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Theorem lem3.3.7i4e2 1070
 Description: Equation 3.7 of [PavMeg1999] p. 9. The variable i in the paper is set to 4, and this is the second part of the equation. (Contributed by Roy F. Longton, 3-Jul-05.)
Assertion
Ref Expression
lem3.3.7i4e2 (a4 (ab)) = ((ab) ≡4 a)

Proof of Theorem lem3.3.7i4e2
StepHypRef Expression
1 lear 161 . . . . . 6 (a ∩ (ab)) ≤ (ab)
2 lea 160 . . . . . . 7 (ab) ≤ a
3 leid 148 . . . . . . 7 (ab) ≤ (ab)
42, 3ler2an 173 . . . . . 6 (ab) ≤ (a ∩ (ab))
51, 4lebi 145 . . . . 5 (a ∩ (ab)) = (ab)
65lor 70 . . . 4 ((ab) ∪ (a ∩ (ab))) = ((ab) ∪ (ab))
76lan 77 . . 3 ((a ∪ (ab)) ∩ ((ab) ∪ (a ∩ (ab)))) = ((a ∪ (ab)) ∩ ((ab) ∪ (ab)))
83sklem 230 . . . 4 ((ab) ∪ (ab)) = 1
98lan 77 . . 3 ((a ∪ (ab)) ∩ ((ab) ∪ (ab))) = ((a ∪ (ab)) ∩ 1)
10 an1 106 . . . 4 ((a ∪ (ab)) ∩ 1) = (a ∪ (ab))
112df2le2 136 . . . . . . 7 ((ab) ∩ a) = (ab)
1211ax-r1 35 . . . . . 6 (ab) = ((ab) ∩ a)
1312lor 70 . . . . 5 (a ∪ (ab)) = (a ∪ ((ab) ∩ a))
14 an1r 107 . . . . . 6 (1 ∩ (a ∪ ((ab) ∩ a))) = (a ∪ ((ab) ∩ a))
1514ax-r1 35 . . . . 5 (a ∪ ((ab) ∩ a)) = (1 ∩ (a ∪ ((ab) ∩ a)))
1613, 15ax-r2 36 . . . 4 (a ∪ (ab)) = (1 ∩ (a ∪ ((ab) ∩ a)))
172sklem 230 . . . . . 6 ((ab)a) = 1
1817ax-r1 35 . . . . 5 1 = ((ab)a)
1918ran 78 . . . 4 (1 ∩ (a ∪ ((ab) ∩ a))) = (((ab)a) ∩ (a ∪ ((ab) ∩ a)))
2010, 16, 193tr 65 . . 3 ((a ∪ (ab)) ∩ 1) = (((ab)a) ∩ (a ∪ ((ab) ∩ a)))
217, 9, 203tr 65 . 2 ((a ∪ (ab)) ∩ ((ab) ∪ (a ∩ (ab)))) = (((ab)a) ∩ (a ∪ ((ab) ∩ a)))
22 df-id4 53 . 2 (a4 (ab)) = ((a ∪ (ab)) ∩ ((ab) ∪ (a ∩ (ab))))
23 df-id4 53 . 2 ((ab) ≡4 a) = (((ab)a) ∩ (a ∪ ((ab) ∩ a)))
2421, 22, 233tr1 63 1 (a4 (ab)) = ((ab) ≡4 a)
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8   ≡4 wid4 21 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-id4 53  df-le1 130  df-le2 131 This theorem is referenced by: (None)
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