Quantum Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  QLE Home  >  Th. List  >  u3lemc4 GIF version

Theorem u3lemc4 703
 Description: Lemma for Kalmbach implication study. (Contributed by NM, 24-Dec-1997.)
Hypothesis
Ref Expression
ulemc3.1 a C b
Assertion
Ref Expression
u3lemc4 (a3 b) = (ab)

Proof of Theorem u3lemc4
StepHypRef Expression
1 df-i3 46 . 2 (a3 b) = (((ab) ∪ (ab )) ∪ (a ∩ (ab)))
2 ulemc3.1 . . . . . . . 8 a C b
32comcom3 454 . . . . . . 7 a C b
42comcom4 455 . . . . . . 7 a C b
53, 4fh1 469 . . . . . 6 (a ∩ (bb )) = ((ab) ∪ (ab ))
65ax-r1 35 . . . . 5 ((ab) ∪ (ab )) = (a ∩ (bb ))
7 df-t 41 . . . . . . . 8 1 = (bb )
87ax-r1 35 . . . . . . 7 (bb ) = 1
98lan 77 . . . . . 6 (a ∩ (bb )) = (a ∩ 1)
10 an1 106 . . . . . 6 (a ∩ 1) = a
119, 10ax-r2 36 . . . . 5 (a ∩ (bb )) = a
126, 11ax-r2 36 . . . 4 ((ab) ∪ (ab )) = a
13 comid 187 . . . . . . 7 a C a
1413comcom2 183 . . . . . 6 a C a
1514, 2fh1 469 . . . . 5 (a ∩ (ab)) = ((aa ) ∪ (ab))
16 ax-a2 31 . . . . . 6 ((aa ) ∪ (ab)) = ((ab) ∪ (aa ))
17 dff 101 . . . . . . . . 9 0 = (aa )
1817ax-r1 35 . . . . . . . 8 (aa ) = 0
1918lor 70 . . . . . . 7 ((ab) ∪ (aa )) = ((ab) ∪ 0)
20 or0 102 . . . . . . 7 ((ab) ∪ 0) = (ab)
2119, 20ax-r2 36 . . . . . 6 ((ab) ∪ (aa )) = (ab)
2216, 21ax-r2 36 . . . . 5 ((aa ) ∪ (ab)) = (ab)
2315, 22ax-r2 36 . . . 4 (a ∩ (ab)) = (ab)
2412, 232or 72 . . 3 (((ab) ∪ (ab )) ∪ (a ∩ (ab))) = (a ∪ (ab))
2514, 2fh4 472 . . . 4 (a ∪ (ab)) = ((aa) ∩ (ab))
26 ancom 74 . . . . 5 ((aa) ∩ (ab)) = ((ab) ∩ (aa))
27 ax-a2 31 . . . . . . . 8 (aa) = (aa )
28 df-t 41 . . . . . . . . 9 1 = (aa )
2928ax-r1 35 . . . . . . . 8 (aa ) = 1
3027, 29ax-r2 36 . . . . . . 7 (aa) = 1
3130lan 77 . . . . . 6 ((ab) ∩ (aa)) = ((ab) ∩ 1)
32 an1 106 . . . . . 6 ((ab) ∩ 1) = (ab)
3331, 32ax-r2 36 . . . . 5 ((ab) ∩ (aa)) = (ab)
3426, 33ax-r2 36 . . . 4 ((aa) ∩ (ab)) = (ab)
3525, 34ax-r2 36 . . 3 (a ∪ (ab)) = (ab)
3624, 35ax-r2 36 . 2 (((ab) ∪ (ab )) ∪ (a ∩ (ab))) = (ab)
371, 36ax-r2 36 1 (a3 b) = (ab)
 Colors of variables: term Syntax hints:   = wb 1   C wc 3  ⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8  0wf 9   →3 wi3 14 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-i3 46  df-le1 130  df-le2 131  df-c1 132  df-c2 133 This theorem is referenced by:  u3lemle1  712  u3lem1  736  u3lem2  746  u3lem5  763  u3lem6  767  u3lem7  774  u3lem8  783  u3lem9  784
 Copyright terms: Public domain W3C validator