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Theorem elo12r 14880
 Description: Sufficient condition for elementhood in the set of eventually bounded functions. (Contributed by Mario Carneiro, 15-Sep-2014.)
Assertion
Ref Expression
elo12r (((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝑀 ∈ ℝ) ∧ ∀𝑥𝐴 (𝐶𝑥 → (abs‘(𝐹𝑥)) ≤ 𝑀)) → 𝐹 ∈ 𝑂(1))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝑥,𝐹   𝑥,𝑀

Proof of Theorem elo12r
Dummy variables 𝑚 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq1 5034 . . . . . . 7 (𝑦 = 𝐶 → (𝑦𝑥𝐶𝑥))
21imbi1d 345 . . . . . 6 (𝑦 = 𝐶 → ((𝑦𝑥 → (abs‘(𝐹𝑥)) ≤ 𝑚) ↔ (𝐶𝑥 → (abs‘(𝐹𝑥)) ≤ 𝑚)))
32ralbidv 3162 . . . . 5 (𝑦 = 𝐶 → (∀𝑥𝐴 (𝑦𝑥 → (abs‘(𝐹𝑥)) ≤ 𝑚) ↔ ∀𝑥𝐴 (𝐶𝑥 → (abs‘(𝐹𝑥)) ≤ 𝑚)))
4 breq2 5035 . . . . . . 7 (𝑚 = 𝑀 → ((abs‘(𝐹𝑥)) ≤ 𝑚 ↔ (abs‘(𝐹𝑥)) ≤ 𝑀))
54imbi2d 344 . . . . . 6 (𝑚 = 𝑀 → ((𝐶𝑥 → (abs‘(𝐹𝑥)) ≤ 𝑚) ↔ (𝐶𝑥 → (abs‘(𝐹𝑥)) ≤ 𝑀)))
65ralbidv 3162 . . . . 5 (𝑚 = 𝑀 → (∀𝑥𝐴 (𝐶𝑥 → (abs‘(𝐹𝑥)) ≤ 𝑚) ↔ ∀𝑥𝐴 (𝐶𝑥 → (abs‘(𝐹𝑥)) ≤ 𝑀)))
73, 6rspc2ev 3583 . . . 4 ((𝐶 ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ ∀𝑥𝐴 (𝐶𝑥 → (abs‘(𝐹𝑥)) ≤ 𝑀)) → ∃𝑦 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑥𝐴 (𝑦𝑥 → (abs‘(𝐹𝑥)) ≤ 𝑚))
873expa 1115 . . 3 (((𝐶 ∈ ℝ ∧ 𝑀 ∈ ℝ) ∧ ∀𝑥𝐴 (𝐶𝑥 → (abs‘(𝐹𝑥)) ≤ 𝑀)) → ∃𝑦 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑥𝐴 (𝑦𝑥 → (abs‘(𝐹𝑥)) ≤ 𝑚))
983adant1 1127 . 2 (((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝑀 ∈ ℝ) ∧ ∀𝑥𝐴 (𝐶𝑥 → (abs‘(𝐹𝑥)) ≤ 𝑀)) → ∃𝑦 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑥𝐴 (𝑦𝑥 → (abs‘(𝐹𝑥)) ≤ 𝑚))
10 elo12 14879 . . 3 ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℝ) → (𝐹 ∈ 𝑂(1) ↔ ∃𝑦 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑥𝐴 (𝑦𝑥 → (abs‘(𝐹𝑥)) ≤ 𝑚)))
11103ad2ant1 1130 . 2 (((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝑀 ∈ ℝ) ∧ ∀𝑥𝐴 (𝐶𝑥 → (abs‘(𝐹𝑥)) ≤ 𝑀)) → (𝐹 ∈ 𝑂(1) ↔ ∃𝑦 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑥𝐴 (𝑦𝑥 → (abs‘(𝐹𝑥)) ≤ 𝑚)))
129, 11mpbird 260 1 (((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝑀 ∈ ℝ) ∧ ∀𝑥𝐴 (𝐶𝑥 → (abs‘(𝐹𝑥)) ≤ 𝑀)) → 𝐹 ∈ 𝑂(1))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107   ⊆ wss 3881   class class class wbr 5031  ⟶wf 6321  ‘cfv 6325  ℂcc 10527  ℝcr 10528   ≤ cle 10668  abscabs 14588  𝑂(1)co1 14838 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444  ax-cnex 10585  ax-resscn 10586  ax-pre-lttri 10603  ax-pre-lttrn 10604 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-po 5439  df-so 5440  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-ov 7139  df-oprab 7140  df-mpo 7141  df-er 8275  df-pm 8395  df-en 8496  df-dom 8497  df-sdom 8498  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-ico 12735  df-o1 14842 This theorem is referenced by:  o1resb  14918  o1of2  14964  o1cxp  25570
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