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Theorem o1co 14616
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 6234 . . . . . 6 (𝜑 → dom 𝐹 = 𝐴)
4 o1dm 14560 . . . . . . 7 (𝐹 ∈ 𝑂(1) → dom 𝐹 ⊆ ℝ)
51, 4syl 17 . . . . . 6 (𝜑 → dom 𝐹 ⊆ ℝ)
63, 5eqsstr3d 3802 . . . . 5 (𝜑𝐴 ⊆ ℝ)
7 elo12 14557 . . . . 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 223 . . 3 (𝜑 → ∃𝑚 ∈ ℝ ∃𝑛 ∈ ℝ ∀𝑧𝐴 (𝑚𝑧 → (abs‘(𝐹𝑧)) ≤ 𝑛))
10 o1co.5 . . . . 5 ((𝜑𝑚 ∈ ℝ) → ∃𝑥 ∈ ℝ ∀𝑦𝐵 (𝑥𝑦𝑚 ≤ (𝐺𝑦)))
11 reeanv 3254 . . . . . 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 6553 . . . . . . . . . . . . . . 15 (((((𝜑𝑚 ∈ ℝ) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℝ) ∧ 𝑦𝐵) → (𝐺𝑦) ∈ 𝐴)
15 breq2 4815 . . . . . . . . . . . . . . . . 17 (𝑧 = (𝐺𝑦) → (𝑚𝑧𝑚 ≤ (𝐺𝑦)))
16 2fveq3 6384 . . . . . . . . . . . . . . . . . 18 (𝑧 = (𝐺𝑦) → (abs‘(𝐹𝑧)) = (abs‘(𝐹‘(𝐺𝑦))))
1716breq1d 4821 . . . . . . . . . . . . . . . . 17 (𝑧 = (𝐺𝑦) → ((abs‘(𝐹𝑧)) ≤ 𝑛 ↔ (abs‘(𝐹‘(𝐺𝑦))) ≤ 𝑛))
1815, 17imbi12d 335 . . . . . . . . . . . . . . . 16 (𝑧 = (𝐺𝑦) → ((𝑚𝑧 → (abs‘(𝐹𝑧)) ≤ 𝑛) ↔ (𝑚 ≤ (𝐺𝑦) → (abs‘(𝐹‘(𝐺𝑦))) ≤ 𝑛)))
1918rspcva 3460 . . . . . . . . . . . . . . 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 472 . . . . . . . . . . . . . . . 16 (((((𝜑𝑚 ∈ ℝ) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℝ) ∧ ∀𝑧𝐴 (𝑚𝑧 → (abs‘(𝐹𝑧)) ≤ 𝑛)) → 𝐺:𝐵𝐴)
23 fvco3 6468 . . . . . . . . . . . . . . . 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 6383 . . . . . . . . . . . . . 14 ((((((𝜑𝑚 ∈ ℝ) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℝ) ∧ ∀𝑧𝐴 (𝑚𝑧 → (abs‘(𝐹𝑧)) ≤ 𝑛)) ∧ 𝑦𝐵) → (abs‘((𝐹𝐺)‘𝑦)) = (abs‘(𝐹‘(𝐺𝑦))))
2625breq1d 4821 . . . . . . . . . . . . 13 ((((((𝜑𝑚 ∈ ℝ) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℝ) ∧ ∀𝑧𝐴 (𝑚𝑧 → (abs‘(𝐹𝑧)) ≤ 𝑛)) ∧ 𝑦𝐵) → ((abs‘((𝐹𝐺)‘𝑦)) ≤ 𝑛 ↔ (abs‘(𝐹‘(𝐺𝑦))) ≤ 𝑛))
2721, 26sylibrd 250 . . . . . . . . . . . 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 3109 . . . . . . . . . 10 (((((𝜑𝑚 ∈ ℝ) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℝ) ∧ ∀𝑧𝐴 (𝑚𝑧 → (abs‘(𝐹𝑧)) ≤ 𝑛)) → (∀𝑦𝐵 (𝑥𝑦𝑚 ≤ (𝐺𝑦)) → ∀𝑦𝐵 (𝑥𝑦 → (abs‘((𝐹𝐺)‘𝑦)) ≤ 𝑛)))
3029expimpd 445 . . . . . . . . 9 ((((𝜑𝑚 ∈ ℝ) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℝ) → ((∀𝑧𝐴 (𝑚𝑧 → (abs‘(𝐹𝑧)) ≤ 𝑛) ∧ ∀𝑦𝐵 (𝑥𝑦𝑚 ≤ (𝐺𝑦))) → ∀𝑦𝐵 (𝑥𝑦 → (abs‘((𝐹𝐺)‘𝑦)) ≤ 𝑛)))
3130ancomsd 457 . . . . . . . 8 ((((𝜑𝑚 ∈ ℝ) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℝ) → ((∀𝑦𝐵 (𝑥𝑦𝑚 ≤ (𝐺𝑦)) ∧ ∀𝑧𝐴 (𝑚𝑧 → (abs‘(𝐹𝑧)) ≤ 𝑛)) → ∀𝑦𝐵 (𝑥𝑦 → (abs‘((𝐹𝐺)‘𝑦)) ≤ 𝑛)))
3231reximdva 3163 . . . . . . 7 (((𝜑𝑚 ∈ ℝ) ∧ 𝑥 ∈ ℝ) → (∃𝑛 ∈ ℝ (∀𝑦𝐵 (𝑥𝑦𝑚 ≤ (𝐺𝑦)) ∧ ∀𝑧𝐴 (𝑚𝑧 → (abs‘(𝐹𝑧)) ≤ 𝑛)) → ∃𝑛 ∈ ℝ ∀𝑦𝐵 (𝑥𝑦 → (abs‘((𝐹𝐺)‘𝑦)) ≤ 𝑛)))
3332reximdva 3163 . . . . . 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 234 . . . . 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 3178 . . 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 6242 . . . 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 14557 . . 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 248 1 (𝜑 → (𝐹𝐺) ∈ 𝑂(1))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wcel 2155  wral 3055  wrex 3056  wss 3734   class class class wbr 4811  dom cdm 5279  ccom 5283  wf 6066  cfv 6070  cc 10191  cr 10192  cle 10333  abscabs 14273  𝑂(1)co1 14516
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7151  ax-cnex 10249  ax-resscn 10250  ax-pre-lttri 10267  ax-pre-lttrn 10268
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-br 4812  df-opab 4874  df-mpt 4891  df-id 5187  df-po 5200  df-so 5201  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-ov 6849  df-oprab 6850  df-mpt2 6851  df-er 7951  df-pm 8067  df-en 8165  df-dom 8166  df-sdom 8167  df-pnf 10334  df-mnf 10335  df-xr 10336  df-ltxr 10337  df-le 10338  df-ico 12388  df-o1 14520
This theorem is referenced by:  o1compt  14617
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