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Theorem o1co 15622
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 (𝜑𝐹:𝐴⟶ℂ)
o1co.2 (𝜑𝐹 ∈ 𝑂(1))
o1co.3 (𝜑𝐺:𝐵𝐴)
o1co.4 (𝜑𝐵 ⊆ ℝ)
o1co.5 ((𝜑𝑚 ∈ ℝ) → ∃𝑥 ∈ ℝ ∀𝑦𝐵 (𝑥𝑦𝑚 ≤ (𝐺𝑦)))
Assertion
Ref Expression
o1co (𝜑 → (𝐹𝐺) ∈ 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.
StepHypRef Expression
1 o1co.2 . . . 4 (𝜑𝐹 ∈ 𝑂(1))
2 o1co.1 . . . . 5 (𝜑𝐹:𝐴⟶ℂ)
32fdmd 6746 . . . . . 6 (𝜑 → dom 𝐹 = 𝐴)
4 o1dm 15566 . . . . . . 7 (𝐹 ∈ 𝑂(1) → dom 𝐹 ⊆ ℝ)
51, 4syl 17 . . . . . 6 (𝜑 → dom 𝐹 ⊆ ℝ)
63, 5eqsstrrd 4019 . . . . 5 (𝜑𝐴 ⊆ ℝ)
7 elo12 15563 . . . . 5 ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℝ) → (𝐹 ∈ 𝑂(1) ↔ ∃𝑚 ∈ ℝ ∃𝑛 ∈ ℝ ∀𝑧𝐴 (𝑚𝑧 → (abs‘(𝐹𝑧)) ≤ 𝑛)))
82, 6, 7syl2anc 584 . . . 4 (𝜑 → (𝐹 ∈ 𝑂(1) ↔ ∃𝑚 ∈ ℝ ∃𝑛 ∈ ℝ ∀𝑧𝐴 (𝑚𝑧 → (abs‘(𝐹𝑧)) ≤ 𝑛)))
91, 8mpbid 232 . . 3 (𝜑 → ∃𝑚 ∈ ℝ ∃𝑛 ∈ ℝ ∀𝑧𝐴 (𝑚𝑧 → (abs‘(𝐹𝑧)) ≤ 𝑛))
10 o1co.5 . . . . 5 ((𝜑𝑚 ∈ ℝ) → ∃𝑥 ∈ ℝ ∀𝑦𝐵 (𝑥𝑦𝑚 ≤ (𝐺𝑦)))
11 reeanv 3229 . . . . . 6 (∃𝑥 ∈ ℝ ∃𝑛 ∈ ℝ (∀𝑦𝐵 (𝑥𝑦𝑚 ≤ (𝐺𝑦)) ∧ ∀𝑧𝐴 (𝑚𝑧 → (abs‘(𝐹𝑧)) ≤ 𝑛)) ↔ (∃𝑥 ∈ ℝ ∀𝑦𝐵 (𝑥𝑦𝑚 ≤ (𝐺𝑦)) ∧ ∃𝑛 ∈ ℝ ∀𝑧𝐴 (𝑚𝑧 → (abs‘(𝐹𝑧)) ≤ 𝑛)))
12 o1co.3 . . . . . . . . . . . . . . . . 17 (𝜑𝐺:𝐵𝐴)
1312ad3antrrr 730 . . . . . . . . . . . . . . . 16 ((((𝜑𝑚 ∈ ℝ) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℝ) → 𝐺:𝐵𝐴)
1413ffvelcdmda 7104 . . . . . . . . . . . . . . 15 (((((𝜑𝑚 ∈ ℝ) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℝ) ∧ 𝑦𝐵) → (𝐺𝑦) ∈ 𝐴)
15 breq2 5147 . . . . . . . . . . . . . . . . 17 (𝑧 = (𝐺𝑦) → (𝑚𝑧𝑚 ≤ (𝐺𝑦)))
16 2fveq3 6911 . . . . . . . . . . . . . . . . . 18 (𝑧 = (𝐺𝑦) → (abs‘(𝐹𝑧)) = (abs‘(𝐹‘(𝐺𝑦))))
1716breq1d 5153 . . . . . . . . . . . . . . . . 17 (𝑧 = (𝐺𝑦) → ((abs‘(𝐹𝑧)) ≤ 𝑛 ↔ (abs‘(𝐹‘(𝐺𝑦))) ≤ 𝑛))
1815, 17imbi12d 344 . . . . . . . . . . . . . . . 16 (𝑧 = (𝐺𝑦) → ((𝑚𝑧 → (abs‘(𝐹𝑧)) ≤ 𝑛) ↔ (𝑚 ≤ (𝐺𝑦) → (abs‘(𝐹‘(𝐺𝑦))) ≤ 𝑛)))
1918rspcva 3620 . . . . . . . . . . . . . . 15 (((𝐺𝑦) ∈ 𝐴 ∧ ∀𝑧𝐴 (𝑚𝑧 → (abs‘(𝐹𝑧)) ≤ 𝑛)) → (𝑚 ≤ (𝐺𝑦) → (abs‘(𝐹‘(𝐺𝑦))) ≤ 𝑛))
2014, 19sylan 580 . . . . . . . . . . . . . 14 ((((((𝜑𝑚 ∈ ℝ) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℝ) ∧ 𝑦𝐵) ∧ ∀𝑧𝐴 (𝑚𝑧 → (abs‘(𝐹𝑧)) ≤ 𝑛)) → (𝑚 ≤ (𝐺𝑦) → (abs‘(𝐹‘(𝐺𝑦))) ≤ 𝑛))
2120an32s 652 . . . . . . . . . . . . 13 ((((((𝜑𝑚 ∈ ℝ) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℝ) ∧ ∀𝑧𝐴 (𝑚𝑧 → (abs‘(𝐹𝑧)) ≤ 𝑛)) ∧ 𝑦𝐵) → (𝑚 ≤ (𝐺𝑦) → (abs‘(𝐹‘(𝐺𝑦))) ≤ 𝑛))
2213adantr 480 . . . . . . . . . . . . . . . 16 (((((𝜑𝑚 ∈ ℝ) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℝ) ∧ ∀𝑧𝐴 (𝑚𝑧 → (abs‘(𝐹𝑧)) ≤ 𝑛)) → 𝐺:𝐵𝐴)
23 fvco3 7008 . . . . . . . . . . . . . . . 16 ((𝐺:𝐵𝐴𝑦𝐵) → ((𝐹𝐺)‘𝑦) = (𝐹‘(𝐺𝑦)))
2422, 23sylan 580 . . . . . . . . . . . . . . 15 ((((((𝜑𝑚 ∈ ℝ) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℝ) ∧ ∀𝑧𝐴 (𝑚𝑧 → (abs‘(𝐹𝑧)) ≤ 𝑛)) ∧ 𝑦𝐵) → ((𝐹𝐺)‘𝑦) = (𝐹‘(𝐺𝑦)))
2524fveq2d 6910 . . . . . . . . . . . . . 14 ((((((𝜑𝑚 ∈ ℝ) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℝ) ∧ ∀𝑧𝐴 (𝑚𝑧 → (abs‘(𝐹𝑧)) ≤ 𝑛)) ∧ 𝑦𝐵) → (abs‘((𝐹𝐺)‘𝑦)) = (abs‘(𝐹‘(𝐺𝑦))))
2625breq1d 5153 . . . . . . . . . . . . 13 ((((((𝜑𝑚 ∈ ℝ) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℝ) ∧ ∀𝑧𝐴 (𝑚𝑧 → (abs‘(𝐹𝑧)) ≤ 𝑛)) ∧ 𝑦𝐵) → ((abs‘((𝐹𝐺)‘𝑦)) ≤ 𝑛 ↔ (abs‘(𝐹‘(𝐺𝑦))) ≤ 𝑛))
2721, 26sylibrd 259 . . . . . . . . . . . 12 ((((((𝜑𝑚 ∈ ℝ) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℝ) ∧ ∀𝑧𝐴 (𝑚𝑧 → (abs‘(𝐹𝑧)) ≤ 𝑛)) ∧ 𝑦𝐵) → (𝑚 ≤ (𝐺𝑦) → (abs‘((𝐹𝐺)‘𝑦)) ≤ 𝑛))
2827imim2d 57 . . . . . . . . . . 11 ((((((𝜑𝑚 ∈ ℝ) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℝ) ∧ ∀𝑧𝐴 (𝑚𝑧 → (abs‘(𝐹𝑧)) ≤ 𝑛)) ∧ 𝑦𝐵) → ((𝑥𝑦𝑚 ≤ (𝐺𝑦)) → (𝑥𝑦 → (abs‘((𝐹𝐺)‘𝑦)) ≤ 𝑛)))
2928ralimdva 3167 . . . . . . . . . 10 (((((𝜑𝑚 ∈ ℝ) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℝ) ∧ ∀𝑧𝐴 (𝑚𝑧 → (abs‘(𝐹𝑧)) ≤ 𝑛)) → (∀𝑦𝐵 (𝑥𝑦𝑚 ≤ (𝐺𝑦)) → ∀𝑦𝐵 (𝑥𝑦 → (abs‘((𝐹𝐺)‘𝑦)) ≤ 𝑛)))
3029expimpd 453 . . . . . . . . 9 ((((𝜑𝑚 ∈ ℝ) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℝ) → ((∀𝑧𝐴 (𝑚𝑧 → (abs‘(𝐹𝑧)) ≤ 𝑛) ∧ ∀𝑦𝐵 (𝑥𝑦𝑚 ≤ (𝐺𝑦))) → ∀𝑦𝐵 (𝑥𝑦 → (abs‘((𝐹𝐺)‘𝑦)) ≤ 𝑛)))
3130ancomsd 465 . . . . . . . 8 ((((𝜑𝑚 ∈ ℝ) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℝ) → ((∀𝑦𝐵 (𝑥𝑦𝑚 ≤ (𝐺𝑦)) ∧ ∀𝑧𝐴 (𝑚𝑧 → (abs‘(𝐹𝑧)) ≤ 𝑛)) → ∀𝑦𝐵 (𝑥𝑦 → (abs‘((𝐹𝐺)‘𝑦)) ≤ 𝑛)))
3231reximdva 3168 . . . . . . 7 (((𝜑𝑚 ∈ ℝ) ∧ 𝑥 ∈ ℝ) → (∃𝑛 ∈ ℝ (∀𝑦𝐵 (𝑥𝑦𝑚 ≤ (𝐺𝑦)) ∧ ∀𝑧𝐴 (𝑚𝑧 → (abs‘(𝐹𝑧)) ≤ 𝑛)) → ∃𝑛 ∈ ℝ ∀𝑦𝐵 (𝑥𝑦 → (abs‘((𝐹𝐺)‘𝑦)) ≤ 𝑛)))
3332reximdva 3168 . . . . . 6 ((𝜑𝑚 ∈ ℝ) → (∃𝑥 ∈ ℝ ∃𝑛 ∈ ℝ (∀𝑦𝐵 (𝑥𝑦𝑚 ≤ (𝐺𝑦)) ∧ ∀𝑧𝐴 (𝑚𝑧 → (abs‘(𝐹𝑧)) ≤ 𝑛)) → ∃𝑥 ∈ ℝ ∃𝑛 ∈ ℝ ∀𝑦𝐵 (𝑥𝑦 → (abs‘((𝐹𝐺)‘𝑦)) ≤ 𝑛)))
3411, 33biimtrrid 243 . . . . 5 ((𝜑𝑚 ∈ ℝ) → ((∃𝑥 ∈ ℝ ∀𝑦𝐵 (𝑥𝑦𝑚 ≤ (𝐺𝑦)) ∧ ∃𝑛 ∈ ℝ ∀𝑧𝐴 (𝑚𝑧 → (abs‘(𝐹𝑧)) ≤ 𝑛)) → ∃𝑥 ∈ ℝ ∃𝑛 ∈ ℝ ∀𝑦𝐵 (𝑥𝑦 → (abs‘((𝐹𝐺)‘𝑦)) ≤ 𝑛)))
3510, 34mpand 695 . . . 4 ((𝜑𝑚 ∈ ℝ) → (∃𝑛 ∈ ℝ ∀𝑧𝐴 (𝑚𝑧 → (abs‘(𝐹𝑧)) ≤ 𝑛) → ∃𝑥 ∈ ℝ ∃𝑛 ∈ ℝ ∀𝑦𝐵 (𝑥𝑦 → (abs‘((𝐹𝐺)‘𝑦)) ≤ 𝑛)))
3635rexlimdva 3155 . . 3 (𝜑 → (∃𝑚 ∈ ℝ ∃𝑛 ∈ ℝ ∀𝑧𝐴 (𝑚𝑧 → (abs‘(𝐹𝑧)) ≤ 𝑛) → ∃𝑥 ∈ ℝ ∃𝑛 ∈ ℝ ∀𝑦𝐵 (𝑥𝑦 → (abs‘((𝐹𝐺)‘𝑦)) ≤ 𝑛)))
379, 36mpd 15 . 2 (𝜑 → ∃𝑥 ∈ ℝ ∃𝑛 ∈ ℝ ∀𝑦𝐵 (𝑥𝑦 → (abs‘((𝐹𝐺)‘𝑦)) ≤ 𝑛))
38 fco 6760 . . . 4 ((𝐹:𝐴⟶ℂ ∧ 𝐺:𝐵𝐴) → (𝐹𝐺):𝐵⟶ℂ)
392, 12, 38syl2anc 584 . . 3 (𝜑 → (𝐹𝐺):𝐵⟶ℂ)
40 o1co.4 . . 3 (𝜑𝐵 ⊆ ℝ)
41 elo12 15563 . . 3 (((𝐹𝐺):𝐵⟶ℂ ∧ 𝐵 ⊆ ℝ) → ((𝐹𝐺) ∈ 𝑂(1) ↔ ∃𝑥 ∈ ℝ ∃𝑛 ∈ ℝ ∀𝑦𝐵 (𝑥𝑦 → (abs‘((𝐹𝐺)‘𝑦)) ≤ 𝑛)))
4239, 40, 41syl2anc 584 . 2 (𝜑 → ((𝐹𝐺) ∈ 𝑂(1) ↔ ∃𝑥 ∈ ℝ ∃𝑛 ∈ ℝ ∀𝑦𝐵 (𝑥𝑦 → (abs‘((𝐹𝐺)‘𝑦)) ≤ 𝑛)))
4337, 42mpbird 257 1 (𝜑 → (𝐹𝐺) ∈ 𝑂(1))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wral 3061  wrex 3070  wss 3951   class class class wbr 5143  dom cdm 5685  ccom 5689  wf 6557  cfv 6561  cc 11153  cr 11154  cle 11296  abscabs 15273  𝑂(1)co1 15522
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-pre-lttri 11229  ax-pre-lttrn 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-po 5592  df-so 5593  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-er 8745  df-pm 8869  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-ico 13393  df-o1 15526
This theorem is referenced by:  o1compt  15623
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