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Theorem ixxin 12406
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 4043 . . 3 ({𝑧 ∈ ℝ* ∣ (𝐴𝑅𝑧𝑧𝑆𝐵)} ∩ {𝑧 ∈ ℝ* ∣ (𝐶𝑅𝑧𝑧𝑆𝐷)}) = {𝑧 ∈ ℝ* ∣ ((𝐴𝑅𝑧𝑧𝑆𝐵) ∧ (𝐶𝑅𝑧𝑧𝑆𝐷))}
2 ixx.1 . . . . 5 𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧𝑧𝑆𝑦)})
32ixxval 12397 . . . 4 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴𝑂𝐵) = {𝑧 ∈ ℝ* ∣ (𝐴𝑅𝑧𝑧𝑆𝐵)})
42ixxval 12397 . . . 4 ((𝐶 ∈ ℝ*𝐷 ∈ ℝ*) → (𝐶𝑂𝐷) = {𝑧 ∈ ℝ* ∣ (𝐶𝑅𝑧𝑧𝑆𝐷)})
53, 4ineqan12d 3960 . . 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 1112 . . . . . . . 8 (((𝐴 ∈ ℝ*𝐶 ∈ ℝ*) ∧ 𝑧 ∈ ℝ*) → (if(𝐴𝐶, 𝐶, 𝐴)𝑅𝑧 ↔ (𝐴𝑅𝑧𝐶𝑅𝑧)))
87adantlr 753 . . . . . . 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 1114 . . . . . . . . 9 ((𝑧 ∈ ℝ* ∧ (𝐵 ∈ ℝ*𝐷 ∈ ℝ*)) → (𝑧𝑆if(𝐵𝐷, 𝐵, 𝐷) ↔ (𝑧𝑆𝐵𝑧𝑆𝐷)))
1110ancoms 468 . . . . . . . 8 (((𝐵 ∈ ℝ*𝐷 ∈ ℝ*) ∧ 𝑧 ∈ ℝ*) → (𝑧𝑆if(𝐵𝐷, 𝐵, 𝐷) ↔ (𝑧𝑆𝐵𝑧𝑆𝐷)))
1211adantll 752 . . . . . . 7 ((((𝐴 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝐵 ∈ ℝ*𝐷 ∈ ℝ*)) ∧ 𝑧 ∈ ℝ*) → (𝑧𝑆if(𝐵𝐷, 𝐵, 𝐷) ↔ (𝑧𝑆𝐵𝑧𝑆𝐷)))
138, 12anbi12d 749 . . . . . 6 ((((𝐴 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝐵 ∈ ℝ*𝐷 ∈ ℝ*)) ∧ 𝑧 ∈ ℝ*) → ((if(𝐴𝐶, 𝐶, 𝐴)𝑅𝑧𝑧𝑆if(𝐵𝐷, 𝐵, 𝐷)) ↔ ((𝐴𝑅𝑧𝐶𝑅𝑧) ∧ (𝑧𝑆𝐵𝑧𝑆𝐷))))
14 an4 900 . . . . . 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 3329 . . . 4 (((𝐴 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝐵 ∈ ℝ*𝐷 ∈ ℝ*)) → {𝑧 ∈ ℝ* ∣ (if(𝐴𝐶, 𝐶, 𝐴)𝑅𝑧𝑧𝑆if(𝐵𝐷, 𝐵, 𝐷))} = {𝑧 ∈ ℝ* ∣ ((𝐴𝑅𝑧𝑧𝑆𝐵) ∧ (𝐶𝑅𝑧𝑧𝑆𝐷))})
1716an4s 904 . . 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 2821 . 2 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ (𝐶 ∈ ℝ*𝐷 ∈ ℝ*)) → ((𝐴𝑂𝐵) ∩ (𝐶𝑂𝐷)) = {𝑧 ∈ ℝ* ∣ (if(𝐴𝐶, 𝐶, 𝐴)𝑅𝑧𝑧𝑆if(𝐵𝐷, 𝐵, 𝐷))})
19 ifcl 4275 . . . . 5 ((𝐶 ∈ ℝ*𝐴 ∈ ℝ*) → if(𝐴𝐶, 𝐶, 𝐴) ∈ ℝ*)
2019ancoms 468 . . . 4 ((𝐴 ∈ ℝ*𝐶 ∈ ℝ*) → if(𝐴𝐶, 𝐶, 𝐴) ∈ ℝ*)
21 ifcl 4275 . . . 4 ((𝐵 ∈ ℝ*𝐷 ∈ ℝ*) → if(𝐵𝐷, 𝐵, 𝐷) ∈ ℝ*)
222ixxval 12397 . . . 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 495 . . 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 904 . 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 2798 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 383  w3a 1072   = wceq 1632  wcel 2140  {crab 3055  cin 3715  ifcif 4231   class class class wbr 4805  (class class class)co 6815  cmpt2 6817  *cxr 10286  cle 10288
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pr 5056  ax-un 7116  ax-cnex 10205  ax-resscn 10206
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3343  df-sbc 3578  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-br 4806  df-opab 4866  df-id 5175  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-iota 6013  df-fun 6052  df-fv 6058  df-ov 6818  df-oprab 6819  df-mpt2 6820  df-xr 10291
This theorem is referenced by:  iooin  12423  itgspliticc  23823  cvmliftlem10  31605
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