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Theorem elo12r 14253
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 4654 . . . . . . 7 (𝑦 = 𝐶 → (𝑦𝑥𝐶𝑥))
21imbi1d 331 . . . . . 6 (𝑦 = 𝐶 → ((𝑦𝑥 → (abs‘(𝐹𝑥)) ≤ 𝑚) ↔ (𝐶𝑥 → (abs‘(𝐹𝑥)) ≤ 𝑚)))
32ralbidv 2985 . . . . 5 (𝑦 = 𝐶 → (∀𝑥𝐴 (𝑦𝑥 → (abs‘(𝐹𝑥)) ≤ 𝑚) ↔ ∀𝑥𝐴 (𝐶𝑥 → (abs‘(𝐹𝑥)) ≤ 𝑚)))
4 breq2 4655 . . . . . . 7 (𝑚 = 𝑀 → ((abs‘(𝐹𝑥)) ≤ 𝑚 ↔ (abs‘(𝐹𝑥)) ≤ 𝑀))
54imbi2d 330 . . . . . 6 (𝑚 = 𝑀 → ((𝐶𝑥 → (abs‘(𝐹𝑥)) ≤ 𝑚) ↔ (𝐶𝑥 → (abs‘(𝐹𝑥)) ≤ 𝑀)))
65ralbidv 2985 . . . . 5 (𝑚 = 𝑀 → (∀𝑥𝐴 (𝐶𝑥 → (abs‘(𝐹𝑥)) ≤ 𝑚) ↔ ∀𝑥𝐴 (𝐶𝑥 → (abs‘(𝐹𝑥)) ≤ 𝑀)))
73, 6rspc2ev 3322 . . . 4 ((𝐶 ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ ∀𝑥𝐴 (𝐶𝑥 → (abs‘(𝐹𝑥)) ≤ 𝑀)) → ∃𝑦 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑥𝐴 (𝑦𝑥 → (abs‘(𝐹𝑥)) ≤ 𝑚))
873expa 1264 . . 3 (((𝐶 ∈ ℝ ∧ 𝑀 ∈ ℝ) ∧ ∀𝑥𝐴 (𝐶𝑥 → (abs‘(𝐹𝑥)) ≤ 𝑀)) → ∃𝑦 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑥𝐴 (𝑦𝑥 → (abs‘(𝐹𝑥)) ≤ 𝑚))
983adant1 1078 . 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 14252 . . 3 ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℝ) → (𝐹 ∈ 𝑂(1) ↔ ∃𝑦 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑥𝐴 (𝑦𝑥 → (abs‘(𝐹𝑥)) ≤ 𝑚)))
11103ad2ant1 1081 . 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 247 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 196  wa 384  w3a 1037   = wceq 1482  wcel 1989  wral 2911  wrex 2912  wss 3572   class class class wbr 4651  wf 5882  cfv 5886  cc 9931  cr 9932  cle 10072  abscabs 13968  𝑂(1)co1 14211
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946  ax-cnex 9989  ax-resscn 9990  ax-pre-lttri 10007  ax-pre-lttrn 10008
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-nel 2897  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-po 5033  df-so 5034  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-er 7739  df-pm 7857  df-en 7953  df-dom 7954  df-sdom 7955  df-pnf 10073  df-mnf 10074  df-xr 10075  df-ltxr 10076  df-le 10077  df-ico 12178  df-o1 14215
This theorem is referenced by:  o1resb  14291  o1of2  14337  o1cxp  24695
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