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

Theorem mlalem 832
Assertion
Ref Expression
mlalem ((ab) ∩ (b1 c)) ≤ (a1 c)

Proof of Theorem mlalem
StepHypRef Expression
1 comanr2 465 . . . . . 6 b C (ab)
21comcom3 454 . . . . 5 b C (ab)
3 comanr1 464 . . . . . 6 b C (bc)
43comcom3 454 . . . . 5 b C (bc)
52, 4fh2 470 . . . 4 ((ab) ∩ (b ∪ (bc))) = (((ab) ∩ b ) ∪ ((ab) ∩ (bc)))
6 anass 76 . . . . . . 7 ((ab) ∩ b ) = (a ∩ (bb ))
7 dff 101 . . . . . . . . 9 0 = (bb )
87ax-r1 35 . . . . . . . 8 (bb ) = 0
98lan 77 . . . . . . 7 (a ∩ (bb )) = (a ∩ 0)
10 an0 108 . . . . . . 7 (a ∩ 0) = 0
116, 9, 103tr 65 . . . . . 6 ((ab) ∩ b ) = 0
12 le0 147 . . . . . 6 0 ≤ (a ∪ (ac))
1311, 12bltr 138 . . . . 5 ((ab) ∩ b ) ≤ (a ∪ (ac))
14 anass 76 . . . . . . 7 ((ab) ∩ (bc)) = (a ∩ (b ∩ (bc)))
15 an12 81 . . . . . . 7 (a ∩ (b ∩ (bc))) = (b ∩ (a ∩ (bc)))
16 anass 76 . . . . . . . . 9 ((ba) ∩ (bc)) = (b ∩ (a ∩ (bc)))
1716ax-r1 35 . . . . . . . 8 (b ∩ (a ∩ (bc))) = ((ba) ∩ (bc))
18 an4 86 . . . . . . . 8 ((ba) ∩ (bc)) = ((bb) ∩ (ac))
1917, 18ax-r2 36 . . . . . . 7 (b ∩ (a ∩ (bc))) = ((bb) ∩ (ac))
2014, 15, 193tr 65 . . . . . 6 ((ab) ∩ (bc)) = ((bb) ∩ (ac))
21 lear 161 . . . . . . 7 ((bb) ∩ (ac)) ≤ (ac)
22 leor 159 . . . . . . 7 (ac) ≤ (a ∪ (ac))
2321, 22letr 137 . . . . . 6 ((bb) ∩ (ac)) ≤ (a ∪ (ac))
2420, 23bltr 138 . . . . 5 ((ab) ∩ (bc)) ≤ (a ∪ (ac))
2513, 24lel2or 170 . . . 4 (((ab) ∩ b ) ∪ ((ab) ∩ (bc))) ≤ (a ∪ (ac))
265, 25bltr 138 . . 3 ((ab) ∩ (b ∪ (bc))) ≤ (a ∪ (ac))
27 anass 76 . . . 4 ((ab ) ∩ (b ∪ (bc))) = (a ∩ (b ∩ (b ∪ (bc))))
28 lea 160 . . . . 5 (a ∩ (b ∩ (b ∪ (bc)))) ≤ a
29 leo 158 . . . . 5 a ≤ (a ∪ (ac))
3028, 29letr 137 . . . 4 (a ∩ (b ∩ (b ∪ (bc)))) ≤ (a ∪ (ac))
3127, 30bltr 138 . . 3 ((ab ) ∩ (b ∪ (bc))) ≤ (a ∪ (ac))
3226, 31lel2or 170 . 2 (((ab) ∩ (b ∪ (bc))) ∪ ((ab ) ∩ (b ∪ (bc)))) ≤ (a ∪ (ac))
33 dfb 94 . . . 4 (ab) = ((ab) ∪ (ab ))
34 df-i1 44 . . . 4 (b1 c) = (b ∪ (bc))
3533, 342an 79 . . 3 ((ab) ∩ (b1 c)) = (((ab) ∪ (ab )) ∩ (b ∪ (bc)))
36 lear 161 . . . . . 6 (ab ) ≤ b
37 leo 158 . . . . . 6 b ≤ (b ∪ (bc))
3836, 37letr 137 . . . . 5 (ab ) ≤ (b ∪ (bc))
3938lecom 180 . . . 4 (ab ) C (b ∪ (bc))
40 coman1 185 . . . . . . 7 (ab ) C a
41 coman2 186 . . . . . . 7 (ab ) C b
4240, 41com2or 483 . . . . . 6 (ab ) C (ab )
43 oran3 93 . . . . . 6 (ab ) = (ab)
4442, 43cbtr 182 . . . . 5 (ab ) C (ab)
4544comcom7 460 . . . 4 (ab ) C (ab)
4639, 45fh2rc 480 . . 3 (((ab) ∪ (ab )) ∩ (b ∪ (bc))) = (((ab) ∩ (b ∪ (bc))) ∪ ((ab ) ∩ (b ∪ (bc))))
4735, 46ax-r2 36 . 2 ((ab) ∩ (b1 c)) = (((ab) ∩ (b ∪ (bc))) ∪ ((ab ) ∩ (b ∪ (bc))))
48 df-i1 44 . 2 (a1 c) = (a ∪ (ac))
4932, 47, 48le3tr1 140 1 ((ab) ∩ (b1 c)) ≤ (a1 c)
 Colors of variables: term Syntax hints:   ≤ wle 2  ⊥ wn 4   ≡ tb 5   ∪ wo 6   ∩ wa 7  0wf 9   →1 wi1 12 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 This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i1 44  df-le1 130  df-le2 131  df-c1 132  df-c2 133 This theorem is referenced by:  mlaoml  833
 Copyright terms: Public domain W3C validator