Theorem rexanre 15058

Description: Combine two different upper real properties into one. (Contributed by Mario Carneiro, 8-May-2016.)

Assertion

Ref	Expression
rexanre	⊢ (𝐴 ⊆ ℝ → (∃𝑗 ∈ ℝ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (𝜑 ∧ 𝜓)) ↔ (∃𝑗 ∈ ℝ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → 𝜑) ∧ ∃𝑗 ∈ ℝ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → 𝜓))))

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

Allowed substitution hints:   𝜑(𝑘)   𝜓(𝑘)

Proof of Theorem rexanre

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → 𝜑)
2	1	imim2i 16	. . . . 5 ⊢ ((𝑗 ≤ 𝑘 → (𝜑 ∧ 𝜓)) → (𝑗 ≤ 𝑘 → 𝜑))
3	2	ralimi 3087	. . . 4 ⊢ (∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (𝜑 ∧ 𝜓)) → ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → 𝜑))
4	3	reximi 3178	. . 3 ⊢ (∃𝑗 ∈ ℝ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (𝜑 ∧ 𝜓)) → ∃𝑗 ∈ ℝ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → 𝜑))
5		simpr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → 𝜓)
6	5	imim2i 16	. . . . 5 ⊢ ((𝑗 ≤ 𝑘 → (𝜑 ∧ 𝜓)) → (𝑗 ≤ 𝑘 → 𝜓))
7	6	ralimi 3087	. . . 4 ⊢ (∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (𝜑 ∧ 𝜓)) → ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → 𝜓))
8	7	reximi 3178	. . 3 ⊢ (∃𝑗 ∈ ℝ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (𝜑 ∧ 𝜓)) → ∃𝑗 ∈ ℝ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → 𝜓))
9	4, 8	jca 512	. 2 ⊢ (∃𝑗 ∈ ℝ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (𝜑 ∧ 𝜓)) → (∃𝑗 ∈ ℝ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → 𝜑) ∧ ∃𝑗 ∈ ℝ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → 𝜓)))
10		breq1 5077	. . . . . . . 8 ⊢ (𝑗 = 𝑥 → (𝑗 ≤ 𝑘 ↔ 𝑥 ≤ 𝑘))
11	10	imbi1d 342	. . . . . . 7 ⊢ (𝑗 = 𝑥 → ((𝑗 ≤ 𝑘 → 𝜑) ↔ (𝑥 ≤ 𝑘 → 𝜑)))
12	11	ralbidv 3112	. . . . . 6 ⊢ (𝑗 = 𝑥 → (∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → 𝜑) ↔ ∀𝑘 ∈ 𝐴 (𝑥 ≤ 𝑘 → 𝜑)))
13	12	cbvrexvw 3384	. . . . 5 ⊢ (∃𝑗 ∈ ℝ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → 𝜑) ↔ ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝐴 (𝑥 ≤ 𝑘 → 𝜑))
14		breq1 5077	. . . . . . . 8 ⊢ (𝑗 = 𝑦 → (𝑗 ≤ 𝑘 ↔ 𝑦 ≤ 𝑘))
15	14	imbi1d 342	. . . . . . 7 ⊢ (𝑗 = 𝑦 → ((𝑗 ≤ 𝑘 → 𝜓) ↔ (𝑦 ≤ 𝑘 → 𝜓)))
16	15	ralbidv 3112	. . . . . 6 ⊢ (𝑗 = 𝑦 → (∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → 𝜓) ↔ ∀𝑘 ∈ 𝐴 (𝑦 ≤ 𝑘 → 𝜓)))
17	16	cbvrexvw 3384	. . . . 5 ⊢ (∃𝑗 ∈ ℝ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → 𝜓) ↔ ∃𝑦 ∈ ℝ ∀𝑘 ∈ 𝐴 (𝑦 ≤ 𝑘 → 𝜓))
18	13, 17	anbi12i 627	. . . 4 ⊢ ((∃𝑗 ∈ ℝ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → 𝜑) ∧ ∃𝑗 ∈ ℝ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → 𝜓)) ↔ (∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝐴 (𝑥 ≤ 𝑘 → 𝜑) ∧ ∃𝑦 ∈ ℝ ∀𝑘 ∈ 𝐴 (𝑦 ≤ 𝑘 → 𝜓)))
19		reeanv 3294	. . . 4 ⊢ (∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ (∀𝑘 ∈ 𝐴 (𝑥 ≤ 𝑘 → 𝜑) ∧ ∀𝑘 ∈ 𝐴 (𝑦 ≤ 𝑘 → 𝜓)) ↔ (∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝐴 (𝑥 ≤ 𝑘 → 𝜑) ∧ ∃𝑦 ∈ ℝ ∀𝑘 ∈ 𝐴 (𝑦 ≤ 𝑘 → 𝜓)))
20	18, 19	bitr4i 277	. . 3 ⊢ ((∃𝑗 ∈ ℝ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → 𝜑) ∧ ∃𝑗 ∈ ℝ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → 𝜓)) ↔ ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ (∀𝑘 ∈ 𝐴 (𝑥 ≤ 𝑘 → 𝜑) ∧ ∀𝑘 ∈ 𝐴 (𝑦 ≤ 𝑘 → 𝜓)))
21		ifcl 4504	. . . . . . 7 ⊢ ((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) → if(𝑥 ≤ 𝑦, 𝑦, 𝑥) ∈ ℝ)
22	21	ancoms 459	. . . . . 6 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → if(𝑥 ≤ 𝑦, 𝑦, 𝑥) ∈ ℝ)
23	22	adantl 482	. . . . 5 ⊢ ((𝐴 ⊆ ℝ ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → if(𝑥 ≤ 𝑦, 𝑦, 𝑥) ∈ ℝ)
24		r19.26 3095	. . . . . 6 ⊢ (∀𝑘 ∈ 𝐴 ((𝑥 ≤ 𝑘 → 𝜑) ∧ (𝑦 ≤ 𝑘 → 𝜓)) ↔ (∀𝑘 ∈ 𝐴 (𝑥 ≤ 𝑘 → 𝜑) ∧ ∀𝑘 ∈ 𝐴 (𝑦 ≤ 𝑘 → 𝜓)))
25		anim12 806	. . . . . . . 8 ⊢ (((𝑥 ≤ 𝑘 → 𝜑) ∧ (𝑦 ≤ 𝑘 → 𝜓)) → ((𝑥 ≤ 𝑘 ∧ 𝑦 ≤ 𝑘) → (𝜑 ∧ 𝜓)))
26		simplrl 774	. . . . . . . . . 10 ⊢ (((𝐴 ⊆ ℝ ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) ∧ 𝑘 ∈ 𝐴) → 𝑥 ∈ ℝ)
27		simplrr 775	. . . . . . . . . 10 ⊢ (((𝐴 ⊆ ℝ ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) ∧ 𝑘 ∈ 𝐴) → 𝑦 ∈ ℝ)
28		simpl 483	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ ℝ ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → 𝐴 ⊆ ℝ)
29	28	sselda 3921	. . . . . . . . . 10 ⊢ (((𝐴 ⊆ ℝ ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ ℝ)
30		maxle 12925	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑘 ∈ ℝ) → (if(𝑥 ≤ 𝑦, 𝑦, 𝑥) ≤ 𝑘 ↔ (𝑥 ≤ 𝑘 ∧ 𝑦 ≤ 𝑘)))
31	26, 27, 29, 30	syl3anc 1370	. . . . . . . . 9 ⊢ (((𝐴 ⊆ ℝ ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) ∧ 𝑘 ∈ 𝐴) → (if(𝑥 ≤ 𝑦, 𝑦, 𝑥) ≤ 𝑘 ↔ (𝑥 ≤ 𝑘 ∧ 𝑦 ≤ 𝑘)))
32	31	imbi1d 342	. . . . . . . 8 ⊢ (((𝐴 ⊆ ℝ ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) ∧ 𝑘 ∈ 𝐴) → ((if(𝑥 ≤ 𝑦, 𝑦, 𝑥) ≤ 𝑘 → (𝜑 ∧ 𝜓)) ↔ ((𝑥 ≤ 𝑘 ∧ 𝑦 ≤ 𝑘) → (𝜑 ∧ 𝜓))))
33	25, 32	syl5ibr 245	. . . . . . 7 ⊢ (((𝐴 ⊆ ℝ ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) ∧ 𝑘 ∈ 𝐴) → (((𝑥 ≤ 𝑘 → 𝜑) ∧ (𝑦 ≤ 𝑘 → 𝜓)) → (if(𝑥 ≤ 𝑦, 𝑦, 𝑥) ≤ 𝑘 → (𝜑 ∧ 𝜓))))
34	33	ralimdva 3108	. . . . . 6 ⊢ ((𝐴 ⊆ ℝ ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (∀𝑘 ∈ 𝐴 ((𝑥 ≤ 𝑘 → 𝜑) ∧ (𝑦 ≤ 𝑘 → 𝜓)) → ∀𝑘 ∈ 𝐴 (if(𝑥 ≤ 𝑦, 𝑦, 𝑥) ≤ 𝑘 → (𝜑 ∧ 𝜓))))
35	24, 34	syl5bir 242	. . . . 5 ⊢ ((𝐴 ⊆ ℝ ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → ((∀𝑘 ∈ 𝐴 (𝑥 ≤ 𝑘 → 𝜑) ∧ ∀𝑘 ∈ 𝐴 (𝑦 ≤ 𝑘 → 𝜓)) → ∀𝑘 ∈ 𝐴 (if(𝑥 ≤ 𝑦, 𝑦, 𝑥) ≤ 𝑘 → (𝜑 ∧ 𝜓))))
36		breq1 5077	. . . . . 6 ⊢ (𝑗 = if(𝑥 ≤ 𝑦, 𝑦, 𝑥) → (𝑗 ≤ 𝑘 ↔ if(𝑥 ≤ 𝑦, 𝑦, 𝑥) ≤ 𝑘))
37	36	rspceaimv 3565	. . . . 5 ⊢ ((if(𝑥 ≤ 𝑦, 𝑦, 𝑥) ∈ ℝ ∧ ∀𝑘 ∈ 𝐴 (if(𝑥 ≤ 𝑦, 𝑦, 𝑥) ≤ 𝑘 → (𝜑 ∧ 𝜓))) → ∃𝑗 ∈ ℝ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (𝜑 ∧ 𝜓)))
38	23, 35, 37	syl6an 681	. . . 4 ⊢ ((𝐴 ⊆ ℝ ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → ((∀𝑘 ∈ 𝐴 (𝑥 ≤ 𝑘 → 𝜑) ∧ ∀𝑘 ∈ 𝐴 (𝑦 ≤ 𝑘 → 𝜓)) → ∃𝑗 ∈ ℝ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (𝜑 ∧ 𝜓))))
39	38	rexlimdvva 3223	. . 3 ⊢ (𝐴 ⊆ ℝ → (∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ (∀𝑘 ∈ 𝐴 (𝑥 ≤ 𝑘 → 𝜑) ∧ ∀𝑘 ∈ 𝐴 (𝑦 ≤ 𝑘 → 𝜓)) → ∃𝑗 ∈ ℝ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (𝜑 ∧ 𝜓))))
40	20, 39	syl5bi 241	. 2 ⊢ (𝐴 ⊆ ℝ → ((∃𝑗 ∈ ℝ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → 𝜑) ∧ ∃𝑗 ∈ ℝ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → 𝜓)) → ∃𝑗 ∈ ℝ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (𝜑 ∧ 𝜓))))
41	9, 40	impbid2 225	1 ⊢ (𝐴 ⊆ ℝ → (∃𝑗 ∈ ℝ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (𝜑 ∧ 𝜓)) ↔ (∃𝑗 ∈ ℝ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → 𝜑) ∧ ∃𝑗 ∈ ℝ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → 𝜓))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065   ⊆ wss 3887  ifcif 4459   class class class wbr 5074  ℝcr 10870   ≤ cle 11010

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-pre-lttri 10945  ax-pre-lttrn 10946

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-po 5503  df-so 5504  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015

This theorem is referenced by:  o1lo1  15246  rlimuni  15259  lo1add  15336  lo1mul  15337  rlimno1  15365