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Theorem ixxin 12810
 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* ∣ (𝑥𝑅𝑧𝑧𝑆𝑦)})
ixxin.2 ((𝐴 ∈ ℝ*𝐶 ∈ ℝ*𝑧 ∈ ℝ*) → (if(𝐴𝐶, 𝐶, 𝐴)𝑅𝑧 ↔ (𝐴𝑅𝑧𝐶𝑅𝑧)))
ixxin.3 ((𝑧 ∈ ℝ*𝐵 ∈ ℝ*𝐷 ∈ ℝ*) → (𝑧𝑆if(𝐵𝐷, 𝐵, 𝐷) ↔ (𝑧𝑆𝐵𝑧𝑆𝐷)))
Assertion
Ref Expression
ixxin (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ (𝐶 ∈ ℝ*𝐷 ∈ ℝ*)) → ((𝐴𝑂𝐵) ∩ (𝐶𝑂𝐷)) = (if(𝐴𝐶, 𝐶, 𝐴)𝑂if(𝐵𝐷, 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
StepHypRef Expression
1 inrab 4212 . . 3 ({𝑧 ∈ ℝ* ∣ (𝐴𝑅𝑧𝑧𝑆𝐵)} ∩ {𝑧 ∈ ℝ* ∣ (𝐶𝑅𝑧𝑧𝑆𝐷)}) = {𝑧 ∈ ℝ* ∣ ((𝐴𝑅𝑧𝑧𝑆𝐵) ∧ (𝐶𝑅𝑧𝑧𝑆𝐷))}
2 ixx.1 . . . . 5 𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧𝑧𝑆𝑦)})
32ixxval 12801 . . . 4 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴𝑂𝐵) = {𝑧 ∈ ℝ* ∣ (𝐴𝑅𝑧𝑧𝑆𝐵)})
42ixxval 12801 . . . 4 ((𝐶 ∈ ℝ*𝐷 ∈ ℝ*) → (𝐶𝑂𝐷) = {𝑧 ∈ ℝ* ∣ (𝐶𝑅𝑧𝑧𝑆𝐷)})
53, 4ineqan12d 4122 . . 3 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ (𝐶 ∈ ℝ*𝐷 ∈ ℝ*)) → ((𝐴𝑂𝐵) ∩ (𝐶𝑂𝐷)) = ({𝑧 ∈ ℝ* ∣ (𝐴𝑅𝑧𝑧𝑆𝐵)} ∩ {𝑧 ∈ ℝ* ∣ (𝐶𝑅𝑧𝑧𝑆𝐷)}))
6 ixxin.2 . . . . . . . 8 ((𝐴 ∈ ℝ*𝐶 ∈ ℝ*𝑧 ∈ ℝ*) → (if(𝐴𝐶, 𝐶, 𝐴)𝑅𝑧 ↔ (𝐴𝑅𝑧𝐶𝑅𝑧)))
76ad4ant124 1171 . . . . . . 7 ((((𝐴 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝐵 ∈ ℝ*𝐷 ∈ ℝ*)) ∧ 𝑧 ∈ ℝ*) → (if(𝐴𝐶, 𝐶, 𝐴)𝑅𝑧 ↔ (𝐴𝑅𝑧𝐶𝑅𝑧)))
8 ixxin.3 . . . . . . . . . 10 ((𝑧 ∈ ℝ*𝐵 ∈ ℝ*𝐷 ∈ ℝ*) → (𝑧𝑆if(𝐵𝐷, 𝐵, 𝐷) ↔ (𝑧𝑆𝐵𝑧𝑆𝐷)))
983expb 1118 . . . . . . . . 9 ((𝑧 ∈ ℝ* ∧ (𝐵 ∈ ℝ*𝐷 ∈ ℝ*)) → (𝑧𝑆if(𝐵𝐷, 𝐵, 𝐷) ↔ (𝑧𝑆𝐵𝑧𝑆𝐷)))
109ancoms 462 . . . . . . . 8 (((𝐵 ∈ ℝ*𝐷 ∈ ℝ*) ∧ 𝑧 ∈ ℝ*) → (𝑧𝑆if(𝐵𝐷, 𝐵, 𝐷) ↔ (𝑧𝑆𝐵𝑧𝑆𝐷)))
1110adantll 713 . . . . . . 7 ((((𝐴 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝐵 ∈ ℝ*𝐷 ∈ ℝ*)) ∧ 𝑧 ∈ ℝ*) → (𝑧𝑆if(𝐵𝐷, 𝐵, 𝐷) ↔ (𝑧𝑆𝐵𝑧𝑆𝐷)))
127, 11anbi12d 633 . . . . . 6 ((((𝐴 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝐵 ∈ ℝ*𝐷 ∈ ℝ*)) ∧ 𝑧 ∈ ℝ*) → ((if(𝐴𝐶, 𝐶, 𝐴)𝑅𝑧𝑧𝑆if(𝐵𝐷, 𝐵, 𝐷)) ↔ ((𝐴𝑅𝑧𝐶𝑅𝑧) ∧ (𝑧𝑆𝐵𝑧𝑆𝐷))))
13 an4 655 . . . . . 6 (((𝐴𝑅𝑧𝑧𝑆𝐵) ∧ (𝐶𝑅𝑧𝑧𝑆𝐷)) ↔ ((𝐴𝑅𝑧𝐶𝑅𝑧) ∧ (𝑧𝑆𝐵𝑧𝑆𝐷)))
1412, 13bitr4di 292 . . . . 5 ((((𝐴 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝐵 ∈ ℝ*𝐷 ∈ ℝ*)) ∧ 𝑧 ∈ ℝ*) → ((if(𝐴𝐶, 𝐶, 𝐴)𝑅𝑧𝑧𝑆if(𝐵𝐷, 𝐵, 𝐷)) ↔ ((𝐴𝑅𝑧𝑧𝑆𝐵) ∧ (𝐶𝑅𝑧𝑧𝑆𝐷))))
1514rabbidva 3391 . . . 4 (((𝐴 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝐵 ∈ ℝ*𝐷 ∈ ℝ*)) → {𝑧 ∈ ℝ* ∣ (if(𝐴𝐶, 𝐶, 𝐴)𝑅𝑧𝑧𝑆if(𝐵𝐷, 𝐵, 𝐷))} = {𝑧 ∈ ℝ* ∣ ((𝐴𝑅𝑧𝑧𝑆𝐵) ∧ (𝐶𝑅𝑧𝑧𝑆𝐷))})
1615an4s 659 . . 3 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ (𝐶 ∈ ℝ*𝐷 ∈ ℝ*)) → {𝑧 ∈ ℝ* ∣ (if(𝐴𝐶, 𝐶, 𝐴)𝑅𝑧𝑧𝑆if(𝐵𝐷, 𝐵, 𝐷))} = {𝑧 ∈ ℝ* ∣ ((𝐴𝑅𝑧𝑧𝑆𝐵) ∧ (𝐶𝑅𝑧𝑧𝑆𝐷))})
171, 5, 163eqtr4a 2820 . 2 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ (𝐶 ∈ ℝ*𝐷 ∈ ℝ*)) → ((𝐴𝑂𝐵) ∩ (𝐶𝑂𝐷)) = {𝑧 ∈ ℝ* ∣ (if(𝐴𝐶, 𝐶, 𝐴)𝑅𝑧𝑧𝑆if(𝐵𝐷, 𝐵, 𝐷))})
18 ifcl 4469 . . . . 5 ((𝐶 ∈ ℝ*𝐴 ∈ ℝ*) → if(𝐴𝐶, 𝐶, 𝐴) ∈ ℝ*)
1918ancoms 462 . . . 4 ((𝐴 ∈ ℝ*𝐶 ∈ ℝ*) → if(𝐴𝐶, 𝐶, 𝐴) ∈ ℝ*)
20 ifcl 4469 . . . 4 ((𝐵 ∈ ℝ*𝐷 ∈ ℝ*) → if(𝐵𝐷, 𝐵, 𝐷) ∈ ℝ*)
212ixxval 12801 . . . 4 ((if(𝐴𝐶, 𝐶, 𝐴) ∈ ℝ* ∧ if(𝐵𝐷, 𝐵, 𝐷) ∈ ℝ*) → (if(𝐴𝐶, 𝐶, 𝐴)𝑂if(𝐵𝐷, 𝐵, 𝐷)) = {𝑧 ∈ ℝ* ∣ (if(𝐴𝐶, 𝐶, 𝐴)𝑅𝑧𝑧𝑆if(𝐵𝐷, 𝐵, 𝐷))})
2219, 20, 21syl2an 598 . . 3 (((𝐴 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝐵 ∈ ℝ*𝐷 ∈ ℝ*)) → (if(𝐴𝐶, 𝐶, 𝐴)𝑂if(𝐵𝐷, 𝐵, 𝐷)) = {𝑧 ∈ ℝ* ∣ (if(𝐴𝐶, 𝐶, 𝐴)𝑅𝑧𝑧𝑆if(𝐵𝐷, 𝐵, 𝐷))})
2322an4s 659 . 2 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ (𝐶 ∈ ℝ*𝐷 ∈ ℝ*)) → (if(𝐴𝐶, 𝐶, 𝐴)𝑂if(𝐵𝐷, 𝐵, 𝐷)) = {𝑧 ∈ ℝ* ∣ (if(𝐴𝐶, 𝐶, 𝐴)𝑅𝑧𝑧𝑆if(𝐵𝐷, 𝐵, 𝐷))})
2417, 23eqtr4d 2797 1 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ (𝐶 ∈ ℝ*𝐷 ∈ ℝ*)) → ((𝐴𝑂𝐵) ∩ (𝐶𝑂𝐷)) = (if(𝐴𝐶, 𝐶, 𝐴)𝑂if(𝐵𝐷, 𝐵, 𝐷)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1085   = wceq 1539   ∈ wcel 2112  {crab 3075   ∩ cin 3860  ifcif 4424   class class class wbr 5037  (class class class)co 7157   ∈ cmpo 7159  ℝ*cxr 10726   ≤ cle 10728 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 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5174  ax-nul 5181  ax-pr 5303  ax-un 7466  ax-cnex 10645  ax-resscn 10646 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ral 3076  df-rex 3077  df-rab 3080  df-v 3412  df-sbc 3700  df-dif 3864  df-un 3866  df-in 3868  df-ss 3878  df-nul 4229  df-if 4425  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4803  df-br 5038  df-opab 5100  df-id 5435  df-xp 5535  df-rel 5536  df-cnv 5537  df-co 5538  df-dm 5539  df-iota 6300  df-fun 6343  df-fv 6349  df-ov 7160  df-oprab 7161  df-mpo 7162  df-xr 10731 This theorem is referenced by:  iooin  12827  itgspliticc  24551  cvmliftlem10  32786  iccin  45643
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