Theorem ixxin 13497

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 4261	. . 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 13488	. . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴𝑂𝐵) = {𝑧 ∈ ℝ* ∣ (𝐴𝑅𝑧 ∧ 𝑧𝑆𝐵)})
4	2	ixxval 13488	. . . 4 ⊢ ((𝐶 ∈ ℝ* ∧ 𝐷 ∈ ℝ*) → (𝐶𝑂𝐷) = {𝑧 ∈ ℝ* ∣ (𝐶𝑅𝑧 ∧ 𝑧𝑆𝐷)})
5	3, 4	ineqan12d 4119	. . 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 1192	. . . . . . 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 1138	. . . . . . . . 9 ⊢ ((𝑧 ∈ ℝ* ∧ (𝐵 ∈ ℝ* ∧ 𝐷 ∈ ℝ*)) → (𝑧𝑆if(𝐵 ≤ 𝐷, 𝐵, 𝐷) ↔ (𝑧𝑆𝐵 ∧ 𝑧𝑆𝐷)))
10	9	ancoms 464	. . . . . . . 8 ⊢ (((𝐵 ∈ ℝ* ∧ 𝐷 ∈ ℝ*) ∧ 𝑧 ∈ ℝ*) → (𝑧𝑆if(𝐵 ≤ 𝐷, 𝐵, 𝐷) ↔ (𝑧𝑆𝐵 ∧ 𝑧𝑆𝐷)))
11	10	adantll 727	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐵 ∈ ℝ* ∧ 𝐷 ∈ ℝ*)) ∧ 𝑧 ∈ ℝ*) → (𝑧𝑆if(𝐵 ≤ 𝐷, 𝐵, 𝐷) ↔ (𝑧𝑆𝐵 ∧ 𝑧𝑆𝐷)))
12	7, 11	anbi12d 644	. . . . . 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 669	. . . . . 6 ⊢ (((𝐴𝑅𝑧 ∧ 𝑧𝑆𝐵) ∧ (𝐶𝑅𝑧 ∧ 𝑧𝑆𝐷)) ↔ ((𝐴𝑅𝑧 ∧ 𝐶𝑅𝑧) ∧ (𝑧𝑆𝐵 ∧ 𝑧𝑆𝐷)))
14	12, 13	bitr4di 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 3418	. . . 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 673	. . 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 2821	. 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 4527	. . . . 5 ⊢ ((𝐶 ∈ ℝ* ∧ 𝐴 ∈ ℝ*) → if(𝐴 ≤ 𝐶, 𝐶, 𝐴) ∈ ℝ*)
19	18	ancoms 464	. . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → if(𝐴 ≤ 𝐶, 𝐶, 𝐴) ∈ ℝ*)
20		ifcl 4527	. . . 4 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐷 ∈ ℝ*) → if(𝐵 ≤ 𝐷, 𝐵, 𝐷) ∈ ℝ*)
21	2	ixxval 13488	. . . 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 608	. . 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 673	. 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 2798	1 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ (𝐶 ∈ ℝ* ∧ 𝐷 ∈ ℝ*)) → ((𝐴𝑂𝐵) ∩ (𝐶𝑂𝐷)) = (if(𝐴 ≤ 𝐶, 𝐶, 𝐴)𝑂if(𝐵 ≤ 𝐷, 𝐵, 𝐷)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  {crab 3412   ∩ cin 4089  ifcif 4481   class class class wbr 5102  (class class class)co 7421   ∈ cmpo 7423  ℝ^*cxr 11346   ≤ cle 11348

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pr 5390  ax-un 7752  ax-cnex 11260  ax-resscn 11261

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-iota 6494  df-fun 6541  df-fv 6547  df-ov 7424  df-oprab 7425  df-mpo 7426  df-xr 11351

This theorem is used by:  iooin  13514  itgspliticc  26163  cvmliftlem10  36146  iccin  50104