Theorem o1compt 15555

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 7119	. 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 1910	. . . . . . . 8 ⊢ Ⅎ𝑦 𝑥 ≤ 𝑧
8		nfcv 2898	. . . . . . . . 9 ⊢ Ⅎ𝑦𝑚
9		nfcv 2898	. . . . . . . . 9 ⊢ Ⅎ𝑦 ≤
10		nffvmpt1 6902	. . . . . . . . 9 ⊢ Ⅎ𝑦((𝑦 ∈ 𝐵 ↦ 𝐶)‘𝑧)
11	8, 9, 10	nfbr 5189	. . . . . . . 8 ⊢ Ⅎ𝑦 𝑚 ≤ ((𝑦 ∈ 𝐵 ↦ 𝐶)‘𝑧)
12	7, 11	nfim 1892	. . . . . . 7 ⊢ Ⅎ𝑦(𝑥 ≤ 𝑧 → 𝑚 ≤ ((𝑦 ∈ 𝐵 ↦ 𝐶)‘𝑧))
13		nfv 1910	. . . . . . 7 ⊢ Ⅎ𝑧(𝑥 ≤ 𝑦 → 𝑚 ≤ ((𝑦 ∈ 𝐵 ↦ 𝐶)‘𝑦))
14		breq2 5146	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → (𝑥 ≤ 𝑧 ↔ 𝑥 ≤ 𝑦))
15		fveq2 6891	. . . . . . . . 9 ⊢ (𝑧 = 𝑦 → ((𝑦 ∈ 𝐵 ↦ 𝐶)‘𝑧) = ((𝑦 ∈ 𝐵 ↦ 𝐶)‘𝑦))
16	15	breq2d 5154	. . . . . . . 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 3298	. . . . . 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 2727	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑦 ∈ 𝐵 ↦ 𝐶)
21	20	fvmpt2 7010	. . . . . . . . . 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 5154	. . . . . . . 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 3170	. . . . . 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 3173	. . . 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 15554	1 ⊢ (𝜑 → (𝐹 ∘ (𝑦 ∈ 𝐵 ↦ 𝐶)) ∈ 𝑂(1))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1534   ∈ wcel 2099  ∀wral 3056  ∃wrex 3065   ⊆ wss 3944   class class class wbr 5142   ↦ cmpt 5225   ∘ ccom 5676  ⟶wf 6538  ‘cfv 6542  ℂcc 11128  ℝcr 11129   ≤ cle 11271  𝑂(1)co1 15454

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-cnex 11186  ax-resscn 11187  ax-pre-lttri 11204  ax-pre-lttrn 11205

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-po 5584  df-so 5585  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-er 8718  df-pm 8839  df-en 8956  df-dom 8957  df-sdom 8958  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-ico 13354  df-o1 15458

This theorem is referenced by:  dchrisum0  27440