Quantum Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  QLE Home  >  Th. List  >  d3oa GIF version

Theorem d3oa 995
 Description: Derivation of 3-OA from OA distributive law. (Contributed by NM, 30-Dec-1998.)
Hypothesis
Ref Expression
d3oa.1 f = ((ab) ∪ ((a1 c) ∩ (b1 c)))
Assertion
Ref Expression
d3oa ((a1 c) ∩ f) ≤ (b1 c)

Proof of Theorem d3oa
StepHypRef Expression
1 1oai1 821 . . 3 ((a1 c) ∩ ((ab)1 ((a1 c) ∩ (b1 c)))) ≤ (b1 c)
2 2oath1i1 827 . . . 4 ((a1 c) ∩ ((ab)2 ((a1 c) ∩ (b1 c)))) = ((a1 c) ∩ (b1 c))
3 lear 161 . . . 4 ((a1 c) ∩ (b1 c)) ≤ (b1 c)
42, 3bltr 138 . . 3 ((a1 c) ∩ ((ab)2 ((a1 c) ∩ (b1 c)))) ≤ (b1 c)
51, 4le2or 168 . 2 (((a1 c) ∩ ((ab)1 ((a1 c) ∩ (b1 c)))) ∪ ((a1 c) ∩ ((ab)2 ((a1 c) ∩ (b1 c))))) ≤ ((b1 c) ∪ (b1 c))
6 id 59 . . . . 5 (((aa) ∪ ((a1 c) ∩ (a1 c))) ∩ ((ba) ∪ ((b1 c) ∩ (a1 c)))) = (((aa) ∪ ((a1 c) ∩ (a1 c))) ∩ ((ba) ∪ ((b1 c) ∩ (a1 c))))
7 id 59 . . . . 5 (((ab) ∪ ((a1 c) ∩ (b1 c))) ∪ (((aa) ∪ ((a1 c) ∩ (a1 c))) ∩ ((ba) ∪ ((b1 c) ∩ (a1 c))))) = (((ab) ∪ ((a1 c) ∩ (b1 c))) ∪ (((aa) ∪ ((a1 c) ∩ (a1 c))) ∩ ((ba) ∪ ((b1 c) ∩ (a1 c)))))
8 leid 148 . . . . 5 (a1 c) ≤ (a1 c)
9 df-i1 44 . . . . . . 7 ((ab)1 ((a1 c) ∩ (b1 c))) = ((ab) ∪ ((ab) ∩ ((a1 c) ∩ (b1 c))))
10 ax-a1 30 . . . . . . . . . 10 (ab) = (ab)
1110ax-r1 35 . . . . . . . . 9 (ab) = (ab)
1211bile 142 . . . . . . . 8 (ab) ≤ (ab)
13 lear 161 . . . . . . . 8 ((ab) ∩ ((a1 c) ∩ (b1 c))) ≤ ((a1 c) ∩ (b1 c))
1412, 13le2or 168 . . . . . . 7 ((ab) ∪ ((ab) ∩ ((a1 c) ∩ (b1 c)))) ≤ ((ab) ∪ ((a1 c) ∩ (b1 c)))
159, 14bltr 138 . . . . . 6 ((ab)1 ((a1 c) ∩ (b1 c))) ≤ ((ab) ∪ ((a1 c) ∩ (b1 c)))
16 leo 158 . . . . . 6 ((ab) ∪ ((a1 c) ∩ (b1 c))) ≤ (((ab) ∪ ((a1 c) ∩ (b1 c))) ∪ (((aa) ∪ ((a1 c) ∩ (a1 c))) ∩ ((ba) ∪ ((b1 c) ∩ (a1 c)))))
1715, 16letr 137 . . . . 5 ((ab)1 ((a1 c) ∩ (b1 c))) ≤ (((ab) ∪ ((a1 c) ∩ (b1 c))) ∪ (((aa) ∪ ((a1 c) ∩ (a1 c))) ∩ ((ba) ∪ ((b1 c) ∩ (a1 c)))))
18 df-i2 45 . . . . . . . 8 ((ab)2 ((a1 c) ∩ (b1 c))) = (((a1 c) ∩ (b1 c)) ∪ ((ab) ∩ ((a1 c) ∩ (b1 c)) ))
19 ax-a2 31 . . . . . . . 8 (((a1 c) ∩ (b1 c)) ∪ ((ab) ∩ ((a1 c) ∩ (b1 c)) )) = (((ab) ∩ ((a1 c) ∩ (b1 c)) ) ∪ ((a1 c) ∩ (b1 c)))
2018, 19ax-r2 36 . . . . . . 7 ((ab)2 ((a1 c) ∩ (b1 c))) = (((ab) ∩ ((a1 c) ∩ (b1 c)) ) ∪ ((a1 c) ∩ (b1 c)))
21 lea 160 . . . . . . . . 9 ((ab) ∩ ((a1 c) ∩ (b1 c)) ) ≤ (ab)
2221, 11lbtr 139 . . . . . . . 8 ((ab) ∩ ((a1 c) ∩ (b1 c)) ) ≤ (ab)
23 leid 148 . . . . . . . 8 ((a1 c) ∩ (b1 c)) ≤ ((a1 c) ∩ (b1 c))
2422, 23le2or 168 . . . . . . 7 (((ab) ∩ ((a1 c) ∩ (b1 c)) ) ∪ ((a1 c) ∩ (b1 c))) ≤ ((ab) ∪ ((a1 c) ∩ (b1 c)))
2520, 24bltr 138 . . . . . 6 ((ab)2 ((a1 c) ∩ (b1 c))) ≤ ((ab) ∪ ((a1 c) ∩ (b1 c)))
2625, 16letr 137 . . . . 5 ((ab)2 ((a1 c) ∩ (b1 c))) ≤ (((ab) ∪ ((a1 c) ∩ (b1 c))) ∪ (((aa) ∪ ((a1 c) ∩ (a1 c))) ∩ ((ba) ∪ ((b1 c) ∩ (a1 c)))))
27 leo 158 . . . . . 6 ((a1 c) ∩ (b1 c)) ≤ (((a1 c) ∩ (b1 c)) ∪ ((ab) ∩ ((a1 c) ∩ (b1 c)) ))
2818ax-r1 35 . . . . . 6 (((a1 c) ∩ (b1 c)) ∪ ((ab) ∩ ((a1 c) ∩ (b1 c)) )) = ((ab)2 ((a1 c) ∩ (b1 c)))
2927, 28lbtr 139 . . . . 5 ((a1 c) ∩ (b1 c)) ≤ ((ab)2 ((a1 c) ∩ (b1 c)))
306, 7, 8, 17, 26, 29ax-oadist 994 . . . 4 ((a1 c) ∩ (((ab)1 ((a1 c) ∩ (b1 c))) ∪ ((ab)2 ((a1 c) ∩ (b1 c))))) = (((a1 c) ∩ ((ab)1 ((a1 c) ∩ (b1 c)))) ∪ ((a1 c) ∩ ((ab)2 ((a1 c) ∩ (b1 c)))))
3130ax-r1 35 . . 3 (((a1 c) ∩ ((ab)1 ((a1 c) ∩ (b1 c)))) ∪ ((a1 c) ∩ ((ab)2 ((a1 c) ∩ (b1 c))))) = ((a1 c) ∩ (((ab)1 ((a1 c) ∩ (b1 c))) ∪ ((ab)2 ((a1 c) ∩ (b1 c)))))
32 u12lem 771 . . . . . . 7 (((ab)1 ((a1 c) ∩ (b1 c))) ∪ ((ab)2 ((a1 c) ∩ (b1 c)))) = ((ab)0 ((a1 c) ∩ (b1 c)))
33 df-i0 43 . . . . . . 7 ((ab)0 ((a1 c) ∩ (b1 c))) = ((ab) ∪ ((a1 c) ∩ (b1 c)))
3432, 33ax-r2 36 . . . . . 6 (((ab)1 ((a1 c) ∩ (b1 c))) ∪ ((ab)2 ((a1 c) ∩ (b1 c)))) = ((ab) ∪ ((a1 c) ∩ (b1 c)))
3510ax-r5 38 . . . . . . 7 ((ab) ∪ ((a1 c) ∩ (b1 c))) = ((ab) ∪ ((a1 c) ∩ (b1 c)))
3635ax-r1 35 . . . . . 6 ((ab) ∪ ((a1 c) ∩ (b1 c))) = ((ab) ∪ ((a1 c) ∩ (b1 c)))
3734, 36ax-r2 36 . . . . 5 (((ab)1 ((a1 c) ∩ (b1 c))) ∪ ((ab)2 ((a1 c) ∩ (b1 c)))) = ((ab) ∪ ((a1 c) ∩ (b1 c)))
38 d3oa.1 . . . . . 6 f = ((ab) ∪ ((a1 c) ∩ (b1 c)))
3938ax-r1 35 . . . . 5 ((ab) ∪ ((a1 c) ∩ (b1 c))) = f
4037, 39ax-r2 36 . . . 4 (((ab)1 ((a1 c) ∩ (b1 c))) ∪ ((ab)2 ((a1 c) ∩ (b1 c)))) = f
4140lan 77 . . 3 ((a1 c) ∩ (((ab)1 ((a1 c) ∩ (b1 c))) ∪ ((ab)2 ((a1 c) ∩ (b1 c))))) = ((a1 c) ∩ f)
4231, 41ax-r2 36 . 2 (((a1 c) ∩ ((ab)1 ((a1 c) ∩ (b1 c)))) ∪ ((a1 c) ∩ ((ab)2 ((a1 c) ∩ (b1 c))))) = ((a1 c) ∩ f)
43 oridm 110 . 2 ((b1 c) ∪ (b1 c)) = (b1 c)
445, 42, 43le3tr2 141 1 ((a1 c) ∩ f) ≤ (b1 c)
 Colors of variables: term Syntax hints:   = wb 1   ≤ wle 2  ⊥ wn 4   ∪ wo 6   ∩ wa 7   →0 wi0 11   →1 wi1 12   →2 wi2 13 This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a4 33  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-r3 439  ax-oadist 994 This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i0 43  df-i1 44  df-i2 45  df-le1 130  df-le2 131  df-c1 132  df-c2 133 This theorem is referenced by:  d4oa  996
 Copyright terms: Public domain W3C validator