Theorem o1co 15521

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 6680	. . . . . 6 ⊢ (𝜑 → dom 𝐹 = 𝐴)
4		o1dm 15465	. . . . . . 7 ⊢ (𝐹 ∈ 𝑂(1) → dom 𝐹 ⊆ ℝ)
5	1, 4	syl 17	. . . . . 6 ⊢ (𝜑 → dom 𝐹 ⊆ ℝ)
6	3, 5	eqsstrrd 3971	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
7		elo12 15462	. . . . 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 585	. . . 4 ⊢ (𝜑 → (𝐹 ∈ 𝑂(1) ↔ ∃𝑚 ∈ ℝ ∃𝑛 ∈ ℝ ∀𝑧 ∈ 𝐴 (𝑚 ≤ 𝑧 → (abs‘(𝐹‘𝑧)) ≤ 𝑛)))
9	1, 8	mpbid 232	. . 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 3210	. . . . . 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 731	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℝ) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℝ) → 𝐺:𝐵⟶𝐴)
14	13	ffvelcdmda 7038	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑚 ∈ ℝ) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℝ) ∧ 𝑦 ∈ 𝐵) → (𝐺‘𝑦) ∈ 𝐴)
15		breq2 5104	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = (𝐺‘𝑦) → (𝑚 ≤ 𝑧 ↔ 𝑚 ≤ (𝐺‘𝑦)))
16		2fveq3 6847	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = (𝐺‘𝑦) → (abs‘(𝐹‘𝑧)) = (abs‘(𝐹‘(𝐺‘𝑦))))
17	16	breq1d 5110	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = (𝐺‘𝑦) → ((abs‘(𝐹‘𝑧)) ≤ 𝑛 ↔ (abs‘(𝐹‘(𝐺‘𝑦))) ≤ 𝑛))
18	15, 17	imbi12d 344	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (𝐺‘𝑦) → ((𝑚 ≤ 𝑧 → (abs‘(𝐹‘𝑧)) ≤ 𝑛) ↔ (𝑚 ≤ (𝐺‘𝑦) → (abs‘(𝐹‘(𝐺‘𝑦))) ≤ 𝑛)))
19	18	rspcva 3576	. . . . . . . . . . . . . . 15 ⊢ (((𝐺‘𝑦) ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 (𝑚 ≤ 𝑧 → (abs‘(𝐹‘𝑧)) ≤ 𝑛)) → (𝑚 ≤ (𝐺‘𝑦) → (abs‘(𝐹‘(𝐺‘𝑦))) ≤ 𝑛))
20	14, 19	sylan 581	. . . . . . . . . . . . . 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 653	. . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑚 ∈ ℝ) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℝ) ∧ ∀𝑧 ∈ 𝐴 (𝑚 ≤ 𝑧 → (abs‘(𝐹‘𝑧)) ≤ 𝑛)) → 𝐺:𝐵⟶𝐴)
23		fvco3 6941	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺:𝐵⟶𝐴 ∧ 𝑦 ∈ 𝐵) → ((𝐹 ∘ 𝐺)‘𝑦) = (𝐹‘(𝐺‘𝑦)))
24	22, 23	sylan 581	. . . . . . . . . . . . . . 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 6846	. . . . . . . . . . . . . 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 5110	. . . . . . . . . . . . 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 259	. . . . . . . . . . . 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 3150	. . . . . . . . . 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 453	. . . . . . . . 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 465	. . . . . . . 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 3151	. . . . . . 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 3151	. . . . . 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 243	. . . . 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 696	. . . 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 3139	. . 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 6694	. . . 4 ⊢ ((𝐹:𝐴⟶ℂ ∧ 𝐺:𝐵⟶𝐴) → (𝐹 ∘ 𝐺):𝐵⟶ℂ)
39	2, 12, 38	syl2anc 585	. . 3 ⊢ (𝜑 → (𝐹 ∘ 𝐺):𝐵⟶ℂ)
40		o1co.4	. . 3 ⊢ (𝜑 → 𝐵 ⊆ ℝ)
41		elo12 15462	. . 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 585	. 2 ⊢ (𝜑 → ((𝐹 ∘ 𝐺) ∈ 𝑂(1) ↔ ∃𝑥 ∈ ℝ ∃𝑛 ∈ ℝ ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → (abs‘((𝐹 ∘ 𝐺)‘𝑦)) ≤ 𝑛)))
43	37, 42	mpbird 257	1 ⊢ (𝜑 → (𝐹 ∘ 𝐺) ∈ 𝑂(1))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062   ⊆ wss 3903   class class class wbr 5100  dom cdm 5632   ∘ ccom 5636  ⟶wf 6496  ‘cfv 6500  ℂcc 11036  ℝcr 11037   ≤ cle 11179  abscabs 15169  𝑂(1)co1 15421

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-cnex 11094  ax-resscn 11095  ax-pre-lttri 11112  ax-pre-lttrn 11113

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-po 5540  df-so 5541  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-er 8645  df-pm 8778  df-en 8896  df-dom 8897  df-sdom 8898  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-le 11184  df-ico 13279  df-o1 15425

This theorem is referenced by:  o1compt  15522