Quantum Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  QLE Home  >  Th. List  >  wwfh2 GIF version

Theorem wwfh2 217
 Description: Foulis-Holland Theorem (weak).
Hypotheses
Ref Expression
wwfh2.1 a C b
wwfh2.2 c C a
Assertion
Ref Expression
wwfh2 ((b ∩ (ac)) ≡ ((ba) ∪ (bc))) = 1

Proof of Theorem wwfh2
StepHypRef Expression
1 bicom 96 . 2 ((b ∩ (ac)) ≡ ((ba) ∪ (bc))) = (((ba) ∪ (bc)) ≡ (b ∩ (ac)))
2 ledi 174 . . 3 ((ba) ∪ (bc)) ≤ (b ∩ (ac))
3 oran 87 . . . . . . . . . . 11 ((ba) ∪ (bc)) = ((ba) ∩ (bc) )
4 df-a 40 . . . . . . . . . . . . . 14 (ba) = (ba )
54con2 67 . . . . . . . . . . . . 13 (ba) = (ba )
65ran 78 . . . . . . . . . . . 12 ((ba) ∩ (bc) ) = ((ba ) ∩ (bc) )
76ax-r4 37 . . . . . . . . . . 11 ((ba) ∩ (bc) ) = ((ba ) ∩ (bc) )
83, 7ax-r2 36 . . . . . . . . . 10 ((ba) ∪ (bc)) = ((ba ) ∩ (bc) )
98con2 67 . . . . . . . . 9 ((ba) ∪ (bc)) = ((ba ) ∩ (bc) )
109lan 77 . . . . . . . 8 ((b ∩ (ac)) ∩ ((ba) ∪ (bc)) ) = ((b ∩ (ac)) ∩ ((ba ) ∩ (bc) ))
11 an4 86 . . . . . . . . 9 ((b ∩ (ac)) ∩ ((ba ) ∩ (bc) )) = ((b ∩ (ba )) ∩ ((ac) ∩ (bc) ))
12 ax-a1 30 . . . . . . . . . . . . . 14 a = a
1312ax-r1 35 . . . . . . . . . . . . 13 a = a
14 wwfh2.1 . . . . . . . . . . . . 13 a C b
1513, 14bctr 181 . . . . . . . . . . . 12 a C b
1615wwcom3ii 215 . . . . . . . . . . 11 (b ∩ (ba )) = (ba )
17 ancom 74 . . . . . . . . . . 11 (ba ) = (ab)
1816, 17ax-r2 36 . . . . . . . . . 10 (b ∩ (ba )) = (ab)
1918ran 78 . . . . . . . . 9 ((b ∩ (ba )) ∩ ((ac) ∩ (bc) )) = ((ab) ∩ ((ac) ∩ (bc) ))
2011, 19ax-r2 36 . . . . . . . 8 ((b ∩ (ac)) ∩ ((ba ) ∩ (bc) )) = ((ab) ∩ ((ac) ∩ (bc) ))
2110, 20ax-r2 36 . . . . . . 7 ((b ∩ (ac)) ∩ ((ba) ∪ (bc)) ) = ((ab) ∩ ((ac) ∩ (bc) ))
22 an4 86 . . . . . . 7 ((ab) ∩ ((ac) ∩ (bc) )) = ((a ∩ (ac)) ∩ (b ∩ (bc) ))
2321, 22ax-r2 36 . . . . . 6 ((b ∩ (ac)) ∩ ((ba) ∪ (bc)) ) = ((a ∩ (ac)) ∩ (b ∩ (bc) ))
2412ax-r5 38 . . . . . . . . 9 (ac) = (a c)
2524lan 77 . . . . . . . 8 (a ∩ (ac)) = (a ∩ (a c))
26 wwfh2.2 . . . . . . . . . 10 c C a
2726comcom2 183 . . . . . . . . 9 c C a
2827wwcom3ii 215 . . . . . . . 8 (a ∩ (a c)) = (ac)
2925, 28ax-r2 36 . . . . . . 7 (a ∩ (ac)) = (ac)
3029ran 78 . . . . . 6 ((a ∩ (ac)) ∩ (b ∩ (bc) )) = ((ac) ∩ (b ∩ (bc) ))
3123, 30ax-r2 36 . . . . 5 ((b ∩ (ac)) ∩ ((ba) ∪ (bc)) ) = ((ac) ∩ (b ∩ (bc) ))
32 anass 76 . . . . 5 ((ac) ∩ (b ∩ (bc) )) = (a ∩ (c ∩ (b ∩ (bc) )))
3331, 32ax-r2 36 . . . 4 ((b ∩ (ac)) ∩ ((ba) ∪ (bc)) ) = (a ∩ (c ∩ (b ∩ (bc) )))
34 anass 76 . . . . . . . 8 ((bc) ∩ (bc) ) = (b ∩ (c ∩ (bc) ))
3534ax-r1 35 . . . . . . 7 (b ∩ (c ∩ (bc) )) = ((bc) ∩ (bc) )
36 an12 81 . . . . . . 7 (c ∩ (b ∩ (bc) )) = (b ∩ (c ∩ (bc) ))
37 dff 101 . . . . . . 7 0 = ((bc) ∩ (bc) )
3835, 36, 373tr1 63 . . . . . 6 (c ∩ (b ∩ (bc) )) = 0
3938lan 77 . . . . 5 (a ∩ (c ∩ (b ∩ (bc) ))) = (a ∩ 0)
40 an0 108 . . . . 5 (a ∩ 0) = 0
4139, 40ax-r2 36 . . . 4 (a ∩ (c ∩ (b ∩ (bc) ))) = 0
4233, 41ax-r2 36 . . 3 ((b ∩ (ac)) ∩ ((ba) ∪ (bc)) ) = 0
432, 42wwoml3 213 . 2 (((ba) ∪ (bc)) ≡ (b ∩ (ac))) = 1
441, 43ax-r2 36 1 ((b ∩ (ac)) ≡ ((ba) ∪ (bc))) = 1
 Colors of variables: term Syntax hints:   = wb 1   C wc 3  ⊥ wn 4   ≡ tb 5   ∪ wo 6   ∩ wa 7  1wt 8  0wf 9 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 This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-le1 130  df-le2 131  df-c1 132  df-c2 133 This theorem is referenced by:  wwfh4  219
 Copyright terms: Public domain W3C validator