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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  ℂcc 10257  ℝcr 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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