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Theorem lem3.3.7i3e1 1066
 Description: Equation 3.7 of [PavMeg1999] p. 9. The variable i in the paper is set to 3, and this is the first part of the equation. (Contributed by Roy F. Longton, 3-Jul-05.)
Assertion
Ref Expression
lem3.3.7i3e1 (a3 (ab)) = (a3 (ab))

Proof of Theorem lem3.3.7i3e1
StepHypRef Expression
1 anass 76 . . . . . 6 ((aa) ∩ b) = (a ∩ (ab))
21ax-r1 35 . . . . 5 (a ∩ (ab)) = ((aa) ∩ b)
32ax-r5 38 . . . 4 ((a ∩ (ab)) ∪ (a ∩ (ab) )) = (((aa) ∩ b) ∪ (a ∩ (ab) ))
43ax-r5 38 . . 3 (((a ∩ (ab)) ∪ (a ∩ (ab) )) ∪ (a ∩ (a ∪ (ab)))) = ((((aa) ∩ b) ∪ (a ∩ (ab) )) ∪ (a ∩ (a ∪ (ab))))
5 ancom 74 . . . . . 6 (aa) = (aa )
65ran 78 . . . . 5 ((aa) ∩ b) = ((aa ) ∩ b)
76ax-r5 38 . . . 4 (((aa) ∩ b) ∪ (a ∩ (ab) )) = (((aa ) ∩ b) ∪ (a ∩ (ab) ))
87ax-r5 38 . . 3 ((((aa) ∩ b) ∪ (a ∩ (ab) )) ∪ (a ∩ (a ∪ (ab)))) = ((((aa ) ∩ b) ∪ (a ∩ (ab) )) ∪ (a ∩ (a ∪ (ab))))
9 dff 101 . . . . . . . 8 0 = (aa )
109ax-r1 35 . . . . . . 7 (aa ) = 0
1110ran 78 . . . . . 6 ((aa ) ∩ b) = (0 ∩ b)
1211ax-r5 38 . . . . 5 (((aa ) ∩ b) ∪ (a ∩ (ab) )) = ((0 ∩ b) ∪ (a ∩ (ab) ))
1312ax-r5 38 . . . 4 ((((aa ) ∩ b) ∪ (a ∩ (ab) )) ∪ (a ∩ (a ∪ (ab)))) = (((0 ∩ b) ∪ (a ∩ (ab) )) ∪ (a ∩ (a ∪ (ab))))
14 an0r 109 . . . . . 6 (0 ∩ b) = 0
1514ax-r5 38 . . . . 5 ((0 ∩ b) ∪ (a ∩ (ab) )) = (0 ∪ (a ∩ (ab) ))
1615ax-r5 38 . . . 4 (((0 ∩ b) ∪ (a ∩ (ab) )) ∪ (a ∩ (a ∪ (ab)))) = ((0 ∪ (a ∩ (ab) )) ∪ (a ∩ (a ∪ (ab))))
17 or0r 103 . . . . . 6 (0 ∪ (a ∩ (ab) )) = (a ∩ (ab) )
1817ax-r5 38 . . . . 5 ((0 ∪ (a ∩ (ab) )) ∪ (a ∩ (a ∪ (ab)))) = ((a ∩ (ab) ) ∪ (a ∩ (a ∪ (ab))))
19 anor3 90 . . . . . . 7 (a ∩ (ab) ) = (a ∪ (ab))
2019ax-r5 38 . . . . . 6 ((a ∩ (ab) ) ∪ (a ∩ (a ∪ (ab)))) = ((a ∪ (ab)) ∪ (a ∩ (a ∪ (ab))))
21 orabs 120 . . . . . . . 8 (a ∪ (ab)) = a
2221ax-r4 37 . . . . . . 7 (a ∪ (ab)) = a
2322ax-r5 38 . . . . . 6 ((a ∪ (ab)) ∪ (a ∩ (a ∪ (ab)))) = (a ∪ (a ∩ (a ∪ (ab))))
24 womaa 222 . . . . . . . 8 (a ∪ (a ∩ (a ∪ (ab)))) = (a ∪ (ab))
25 an1 106 . . . . . . . . 9 ((a ∪ (ab)) ∩ 1) = (a ∪ (ab))
2625ax-r1 35 . . . . . . . 8 (a ∪ (ab)) = ((a ∪ (ab)) ∩ 1)
27 df-t 41 . . . . . . . . 9 1 = (aa )
2827lan 77 . . . . . . . 8 ((a ∪ (ab)) ∩ 1) = ((a ∪ (ab)) ∩ (aa ))
2924, 26, 283tr 65 . . . . . . 7 (a ∪ (a ∩ (a ∪ (ab)))) = ((a ∪ (ab)) ∩ (aa ))
3021ax-r1 35 . . . . . . . . . 10 a = (a ∪ (ab))
3130ax-r4 37 . . . . . . . . 9 a = (a ∪ (ab))
3231lor 70 . . . . . . . 8 (aa ) = (a ∪ (a ∪ (ab)) )
3332lan 77 . . . . . . 7 ((a ∪ (ab)) ∩ (aa )) = ((a ∪ (ab)) ∩ (a ∪ (a ∪ (ab)) ))
3419ax-r1 35 . . . . . . . . 9 (a ∪ (ab)) = (a ∩ (ab) )
3534lor 70 . . . . . . . 8 (a ∪ (a ∪ (ab)) ) = (a ∪ (a ∩ (ab) ))
3635lan 77 . . . . . . 7 ((a ∪ (ab)) ∩ (a ∪ (a ∪ (ab)) )) = ((a ∪ (ab)) ∩ (a ∪ (a ∩ (ab) )))
3729, 33, 363tr 65 . . . . . 6 (a ∪ (a ∩ (a ∪ (ab)))) = ((a ∪ (ab)) ∩ (a ∪ (a ∩ (ab) )))
3820, 23, 373tr 65 . . . . 5 ((a ∩ (ab) ) ∪ (a ∩ (a ∪ (ab)))) = ((a ∪ (ab)) ∩ (a ∪ (a ∩ (ab) )))
3918, 38ax-r2 36 . . . 4 ((0 ∪ (a ∩ (ab) )) ∪ (a ∩ (a ∪ (ab)))) = ((a ∪ (ab)) ∩ (a ∪ (a ∩ (ab) )))
4013, 16, 393tr 65 . . 3 ((((aa ) ∩ b) ∪ (a ∩ (ab) )) ∪ (a ∩ (a ∪ (ab)))) = ((a ∪ (ab)) ∩ (a ∪ (a ∩ (ab) )))
414, 8, 403tr 65 . 2 (((a ∩ (ab)) ∪ (a ∩ (ab) )) ∪ (a ∩ (a ∪ (ab)))) = ((a ∪ (ab)) ∩ (a ∪ (a ∩ (ab) )))
42 df-i3 46 . 2 (a3 (ab)) = (((a ∩ (ab)) ∪ (a ∩ (ab) )) ∪ (a ∩ (a ∪ (ab))))
43 df-id3 52 . 2 (a3 (ab)) = ((a ∪ (ab)) ∩ (a ∪ (a ∩ (ab) )))
4441, 42, 433tr1 63 1 (a3 (ab)) = (a3 (ab))
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8  0wf 9   →3 wi3 14   ≡3 wid3 20 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-a 40  df-t 41  df-f 42  df-i3 46  df-id3 52  df-le1 130  df-le2 131 This theorem is referenced by: (None)
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