Theorem ixxin 13360

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 4266	. . 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 13351	. . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴𝑂𝐵) = {𝑧 ∈ ℝ* ∣ (𝐴𝑅𝑧 ∧ 𝑧𝑆𝐵)})
4	2	ixxval 13351	. . . 4 ⊢ ((𝐶 ∈ ℝ* ∧ 𝐷 ∈ ℝ*) → (𝐶𝑂𝐷) = {𝑧 ∈ ℝ* ∣ (𝐶𝑅𝑧 ∧ 𝑧𝑆𝐷)})
5	3, 4	ineqan12d 4172	. . 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 1186	. . . . . . 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 1132	. . . . . . . . 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 724	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐵 ∈ ℝ* ∧ 𝐷 ∈ ℝ*)) ∧ 𝑧 ∈ ℝ*) → (𝑧𝑆if(𝐵 ≤ 𝐷, 𝐵, 𝐷) ↔ (𝑧𝑆𝐵 ∧ 𝑧𝑆𝐷)))
12	7, 11	anbi12d 641	. . . . . 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 666	. . . . . 6 ⊢ (((𝐴𝑅𝑧 ∧ 𝑧𝑆𝐵) ∧ (𝐶𝑅𝑧 ∧ 𝑧𝑆𝐷)) ↔ ((𝐴𝑅𝑧 ∧ 𝐶𝑅𝑧) ∧ (𝑧𝑆𝐵 ∧ 𝑧𝑆𝐷)))
14	12, 13	bitr4di 291	. . . . 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 3419	. . . 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 670	. . 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 2822	. 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 4523	. . . . 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 4523	. . . 4 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐷 ∈ ℝ*) → if(𝐵 ≤ 𝐷, 𝐵, 𝐷) ∈ ℝ*)
21	2	ixxval 13351	. . . 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 605	. . 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 670	. 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 2799	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 208   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141  {crab 3413   ∩ cin 3901  ifcif 4477   class class class wbr 5097  (class class class)co 7391   ∈ cmpo 7393  ℝ^*cxr 11209   ≤ cle 11211

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-pr 5387  ax-un 7713  ax-cnex 11123  ax-resscn 11124

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-sbc 3743  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-iota 6472  df-fun 6518  df-fv 6524  df-ov 7394  df-oprab 7395  df-mpo 7396  df-xr 11214

This theorem is referenced by:  iooin  13377  itgspliticc  25887  cvmliftlem10  35605  iccin  49478