Theorem ixxin 13401

Description: Intersection of two intervals of extended reals. (Contributed by Mario Carneiro, 3-Nov-2013.)

Hypotheses

Ref	Expression
ixx.1	⊢ 𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)})
ixxin.2	⊢ ((𝐴 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ∧ 𝑧 ∈ ℝ*) → (if(𝐴 ≤ 𝐶, 𝐶, 𝐴)𝑅𝑧 ↔ (𝐴𝑅𝑧 ∧ 𝐶𝑅𝑧)))
ixxin.3	⊢ ((𝑧 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐷 ∈ ℝ*) → (𝑧𝑆if(𝐵 ≤ 𝐷, 𝐵, 𝐷) ↔ (𝑧𝑆𝐵 ∧ 𝑧𝑆𝐷)))

Assertion

Ref	Expression
ixxin	⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ (𝐶 ∈ ℝ* ∧ 𝐷 ∈ ℝ*)) → ((𝐴𝑂𝐵) ∩ (𝐶𝑂𝐷)) = (if(𝐴 ≤ 𝐶, 𝐶, 𝐴)𝑂if(𝐵 ≤ 𝐷, 𝐵, 𝐷)))

Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐶,𝑦,𝑧   𝑥,𝐷,𝑦,𝑧   𝑥,𝐵,𝑦,𝑧   𝑥,𝑅,𝑦,𝑧   𝑥,𝑆,𝑦,𝑧

Allowed substitution hints:   𝑂(𝑥,𝑦,𝑧)

Proof of Theorem ixxin

Step	Hyp	Ref	Expression
1		inrab 4322	. . 3 ⊢ ({𝑧 ∈ ℝ* ∣ (𝐴𝑅𝑧 ∧ 𝑧𝑆𝐵)} ∩ {𝑧 ∈ ℝ* ∣ (𝐶𝑅𝑧 ∧ 𝑧𝑆𝐷)}) = {𝑧 ∈ ℝ* ∣ ((𝐴𝑅𝑧 ∧ 𝑧𝑆𝐵) ∧ (𝐶𝑅𝑧 ∧ 𝑧𝑆𝐷))}
2		ixx.1	. . . . 5 ⊢ 𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)})
3	2	ixxval 13392	. . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴𝑂𝐵) = {𝑧 ∈ ℝ* ∣ (𝐴𝑅𝑧 ∧ 𝑧𝑆𝐵)})
4	2	ixxval 13392	. . . 4 ⊢ ((𝐶 ∈ ℝ* ∧ 𝐷 ∈ ℝ*) → (𝐶𝑂𝐷) = {𝑧 ∈ ℝ* ∣ (𝐶𝑅𝑧 ∧ 𝑧𝑆𝐷)})
5	3, 4	ineqan12d 4230	. . 3 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ (𝐶 ∈ ℝ* ∧ 𝐷 ∈ ℝ*)) → ((𝐴𝑂𝐵) ∩ (𝐶𝑂𝐷)) = ({𝑧 ∈ ℝ* ∣ (𝐴𝑅𝑧 ∧ 𝑧𝑆𝐵)} ∩ {𝑧 ∈ ℝ* ∣ (𝐶𝑅𝑧 ∧ 𝑧𝑆𝐷)}))
6		ixxin.2	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ∧ 𝑧 ∈ ℝ*) → (if(𝐴 ≤ 𝐶, 𝐶, 𝐴)𝑅𝑧 ↔ (𝐴𝑅𝑧 ∧ 𝐶𝑅𝑧)))
7	6	ad4ant124 1172	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐵 ∈ ℝ* ∧ 𝐷 ∈ ℝ*)) ∧ 𝑧 ∈ ℝ*) → (if(𝐴 ≤ 𝐶, 𝐶, 𝐴)𝑅𝑧 ↔ (𝐴𝑅𝑧 ∧ 𝐶𝑅𝑧)))
8		ixxin.3	. . . . . . . . . 10 ⊢ ((𝑧 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐷 ∈ ℝ*) → (𝑧𝑆if(𝐵 ≤ 𝐷, 𝐵, 𝐷) ↔ (𝑧𝑆𝐵 ∧ 𝑧𝑆𝐷)))
9	8	3expb 1119	. . . . . . . . 9 ⊢ ((𝑧 ∈ ℝ* ∧ (𝐵 ∈ ℝ* ∧ 𝐷 ∈ ℝ*)) → (𝑧𝑆if(𝐵 ≤ 𝐷, 𝐵, 𝐷) ↔ (𝑧𝑆𝐵 ∧ 𝑧𝑆𝐷)))
10	9	ancoms 458	. . . . . . . 8 ⊢ (((𝐵 ∈ ℝ* ∧ 𝐷 ∈ ℝ*) ∧ 𝑧 ∈ ℝ*) → (𝑧𝑆if(𝐵 ≤ 𝐷, 𝐵, 𝐷) ↔ (𝑧𝑆𝐵 ∧ 𝑧𝑆𝐷)))
11	10	adantll 714	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐵 ∈ ℝ* ∧ 𝐷 ∈ ℝ*)) ∧ 𝑧 ∈ ℝ*) → (𝑧𝑆if(𝐵 ≤ 𝐷, 𝐵, 𝐷) ↔ (𝑧𝑆𝐵 ∧ 𝑧𝑆𝐷)))
12	7, 11	anbi12d 632	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐵 ∈ ℝ* ∧ 𝐷 ∈ ℝ*)) ∧ 𝑧 ∈ ℝ*) → ((if(𝐴 ≤ 𝐶, 𝐶, 𝐴)𝑅𝑧 ∧ 𝑧𝑆if(𝐵 ≤ 𝐷, 𝐵, 𝐷)) ↔ ((𝐴𝑅𝑧 ∧ 𝐶𝑅𝑧) ∧ (𝑧𝑆𝐵 ∧ 𝑧𝑆𝐷))))
13		an4 656	. . . . . 6 ⊢ (((𝐴𝑅𝑧 ∧ 𝑧𝑆𝐵) ∧ (𝐶𝑅𝑧 ∧ 𝑧𝑆𝐷)) ↔ ((𝐴𝑅𝑧 ∧ 𝐶𝑅𝑧) ∧ (𝑧𝑆𝐵 ∧ 𝑧𝑆𝐷)))
14	12, 13	bitr4di 289	. . . . 5 ⊢ ((((𝐴 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐵 ∈ ℝ* ∧ 𝐷 ∈ ℝ*)) ∧ 𝑧 ∈ ℝ*) → ((if(𝐴 ≤ 𝐶, 𝐶, 𝐴)𝑅𝑧 ∧ 𝑧𝑆if(𝐵 ≤ 𝐷, 𝐵, 𝐷)) ↔ ((𝐴𝑅𝑧 ∧ 𝑧𝑆𝐵) ∧ (𝐶𝑅𝑧 ∧ 𝑧𝑆𝐷))))
15	14	rabbidva 3440	. . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐵 ∈ ℝ* ∧ 𝐷 ∈ ℝ*)) → {𝑧 ∈ ℝ* ∣ (if(𝐴 ≤ 𝐶, 𝐶, 𝐴)𝑅𝑧 ∧ 𝑧𝑆if(𝐵 ≤ 𝐷, 𝐵, 𝐷))} = {𝑧 ∈ ℝ* ∣ ((𝐴𝑅𝑧 ∧ 𝑧𝑆𝐵) ∧ (𝐶𝑅𝑧 ∧ 𝑧𝑆𝐷))})
16	15	an4s 660	. . 3 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ (𝐶 ∈ ℝ* ∧ 𝐷 ∈ ℝ*)) → {𝑧 ∈ ℝ* ∣ (if(𝐴 ≤ 𝐶, 𝐶, 𝐴)𝑅𝑧 ∧ 𝑧𝑆if(𝐵 ≤ 𝐷, 𝐵, 𝐷))} = {𝑧 ∈ ℝ* ∣ ((𝐴𝑅𝑧 ∧ 𝑧𝑆𝐵) ∧ (𝐶𝑅𝑧 ∧ 𝑧𝑆𝐷))})
17	1, 5, 16	3eqtr4a 2801	. 2 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ (𝐶 ∈ ℝ* ∧ 𝐷 ∈ ℝ*)) → ((𝐴𝑂𝐵) ∩ (𝐶𝑂𝐷)) = {𝑧 ∈ ℝ* ∣ (if(𝐴 ≤ 𝐶, 𝐶, 𝐴)𝑅𝑧 ∧ 𝑧𝑆if(𝐵 ≤ 𝐷, 𝐵, 𝐷))})
18		ifcl 4576	. . . . 5 ⊢ ((𝐶 ∈ ℝ* ∧ 𝐴 ∈ ℝ*) → if(𝐴 ≤ 𝐶, 𝐶, 𝐴) ∈ ℝ*)
19	18	ancoms 458	. . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → if(𝐴 ≤ 𝐶, 𝐶, 𝐴) ∈ ℝ*)
20		ifcl 4576	. . . 4 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐷 ∈ ℝ*) → if(𝐵 ≤ 𝐷, 𝐵, 𝐷) ∈ ℝ*)
21	2	ixxval 13392	. . . 4 ⊢ ((if(𝐴 ≤ 𝐶, 𝐶, 𝐴) ∈ ℝ* ∧ if(𝐵 ≤ 𝐷, 𝐵, 𝐷) ∈ ℝ*) → (if(𝐴 ≤ 𝐶, 𝐶, 𝐴)𝑂if(𝐵 ≤ 𝐷, 𝐵, 𝐷)) = {𝑧 ∈ ℝ* ∣ (if(𝐴 ≤ 𝐶, 𝐶, 𝐴)𝑅𝑧 ∧ 𝑧𝑆if(𝐵 ≤ 𝐷, 𝐵, 𝐷))})
22	19, 20, 21	syl2an 596	. . 3 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐵 ∈ ℝ* ∧ 𝐷 ∈ ℝ*)) → (if(𝐴 ≤ 𝐶, 𝐶, 𝐴)𝑂if(𝐵 ≤ 𝐷, 𝐵, 𝐷)) = {𝑧 ∈ ℝ* ∣ (if(𝐴 ≤ 𝐶, 𝐶, 𝐴)𝑅𝑧 ∧ 𝑧𝑆if(𝐵 ≤ 𝐷, 𝐵, 𝐷))})
23	22	an4s 660	. 2 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ (𝐶 ∈ ℝ* ∧ 𝐷 ∈ ℝ*)) → (if(𝐴 ≤ 𝐶, 𝐶, 𝐴)𝑂if(𝐵 ≤ 𝐷, 𝐵, 𝐷)) = {𝑧 ∈ ℝ* ∣ (if(𝐴 ≤ 𝐶, 𝐶, 𝐴)𝑅𝑧 ∧ 𝑧𝑆if(𝐵 ≤ 𝐷, 𝐵, 𝐷))})
24	17, 23	eqtr4d 2778	1 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ (𝐶 ∈ ℝ* ∧ 𝐷 ∈ ℝ*)) → ((𝐴𝑂𝐵) ∩ (𝐶𝑂𝐷)) = (if(𝐴 ≤ 𝐶, 𝐶, 𝐴)𝑂if(𝐵 ≤ 𝐷, 𝐵, 𝐷)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106  {crab 3433   ∩ cin 3962  ifcif 4531   class class class wbr 5148  (class class class)co 7431   ∈ cmpo 7433  ℝ^*cxr 11292   ≤ cle 11294

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-xr 11297

This theorem is referenced by:  iooin  13418  itgspliticc  25887  cvmliftlem10  35279  iccin  48693