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Theorem lem3.3.7i5e1 1072
 Description: Equation 3.7 of [PavMeg1999] p. 9. The variable i in the paper is set to 5, and this is the first part of the equation. (Contributed by Roy F. Longton, 3-Jul-05.)
Assertion
Ref Expression
lem3.3.7i5e1 (a5 (ab)) = (a5 (ab))

Proof of Theorem lem3.3.7i5e1
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)
62lecon 154 . . . . . 6 a ≤ (ab)
76ortha 438 . . . . 5 (a ∩ (ab)) = 0
85, 72or 72 . . . 4 ((a ∩ (ab)) ∪ (a ∩ (ab))) = ((ab) ∪ 0)
98ax-r5 38 . . 3 (((a ∩ (ab)) ∪ (a ∩ (ab))) ∪ (a ∩ (ab) )) = (((ab) ∪ 0) ∪ (a ∩ (ab) ))
10 or0 102 . . . 4 ((ab) ∪ 0) = (ab)
116df2le2 136 . . . 4 (a ∩ (ab) ) = a
1210, 112or 72 . . 3 (((ab) ∪ 0) ∪ (a ∩ (ab) )) = ((ab) ∪ a )
134, 1lebi 145 . . . 4 (ab) = (a ∩ (ab))
1411ax-r1 35 . . . 4 a = (a ∩ (ab) )
1513, 142or 72 . . 3 ((ab) ∪ a ) = ((a ∩ (ab)) ∪ (a ∩ (ab) ))
169, 12, 153tr 65 . 2 (((a ∩ (ab)) ∪ (a ∩ (ab))) ∪ (a ∩ (ab) )) = ((a ∩ (ab)) ∪ (a ∩ (ab) ))
17 df-i5 48 . 2 (a5 (ab)) = (((a ∩ (ab)) ∪ (a ∩ (ab))) ∪ (a ∩ (ab) ))
18 df-id5 1047 . 2 (a5 (ab)) = ((a ∩ (ab)) ∪ (a ∩ (ab) ))
1916, 17, 183tr1 63 1 (a5 (ab)) = (a5 (ab))
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7  0wf 9   →5 wi5 16   ≡5 wid5 22 This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  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-i5 48  df-le1 130  df-le2 131  df-id5 1047 This theorem is referenced by: (None)
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