Theorem o1co 15746

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 6718	. . . . . 6 ⊢ (𝜑 → dom 𝐹 = 𝐴)
4		o1dm 15690	. . . . . . 7 ⊢ (𝐹 ∈ 𝑂(1) → dom 𝐹 ⊆ ℝ)
5	1, 4	syl 18	. . . . . 6 ⊢ (𝜑 → dom 𝐹 ⊆ ℝ)
6	3, 5	eqsstrrd 3966	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
7		elo12 15687	. . . . 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 596	. . . 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 3235	. . . . . 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 743	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℝ) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℝ) → 𝐺:𝐵⟶𝐴)
14	13	ffvelcdmda 7082	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑚 ∈ ℝ) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℝ) ∧ 𝑦 ∈ 𝐵) → (𝐺‘𝑦) ∈ 𝐴)
15		breq2 5107	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = (𝐺‘𝑦) → (𝑚 ≤ 𝑧 ↔ 𝑚 ≤ (𝐺‘𝑦)))
16		2fveq3 6888	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = (𝐺‘𝑦) → (abs‘(𝐹‘𝑧)) = (abs‘(𝐹‘(𝐺‘𝑦))))
17	16	breq1d 5113	. . . . . . . . . . . . . . . . 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 3575	. . . . . . . . . . . . . . 15 ⊢ (((𝐺‘𝑦) ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 (𝑚 ≤ 𝑧 → (abs‘(𝐹‘𝑧)) ≤ 𝑛)) → (𝑚 ≤ (𝐺‘𝑦) → (abs‘(𝐹‘(𝐺‘𝑦))) ≤ 𝑛))
20	14, 19	sylan 592	. . . . . . . . . . . . . 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 665	. . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑚 ∈ ℝ) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℝ) ∧ ∀𝑧 ∈ 𝐴 (𝑚 ≤ 𝑧 → (abs‘(𝐹‘𝑧)) ≤ 𝑛)) → 𝐺:𝐵⟶𝐴)
23		fvco3 6983	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺:𝐵⟶𝐴 ∧ 𝑦 ∈ 𝐵) → ((𝐹 ∘ 𝐺)‘𝑦) = (𝐹‘(𝐺‘𝑦)))
24	22, 23	sylan 592	. . . . . . . . . . . . . . 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 6887	. . . . . . . . . . . . . 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 5113	. . . . . . . . . . . . 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 3175	. . . . . . . . . 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 459	. . . . . . . . 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 471	. . . . . . . 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 3176	. . . . . . 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 3176	. . . . . 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 708	. . . 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 3164	. . 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 6732	. . . 4 ⊢ ((𝐹:𝐴⟶ℂ ∧ 𝐺:𝐵⟶𝐴) → (𝐹 ∘ 𝐺):𝐵⟶ℂ)
39	2, 12, 38	syl2anc 596	. . 3 ⊢ (𝜑 → (𝐹 ∘ 𝐺):𝐵⟶ℂ)
40		o1co.4	. . 3 ⊢ (𝜑 → 𝐵 ⊆ ℝ)
41		elo12 15687	. . 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 596	. 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))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3077  ∃wrex 3087   ⊆ wss 3899   class class class wbr 5103  dom cdm 5651   ∘ ccom 5655  ⟶wf 6533  ‘cfv 6537  ℂcc 11191  ℝcr 11192   ≤ cle 11337  abscabs 15394  𝑂(1)co1 15646

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749  ax-cnex 11249  ax-resscn 11250  ax-pre-lttri 11267  ax-pre-lttrn 11268

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-po 5559  df-so 5560  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7421  df-oprab 7422  df-mpo 7423  df-er 8710  df-pm 8843  df-en 8967  df-dom 8968  df-sdom 8969  df-pnf 11338  df-mnf 11339  df-xr 11340  df-ltxr 11341  df-le 11342  df-ico 13475  df-o1 15650

This theorem is used by:  o1compt  15747