Theorem ixxin 12743

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 4227	. . 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 12734	. . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴𝑂𝐵) = {𝑧 ∈ ℝ* ∣ (𝐴𝑅𝑧 ∧ 𝑧𝑆𝐵)})
4	2	ixxval 12734	. . . 4 ⊢ ((𝐶 ∈ ℝ* ∧ 𝐷 ∈ ℝ*) → (𝐶𝑂𝐷) = {𝑧 ∈ ℝ* ∣ (𝐶𝑅𝑧 ∧ 𝑧𝑆𝐷)})
5	3, 4	ineqan12d 4141	. . 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 1170	. . . . . . 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 1117	. . . . . . . . 9 ⊢ ((𝑧 ∈ ℝ* ∧ (𝐵 ∈ ℝ* ∧ 𝐷 ∈ ℝ*)) → (𝑧𝑆if(𝐵 ≤ 𝐷, 𝐵, 𝐷) ↔ (𝑧𝑆𝐵 ∧ 𝑧𝑆𝐷)))
10	9	ancoms 462	. . . . . . . 8 ⊢ (((𝐵 ∈ ℝ* ∧ 𝐷 ∈ ℝ*) ∧ 𝑧 ∈ ℝ*) → (𝑧𝑆if(𝐵 ≤ 𝐷, 𝐵, 𝐷) ↔ (𝑧𝑆𝐵 ∧ 𝑧𝑆𝐷)))
11	10	adantll 713	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐵 ∈ ℝ* ∧ 𝐷 ∈ ℝ*)) ∧ 𝑧 ∈ ℝ*) → (𝑧𝑆if(𝐵 ≤ 𝐷, 𝐵, 𝐷) ↔ (𝑧𝑆𝐵 ∧ 𝑧𝑆𝐷)))
12	7, 11	anbi12d 633	. . . . . 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 655	. . . . . 6 ⊢ (((𝐴𝑅𝑧 ∧ 𝑧𝑆𝐵) ∧ (𝐶𝑅𝑧 ∧ 𝑧𝑆𝐷)) ↔ ((𝐴𝑅𝑧 ∧ 𝐶𝑅𝑧) ∧ (𝑧𝑆𝐵 ∧ 𝑧𝑆𝐷)))
14	12, 13	syl6bbr 292	. . . . 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 3425	. . . 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 659	. . 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 2859	. 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 4469	. . . . 5 ⊢ ((𝐶 ∈ ℝ* ∧ 𝐴 ∈ ℝ*) → if(𝐴 ≤ 𝐶, 𝐶, 𝐴) ∈ ℝ*)
19	18	ancoms 462	. . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → if(𝐴 ≤ 𝐶, 𝐶, 𝐴) ∈ ℝ*)
20		ifcl 4469	. . . 4 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐷 ∈ ℝ*) → if(𝐵 ≤ 𝐷, 𝐵, 𝐷) ∈ ℝ*)
21	2	ixxval 12734	. . . 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 598	. . 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 659	. 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 2836	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 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  {crab 3110   ∩ cin 3880  ifcif 4425   class class class wbr 5030  (class class class)co 7135   ∈ cmpo 7137  ℝ^*cxr 10663   ≤ cle 10665

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-iota 6283  df-fun 6326  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-xr 10668

This theorem is referenced by:  iooin  12760  itgspliticc  24440  cvmliftlem10  32654