Theorem lo1le 14382

Description: Transfer eventual upper boundedness from a larger function to a smaller function. (Contributed by Mario Carneiro, 26-May-2016.)

Hypotheses

Ref	Expression
lo1le.1	⊢ (𝜑 → 𝑀 ∈ ℝ)
lo1le.2	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ ≤𝑂(1))
lo1le.3	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
lo1le.4	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℝ)
lo1le.5	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑀 ≤ 𝑥)) → 𝐶 ≤ 𝐵)

Assertion

Ref	Expression
lo1le	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ ≤𝑂(1))

Distinct variable groups:   𝑥,𝐴   𝑥,𝑀   𝜑,𝑥

Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥)   𝑉(𝑥)

Proof of Theorem lo1le

Dummy variables 𝑚 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lo1le.2	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ ≤𝑂(1))
2		simpr 477	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ)
3		lo1le.1	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℝ)
4	3	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝑀 ∈ ℝ)
5	2, 4	ifcld 4131	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → if(𝑀 ≤ 𝑦, 𝑦, 𝑀) ∈ ℝ)
6	3	ad2antrr 762	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ (𝑚 ∈ ℝ ∧ 𝑥 ∈ 𝐴)) → 𝑀 ∈ ℝ)
7		simplr 792	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ (𝑚 ∈ ℝ ∧ 𝑥 ∈ 𝐴)) → 𝑦 ∈ ℝ)
8		lo1le.3	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
9	8	ralrimiva 2966	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉)
10		dmmptg 5632	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → dom (𝑥 ∈ 𝐴 ↦ 𝐵) = 𝐴)
11	9, 10	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → dom (𝑥 ∈ 𝐴 ↦ 𝐵) = 𝐴)
12		lo1dm 14250	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ ≤𝑂(1) → dom (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ ℝ)
13	1, 12	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → dom (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ ℝ)
14	11, 13	eqsstr3d 3640	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
15	14	ad2antrr 762	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ (𝑚 ∈ ℝ ∧ 𝑥 ∈ 𝐴)) → 𝐴 ⊆ ℝ)
16		simprr 796	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ (𝑚 ∈ ℝ ∧ 𝑥 ∈ 𝐴)) → 𝑥 ∈ 𝐴)
17	15, 16	sseldd 3604	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ (𝑚 ∈ ℝ ∧ 𝑥 ∈ 𝐴)) → 𝑥 ∈ ℝ)
18		maxle 12022	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (if(𝑀 ≤ 𝑦, 𝑦, 𝑀) ≤ 𝑥 ↔ (𝑀 ≤ 𝑥 ∧ 𝑦 ≤ 𝑥)))
19	6, 7, 17, 18	syl3anc 1326	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ (𝑚 ∈ ℝ ∧ 𝑥 ∈ 𝐴)) → (if(𝑀 ≤ 𝑦, 𝑦, 𝑀) ≤ 𝑥 ↔ (𝑀 ≤ 𝑥 ∧ 𝑦 ≤ 𝑥)))
20		simpr 477	. . . . . . . . . . 11 ⊢ ((𝑀 ≤ 𝑥 ∧ 𝑦 ≤ 𝑥) → 𝑦 ≤ 𝑥)
21	19, 20	syl6bi 243	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ (𝑚 ∈ ℝ ∧ 𝑥 ∈ 𝐴)) → (if(𝑀 ≤ 𝑦, 𝑦, 𝑀) ≤ 𝑥 → 𝑦 ≤ 𝑥))
22	21	imim1d 82	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ (𝑚 ∈ ℝ ∧ 𝑥 ∈ 𝐴)) → ((𝑦 ≤ 𝑥 → 𝐵 ≤ 𝑚) → (if(𝑀 ≤ 𝑦, 𝑦, 𝑀) ≤ 𝑥 → 𝐵 ≤ 𝑚)))
23		lo1le.5	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑀 ≤ 𝑥)) → 𝐶 ≤ 𝐵)
24	23	adantlr 751	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑀 ≤ 𝑥)) → 𝐶 ≤ 𝐵)
25	24	adantrll 758	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ((𝑚 ∈ ℝ ∧ 𝑥 ∈ 𝐴) ∧ 𝑀 ≤ 𝑥)) → 𝐶 ≤ 𝐵)
26		simpl 473	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝜑)
27		simplr 792	. . . . . . . . . . . . . . . 16 ⊢ (((𝑚 ∈ ℝ ∧ 𝑥 ∈ 𝐴) ∧ 𝑀 ≤ 𝑥) → 𝑥 ∈ 𝐴)
28		lo1le.4	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℝ)
29	26, 27, 28	syl2an 494	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ((𝑚 ∈ ℝ ∧ 𝑥 ∈ 𝐴) ∧ 𝑀 ≤ 𝑥)) → 𝐶 ∈ ℝ)
30	8, 1	lo1mptrcl 14352	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)
31	26, 27, 30	syl2an 494	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ((𝑚 ∈ ℝ ∧ 𝑥 ∈ 𝐴) ∧ 𝑀 ≤ 𝑥)) → 𝐵 ∈ ℝ)
32		simprll 802	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ((𝑚 ∈ ℝ ∧ 𝑥 ∈ 𝐴) ∧ 𝑀 ≤ 𝑥)) → 𝑚 ∈ ℝ)
33		letr 10131	. . . . . . . . . . . . . . 15 ⊢ ((𝐶 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑚 ∈ ℝ) → ((𝐶 ≤ 𝐵 ∧ 𝐵 ≤ 𝑚) → 𝐶 ≤ 𝑚))
34	29, 31, 32, 33	syl3anc 1326	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ((𝑚 ∈ ℝ ∧ 𝑥 ∈ 𝐴) ∧ 𝑀 ≤ 𝑥)) → ((𝐶 ≤ 𝐵 ∧ 𝐵 ≤ 𝑚) → 𝐶 ≤ 𝑚))
35	25, 34	mpand 711	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ((𝑚 ∈ ℝ ∧ 𝑥 ∈ 𝐴) ∧ 𝑀 ≤ 𝑥)) → (𝐵 ≤ 𝑚 → 𝐶 ≤ 𝑚))
36	35	expr 643	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ (𝑚 ∈ ℝ ∧ 𝑥 ∈ 𝐴)) → (𝑀 ≤ 𝑥 → (𝐵 ≤ 𝑚 → 𝐶 ≤ 𝑚)))
37	36	adantrd 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ (𝑚 ∈ ℝ ∧ 𝑥 ∈ 𝐴)) → ((𝑀 ≤ 𝑥 ∧ 𝑦 ≤ 𝑥) → (𝐵 ≤ 𝑚 → 𝐶 ≤ 𝑚)))
38	19, 37	sylbid 230	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ (𝑚 ∈ ℝ ∧ 𝑥 ∈ 𝐴)) → (if(𝑀 ≤ 𝑦, 𝑦, 𝑀) ≤ 𝑥 → (𝐵 ≤ 𝑚 → 𝐶 ≤ 𝑚)))
39	38	a2d 29	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ (𝑚 ∈ ℝ ∧ 𝑥 ∈ 𝐴)) → ((if(𝑀 ≤ 𝑦, 𝑦, 𝑀) ≤ 𝑥 → 𝐵 ≤ 𝑚) → (if(𝑀 ≤ 𝑦, 𝑦, 𝑀) ≤ 𝑥 → 𝐶 ≤ 𝑚)))
40	22, 39	syld 47	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ (𝑚 ∈ ℝ ∧ 𝑥 ∈ 𝐴)) → ((𝑦 ≤ 𝑥 → 𝐵 ≤ 𝑚) → (if(𝑀 ≤ 𝑦, 𝑦, 𝑀) ≤ 𝑥 → 𝐶 ≤ 𝑚)))
41	40	anassrs 680	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑚 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → ((𝑦 ≤ 𝑥 → 𝐵 ≤ 𝑚) → (if(𝑀 ≤ 𝑦, 𝑦, 𝑀) ≤ 𝑥 → 𝐶 ≤ 𝑚)))
42	41	ralimdva 2962	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑚 ∈ ℝ) → (∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → 𝐵 ≤ 𝑚) → ∀𝑥 ∈ 𝐴 (if(𝑀 ≤ 𝑦, 𝑦, 𝑀) ≤ 𝑥 → 𝐶 ≤ 𝑚)))
43	42	reximdva 3017	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (∃𝑚 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → 𝐵 ≤ 𝑚) → ∃𝑚 ∈ ℝ ∀𝑥 ∈ 𝐴 (if(𝑀 ≤ 𝑦, 𝑦, 𝑀) ≤ 𝑥 → 𝐶 ≤ 𝑚)))
44		breq1 4656	. . . . . . . 8 ⊢ (𝑧 = if(𝑀 ≤ 𝑦, 𝑦, 𝑀) → (𝑧 ≤ 𝑥 ↔ if(𝑀 ≤ 𝑦, 𝑦, 𝑀) ≤ 𝑥))
45	44	imbi1d 331	. . . . . . 7 ⊢ (𝑧 = if(𝑀 ≤ 𝑦, 𝑦, 𝑀) → ((𝑧 ≤ 𝑥 → 𝐶 ≤ 𝑚) ↔ (if(𝑀 ≤ 𝑦, 𝑦, 𝑀) ≤ 𝑥 → 𝐶 ≤ 𝑚)))
46	45	rexralbidv 3058	. . . . . 6 ⊢ (𝑧 = if(𝑀 ≤ 𝑦, 𝑦, 𝑀) → (∃𝑚 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑧 ≤ 𝑥 → 𝐶 ≤ 𝑚) ↔ ∃𝑚 ∈ ℝ ∀𝑥 ∈ 𝐴 (if(𝑀 ≤ 𝑦, 𝑦, 𝑀) ≤ 𝑥 → 𝐶 ≤ 𝑚)))
47	46	rspcev 3309	. . . . 5 ⊢ ((if(𝑀 ≤ 𝑦, 𝑦, 𝑀) ∈ ℝ ∧ ∃𝑚 ∈ ℝ ∀𝑥 ∈ 𝐴 (if(𝑀 ≤ 𝑦, 𝑦, 𝑀) ≤ 𝑥 → 𝐶 ≤ 𝑚)) → ∃𝑧 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑧 ≤ 𝑥 → 𝐶 ≤ 𝑚))
48	5, 43, 47	syl6an 568	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (∃𝑚 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → 𝐵 ≤ 𝑚) → ∃𝑧 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑧 ≤ 𝑥 → 𝐶 ≤ 𝑚)))
49	48	rexlimdva 3031	. . 3 ⊢ (𝜑 → (∃𝑦 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → 𝐵 ≤ 𝑚) → ∃𝑧 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑧 ≤ 𝑥 → 𝐶 ≤ 𝑚)))
50	14, 30	ello1mpt 14252	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ ≤𝑂(1) ↔ ∃𝑦 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → 𝐵 ≤ 𝑚)))
51	14, 28	ello1mpt 14252	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ∈ ≤𝑂(1) ↔ ∃𝑧 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑧 ≤ 𝑥 → 𝐶 ≤ 𝑚)))
52	49, 50, 51	3imtr4d 283	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ ≤𝑂(1) → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ ≤𝑂(1)))
53	1, 52	mpd 15	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ ≤𝑂(1))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913   ⊆ wss 3574  ifcif 4086   class class class wbr 4653   ↦ cmpt 4729  dom cdm 5114  ℝcr 9935   ≤ cle 10075  ≤𝑂(1)clo1 14218

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-pre-lttri 10010  ax-pre-lttrn 10011

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-er 7742  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-ico 12181  df-lo1 14222

This theorem is referenced by:  o1le  14383  vmalogdivsum2  25227  pntrlog2bndlem1  25266  pntrlog2bndlem5  25270