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Theorem elbigo 42143
 Description: Properties of a function of order G(x). (Contributed by AV, 16-May-2020.)
Assertion
Ref Expression
elbigo (𝐹 ∈ (Ο‘𝐺) ↔ (𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐺 ∈ (ℝ ↑pm ℝ) ∧ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝐹 ∩ (𝑥[,)+∞))(𝐹𝑦) ≤ (𝑚 · (𝐺𝑦))))
Distinct variable groups:   𝑥,𝐺,𝑚,𝑦   𝑚,𝐹,𝑥,𝑦

Proof of Theorem elbigo
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 bigoval 42141 . . . . 5 (𝐺 ∈ (ℝ ↑pm ℝ) → (Ο‘𝐺) = {𝑓 ∈ (ℝ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓𝑦) ≤ (𝑚 · (𝐺𝑦))})
21eleq2d 2673 . . . 4 (𝐺 ∈ (ℝ ↑pm ℝ) → (𝐹 ∈ (Ο‘𝐺) ↔ 𝐹 ∈ {𝑓 ∈ (ℝ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓𝑦) ≤ (𝑚 · (𝐺𝑦))}))
3 dmeq 5246 . . . . . . . 8 (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
43ineq1d 3775 . . . . . . 7 (𝑓 = 𝐹 → (dom 𝑓 ∩ (𝑥[,)+∞)) = (dom 𝐹 ∩ (𝑥[,)+∞)))
5 fveq1 6102 . . . . . . . 8 (𝑓 = 𝐹 → (𝑓𝑦) = (𝐹𝑦))
65breq1d 4593 . . . . . . 7 (𝑓 = 𝐹 → ((𝑓𝑦) ≤ (𝑚 · (𝐺𝑦)) ↔ (𝐹𝑦) ≤ (𝑚 · (𝐺𝑦))))
74, 6raleqbidv 3129 . . . . . 6 (𝑓 = 𝐹 → (∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓𝑦) ≤ (𝑚 · (𝐺𝑦)) ↔ ∀𝑦 ∈ (dom 𝐹 ∩ (𝑥[,)+∞))(𝐹𝑦) ≤ (𝑚 · (𝐺𝑦))))
872rexbidv 3039 . . . . 5 (𝑓 = 𝐹 → (∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓𝑦) ≤ (𝑚 · (𝐺𝑦)) ↔ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝐹 ∩ (𝑥[,)+∞))(𝐹𝑦) ≤ (𝑚 · (𝐺𝑦))))
98elrab 3331 . . . 4 (𝐹 ∈ {𝑓 ∈ (ℝ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓𝑦) ≤ (𝑚 · (𝐺𝑦))} ↔ (𝐹 ∈ (ℝ ↑pm ℝ) ∧ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝐹 ∩ (𝑥[,)+∞))(𝐹𝑦) ≤ (𝑚 · (𝐺𝑦))))
102, 9syl6bb 275 . . 3 (𝐺 ∈ (ℝ ↑pm ℝ) → (𝐹 ∈ (Ο‘𝐺) ↔ (𝐹 ∈ (ℝ ↑pm ℝ) ∧ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝐹 ∩ (𝑥[,)+∞))(𝐹𝑦) ≤ (𝑚 · (𝐺𝑦)))))
1110pm5.32i 667 . 2 ((𝐺 ∈ (ℝ ↑pm ℝ) ∧ 𝐹 ∈ (Ο‘𝐺)) ↔ (𝐺 ∈ (ℝ ↑pm ℝ) ∧ (𝐹 ∈ (ℝ ↑pm ℝ) ∧ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝐹 ∩ (𝑥[,)+∞))(𝐹𝑦) ≤ (𝑚 · (𝐺𝑦)))))
12 elbigofrcl 42142 . . 3 (𝐹 ∈ (Ο‘𝐺) → 𝐺 ∈ (ℝ ↑pm ℝ))
1312pm4.71ri 663 . 2 (𝐹 ∈ (Ο‘𝐺) ↔ (𝐺 ∈ (ℝ ↑pm ℝ) ∧ 𝐹 ∈ (Ο‘𝐺)))
14 3anan12 1044 . 2 ((𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐺 ∈ (ℝ ↑pm ℝ) ∧ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝐹 ∩ (𝑥[,)+∞))(𝐹𝑦) ≤ (𝑚 · (𝐺𝑦))) ↔ (𝐺 ∈ (ℝ ↑pm ℝ) ∧ (𝐹 ∈ (ℝ ↑pm ℝ) ∧ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝐹 ∩ (𝑥[,)+∞))(𝐹𝑦) ≤ (𝑚 · (𝐺𝑦)))))
1511, 13, 143bitr4i 291 1 (𝐹 ∈ (Ο‘𝐺) ↔ (𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐺 ∈ (ℝ ↑pm ℝ) ∧ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝐹 ∩ (𝑥[,)+∞))(𝐹𝑦) ≤ (𝑚 · (𝐺𝑦))))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  {crab 2900   ∩ cin 3539   class class class wbr 4583  dom cdm 5038  ‘cfv 5804  (class class class)co 6549   ↑pm cpm 7745  ℝcr 9814   · cmul 9820  +∞cpnf 9950   ≤ cle 9954  [,)cico 12048  Οcbigo 42139 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-bigo 42140 This theorem is referenced by:  elbigo2  42144  elbigof  42146  elbigodm  42147
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