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Theorem lem3.3.7i3e2 1067
 Description: Equation 3.7 of [PavMeg1999] p. 9. The variable i in the paper is set to 3, and this is the second part of the equation. (Contributed by Roy F. Longton, 3-Jul-05.)
Assertion
Ref Expression
lem3.3.7i3e2 (a3 (ab)) = ((ab) ≡3 a)

Proof of Theorem lem3.3.7i3e2
StepHypRef Expression
1 anor3 90 . . . . . . . . . 10 (a ∩ (ab) ) = (a ∪ (ab))
21lor 70 . . . . . . . . 9 (a ∪ (a ∩ (ab) )) = (a ∪ (a ∪ (ab)) )
32lan 77 . . . . . . . 8 ((a ∪ (ab)) ∩ (a ∪ (a ∩ (ab) ))) = ((a ∪ (ab)) ∩ (a ∪ (a ∪ (ab)) ))
4 orabs 120 . . . . . . . . . . 11 (a ∪ (ab)) = a
54ax-r4 37 . . . . . . . . . 10 (a ∪ (ab)) = a
65lor 70 . . . . . . . . 9 (a ∪ (a ∪ (ab)) ) = (aa )
76lan 77 . . . . . . . 8 ((a ∪ (ab)) ∩ (a ∪ (a ∪ (ab)) )) = ((a ∪ (ab)) ∩ (aa ))
8 df-t 41 . . . . . . . . . . 11 1 = (aa )
98ax-r1 35 . . . . . . . . . 10 (aa ) = 1
109lan 77 . . . . . . . . 9 ((a ∪ (ab)) ∩ (aa )) = ((a ∪ (ab)) ∩ 1)
11 an1 106 . . . . . . . . 9 ((a ∪ (ab)) ∩ 1) = (a ∪ (ab))
12 ax-a2 31 . . . . . . . . 9 (a ∪ (ab)) = ((ab) ∪ a )
1310, 11, 123tr 65 . . . . . . . 8 ((a ∪ (ab)) ∩ (aa )) = ((ab) ∪ a )
143, 7, 133tr 65 . . . . . . 7 ((a ∪ (ab)) ∩ (a ∪ (a ∩ (ab) ))) = ((ab) ∪ a )
15 lea 160 . . . . . . . . . . 11 (ab) ≤ a
1615df-le2 131 . . . . . . . . . 10 ((ab) ∪ a) = a
1716ax-r1 35 . . . . . . . . 9 a = ((ab) ∪ a)
1817ax-r4 37 . . . . . . . 8 a = ((ab) ∪ a)
1918lor 70 . . . . . . 7 ((ab) ∪ a ) = ((ab) ∪ ((ab) ∪ a) )
20 anor3 90 . . . . . . . . 9 ((ab)a ) = ((ab) ∪ a)
2120ax-r1 35 . . . . . . . 8 ((ab) ∪ a) = ((ab)a )
2221lor 70 . . . . . . 7 ((ab) ∪ ((ab) ∪ a) ) = ((ab) ∪ ((ab)a ))
2314, 19, 223tr 65 . . . . . 6 ((a ∪ (ab)) ∩ (a ∪ (a ∩ (ab) ))) = ((ab) ∪ ((ab)a ))
24 an1r 107 . . . . . . 7 (1 ∩ ((ab) ∪ ((ab)a ))) = ((ab) ∪ ((ab)a ))
2524ax-r1 35 . . . . . 6 ((ab) ∪ ((ab)a )) = (1 ∩ ((ab) ∪ ((ab)a )))
2623, 25ax-r2 36 . . . . 5 ((a ∪ (ab)) ∩ (a ∪ (a ∩ (ab) ))) = (1 ∩ ((ab) ∪ ((ab)a )))
27 or1 104 . . . . . . 7 (b ∪ 1) = 1
2827ax-r1 35 . . . . . 6 1 = (b ∪ 1)
2928ran 78 . . . . 5 (1 ∩ ((ab) ∪ ((ab)a ))) = ((b ∪ 1) ∩ ((ab) ∪ ((ab)a )))
308lor 70 . . . . . 6 (b ∪ 1) = (b ∪ (aa ))
3130ran 78 . . . . 5 ((b ∪ 1) ∩ ((ab) ∪ ((ab)a ))) = ((b ∪ (aa )) ∩ ((ab) ∪ ((ab)a )))
3226, 29, 313tr 65 . . . 4 ((a ∪ (ab)) ∩ (a ∪ (a ∩ (ab) ))) = ((b ∪ (aa )) ∩ ((ab) ∪ ((ab)a )))
33 ax-a2 31 . . . . . 6 (aa ) = (aa)
3433lor 70 . . . . 5 (b ∪ (aa )) = (b ∪ (aa))
3534ran 78 . . . 4 ((b ∪ (aa )) ∩ ((ab) ∪ ((ab)a ))) = ((b ∪ (aa)) ∩ ((ab) ∪ ((ab)a )))
36 ax-a3 32 . . . . . 6 ((ba ) ∪ a) = (b ∪ (aa))
3736ax-r1 35 . . . . 5 (b ∪ (aa)) = ((ba ) ∪ a)
3837ran 78 . . . 4 ((b ∪ (aa)) ∩ ((ab) ∪ ((ab)a ))) = (((ba ) ∪ a) ∩ ((ab) ∪ ((ab)a )))
3932, 35, 383tr 65 . . 3 ((a ∪ (ab)) ∩ (a ∪ (a ∩ (ab) ))) = (((ba ) ∪ a) ∩ ((ab) ∪ ((ab)a )))
40 ax-a2 31 . . . . 5 (ba ) = (ab )
4140ax-r5 38 . . . 4 ((ba ) ∪ a) = ((ab ) ∪ a)
4241ran 78 . . 3 (((ba ) ∪ a) ∩ ((ab) ∪ ((ab)a ))) = (((ab ) ∪ a) ∩ ((ab) ∪ ((ab)a )))
43 oran3 93 . . . . 5 (ab ) = (ab)
4443ax-r5 38 . . . 4 ((ab ) ∪ a) = ((ab)a)
4544ran 78 . . 3 (((ab ) ∪ a) ∩ ((ab) ∪ ((ab)a ))) = (((ab)a) ∩ ((ab) ∪ ((ab)a )))
4639, 42, 453tr 65 . 2 ((a ∪ (ab)) ∩ (a ∪ (a ∩ (ab) ))) = (((ab)a) ∩ ((ab) ∪ ((ab)a )))
47 df-id3 52 . 2 (a3 (ab)) = ((a ∪ (ab)) ∩ (a ∪ (a ∩ (ab) )))
48 df-id3 52 . 2 ((ab) ≡3 a) = (((ab)a) ∩ ((ab) ∪ ((ab)a )))
4946, 47, 483tr1 63 1 (a3 (ab)) = ((ab) ≡3 a)
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8   ≡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-id3 52  df-le1 130  df-le2 131 This theorem is referenced by: (None)
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