Theorem o1compt 15534

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 7109	. 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 1909	. . . . . . . 8 ⊢ Ⅎ𝑦 𝑥 ≤ 𝑧
8		nfcv 2897	. . . . . . . . 9 ⊢ Ⅎ𝑦𝑚
9		nfcv 2897	. . . . . . . . 9 ⊢ Ⅎ𝑦 ≤
10		nffvmpt1 6895	. . . . . . . . 9 ⊢ Ⅎ𝑦((𝑦 ∈ 𝐵 ↦ 𝐶)‘𝑧)
11	8, 9, 10	nfbr 5188	. . . . . . . 8 ⊢ Ⅎ𝑦 𝑚 ≤ ((𝑦 ∈ 𝐵 ↦ 𝐶)‘𝑧)
12	7, 11	nfim 1891	. . . . . . 7 ⊢ Ⅎ𝑦(𝑥 ≤ 𝑧 → 𝑚 ≤ ((𝑦 ∈ 𝐵 ↦ 𝐶)‘𝑧))
13		nfv 1909	. . . . . . 7 ⊢ Ⅎ𝑧(𝑥 ≤ 𝑦 → 𝑚 ≤ ((𝑦 ∈ 𝐵 ↦ 𝐶)‘𝑦))
14		breq2 5145	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → (𝑥 ≤ 𝑧 ↔ 𝑥 ≤ 𝑦))
15		fveq2 6884	. . . . . . . . 9 ⊢ (𝑧 = 𝑦 → ((𝑦 ∈ 𝐵 ↦ 𝐶)‘𝑧) = ((𝑦 ∈ 𝐵 ↦ 𝐶)‘𝑦))
16	15	breq2d 5153	. . . . . . . 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 3297	. . . . . 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 2726	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑦 ∈ 𝐵 ↦ 𝐶)
21	20	fvmpt2 7002	. . . . . . . . . 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 5153	. . . . . . . 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 3169	. . . . . 6 ⊢ (𝜑 → (∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → 𝑚 ≤ ((𝑦 ∈ 𝐵 ↦ 𝐶)‘𝑦)) ↔ ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → 𝑚 ≤ 𝐶)))
26	18, 25	bitrid 283	. . . . 5 ⊢ (𝜑 → (∀𝑧 ∈ 𝐵 (𝑥 ≤ 𝑧 → 𝑚 ≤ ((𝑦 ∈ 𝐵 ↦ 𝐶)‘𝑧)) ↔ ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → 𝑚 ≤ 𝐶)))
27	26	rexbidv 3172	. . . 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 257	. 2 ⊢ ((𝜑 ∧ 𝑚 ∈ ℝ) → ∃𝑥 ∈ ℝ ∀𝑧 ∈ 𝐵 (𝑥 ≤ 𝑧 → 𝑚 ≤ ((𝑦 ∈ 𝐵 ↦ 𝐶)‘𝑧)))
30	1, 2, 4, 5, 29	o1co 15533	1 ⊢ (𝜑 → (𝐹 ∘ (𝑦 ∈ 𝐵 ↦ 𝐶)) ∈ 𝑂(1))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098  ∀wral 3055  ∃wrex 3064   ⊆ wss 3943   class class class wbr 5141   ↦ cmpt 5224   ∘ ccom 5673  ⟶wf 6532  ‘cfv 6536  ℂcc 11107  ℝcr 11108   ≤ cle 11250  𝑂(1)co1 15433

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721  ax-cnex 11165  ax-resscn 11166  ax-pre-lttri 11183  ax-pre-lttrn 11184

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-po 5581  df-so 5582  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-er 8702  df-pm 8822  df-en 8939  df-dom 8940  df-sdom 8941  df-pnf 11251  df-mnf 11252  df-xr 11253  df-ltxr 11254  df-le 11255  df-ico 13333  df-o1 15437

This theorem is referenced by:  dchrisum0  27403