Theorem o1compt 15761

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 7118	. 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 1946	. . . . . . . 8 ⊢ Ⅎ𝑦 𝑥 ≤ 𝑧
8		nfcv 2922	. . . . . . . . 9 ⊢ Ⅎ𝑦𝑚
9		nfcv 2922	. . . . . . . . 9 ⊢ Ⅎ𝑦 ≤
10		nffvmpt1 6899	. . . . . . . . 9 ⊢ Ⅎ𝑦((𝑦 ∈ 𝐵 ↦ 𝐶)‘𝑧)
11	8, 9, 10	nfbr 5151	. . . . . . . 8 ⊢ Ⅎ𝑦 𝑚 ≤ ((𝑦 ∈ 𝐵 ↦ 𝐶)‘𝑧)
12	7, 11	nfim 1928	. . . . . . 7 ⊢ Ⅎ𝑦(𝑥 ≤ 𝑧 → 𝑚 ≤ ((𝑦 ∈ 𝐵 ↦ 𝐶)‘𝑧))
13		nfv 1946	. . . . . . 7 ⊢ Ⅎ𝑧(𝑥 ≤ 𝑦 → 𝑚 ≤ ((𝑦 ∈ 𝐵 ↦ 𝐶)‘𝑦))
14		breq2 5106	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → (𝑥 ≤ 𝑧 ↔ 𝑥 ≤ 𝑦))
15		fveq2 6888	. . . . . . . . 9 ⊢ (𝑧 = 𝑦 → ((𝑦 ∈ 𝐵 ↦ 𝐶)‘𝑧) = ((𝑦 ∈ 𝐵 ↦ 𝐶)‘𝑦))
16	15	breq2d 5114	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → (𝑚 ≤ ((𝑦 ∈ 𝐵 ↦ 𝐶)‘𝑧) ↔ 𝑚 ≤ ((𝑦 ∈ 𝐵 ↦ 𝐶)‘𝑦)))
17	14, 16	imbi12d 347	. . . . . . 7 ⊢ (𝑧 = 𝑦 → ((𝑥 ≤ 𝑧 → 𝑚 ≤ ((𝑦 ∈ 𝐵 ↦ 𝐶)‘𝑧)) ↔ (𝑥 ≤ 𝑦 → 𝑚 ≤ ((𝑦 ∈ 𝐵 ↦ 𝐶)‘𝑦))))
18	12, 13, 17	cbvralw 3304	. . . . . 6 ⊢ (∀𝑧 ∈ 𝐵 (𝑥 ≤ 𝑧 → 𝑚 ≤ ((𝑦 ∈ 𝐵 ↦ 𝐶)‘𝑧)) ↔ ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → 𝑚 ≤ ((𝑦 ∈ 𝐵 ↦ 𝐶)‘𝑦)))
19		simpr 490	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵)
20		eqid 2760	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑦 ∈ 𝐵 ↦ 𝐶)
21	20	fvmpt2 7008	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝐶 ∈ 𝐴) → ((𝑦 ∈ 𝐵 ↦ 𝐶)‘𝑦) = 𝐶)
22	19, 3, 21	syl2anc 596	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ((𝑦 ∈ 𝐵 ↦ 𝐶)‘𝑦) = 𝐶)
23	22	breq2d 5114	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝑚 ≤ ((𝑦 ∈ 𝐵 ↦ 𝐶)‘𝑦) ↔ 𝑚 ≤ 𝐶))
24	23	imbi2d 343	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ((𝑥 ≤ 𝑦 → 𝑚 ≤ ((𝑦 ∈ 𝐵 ↦ 𝐶)‘𝑦)) ↔ (𝑥 ≤ 𝑦 → 𝑚 ≤ 𝐶)))
25	24	ralbidva 3183	. . . . . 6 ⊢ (𝜑 → (∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → 𝑚 ≤ ((𝑦 ∈ 𝐵 ↦ 𝐶)‘𝑦)) ↔ ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → 𝑚 ≤ 𝐶)))
26	18, 25	bitrid 286	. . . . 5 ⊢ (𝜑 → (∀𝑧 ∈ 𝐵 (𝑥 ≤ 𝑧 → 𝑚 ≤ ((𝑦 ∈ 𝐵 ↦ 𝐶)‘𝑧)) ↔ ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → 𝑚 ≤ 𝐶)))
27	26	rexbidv 3186	. . . 4 ⊢ (𝜑 → (∃𝑥 ∈ ℝ ∀𝑧 ∈ 𝐵 (𝑥 ≤ 𝑧 → 𝑚 ≤ ((𝑦 ∈ 𝐵 ↦ 𝐶)‘𝑧)) ↔ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → 𝑚 ≤ 𝐶)))
28	27	adantr 486	. . 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 260	. 2 ⊢ ((𝜑 ∧ 𝑚 ∈ ℝ) → ∃𝑥 ∈ ℝ ∀𝑧 ∈ 𝐵 (𝑥 ≤ 𝑧 → 𝑚 ≤ ((𝑦 ∈ 𝐵 ↦ 𝐶)‘𝑧)))
30	1, 2, 4, 5, 29	o1co 15760	1 ⊢ (𝜑 → (𝐹 ∘ (𝑦 ∈ 𝐵 ↦ 𝐶)) ∈ 𝑂(1))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086   ⊆ wss 3896   class class class wbr 5102   ↦ cmpt 5185   ∘ ccom 5654  ⟶wf 6537  ‘cfv 6541  ℂcc 11203  ℝcr 11204   ≤ cle 11349  𝑂(1)co1 15660

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7753  ax-cnex 11261  ax-resscn 11262  ax-pre-lttri 11279  ax-pre-lttrn 11280

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-po 5558  df-so 5559  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6496  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-ov 7425  df-oprab 7426  df-mpo 7427  df-er 8722  df-pm 8855  df-en 8979  df-dom 8980  df-sdom 8981  df-pnf 11350  df-mnf 11351  df-xr 11352  df-ltxr 11353  df-le 11354  df-ico 13487  df-o1 15664

This theorem is used by:  dchrisum0  27854