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Theorem ixxin 12134
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 3875 . . 3 ({𝑧 ∈ ℝ* ∣ (𝐴𝑅𝑧𝑧𝑆𝐵)} ∩ {𝑧 ∈ ℝ* ∣ (𝐶𝑅𝑧𝑧𝑆𝐷)}) = {𝑧 ∈ ℝ* ∣ ((𝐴𝑅𝑧𝑧𝑆𝐵) ∧ (𝐶𝑅𝑧𝑧𝑆𝐷))}
2 ixx.1 . . . . 5 𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧𝑧𝑆𝑦)})
32ixxval 12125 . . . 4 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴𝑂𝐵) = {𝑧 ∈ ℝ* ∣ (𝐴𝑅𝑧𝑧𝑆𝐵)})
42ixxval 12125 . . . 4 ((𝐶 ∈ ℝ*𝐷 ∈ ℝ*) → (𝐶𝑂𝐷) = {𝑧 ∈ ℝ* ∣ (𝐶𝑅𝑧𝑧𝑆𝐷)})
53, 4ineqan12d 3794 . . 3 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ (𝐶 ∈ ℝ*𝐷 ∈ ℝ*)) → ((𝐴𝑂𝐵) ∩ (𝐶𝑂𝐷)) = ({𝑧 ∈ ℝ* ∣ (𝐴𝑅𝑧𝑧𝑆𝐵)} ∩ {𝑧 ∈ ℝ* ∣ (𝐶𝑅𝑧𝑧𝑆𝐷)}))
6 ixxin.2 . . . . . . . . 9 ((𝐴 ∈ ℝ*𝐶 ∈ ℝ*𝑧 ∈ ℝ*) → (if(𝐴𝐶, 𝐶, 𝐴)𝑅𝑧 ↔ (𝐴𝑅𝑧𝐶𝑅𝑧)))
763expa 1262 . . . . . . . 8 (((𝐴 ∈ ℝ*𝐶 ∈ ℝ*) ∧ 𝑧 ∈ ℝ*) → (if(𝐴𝐶, 𝐶, 𝐴)𝑅𝑧 ↔ (𝐴𝑅𝑧𝐶𝑅𝑧)))
87adantlr 750 . . . . . . 7 ((((𝐴 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝐵 ∈ ℝ*𝐷 ∈ ℝ*)) ∧ 𝑧 ∈ ℝ*) → (if(𝐴𝐶, 𝐶, 𝐴)𝑅𝑧 ↔ (𝐴𝑅𝑧𝐶𝑅𝑧)))
9 ixxin.3 . . . . . . . . . 10 ((𝑧 ∈ ℝ*𝐵 ∈ ℝ*𝐷 ∈ ℝ*) → (𝑧𝑆if(𝐵𝐷, 𝐵, 𝐷) ↔ (𝑧𝑆𝐵𝑧𝑆𝐷)))
1093expb 1263 . . . . . . . . 9 ((𝑧 ∈ ℝ* ∧ (𝐵 ∈ ℝ*𝐷 ∈ ℝ*)) → (𝑧𝑆if(𝐵𝐷, 𝐵, 𝐷) ↔ (𝑧𝑆𝐵𝑧𝑆𝐷)))
1110ancoms 469 . . . . . . . 8 (((𝐵 ∈ ℝ*𝐷 ∈ ℝ*) ∧ 𝑧 ∈ ℝ*) → (𝑧𝑆if(𝐵𝐷, 𝐵, 𝐷) ↔ (𝑧𝑆𝐵𝑧𝑆𝐷)))
1211adantll 749 . . . . . . 7 ((((𝐴 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝐵 ∈ ℝ*𝐷 ∈ ℝ*)) ∧ 𝑧 ∈ ℝ*) → (𝑧𝑆if(𝐵𝐷, 𝐵, 𝐷) ↔ (𝑧𝑆𝐵𝑧𝑆𝐷)))
138, 12anbi12d 746 . . . . . 6 ((((𝐴 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝐵 ∈ ℝ*𝐷 ∈ ℝ*)) ∧ 𝑧 ∈ ℝ*) → ((if(𝐴𝐶, 𝐶, 𝐴)𝑅𝑧𝑧𝑆if(𝐵𝐷, 𝐵, 𝐷)) ↔ ((𝐴𝑅𝑧𝐶𝑅𝑧) ∧ (𝑧𝑆𝐵𝑧𝑆𝐷))))
14 an4 864 . . . . . 6 (((𝐴𝑅𝑧𝑧𝑆𝐵) ∧ (𝐶𝑅𝑧𝑧𝑆𝐷)) ↔ ((𝐴𝑅𝑧𝐶𝑅𝑧) ∧ (𝑧𝑆𝐵𝑧𝑆𝐷)))
1513, 14syl6bbr 278 . . . . 5 ((((𝐴 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝐵 ∈ ℝ*𝐷 ∈ ℝ*)) ∧ 𝑧 ∈ ℝ*) → ((if(𝐴𝐶, 𝐶, 𝐴)𝑅𝑧𝑧𝑆if(𝐵𝐷, 𝐵, 𝐷)) ↔ ((𝐴𝑅𝑧𝑧𝑆𝐵) ∧ (𝐶𝑅𝑧𝑧𝑆𝐷))))
1615rabbidva 3176 . . . 4 (((𝐴 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝐵 ∈ ℝ*𝐷 ∈ ℝ*)) → {𝑧 ∈ ℝ* ∣ (if(𝐴𝐶, 𝐶, 𝐴)𝑅𝑧𝑧𝑆if(𝐵𝐷, 𝐵, 𝐷))} = {𝑧 ∈ ℝ* ∣ ((𝐴𝑅𝑧𝑧𝑆𝐵) ∧ (𝐶𝑅𝑧𝑧𝑆𝐷))})
1716an4s 868 . . 3 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ (𝐶 ∈ ℝ*𝐷 ∈ ℝ*)) → {𝑧 ∈ ℝ* ∣ (if(𝐴𝐶, 𝐶, 𝐴)𝑅𝑧𝑧𝑆if(𝐵𝐷, 𝐵, 𝐷))} = {𝑧 ∈ ℝ* ∣ ((𝐴𝑅𝑧𝑧𝑆𝐵) ∧ (𝐶𝑅𝑧𝑧𝑆𝐷))})
181, 5, 173eqtr4a 2681 . 2 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ (𝐶 ∈ ℝ*𝐷 ∈ ℝ*)) → ((𝐴𝑂𝐵) ∩ (𝐶𝑂𝐷)) = {𝑧 ∈ ℝ* ∣ (if(𝐴𝐶, 𝐶, 𝐴)𝑅𝑧𝑧𝑆if(𝐵𝐷, 𝐵, 𝐷))})
19 ifcl 4102 . . . . 5 ((𝐶 ∈ ℝ*𝐴 ∈ ℝ*) → if(𝐴𝐶, 𝐶, 𝐴) ∈ ℝ*)
2019ancoms 469 . . . 4 ((𝐴 ∈ ℝ*𝐶 ∈ ℝ*) → if(𝐴𝐶, 𝐶, 𝐴) ∈ ℝ*)
21 ifcl 4102 . . . 4 ((𝐵 ∈ ℝ*𝐷 ∈ ℝ*) → if(𝐵𝐷, 𝐵, 𝐷) ∈ ℝ*)
222ixxval 12125 . . . 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))})
2320, 21, 22syl2an 494 . . 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))})
2423an4s 868 . 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))})
2518, 24eqtr4d 2658 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 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  {crab 2911  cin 3554  ifcif 4058   class class class wbr 4613  (class class class)co 6604  cmpt2 6606  *cxr 10017  cle 10019
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-iota 5810  df-fun 5849  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-xr 10022
This theorem is referenced by:  iooin  12151  itgspliticc  23509  cvmliftlem10  30984
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