Theorem elo12r 14259

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) ∧ ∀𝑥 ∈ 𝐴 (𝐶 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 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.

Step	Hyp	Ref	Expression
1		breq1 4656	. . . . . . 7 ⊢ (𝑦 = 𝐶 → (𝑦 ≤ 𝑥 ↔ 𝐶 ≤ 𝑥))
2	1	imbi1d 331	. . . . . 6 ⊢ (𝑦 = 𝐶 → ((𝑦 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑚) ↔ (𝐶 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑚)))
3	2	ralbidv 2986	. . . . 5 ⊢ (𝑦 = 𝐶 → (∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑚) ↔ ∀𝑥 ∈ 𝐴 (𝐶 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑚)))
4		breq2 4657	. . . . . . 7 ⊢ (𝑚 = 𝑀 → ((abs‘(𝐹‘𝑥)) ≤ 𝑚 ↔ (abs‘(𝐹‘𝑥)) ≤ 𝑀))
5	4	imbi2d 330	. . . . . 6 ⊢ (𝑚 = 𝑀 → ((𝐶 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑚) ↔ (𝐶 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑀)))
6	5	ralbidv 2986	. . . . 5 ⊢ (𝑚 = 𝑀 → (∀𝑥 ∈ 𝐴 (𝐶 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑚) ↔ ∀𝑥 ∈ 𝐴 (𝐶 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑀)))
7	3, 6	rspc2ev 3324	. . . 4 ⊢ ((𝐶 ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ ∀𝑥 ∈ 𝐴 (𝐶 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑀)) → ∃𝑦 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑚))
8	7	3expa 1265	. . 3 ⊢ (((𝐶 ∈ ℝ ∧ 𝑀 ∈ ℝ) ∧ ∀𝑥 ∈ 𝐴 (𝐶 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑀)) → ∃𝑦 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑚))
9	8	3adant1 1079	. 2 ⊢ (((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝑀 ∈ ℝ) ∧ ∀𝑥 ∈ 𝐴 (𝐶 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑀)) → ∃𝑦 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑚))
10		elo12 14258	. . 3 ⊢ ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℝ) → (𝐹 ∈ 𝑂(1) ↔ ∃𝑦 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑚)))
11	10	3ad2ant1 1082	. 2 ⊢ (((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝑀 ∈ ℝ) ∧ ∀𝑥 ∈ 𝐴 (𝐶 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑀)) → (𝐹 ∈ 𝑂(1) ↔ ∃𝑦 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑚)))
12	9, 11	mpbird 247	1 ⊢ (((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝑀 ∈ ℝ) ∧ ∀𝑥 ∈ 𝐴 (𝐶 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑀)) → 𝐹 ∈ 𝑂(1))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913   ⊆ wss 3574   class class class wbr 4653  ⟶wf 5884  ‘cfv 5888  ℂcc 9934  ℝcr 9935   ≤ cle 10075  abscabs 13974  𝑂(1)co1 14217

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-pre-lttri 10010  ax-pre-lttrn 10011

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-er 7742  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-ico 12181  df-o1 14221

This theorem is referenced by:  o1resb  14297  o1of2  14343  o1cxp  24701