Theorem lo1le 15000

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 488	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ)
3		lo1le.1	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℝ)
4	3	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝑀 ∈ ℝ)
5	2, 4	ifcld 4470	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → if(𝑀 ≤ 𝑦, 𝑦, 𝑀) ∈ ℝ)
6	3	ad2antrr 725	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ (𝑚 ∈ ℝ ∧ 𝑥 ∈ 𝐴)) → 𝑀 ∈ ℝ)
7		simplr 768	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ (𝑚 ∈ ℝ ∧ 𝑥 ∈ 𝐴)) → 𝑦 ∈ ℝ)
8		lo1le.3	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
9	8	ralrimiva 3149	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉)
10		dmmptg 6063	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → dom (𝑥 ∈ 𝐴 ↦ 𝐵) = 𝐴)
11	9, 10	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → dom (𝑥 ∈ 𝐴 ↦ 𝐵) = 𝐴)
12		lo1dm 14868	. . . . . . . . . . . . . . . 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	eqsstrrd 3954	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
15	14	ad2antrr 725	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ (𝑚 ∈ ℝ ∧ 𝑥 ∈ 𝐴)) → 𝐴 ⊆ ℝ)
16		simprr 772	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ (𝑚 ∈ ℝ ∧ 𝑥 ∈ 𝐴)) → 𝑥 ∈ 𝐴)
17	15, 16	sseldd 3916	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ (𝑚 ∈ ℝ ∧ 𝑥 ∈ 𝐴)) → 𝑥 ∈ ℝ)
18		maxle 12572	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (if(𝑀 ≤ 𝑦, 𝑦, 𝑀) ≤ 𝑥 ↔ (𝑀 ≤ 𝑥 ∧ 𝑦 ≤ 𝑥)))
19	6, 7, 17, 18	syl3anc 1368	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ (𝑚 ∈ ℝ ∧ 𝑥 ∈ 𝐴)) → (if(𝑀 ≤ 𝑦, 𝑦, 𝑀) ≤ 𝑥 ↔ (𝑀 ≤ 𝑥 ∧ 𝑦 ≤ 𝑥)))
20		simpr 488	. . . . . . . . . . 11 ⊢ ((𝑀 ≤ 𝑥 ∧ 𝑦 ≤ 𝑥) → 𝑦 ≤ 𝑥)
21	19, 20	syl6bi 256	. . . . . . . . . 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 714	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑀 ≤ 𝑥)) → 𝐶 ≤ 𝐵)
25	24	adantrll 721	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ((𝑚 ∈ ℝ ∧ 𝑥 ∈ 𝐴) ∧ 𝑀 ≤ 𝑥)) → 𝐶 ≤ 𝐵)
26		simpl 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝜑)
27		simplr 768	. . . . . . . . . . . . . . . 16 ⊢ (((𝑚 ∈ ℝ ∧ 𝑥 ∈ 𝐴) ∧ 𝑀 ≤ 𝑥) → 𝑥 ∈ 𝐴)
28		lo1le.4	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℝ)
29	26, 27, 28	syl2an 598	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ((𝑚 ∈ ℝ ∧ 𝑥 ∈ 𝐴) ∧ 𝑀 ≤ 𝑥)) → 𝐶 ∈ ℝ)
30	8, 1	lo1mptrcl 14970	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)
31	26, 27, 30	syl2an 598	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ((𝑚 ∈ ℝ ∧ 𝑥 ∈ 𝐴) ∧ 𝑀 ≤ 𝑥)) → 𝐵 ∈ ℝ)
32		simprll 778	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ((𝑚 ∈ ℝ ∧ 𝑥 ∈ 𝐴) ∧ 𝑀 ≤ 𝑥)) → 𝑚 ∈ ℝ)
33		letr 10723	. . . . . . . . . . . . . . 15 ⊢ ((𝐶 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑚 ∈ ℝ) → ((𝐶 ≤ 𝐵 ∧ 𝐵 ≤ 𝑚) → 𝐶 ≤ 𝑚))
34	29, 31, 32, 33	syl3anc 1368	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ((𝑚 ∈ ℝ ∧ 𝑥 ∈ 𝐴) ∧ 𝑀 ≤ 𝑥)) → ((𝐶 ≤ 𝐵 ∧ 𝐵 ≤ 𝑚) → 𝐶 ≤ 𝑚))
35	25, 34	mpand 694	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ((𝑚 ∈ ℝ ∧ 𝑥 ∈ 𝐴) ∧ 𝑀 ≤ 𝑥)) → (𝐵 ≤ 𝑚 → 𝐶 ≤ 𝑚))
36	35	expr 460	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ (𝑚 ∈ ℝ ∧ 𝑥 ∈ 𝐴)) → (𝑀 ≤ 𝑥 → (𝐵 ≤ 𝑚 → 𝐶 ≤ 𝑚)))
37	36	adantrd 495	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ (𝑚 ∈ ℝ ∧ 𝑥 ∈ 𝐴)) → ((𝑀 ≤ 𝑥 ∧ 𝑦 ≤ 𝑥) → (𝐵 ≤ 𝑚 → 𝐶 ≤ 𝑚)))
38	19, 37	sylbid 243	. . . . . . . . . 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 471	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑚 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → ((𝑦 ≤ 𝑥 → 𝐵 ≤ 𝑚) → (if(𝑀 ≤ 𝑦, 𝑦, 𝑀) ≤ 𝑥 → 𝐶 ≤ 𝑚)))
42	41	ralimdva 3144	. . . . . 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 3233	. . . . 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 5033	. . . . . . . 8 ⊢ (𝑧 = if(𝑀 ≤ 𝑦, 𝑦, 𝑀) → (𝑧 ≤ 𝑥 ↔ if(𝑀 ≤ 𝑦, 𝑦, 𝑀) ≤ 𝑥))
45	44	imbi1d 345	. . . . . . 7 ⊢ (𝑧 = if(𝑀 ≤ 𝑦, 𝑦, 𝑀) → ((𝑧 ≤ 𝑥 → 𝐶 ≤ 𝑚) ↔ (if(𝑀 ≤ 𝑦, 𝑦, 𝑀) ≤ 𝑥 → 𝐶 ≤ 𝑚)))
46	45	rexralbidv 3260	. . . . . 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 3571	. . . . 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 683	. . . 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 3243	. . 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 14870	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ ≤𝑂(1) ↔ ∃𝑦 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → 𝐵 ≤ 𝑚)))
51	14, 28	ello1mpt 14870	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ∈ ≤𝑂(1) ↔ ∃𝑧 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑧 ≤ 𝑥 → 𝐶 ≤ 𝑚)))
52	49, 50, 51	3imtr4d 297	. 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 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107   ⊆ wss 3881  ifcif 4425   class class class wbr 5030   ↦ cmpt 5110  dom cdm 5519  ℝcr 10525   ≤ cle 10665  ≤𝑂(1)clo1 14836

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583  ax-pre-lttri 10600  ax-pre-lttrn 10601

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-po 5438  df-so 5439  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-er 8272  df-pm 8392  df-en 8493  df-dom 8494  df-sdom 8495  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-ico 12732  df-lo1 14840

This theorem is referenced by:  o1le  15001  vmalogdivsum2  26122  pntrlog2bndlem1  26161  pntrlog2bndlem5  26165