Theorem ixxin 13306

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 4244	. . 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 13297	. . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴𝑂𝐵) = {𝑧 ∈ ℝ* ∣ (𝐴𝑅𝑧 ∧ 𝑧𝑆𝐵)})
4	2	ixxval 13297	. . . 4 ⊢ ((𝐶 ∈ ℝ* ∧ 𝐷 ∈ ℝ*) → (𝐶𝑂𝐷) = {𝑧 ∈ ℝ* ∣ (𝐶𝑅𝑧 ∧ 𝑧𝑆𝐷)})
5	3, 4	ineqan12d 4151	. . 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 1180	. . . . . . 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 1126	. . . . . . . . 9 ⊢ ((𝑧 ∈ ℝ* ∧ (𝐵 ∈ ℝ* ∧ 𝐷 ∈ ℝ*)) → (𝑧𝑆if(𝐵 ≤ 𝐷, 𝐵, 𝐷) ↔ (𝑧𝑆𝐵 ∧ 𝑧𝑆𝐷)))
10	9	ancoms 459	. . . . . . . 8 ⊢ (((𝐵 ∈ ℝ* ∧ 𝐷 ∈ ℝ*) ∧ 𝑧 ∈ ℝ*) → (𝑧𝑆if(𝐵 ≤ 𝐷, 𝐵, 𝐷) ↔ (𝑧𝑆𝐵 ∧ 𝑧𝑆𝐷)))
11	10	adantll 720	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐵 ∈ ℝ* ∧ 𝐷 ∈ ℝ*)) ∧ 𝑧 ∈ ℝ*) → (𝑧𝑆if(𝐵 ≤ 𝐷, 𝐵, 𝐷) ↔ (𝑧𝑆𝐵 ∧ 𝑧𝑆𝐷)))
12	7, 11	anbi12d 638	. . . . . 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 662	. . . . . 6 ⊢ (((𝐴𝑅𝑧 ∧ 𝑧𝑆𝐵) ∧ (𝐶𝑅𝑧 ∧ 𝑧𝑆𝐷)) ↔ ((𝐴𝑅𝑧 ∧ 𝐶𝑅𝑧) ∧ (𝑧𝑆𝐵 ∧ 𝑧𝑆𝐷)))
14	12, 13	bitr4di 290	. . . . 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 3397	. . . 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 666	. . 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 2800	. 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 4500	. . . . 5 ⊢ ((𝐶 ∈ ℝ* ∧ 𝐴 ∈ ℝ*) → if(𝐴 ≤ 𝐶, 𝐶, 𝐴) ∈ ℝ*)
19	18	ancoms 459	. . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → if(𝐴 ≤ 𝐶, 𝐶, 𝐴) ∈ ℝ*)
20		ifcl 4500	. . . 4 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐷 ∈ ℝ*) → if(𝐵 ≤ 𝐷, 𝐵, 𝐷) ∈ ℝ*)
21	2	ixxval 13297	. . . 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 602	. . 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 666	. 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 2777	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 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  {crab 3391   ∩ cin 3882  ifcif 4454   class class class wbr 5072  (class class class)co 7356   ∈ cmpo 7358  ℝ^*cxr 11169   ≤ cle 11171

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-pr 5362  ax-un 7678  ax-cnex 11085  ax-resscn 11086

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-sbc 3724  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-iota 6441  df-fun 6487  df-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-xr 11174

This theorem is referenced by:  iooin  13323  itgspliticc  25822  cvmliftlem10  35522  iccin  49386