Theorem rexico 15307

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 484	. . . 4 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ) → 𝐵 ∈ ℝ)
2		pnfxr 11190	. . . 4 ⊢ +∞ ∈ ℝ*
3		icossre 13372	. . . 4 ⊢ ((𝐵 ∈ ℝ ∧ +∞ ∈ ℝ*) → (𝐵[,)+∞) ⊆ ℝ)
4	1, 2, 3	sylancl 587	. . 3 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ) → (𝐵[,)+∞) ⊆ ℝ)
5		ssrexv 3992	. . 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 484	. . . . . . 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 4514	. . . . . 6 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑗 ∈ ℝ) → if(𝐵 ≤ 𝑗, 𝑗, 𝐵) ∈ ℝ)
10		max1 13128	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ ∧ 𝑗 ∈ ℝ) → 𝐵 ≤ if(𝐵 ≤ 𝑗, 𝑗, 𝐵))
11	10	adantll 715	. . . . . 6 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑗 ∈ ℝ) → 𝐵 ≤ if(𝐵 ≤ 𝑗, 𝑗, 𝐵))
12		elicopnf 13389	. . . . . . 7 ⊢ (𝐵 ∈ ℝ → (if(𝐵 ≤ 𝑗, 𝑗, 𝐵) ∈ (𝐵[,)+∞) ↔ (if(𝐵 ≤ 𝑗, 𝑗, 𝐵) ∈ ℝ ∧ 𝐵 ≤ if(𝐵 ≤ 𝑗, 𝑗, 𝐵))))
13	12	ad2antlr 728	. . . . . 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 714	. . . . 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 3922	. . . . . . . . 9 ⊢ ((((𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑗 ∈ ℝ) ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ ℝ)
19		maxle 13134	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℝ ∧ 𝑗 ∈ ℝ ∧ 𝑘 ∈ ℝ) → (if(𝐵 ≤ 𝑗, 𝑗, 𝐵) ≤ 𝑘 ↔ (𝐵 ≤ 𝑘 ∧ 𝑗 ≤ 𝑘)))
20	15, 16, 18, 19	syl3anc 1374	. . . . . . . 8 ⊢ ((((𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑗 ∈ ℝ) ∧ 𝑘 ∈ 𝐴) → (if(𝐵 ≤ 𝑗, 𝑗, 𝐵) ≤ 𝑘 ↔ (𝐵 ≤ 𝑘 ∧ 𝑗 ≤ 𝑘)))
21		simpr 484	. . . . . . . 8 ⊢ ((𝐵 ≤ 𝑘 ∧ 𝑗 ≤ 𝑘) → 𝑗 ≤ 𝑘)
22	20, 21	biimtrdi 253	. . . . . . 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 3150	. . . . 5 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑗 ∈ ℝ) → (∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → 𝜑) → ∀𝑘 ∈ 𝐴 (if(𝐵 ≤ 𝑗, 𝑗, 𝐵) ≤ 𝑘 → 𝜑)))
25		breq1 5089	. . . . . 6 ⊢ (𝑛 = if(𝐵 ≤ 𝑗, 𝑗, 𝐵) → (𝑛 ≤ 𝑘 ↔ if(𝐵 ≤ 𝑗, 𝑗, 𝐵) ≤ 𝑘))
26	25	rspceaimv 3571	. . . . 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 685	. . . 4 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑗 ∈ ℝ) → (∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → 𝜑) → ∃𝑛 ∈ (𝐵[,)+∞)∀𝑘 ∈ 𝐴 (𝑛 ≤ 𝑘 → 𝜑)))
28	27	rexlimdva 3139	. . 3 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ) → (∃𝑗 ∈ ℝ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → 𝜑) → ∃𝑛 ∈ (𝐵[,)+∞)∀𝑘 ∈ 𝐴 (𝑛 ≤ 𝑘 → 𝜑)))
29		breq1 5089	. . . . . 6 ⊢ (𝑛 = 𝑗 → (𝑛 ≤ 𝑘 ↔ 𝑗 ≤ 𝑘))
30	29	imbi1d 341	. . . . 5 ⊢ (𝑛 = 𝑗 → ((𝑛 ≤ 𝑘 → 𝜑) ↔ (𝑗 ≤ 𝑘 → 𝜑)))
31	30	ralbidv 3161	. . . 4 ⊢ (𝑛 = 𝑗 → (∀𝑘 ∈ 𝐴 (𝑛 ≤ 𝑘 → 𝜑) ↔ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → 𝜑)))
32	31	cbvrexvw 3217	. . 3 ⊢ (∃𝑛 ∈ (𝐵[,)+∞)∀𝑘 ∈ 𝐴 (𝑛 ≤ 𝑘 → 𝜑) ↔ ∃𝑗 ∈ (𝐵[,)+∞)∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → 𝜑))
33	28, 32	imbitrdi 251	. 2 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ) → (∃𝑗 ∈ ℝ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → 𝜑) → ∃𝑗 ∈ (𝐵[,)+∞)∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → 𝜑)))
34	6, 33	impbid 212	1 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ) → (∃𝑗 ∈ (𝐵[,)+∞)∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → 𝜑) ↔ ∃𝑗 ∈ ℝ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → 𝜑)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062   ⊆ wss 3890  ifcif 4467   class class class wbr 5086  (class class class)co 7360  ℝcr 11028  +∞cpnf 11167  ℝ^*cxr 11169   ≤ cle 11171  [,)cico 13291

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-cnex 11085  ax-resscn 11086  ax-pre-lttri 11103  ax-pre-lttrn 11104

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-po 5532  df-so 5533  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7363  df-oprab 7364  df-mpo 7365  df-er 8636  df-en 8887  df-dom 8888  df-sdom 8889  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-ico 13295

This theorem is referenced by:  rlimi2  15467  ello1mpt2  15475  dvfsumrlim  26008