Theorem rexico 15326

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 11292	. . . 4 ⊢ +∞ ∈ ℝ*
3		icossre 13431	. . . 4 ⊢ ((𝐵 ∈ ℝ ∧ +∞ ∈ ℝ*) → (𝐵[,)+∞) ⊆ ℝ)
4	1, 2, 3	sylancl 585	. . 3 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ) → (𝐵[,)+∞) ⊆ ℝ)
5		ssrexv 4047	. . 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 768	. . . . . . 7 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑗 ∈ ℝ) → 𝐵 ∈ ℝ)
9	7, 8	ifcld 4570	. . . . . 6 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑗 ∈ ℝ) → if(𝐵 ≤ 𝑗, 𝑗, 𝐵) ∈ ℝ)
10		max1 13190	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ ∧ 𝑗 ∈ ℝ) → 𝐵 ≤ if(𝐵 ≤ 𝑗, 𝑗, 𝐵))
11	10	adantll 713	. . . . . 6 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑗 ∈ ℝ) → 𝐵 ≤ if(𝐵 ≤ 𝑗, 𝑗, 𝐵))
12		elicopnf 13448	. . . . . . 7 ⊢ (𝐵 ∈ ℝ → (if(𝐵 ≤ 𝑗, 𝑗, 𝐵) ∈ (𝐵[,)+∞) ↔ (if(𝐵 ≤ 𝑗, 𝑗, 𝐵) ∈ ℝ ∧ 𝐵 ≤ if(𝐵 ≤ 𝑗, 𝑗, 𝐵))))
13	12	ad2antlr 726	. . . . . 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 712	. . . . 5 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑗 ∈ ℝ) → if(𝐵 ≤ 𝑗, 𝑗, 𝐵) ∈ (𝐵[,)+∞))
15		simpllr 775	. . . . . . . . 9 ⊢ ((((𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑗 ∈ ℝ) ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ)
16		simplr 768	. . . . . . . . 9 ⊢ ((((𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑗 ∈ ℝ) ∧ 𝑘 ∈ 𝐴) → 𝑗 ∈ ℝ)
17		simpll 766	. . . . . . . . . 10 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑗 ∈ ℝ) → 𝐴 ⊆ ℝ)
18	17	sselda 3978	. . . . . . . . 9 ⊢ ((((𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑗 ∈ ℝ) ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ ℝ)
19		maxle 13196	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℝ ∧ 𝑗 ∈ ℝ ∧ 𝑘 ∈ ℝ) → (if(𝐵 ≤ 𝑗, 𝑗, 𝐵) ≤ 𝑘 ↔ (𝐵 ≤ 𝑘 ∧ 𝑗 ≤ 𝑘)))
20	15, 16, 18, 19	syl3anc 1369	. . . . . . . 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 252	. . . . . . 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 3162	. . . . 5 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑗 ∈ ℝ) → (∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → 𝜑) → ∀𝑘 ∈ 𝐴 (if(𝐵 ≤ 𝑗, 𝑗, 𝐵) ≤ 𝑘 → 𝜑)))
25		breq1 5145	. . . . . 6 ⊢ (𝑛 = if(𝐵 ≤ 𝑗, 𝑗, 𝐵) → (𝑛 ≤ 𝑘 ↔ if(𝐵 ≤ 𝑗, 𝑗, 𝐵) ≤ 𝑘))
26	25	rspceaimv 3613	. . . . 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 683	. . . 4 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑗 ∈ ℝ) → (∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → 𝜑) → ∃𝑛 ∈ (𝐵[,)+∞)∀𝑘 ∈ 𝐴 (𝑛 ≤ 𝑘 → 𝜑)))
28	27	rexlimdva 3150	. . 3 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ) → (∃𝑗 ∈ ℝ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → 𝜑) → ∃𝑛 ∈ (𝐵[,)+∞)∀𝑘 ∈ 𝐴 (𝑛 ≤ 𝑘 → 𝜑)))
29		breq1 5145	. . . . . 6 ⊢ (𝑛 = 𝑗 → (𝑛 ≤ 𝑘 ↔ 𝑗 ≤ 𝑘))
30	29	imbi1d 341	. . . . 5 ⊢ (𝑛 = 𝑗 → ((𝑛 ≤ 𝑘 → 𝜑) ↔ (𝑗 ≤ 𝑘 → 𝜑)))
31	30	ralbidv 3172	. . . 4 ⊢ (𝑛 = 𝑗 → (∀𝑘 ∈ 𝐴 (𝑛 ≤ 𝑘 → 𝜑) ↔ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → 𝜑)))
32	31	cbvrexvw 3230	. . 3 ⊢ (∃𝑛 ∈ (𝐵[,)+∞)∀𝑘 ∈ 𝐴 (𝑛 ≤ 𝑘 → 𝜑) ↔ ∃𝑗 ∈ (𝐵[,)+∞)∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → 𝜑))
33	28, 32	imbitrdi 250	. 2 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ) → (∃𝑗 ∈ ℝ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → 𝜑) → ∃𝑗 ∈ (𝐵[,)+∞)∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → 𝜑)))
34	6, 33	impbid 211	1 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ) → (∃𝑗 ∈ (𝐵[,)+∞)∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → 𝜑) ↔ ∃𝑗 ∈ ℝ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → 𝜑)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∈ wcel 2099  ∀wral 3056  ∃wrex 3065   ⊆ wss 3944  ifcif 4524   class class class wbr 5142  (class class class)co 7414  ℝcr 11131  +∞cpnf 11269  ℝ^*cxr 11271   ≤ cle 11273  [,)cico 13352

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-cnex 11188  ax-resscn 11189  ax-pre-lttri 11206  ax-pre-lttrn 11207

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-po 5584  df-so 5585  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-er 8718  df-en 8958  df-dom 8959  df-sdom 8960  df-pnf 11274  df-mnf 11275  df-xr 11276  df-ltxr 11277  df-le 11278  df-ico 13356

This theorem is referenced by:  rlimi2  15484  ello1mpt2  15492  dvfsumrlim  25959