Theorem lo1le 15679

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 4527	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → if(𝑀 ≤ 𝑦, 𝑦, 𝑀) ∈ ℝ)
6	3	ad2antrr 736	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ (𝑚 ∈ ℝ ∧ 𝑥 ∈ 𝐴)) → 𝑀 ∈ ℝ)
7		simplr 778	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ (𝑚 ∈ ℝ ∧ 𝑥 ∈ 𝐴)) → 𝑦 ∈ ℝ)
8		lo1le.3	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
9	8	ralrimiva 3154	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉)
10		dmmptg 6229	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → dom (𝑥 ∈ 𝐴 ↦ 𝐵) = 𝐴)
11	9, 10	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → dom (𝑥 ∈ 𝐴 ↦ 𝐵) = 𝐴)
12		lo1dm 15546	. . . . . . . . . . . . . . . 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 3971	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
15	14	ad2antrr 736	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ (𝑚 ∈ ℝ ∧ 𝑥 ∈ 𝐴)) → 𝐴 ⊆ ℝ)
16		simprr 782	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ (𝑚 ∈ ℝ ∧ 𝑥 ∈ 𝐴)) → 𝑥 ∈ 𝐴)
17	15, 16	sseldd 3937	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ (𝑚 ∈ ℝ ∧ 𝑥 ∈ 𝐴)) → 𝑥 ∈ ℝ)
18		maxle 13194	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (if(𝑀 ≤ 𝑦, 𝑦, 𝑀) ≤ 𝑥 ↔ (𝑀 ≤ 𝑥 ∧ 𝑦 ≤ 𝑥)))
19	6, 7, 17, 18	syl3anc 1390	. . . . . . . . . . 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	biimtrdi 255	. . . . . . . . . 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 725	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑀 ≤ 𝑥)) → 𝐶 ≤ 𝐵)
25	24	adantrll 732	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ((𝑚 ∈ ℝ ∧ 𝑥 ∈ 𝐴) ∧ 𝑀 ≤ 𝑥)) → 𝐶 ≤ 𝐵)
26		simpl 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝜑)
27		simplr 778	. . . . . . . . . . . . . . . 16 ⊢ (((𝑚 ∈ ℝ ∧ 𝑥 ∈ 𝐴) ∧ 𝑀 ≤ 𝑥) → 𝑥 ∈ 𝐴)
28		lo1le.4	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℝ)
29	26, 27, 28	syl2an 605	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ((𝑚 ∈ ℝ ∧ 𝑥 ∈ 𝐴) ∧ 𝑀 ≤ 𝑥)) → 𝐶 ∈ ℝ)
30	8, 1	lo1mptrcl 15649	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)
31	26, 27, 30	syl2an 605	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ((𝑚 ∈ ℝ ∧ 𝑥 ∈ 𝐴) ∧ 𝑀 ≤ 𝑥)) → 𝐵 ∈ ℝ)
32		simprll 788	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ((𝑚 ∈ ℝ ∧ 𝑥 ∈ 𝐴) ∧ 𝑀 ≤ 𝑥)) → 𝑚 ∈ ℝ)
33		letr 11277	. . . . . . . . . . . . . . 15 ⊢ ((𝐶 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑚 ∈ ℝ) → ((𝐶 ≤ 𝐵 ∧ 𝐵 ≤ 𝑚) → 𝐶 ≤ 𝑚))
34	29, 31, 32, 33	syl3anc 1390	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ((𝑚 ∈ ℝ ∧ 𝑥 ∈ 𝐴) ∧ 𝑀 ≤ 𝑥)) → ((𝐶 ≤ 𝐵 ∧ 𝐵 ≤ 𝑚) → 𝐶 ≤ 𝑚))
35	25, 34	mpand 705	. . . . . . . . . . . . 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 242	. . . . . . . . . 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 3174	. . . . . 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 3175	. . . . 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 5103	. . . . . . . 8 ⊢ (𝑧 = if(𝑀 ≤ 𝑦, 𝑦, 𝑀) → (𝑧 ≤ 𝑥 ↔ if(𝑀 ≤ 𝑦, 𝑦, 𝑀) ≤ 𝑥))
45	44	imbi1d 343	. . . . . . 7 ⊢ (𝑧 = if(𝑀 ≤ 𝑦, 𝑦, 𝑀) → ((𝑧 ≤ 𝑥 → 𝐶 ≤ 𝑚) ↔ (if(𝑀 ≤ 𝑦, 𝑦, 𝑀) ≤ 𝑥 → 𝐶 ≤ 𝑚)))
46	45	rexralbidv 3228	. . . . . 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 3581	. . . . 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 694	. . . 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 3163	. . 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 15548	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ ≤𝑂(1) ↔ ∃𝑦 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → 𝐵 ≤ 𝑚)))
51	14, 28	ello1mpt 15548	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ∈ ≤𝑂(1) ↔ ∃𝑧 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑧 ≤ 𝑥 → 𝐶 ≤ 𝑚)))
52	49, 50, 51	3imtr4d 296	. 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 208   ∧ wa 399   = wceq 1560   ∈ wcel 2142  ∀wral 3076  ∃wrex 3086   ⊆ wss 3904  ifcif 4480   class class class wbr 5100   ↦ cmpt 5181  dom cdm 5647  ℝcr 11072   ≤ cle 11217  ≤𝑂(1)clo1 15514

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718  ax-cnex 11129  ax-resscn 11130  ax-pre-lttri 11147  ax-pre-lttrn 11148

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3062  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-po 5555  df-so 5556  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-ov 7399  df-oprab 7400  df-mpo 7401  df-er 8678  df-pm 8811  df-en 8928  df-dom 8929  df-sdom 8930  df-pnf 11218  df-mnf 11219  df-xr 11220  df-ltxr 11221  df-le 11222  df-ico 13355  df-lo1 15518

This theorem is referenced by:  o1le  15680  vmalogdivsum2  27599  pntrlog2bndlem1  27638  pntrlog2bndlem5  27642