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Theorem com3ii 457
 Description: Lemma 3(ii) of Kalmbach 83 p. 23.
Hypothesis
Ref Expression
comcom.1 a C b
Assertion
Ref Expression
com3ii (a ∩ (ab)) = (ab)

Proof of Theorem com3ii
StepHypRef Expression
1 comcom.1 . . . . . 6 a C b
21comcom 453 . . . . 5 b C a
32comd 456 . . . 4 b = ((ba) ∩ (ba ))
43lan 77 . . 3 (ab) = (a ∩ ((ba) ∩ (ba )))
5 anass 76 . . . . 5 ((a ∩ (ba)) ∩ (ba )) = (a ∩ ((ba) ∩ (ba )))
65ax-r1 35 . . . 4 (a ∩ ((ba) ∩ (ba ))) = ((a ∩ (ba)) ∩ (ba ))
7 ax-a2 31 . . . . . . 7 (ba) = (ab)
87lan 77 . . . . . 6 (a ∩ (ba)) = (a ∩ (ab))
9 anabs 121 . . . . . 6 (a ∩ (ab)) = a
108, 9ax-r2 36 . . . . 5 (a ∩ (ba)) = a
11 ax-a2 31 . . . . 5 (ba ) = (ab)
1210, 112an 79 . . . 4 ((a ∩ (ba)) ∩ (ba )) = (a ∩ (ab))
136, 12ax-r2 36 . . 3 (a ∩ ((ba) ∩ (ba ))) = (a ∩ (ab))
144, 13ax-r2 36 . 2 (ab) = (a ∩ (ab))
1514ax-r1 35 1 (a ∩ (ab)) = (ab)
 Colors of variables: term Syntax hints:   = wb 1   C wc 3  ⊥ wn 4   ∪ wo 6   ∩ wa 7 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  ax-r3 439 This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-le1 130  df-le2 131  df-c1 132  df-c2 133 This theorem is referenced by:  fh1  469  fh2  470
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