ∘ 𝐺)‘𝑦)) ≤ 𝑛)))
36	35	rexlimdva 3243	. . 3 ⊢ (𝜑 → (∃𝑚 ∈ ℝ ∃𝑛 ∈ ℝ ∀𝑧 ∈ 𝐴 (𝑚 ≤ 𝑧 → (abs‘(𝐹‘𝑧)) ≤ 𝑛) → ∃𝑥 ∈ ℝ ∃𝑛 ∈ ℝ ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → (abs‘((𝐹 ∘ 𝐺)‘𝑦)) ≤ 𝑛)))
37	9, 36	mpd 15	. 2 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∃𝑛 ∈ ℝ ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → (abs‘((𝐹 ∘ 𝐺)‘𝑦)) ≤ 𝑛))
38		fco 6505	. . . 4 ⊢ ((𝐹:𝐴⟶ℂ ∧ 𝐺:𝐵⟶𝐴) → (𝐹 ∘ 𝐺):𝐵⟶ℂ)
39	2, 12, 38	syl2anc 587	. . 3 ⊢ (𝜑 → (𝐹 ∘ 𝐺):𝐵⟶ℂ)
40		o1co.4	. . 3 ⊢ (𝜑 → 𝐵 ⊆ ℝ)
41		elo12 14876	. . 3 ⊢ (((𝐹 ∘ 𝐺):𝐵⟶ℂ ∧ 𝐵 ⊆ ℝ) → ((𝐹 ∘ 𝐺) ∈ 𝑂(1) ↔ ∃𝑥 ∈ ℝ ∃𝑛 ∈ ℝ ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → (abs‘((𝐹 ∘ 𝐺)‘𝑦)) ≤ 𝑛)))
42	39, 40, 41	syl2anc 587	. 2 ⊢ (𝜑 → ((𝐹 ∘ 𝐺) ∈ 𝑂(1) ↔ ∃𝑥 ∈ ℝ ∃𝑛 ∈ ℝ ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → (abs‘((𝐹 ∘ 𝐺)‘𝑦)) ≤ 𝑛)))
43	37, 42	mpbird 260	1 ⊢ (𝜑 → (𝐹 ∘ 𝐺) ∈ 𝑂(1))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107   ⊆ wss 3881   class class class wbr 5030  dom cdm 5519   ∘ ccom 5523  ⟶wf 6320  ‘cfv 6324  ℂcc 10524  ℝcr 10525   ≤ cle 10665  abscabs 14585  𝑂(1)co1 14835

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-o1 14839

This theorem is referenced by:  o1compt  14936