Theorem rexico 14917

Description: Restrict the base of an upper real quantifier to an upper real set. (Contributed by Mario Carneiro, 12-May-2016.)

Assertion

Ref	Expression
rexico	⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ) → (∃𝑗 ∈ (𝐵[,)+∞)∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → 𝜑) ↔ ∃𝑗 ∈ ℝ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → 𝜑)))

Distinct variable groups:   𝑗,𝑘,𝐴   𝐵,𝑗,𝑘   𝜑,𝑗

Allowed substitution hint:   𝜑(𝑘)

Proof of Theorem rexico

Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 488	. . . 4 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ) → 𝐵 ∈ ℝ)
2		pnfxr 10887	. . . 4 ⊢ +∞ ∈ ℝ*
3		icossre 13016	. . . 4 ⊢ ((𝐵 ∈ ℝ ∧ +∞ ∈ ℝ*) → (𝐵[,)+∞) ⊆ ℝ)
4	1, 2, 3	sylancl 589	. . 3 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ) → (𝐵[,)+∞) ⊆ ℝ)
5		ssrexv 3968	. . 3 ⊢ ((𝐵[,)+∞) ⊆ ℝ → (∃𝑗 ∈ (𝐵[,)+∞)∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → 𝜑) → ∃𝑗 ∈ ℝ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → 𝜑)))
6	4, 5	syl 17	. 2 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ) → (∃𝑗 ∈ (𝐵[,)+∞)∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → 𝜑) → ∃𝑗 ∈ ℝ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → 𝜑)))
7		simpr 488	. . . . . . 7 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑗 ∈ ℝ) → 𝑗 ∈ ℝ)
8		simplr 769	. . . . . . 7 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑗 ∈ ℝ) → 𝐵 ∈ ℝ)
9	7, 8	ifcld 4485	. . . . . 6 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑗 ∈ ℝ) → if(𝐵 ≤ 𝑗, 𝑗, 𝐵) ∈ ℝ)
10		max1 12775	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ ∧ 𝑗 ∈ ℝ) → 𝐵 ≤ if(𝐵 ≤ 𝑗, 𝑗, 𝐵))
11	10	adantll 714	. . . . . 6 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑗 ∈ ℝ) → 𝐵 ≤ if(𝐵 ≤ 𝑗, 𝑗, 𝐵))
12		elicopnf 13033	. . . . . . 7 ⊢ (𝐵 ∈ ℝ → (if(𝐵 ≤ 𝑗, 𝑗, 𝐵) ∈ (𝐵[,)+∞) ↔ (if(𝐵 ≤ 𝑗, 𝑗, 𝐵) ∈ ℝ ∧ 𝐵 ≤ if(𝐵 ≤ 𝑗, 𝑗, 𝐵))))
13	12	ad2antlr 727	. . . . . 6 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑗 ∈ ℝ) → (if(𝐵 ≤ 𝑗, 𝑗, 𝐵) ∈ (𝐵[,)+∞) ↔ (if(𝐵 ≤ 𝑗, 𝑗, 𝐵) ∈ ℝ ∧ 𝐵 ≤ if(𝐵 ≤ 𝑗, 𝑗, 𝐵))))
14	9, 11, 13	mpbir2and 713	. . . . 5 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑗 ∈ ℝ) → if(𝐵 ≤ 𝑗, 𝑗, 𝐵) ∈ (𝐵[,)+∞))
15		simpllr 776	. . . . . . . . 9 ⊢ ((((𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑗 ∈ ℝ) ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ)
16		simplr 769	. . . . . . . . 9 ⊢ ((((𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑗 ∈ ℝ) ∧ 𝑘 ∈ 𝐴) → 𝑗 ∈ ℝ)
17		simpll 767	. . . . . . . . . 10 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑗 ∈ ℝ) → 𝐴 ⊆ ℝ)
18	17	sselda 3901	. . . . . . . . 9 ⊢ ((((𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑗 ∈ ℝ) ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ ℝ)
19		maxle 12781	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℝ ∧ 𝑗 ∈ ℝ ∧ 𝑘 ∈ ℝ) → (if(𝐵 ≤ 𝑗, 𝑗, 𝐵) ≤ 𝑘 ↔ (𝐵 ≤ 𝑘 ∧ 𝑗 ≤ 𝑘)))
20	15, 16, 18, 19	syl3anc 1373	. . . . . . . 8 ⊢ ((((𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑗 ∈ ℝ) ∧ 𝑘 ∈ 𝐴) → (if(𝐵 ≤ 𝑗, 𝑗, 𝐵) ≤ 𝑘 ↔ (𝐵 ≤ 𝑘 ∧ 𝑗 ≤ 𝑘)))
21		simpr 488	. . . . . . . 8 ⊢ ((𝐵 ≤ 𝑘 ∧ 𝑗 ≤ 𝑘) → 𝑗 ≤ 𝑘)
22	20, 21	syl6bi 256	. . . . . . 7 ⊢ ((((𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑗 ∈ ℝ) ∧ 𝑘 ∈ 𝐴) → (if(𝐵 ≤ 𝑗, 𝑗, 𝐵) ≤ 𝑘 → 𝑗 ≤ 𝑘))
23	22	imim1d 82	. . . . . 6 ⊢ ((((𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑗 ∈ ℝ) ∧ 𝑘 ∈ 𝐴) → ((𝑗 ≤ 𝑘 → 𝜑) → (if(𝐵 ≤ 𝑗, 𝑗, 𝐵) ≤ 𝑘 → 𝜑)))
24	23	ralimdva 3100	. . . . 5 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑗 ∈ ℝ) → (∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → 𝜑) → ∀𝑘 ∈ 𝐴 (if(𝐵 ≤ 𝑗, 𝑗, 𝐵) ≤ 𝑘 → 𝜑)))
25		breq1 5056	. . . . . 6 ⊢ (𝑛 = if(𝐵 ≤ 𝑗, 𝑗, 𝐵) → (𝑛 ≤ 𝑘 ↔ if(𝐵 ≤ 𝑗, 𝑗, 𝐵) ≤ 𝑘))
26	25	rspceaimv 3542	. . . . 5 ⊢ ((if(𝐵 ≤ 𝑗, 𝑗, 𝐵) ∈ (𝐵[,)+∞) ∧ ∀𝑘 ∈ 𝐴 (if(𝐵 ≤ 𝑗, 𝑗, 𝐵) ≤ 𝑘 → 𝜑)) → ∃𝑛 ∈ (𝐵[,)+∞)∀𝑘 ∈ 𝐴 (𝑛 ≤ 𝑘 → 𝜑))
27	14, 24, 26	syl6an 684	. . . 4 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑗 ∈ ℝ) → (∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → 𝜑) → ∃𝑛 ∈ (𝐵[,)+∞)∀𝑘 ∈ 𝐴 (𝑛 ≤ 𝑘 → 𝜑)))
28	27	rexlimdva 3203	. . 3 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ) → (∃𝑗 ∈ ℝ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → 𝜑) → ∃𝑛 ∈ (𝐵[,)+∞)∀𝑘 ∈ 𝐴 (𝑛 ≤ 𝑘 → 𝜑)))
29		breq1 5056	. . . . . 6 ⊢ (𝑛 = 𝑗 → (𝑛 ≤ 𝑘 ↔ 𝑗 ≤ 𝑘))
30	29	imbi1d 345	. . . . 5 ⊢ (𝑛 = 𝑗 → ((𝑛 ≤ 𝑘 → 𝜑) ↔ (𝑗 ≤ 𝑘 → 𝜑)))
31	30	ralbidv 3118	. . . 4 ⊢ (𝑛 = 𝑗 → (∀𝑘 ∈ 𝐴 (𝑛 ≤ 𝑘 → 𝜑) ↔ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → 𝜑)))
32	31	cbvrexvw 3359	. . 3 ⊢ (∃𝑛 ∈ (𝐵[,)+∞)∀𝑘 ∈ 𝐴 (𝑛 ≤ 𝑘 → 𝜑) ↔ ∃𝑗 ∈ (𝐵[,)+∞)∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → 𝜑))
33	28, 32	syl6ib 254	. 2 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ) → (∃𝑗 ∈ ℝ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → 𝜑) → ∃𝑗 ∈ (𝐵[,)+∞)∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → 𝜑)))
34	6, 33	impbid 215	1 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ) → (∃𝑗 ∈ (𝐵[,)+∞)∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → 𝜑) ↔ ∃𝑗 ∈ ℝ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → 𝜑)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∈ wcel 2110  ∀wral 3061  ∃wrex 3062   ⊆ wss 3866  ifcif 4439   class class class wbr 5053  (class class class)co 7213  ℝcr 10728  +∞cpnf 10864  ℝ^*cxr 10866   ≤ cle 10868  [,)cico 12937

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-cnex 10785  ax-resscn 10786  ax-pre-lttri 10803  ax-pre-lttrn 10804

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-po 5468  df-so 5469  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-ov 7216  df-oprab 7217  df-mpo 7218  df-er 8391  df-en 8627  df-dom 8628  df-sdom 8629  df-pnf 10869  df-mnf 10870  df-xr 10871  df-ltxr 10872  df-le 10873  df-ico 12941

This theorem is referenced by:  rlimi2  15075  ello1mpt2  15083  dvfsumrlim  24928