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Theorem i3lem3 506
Description: Lemma for Kalmbach implication. (Contributed by NM, 7-Nov-1997.)
Hypothesis
Ref Expression
i3lem.1 (a3 b) = 1
Assertion
Ref Expression
i3lem3 ((ab) ∩ b ) = (ab )

Proof of Theorem i3lem3
StepHypRef Expression
1 omlan 448 . 2 (b ∩ (b ∪ (ba ))) = (ba )
2 ancom 74 . . 3 ((ab) ∩ b ) = (b ∩ (ab))
3 ax-a2 31 . . . . 5 (ab) = (ba )
4 ax-a3 32 . . . . . . 7 ((b ∪ (ab)) ∪ (ab )) = (b ∪ ((ab) ∪ (ab )))
54ax-r1 35 . . . . . 6 (b ∪ ((ab) ∪ (ab ))) = ((b ∪ (ab)) ∪ (ab ))
6 i3lem.1 . . . . . . . 8 (a3 b) = 1
76i3lem1 504 . . . . . . 7 ((ab) ∪ (ab )) = a
87lor 70 . . . . . 6 (b ∪ ((ab) ∪ (ab ))) = (ba )
9 ancom 74 . . . . . . . . 9 (ab) = (ba )
109lor 70 . . . . . . . 8 (b ∪ (ab)) = (b ∪ (ba ))
11 orabs 120 . . . . . . . 8 (b ∪ (ba )) = b
1210, 11ax-r2 36 . . . . . . 7 (b ∪ (ab)) = b
13 ancom 74 . . . . . . 7 (ab ) = (ba )
1412, 132or 72 . . . . . 6 ((b ∪ (ab)) ∪ (ab )) = (b ∪ (ba ))
155, 8, 143tr2 64 . . . . 5 (ba ) = (b ∪ (ba ))
163, 15ax-r2 36 . . . 4 (ab) = (b ∪ (ba ))
1716lan 77 . . 3 (b ∩ (ab)) = (b ∩ (b ∪ (ba )))
182, 17ax-r2 36 . 2 ((ab) ∩ b ) = (b ∩ (b ∪ (ba )))
191, 18, 133tr1 63 1 ((ab) ∩ b ) = (ab )
Colors of variables: term
Syntax hints:   = wb 1   wn 4  wo 6  wa 7  1wt 8  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:  i3le  515
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