Theorem elo12r 15579

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 5111	. . . . . . 7 ⊢ (𝑦 = 𝐶 → (𝑦 ≤ 𝑥 ↔ 𝐶 ≤ 𝑥))
2	1	imbi1d 344	. . . . . 6 ⊢ (𝑦 = 𝐶 → ((𝑦 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑚) ↔ (𝐶 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑚)))
3	2	ralbidv 3186	. . . . 5 ⊢ (𝑦 = 𝐶 → (∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑚) ↔ ∀𝑥 ∈ 𝐴 (𝐶 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑚)))
4		breq2 5112	. . . . . . 7 ⊢ (𝑚 = 𝑀 → ((abs‘(𝐹‘𝑥)) ≤ 𝑚 ↔ (abs‘(𝐹‘𝑥)) ≤ 𝑀))
5	4	imbi2d 343	. . . . . 6 ⊢ (𝑚 = 𝑀 → ((𝐶 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑚) ↔ (𝐶 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑀)))
6	5	ralbidv 3186	. . . . 5 ⊢ (𝑚 = 𝑀 → (∀𝑥 ∈ 𝐴 (𝐶 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑚) ↔ ∀𝑥 ∈ 𝐴 (𝐶 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑀)))
7	3, 6	rspc2ev 3593	. . . 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 1134	. . 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 1146	. 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 15578	. . 3 ⊢ ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℝ) → (𝐹 ∈ 𝑂(1) ↔ ∃𝑦 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑚)))
11	10	3ad2ant1 1149	. 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 260	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 209   ∧ wa 400   ∧ w3a 1101   = wceq 1568   ∈ wcel 2141  ∀wral 3077  ∃wrex 3087   ⊆ wss 3904   class class class wbr 5108  ⟶wf 6532  ‘cfv 6536  ℂcc 11097  ℝcr 11098   ≤ cle 11243  abscabs 15285  𝑂(1)co1 15537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5256  ax-nul 5268  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11155  ax-resscn 11156  ax-pre-lttri 11173  ax-pre-lttrn 11174

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5556  df-po 5569  df-so 5570  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-er 8693  df-pm 8826  df-en 8943  df-dom 8944  df-sdom 8945  df-pnf 11244  df-mnf 11245  df-xr 11246  df-ltxr 11247  df-le 11248  df-ico 13377  df-o1 15541

This theorem is referenced by:  o1resb  15617  o1of2  15664  o1cxp  27115