Theorem elbigofrcl 49584

Description: Reverse closure of the "big-O" function. (Contributed by AV, 16-May-2020.)

Assertion

Ref	Expression
elbigofrcl	⊢ (𝐹 ∈ (Ο‘𝐺) → 𝐺 ∈ (ℝ ↑pm ℝ))

Proof of Theorem elbigofrcl

Dummy variables 𝑔 𝑓 𝑥 𝑚 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfvdm 6907	. 2 ⊢ (𝐹 ∈ (Ο‘𝐺) → 𝐺 ∈ dom Ο)
2		df-bigo 49582	. . . 4 ⊢ Ο = (𝑔 ∈ (ℝ ↑pm ℝ) ↦ {𝑓 ∈ (ℝ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓‘𝑦) ≤ (𝑚 · (𝑔‘𝑦))})
3	2	dmeqi 5882	. . 3 ⊢ dom Ο = dom (𝑔 ∈ (ℝ ↑pm ℝ) ↦ {𝑓 ∈ (ℝ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓‘𝑦) ≤ (𝑚 · (𝑔‘𝑦))})
4		dmmptg 6232	. . . 4 ⊢ (∀𝑔 ∈ (ℝ ↑pm ℝ){𝑓 ∈ (ℝ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓‘𝑦) ≤ (𝑚 · (𝑔‘𝑦))} ∈ V → dom (𝑔 ∈ (ℝ ↑pm ℝ) ↦ {𝑓 ∈ (ℝ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓‘𝑦) ≤ (𝑚 · (𝑔‘𝑦))}) = (ℝ ↑pm ℝ))
5		ovex 7441	. . . . . 6 ⊢ (ℝ ↑pm ℝ) ∈ V
6	5	rabex 5299	. . . . 5 ⊢ {𝑓 ∈ (ℝ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓‘𝑦) ≤ (𝑚 · (𝑔‘𝑦))} ∈ V
7	6	a1i 11	. . . 4 ⊢ (𝑔 ∈ (ℝ ↑pm ℝ) → {𝑓 ∈ (ℝ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓‘𝑦) ≤ (𝑚 · (𝑔‘𝑦))} ∈ V)
8	4, 7	mprg 3082	. . 3 ⊢ dom (𝑔 ∈ (ℝ ↑pm ℝ) ↦ {𝑓 ∈ (ℝ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓‘𝑦) ≤ (𝑚 · (𝑔‘𝑦))}) = (ℝ ↑pm ℝ)
9	3, 8	eqtri 2783	. 2 ⊢ dom Ο = (ℝ ↑pm ℝ)
10	1, 9	eleqtrdi 2870	1 ⊢ (𝐹 ∈ (Ο‘𝐺) → 𝐺 ∈ (ℝ ↑pm ℝ))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  ∀wral 3076  ∃wrex 3086  {crab 3412  Vcvv 3450   ∩ cin 3897   class class class wbr 5102   ↦ cmpt 5185  dom cdm 5647  ‘cfv 6527  (class class class)co 7408   ↑_pm cpm 8826  ℝcr 11170   · cmul 11176  +∞cpnf 11311   ≤ cle 11315  [,)cico 13447  Οcbigo 49581

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rab 3413  df-v 3452  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-xp 5653  df-rel 5654  df-cnv 5655  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6483  df-fv 6535  df-ov 7411  df-bigo 49582

This theorem is used by:  elbigo  49585  elbigoimp  49590