Theorem o1co 15633

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 6716	. . . . . 6 ⊢ (𝜑 → dom 𝐹 = 𝐴)
4		o1dm 15577	. . . . . . 7 ⊢ (𝐹 ∈ 𝑂(1) → dom 𝐹 ⊆ ℝ)
5	1, 4	syl 18	. . . . . 6 ⊢ (𝜑 → dom 𝐹 ⊆ ℝ)
6	3, 5	eqsstrrd 3972	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
7		elo12 15574	. . . . 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 595	. . . 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 3237	. . . . . 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 742	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℝ) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℝ) → 𝐺:𝐵⟶𝐴)
14	13	ffvelcdmda 7079	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑚 ∈ ℝ) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℝ) ∧ 𝑦 ∈ 𝐵) → (𝐺‘𝑦) ∈ 𝐴)
15		breq2 5113	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = (𝐺‘𝑦) → (𝑚 ≤ 𝑧 ↔ 𝑚 ≤ (𝐺‘𝑦)))
16		2fveq3 6886	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = (𝐺‘𝑦) → (abs‘(𝐹‘𝑧)) = (abs‘(𝐹‘(𝐺‘𝑦))))
17	16	breq1d 5119	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = (𝐺‘𝑦) → ((abs‘(𝐹‘𝑧)) ≤ 𝑛 ↔ (abs‘(𝐹‘(𝐺‘𝑦))) ≤ 𝑛))
18	15, 17	imbi12d 347	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (𝐺‘𝑦) → ((𝑚 ≤ 𝑧 → (abs‘(𝐹‘𝑧)) ≤ 𝑛) ↔ (𝑚 ≤ (𝐺‘𝑦) → (abs‘(𝐹‘(𝐺‘𝑦))) ≤ 𝑛)))
19	18	rspcva 3579	. . . . . . . . . . . . . . 15 ⊢ (((𝐺‘𝑦) ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 (𝑚 ≤ 𝑧 → (abs‘(𝐹‘𝑧)) ≤ 𝑛)) → (𝑚 ≤ (𝐺‘𝑦) → (abs‘(𝐹‘(𝐺‘𝑦))) ≤ 𝑛))
20	14, 19	sylan 591	. . . . . . . . . . . . . 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 664	. . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑚 ∈ ℝ) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℝ) ∧ ∀𝑧 ∈ 𝐴 (𝑚 ≤ 𝑧 → (abs‘(𝐹‘𝑧)) ≤ 𝑛)) → 𝐺:𝐵⟶𝐴)
23		fvco3 6981	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺:𝐵⟶𝐴 ∧ 𝑦 ∈ 𝐵) → ((𝐹 ∘ 𝐺)‘𝑦) = (𝐹‘(𝐺‘𝑦)))
24	22, 23	sylan 591	. . . . . . . . . . . . . . 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 6885	. . . . . . . . . . . . . 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 5119	. . . . . . . . . . . . 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 58	. . . . . . . . . . 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 3177	. . . . . . . . . 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 458	. . . . . . . . 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 470	. . . . . . . 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 3178	. . . . . . 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 3178	. . . . . 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	biimtrrid 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 707	. . . 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 3166	. . 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 16	. 2 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∃𝑛 ∈ ℝ ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → (abs‘((𝐹 ∘ 𝐺)‘𝑦)) ≤ 𝑛))
38		fco 6730	. . . 4 ⊢ ((𝐹:𝐴⟶ℂ ∧ 𝐺:𝐵⟶𝐴) → (𝐹 ∘ 𝐺):𝐵⟶ℂ)
39	2, 12, 38	syl2anc 595	. . 3 ⊢ (𝜑 → (𝐹 ∘ 𝐺):𝐵⟶ℂ)
40		o1co.4	. . 3 ⊢ (𝜑 → 𝐵 ⊆ ℝ)
41		elo12 15574	. . 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 595	. 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 400   = wceq 1570   ∈ wcel 2143  ∀wral 3079  ∃wrex 3089   ⊆ wss 3905   class class class wbr 5109  dom cdm 5661   ∘ ccom 5665  ⟶wf 6532  ‘cfv 6536  ℂcc 11093  ℝcr 11094   ≤ cle 11239  abscabs 15281  𝑂(1)co1 15533

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-o1 15537

This theorem is referenced by:  o1compt  15634