Theorem o1compt 15224

Description: Sufficient condition for transforming the index set of an eventually bounded function. (Contributed by Mario Carneiro, 12-May-2016.)

Hypotheses

Ref	Expression
o1compt.1	⊢ (𝜑 → 𝐹:𝐴⟶ℂ)
o1compt.2	⊢ (𝜑 → 𝐹 ∈ 𝑂(1))
o1compt.3	⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐶 ∈ 𝐴)
o1compt.4	⊢ (𝜑 → 𝐵 ⊆ ℝ)
o1compt.5	⊢ ((𝜑 ∧ 𝑚 ∈ ℝ) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → 𝑚 ≤ 𝐶))

Assertion

Ref	Expression
o1compt	⊢ (𝜑 → (𝐹 ∘ (𝑦 ∈ 𝐵 ↦ 𝐶)) ∈ 𝑂(1))

Distinct variable groups:   𝑥,𝑚,𝑦,𝐴   𝐵,𝑚,𝑥,𝑦   𝐶,𝑚,𝑥   𝜑,𝑚,𝑥,𝑦   𝑚,𝐹,𝑥

Allowed substitution hints:   𝐶(𝑦)   𝐹(𝑦)

Proof of Theorem o1compt

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		o1compt.1	. 2 ⊢ (𝜑 → 𝐹:𝐴⟶ℂ)
2		o1compt.2	. 2 ⊢ (𝜑 → 𝐹 ∈ 𝑂(1))
3		o1compt.3	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐶 ∈ 𝐴)
4	3	fmpttd 6971	. 2 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↦ 𝐶):𝐵⟶𝐴)
5		o1compt.4	. 2 ⊢ (𝜑 → 𝐵 ⊆ ℝ)
6		o1compt.5	. . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ ℝ) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → 𝑚 ≤ 𝐶))
7		nfv 1918	. . . . . . . 8 ⊢ Ⅎ𝑦 𝑥 ≤ 𝑧
8		nfcv 2906	. . . . . . . . 9 ⊢ Ⅎ𝑦𝑚
9		nfcv 2906	. . . . . . . . 9 ⊢ Ⅎ𝑦 ≤
10		nffvmpt1 6767	. . . . . . . . 9 ⊢ Ⅎ𝑦((𝑦 ∈ 𝐵 ↦ 𝐶)‘𝑧)
11	8, 9, 10	nfbr 5117	. . . . . . . 8 ⊢ Ⅎ𝑦 𝑚 ≤ ((𝑦 ∈ 𝐵 ↦ 𝐶)‘𝑧)
12	7, 11	nfim 1900	. . . . . . 7 ⊢ Ⅎ𝑦(𝑥 ≤ 𝑧 → 𝑚 ≤ ((𝑦 ∈ 𝐵 ↦ 𝐶)‘𝑧))
13		nfv 1918	. . . . . . 7 ⊢ Ⅎ𝑧(𝑥 ≤ 𝑦 → 𝑚 ≤ ((𝑦 ∈ 𝐵 ↦ 𝐶)‘𝑦))
14		breq2 5074	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → (𝑥 ≤ 𝑧 ↔ 𝑥 ≤ 𝑦))
15		fveq2 6756	. . . . . . . . 9 ⊢ (𝑧 = 𝑦 → ((𝑦 ∈ 𝐵 ↦ 𝐶)‘𝑧) = ((𝑦 ∈ 𝐵 ↦ 𝐶)‘𝑦))
16	15	breq2d 5082	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → (𝑚 ≤ ((𝑦 ∈ 𝐵 ↦ 𝐶)‘𝑧) ↔ 𝑚 ≤ ((𝑦 ∈ 𝐵 ↦ 𝐶)‘𝑦)))
17	14, 16	imbi12d 344	. . . . . . 7 ⊢ (𝑧 = 𝑦 → ((𝑥 ≤ 𝑧 → 𝑚 ≤ ((𝑦 ∈ 𝐵 ↦ 𝐶)‘𝑧)) ↔ (𝑥 ≤ 𝑦 → 𝑚 ≤ ((𝑦 ∈ 𝐵 ↦ 𝐶)‘𝑦))))
18	12, 13, 17	cbvralw 3363	. . . . . 6 ⊢ (∀𝑧 ∈ 𝐵 (𝑥 ≤ 𝑧 → 𝑚 ≤ ((𝑦 ∈ 𝐵 ↦ 𝐶)‘𝑧)) ↔ ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → 𝑚 ≤ ((𝑦 ∈ 𝐵 ↦ 𝐶)‘𝑦)))
19		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵)
20		eqid 2738	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑦 ∈ 𝐵 ↦ 𝐶)
21	20	fvmpt2 6868	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝐶 ∈ 𝐴) → ((𝑦 ∈ 𝐵 ↦ 𝐶)‘𝑦) = 𝐶)
22	19, 3, 21	syl2anc 583	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ((𝑦 ∈ 𝐵 ↦ 𝐶)‘𝑦) = 𝐶)
23	22	breq2d 5082	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝑚 ≤ ((𝑦 ∈ 𝐵 ↦ 𝐶)‘𝑦) ↔ 𝑚 ≤ 𝐶))
24	23	imbi2d 340	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ((𝑥 ≤ 𝑦 → 𝑚 ≤ ((𝑦 ∈ 𝐵 ↦ 𝐶)‘𝑦)) ↔ (𝑥 ≤ 𝑦 → 𝑚 ≤ 𝐶)))
25	24	ralbidva 3119	. . . . . 6 ⊢ (𝜑 → (∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → 𝑚 ≤ ((𝑦 ∈ 𝐵 ↦ 𝐶)‘𝑦)) ↔ ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → 𝑚 ≤ 𝐶)))
26	18, 25	syl5bb 282	. . . . 5 ⊢ (𝜑 → (∀𝑧 ∈ 𝐵 (𝑥 ≤ 𝑧 → 𝑚 ≤ ((𝑦 ∈ 𝐵 ↦ 𝐶)‘𝑧)) ↔ ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → 𝑚 ≤ 𝐶)))
27	26	rexbidv 3225	. . . 4 ⊢ (𝜑 → (∃𝑥 ∈ ℝ ∀𝑧 ∈ 𝐵 (𝑥 ≤ 𝑧 → 𝑚 ≤ ((𝑦 ∈ 𝐵 ↦ 𝐶)‘𝑧)) ↔ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → 𝑚 ≤ 𝐶)))
28	27	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ ℝ) → (∃𝑥 ∈ ℝ ∀𝑧 ∈ 𝐵 (𝑥 ≤ 𝑧 → 𝑚 ≤ ((𝑦 ∈ 𝐵 ↦ 𝐶)‘𝑧)) ↔ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → 𝑚 ≤ 𝐶)))
29	6, 28	mpbird 256	. 2 ⊢ ((𝜑 ∧ 𝑚 ∈ ℝ) → ∃𝑥 ∈ ℝ ∀𝑧 ∈ 𝐵 (𝑥 ≤ 𝑧 → 𝑚 ≤ ((𝑦 ∈ 𝐵 ↦ 𝐶)‘𝑧)))
30	1, 2, 4, 5, 29	o1co 15223	1 ⊢ (𝜑 → (𝐹 ∘ (𝑦 ∈ 𝐵 ↦ 𝐶)) ∈ 𝑂(1))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∀wral 3063  ∃wrex 3064   ⊆ wss 3883   class class class wbr 5070   ↦ cmpt 5153   ∘ ccom 5584  ⟶wf 6414  ‘cfv 6418  ℂcc 10800  ℝcr 10801   ≤ cle 10941  𝑂(1)co1 15123

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-pre-lttri 10876  ax-pre-lttrn 10877

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-po 5494  df-so 5495  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-er 8456  df-pm 8576  df-en 8692  df-dom 8693  df-sdom 8694  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-ico 13014  df-o1 15127

This theorem is referenced by:  dchrisum0  26573