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Theorem o1co 14701
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 6291 . . . . . 6 (𝜑 → dom 𝐹 = 𝐴)
4 o1dm 14645 . . . . . . 7 (𝐹 ∈ 𝑂(1) → dom 𝐹 ⊆ ℝ)
51, 4syl 17 . . . . . 6 (𝜑 → dom 𝐹 ⊆ ℝ)
63, 5eqsstr3d 3865 . . . . 5 (𝜑𝐴 ⊆ ℝ)
7 elo12 14642 . . . . 5 ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℝ) → (𝐹 ∈ 𝑂(1) ↔ ∃𝑚 ∈ ℝ ∃𝑛 ∈ ℝ ∀𝑧𝐴 (𝑚𝑧 → (abs‘(𝐹𝑧)) ≤ 𝑛)))
82, 6, 7syl2anc 579 . . . 4 (𝜑 → (𝐹 ∈ 𝑂(1) ↔ ∃𝑚 ∈ ℝ ∃𝑛 ∈ ℝ ∀𝑧𝐴 (𝑚𝑧 → (abs‘(𝐹𝑧)) ≤ 𝑛)))
91, 8mpbid 224 . . 3 (𝜑 → ∃𝑚 ∈ ℝ ∃𝑛 ∈ ℝ ∀𝑧𝐴 (𝑚𝑧 → (abs‘(𝐹𝑧)) ≤ 𝑛))
10 o1co.5 . . . . 5 ((𝜑𝑚 ∈ ℝ) → ∃𝑥 ∈ ℝ ∀𝑦𝐵 (𝑥𝑦𝑚 ≤ (𝐺𝑦)))
11 reeanv 3317 . . . . . 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 721 . . . . . . . . . . . . . . . 16 ((((𝜑𝑚 ∈ ℝ) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℝ) → 𝐺:𝐵𝐴)
1413ffvelrnda 6613 . . . . . . . . . . . . . . 15 (((((𝜑𝑚 ∈ ℝ) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℝ) ∧ 𝑦𝐵) → (𝐺𝑦) ∈ 𝐴)
15 breq2 4879 . . . . . . . . . . . . . . . . 17 (𝑧 = (𝐺𝑦) → (𝑚𝑧𝑚 ≤ (𝐺𝑦)))
16 2fveq3 6442 . . . . . . . . . . . . . . . . . 18 (𝑧 = (𝐺𝑦) → (abs‘(𝐹𝑧)) = (abs‘(𝐹‘(𝐺𝑦))))
1716breq1d 4885 . . . . . . . . . . . . . . . . 17 (𝑧 = (𝐺𝑦) → ((abs‘(𝐹𝑧)) ≤ 𝑛 ↔ (abs‘(𝐹‘(𝐺𝑦))) ≤ 𝑛))
1815, 17imbi12d 336 . . . . . . . . . . . . . . . 16 (𝑧 = (𝐺𝑦) → ((𝑚𝑧 → (abs‘(𝐹𝑧)) ≤ 𝑛) ↔ (𝑚 ≤ (𝐺𝑦) → (abs‘(𝐹‘(𝐺𝑦))) ≤ 𝑛)))
1918rspcva 3524 . . . . . . . . . . . . . . 15 (((𝐺𝑦) ∈ 𝐴 ∧ ∀𝑧𝐴 (𝑚𝑧 → (abs‘(𝐹𝑧)) ≤ 𝑛)) → (𝑚 ≤ (𝐺𝑦) → (abs‘(𝐹‘(𝐺𝑦))) ≤ 𝑛))
2014, 19sylan 575 . . . . . . . . . . . . . 14 ((((((𝜑𝑚 ∈ ℝ) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℝ) ∧ 𝑦𝐵) ∧ ∀𝑧𝐴 (𝑚𝑧 → (abs‘(𝐹𝑧)) ≤ 𝑛)) → (𝑚 ≤ (𝐺𝑦) → (abs‘(𝐹‘(𝐺𝑦))) ≤ 𝑛))
2120an32s 642 . . . . . . . . . . . . 13 ((((((𝜑𝑚 ∈ ℝ) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℝ) ∧ ∀𝑧𝐴 (𝑚𝑧 → (abs‘(𝐹𝑧)) ≤ 𝑛)) ∧ 𝑦𝐵) → (𝑚 ≤ (𝐺𝑦) → (abs‘(𝐹‘(𝐺𝑦))) ≤ 𝑛))
2213adantr 474 . . . . . . . . . . . . . . . 16 (((((𝜑𝑚 ∈ ℝ) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℝ) ∧ ∀𝑧𝐴 (𝑚𝑧 → (abs‘(𝐹𝑧)) ≤ 𝑛)) → 𝐺:𝐵𝐴)
23 fvco3 6526 . . . . . . . . . . . . . . . 16 ((𝐺:𝐵𝐴𝑦𝐵) → ((𝐹𝐺)‘𝑦) = (𝐹‘(𝐺𝑦)))
2422, 23sylan 575 . . . . . . . . . . . . . . 15 ((((((𝜑𝑚 ∈ ℝ) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℝ) ∧ ∀𝑧𝐴 (𝑚𝑧 → (abs‘(𝐹𝑧)) ≤ 𝑛)) ∧ 𝑦𝐵) → ((𝐹𝐺)‘𝑦) = (𝐹‘(𝐺𝑦)))
2524fveq2d 6441 . . . . . . . . . . . . . 14 ((((((𝜑𝑚 ∈ ℝ) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℝ) ∧ ∀𝑧𝐴 (𝑚𝑧 → (abs‘(𝐹𝑧)) ≤ 𝑛)) ∧ 𝑦𝐵) → (abs‘((𝐹𝐺)‘𝑦)) = (abs‘(𝐹‘(𝐺𝑦))))
2625breq1d 4885 . . . . . . . . . . . . 13 ((((((𝜑𝑚 ∈ ℝ) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℝ) ∧ ∀𝑧𝐴 (𝑚𝑧 → (abs‘(𝐹𝑧)) ≤ 𝑛)) ∧ 𝑦𝐵) → ((abs‘((𝐹𝐺)‘𝑦)) ≤ 𝑛 ↔ (abs‘(𝐹‘(𝐺𝑦))) ≤ 𝑛))
2721, 26sylibrd 251 . . . . . . . . . . . 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 3171 . . . . . . . . . 10 (((((𝜑𝑚 ∈ ℝ) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℝ) ∧ ∀𝑧𝐴 (𝑚𝑧 → (abs‘(𝐹𝑧)) ≤ 𝑛)) → (∀𝑦𝐵 (𝑥𝑦𝑚 ≤ (𝐺𝑦)) → ∀𝑦𝐵 (𝑥𝑦 → (abs‘((𝐹𝐺)‘𝑦)) ≤ 𝑛)))
3029expimpd 447 . . . . . . . . 9 ((((𝜑𝑚 ∈ ℝ) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℝ) → ((∀𝑧𝐴 (𝑚𝑧 → (abs‘(𝐹𝑧)) ≤ 𝑛) ∧ ∀𝑦𝐵 (𝑥𝑦𝑚 ≤ (𝐺𝑦))) → ∀𝑦𝐵 (𝑥𝑦 → (abs‘((𝐹𝐺)‘𝑦)) ≤ 𝑛)))
3130ancomsd 459 . . . . . . . 8 ((((𝜑𝑚 ∈ ℝ) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℝ) → ((∀𝑦𝐵 (𝑥𝑦𝑚 ≤ (𝐺𝑦)) ∧ ∀𝑧𝐴 (𝑚𝑧 → (abs‘(𝐹𝑧)) ≤ 𝑛)) → ∀𝑦𝐵 (𝑥𝑦 → (abs‘((𝐹𝐺)‘𝑦)) ≤ 𝑛)))
3231reximdva 3225 . . . . . . 7 (((𝜑𝑚 ∈ ℝ) ∧ 𝑥 ∈ ℝ) → (∃𝑛 ∈ ℝ (∀𝑦𝐵 (𝑥𝑦𝑚 ≤ (𝐺𝑦)) ∧ ∀𝑧𝐴 (𝑚𝑧 → (abs‘(𝐹𝑧)) ≤ 𝑛)) → ∃𝑛 ∈ ℝ ∀𝑦𝐵 (𝑥𝑦 → (abs‘((𝐹𝐺)‘𝑦)) ≤ 𝑛)))
3332reximdva 3225 . . . . . 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, 33syl5bir 235 . . . . 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 686 . . . 4 ((𝜑𝑚 ∈ ℝ) → (∃𝑛 ∈ ℝ ∀𝑧𝐴 (𝑚𝑧 → (abs‘(𝐹𝑧)) ≤ 𝑛) → ∃𝑥 ∈ ℝ ∃𝑛 ∈ ℝ ∀𝑦𝐵 (𝑥𝑦 → (abs‘((𝐹𝐺)‘𝑦)) ≤ 𝑛)))
3635rexlimdva 3240 . . 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 6299 . . . 4 ((𝐹:𝐴⟶ℂ ∧ 𝐺:𝐵𝐴) → (𝐹𝐺):𝐵⟶ℂ)
392, 12, 38syl2anc 579 . . 3 (𝜑 → (𝐹𝐺):𝐵⟶ℂ)
40 o1co.4 . . 3 (𝜑𝐵 ⊆ ℝ)
41 elo12 14642 . . 3 (((𝐹𝐺):𝐵⟶ℂ ∧ 𝐵 ⊆ ℝ) → ((𝐹𝐺) ∈ 𝑂(1) ↔ ∃𝑥 ∈ ℝ ∃𝑛 ∈ ℝ ∀𝑦𝐵 (𝑥𝑦 → (abs‘((𝐹𝐺)‘𝑦)) ≤ 𝑛)))
4239, 40, 41syl2anc 579 . 2 (𝜑 → ((𝐹𝐺) ∈ 𝑂(1) ↔ ∃𝑥 ∈ ℝ ∃𝑛 ∈ ℝ ∀𝑦𝐵 (𝑥𝑦 → (abs‘((𝐹𝐺)‘𝑦)) ≤ 𝑛)))
4337, 42mpbird 249 1 (𝜑 → (𝐹𝐺) ∈ 𝑂(1))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386   = wceq 1656  wcel 2164  wral 3117  wrex 3118  wss 3798   class class class wbr 4875  dom cdm 5346  ccom 5350  wf 6123  cfv 6127  cc 10257  cr 10258  cle 10399  abscabs 14358  𝑂(1)co1 14601
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214  ax-cnex 10315  ax-resscn 10316  ax-pre-lttri 10333  ax-pre-lttrn 10334
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-po 5265  df-so 5266  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-er 8014  df-pm 8130  df-en 8229  df-dom 8230  df-sdom 8231  df-pnf 10400  df-mnf 10401  df-xr 10402  df-ltxr 10403  df-le 10404  df-ico 12476  df-o1 14605
This theorem is referenced by:  o1compt  14702
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