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Theorem com3i 467
 Description: Lemma 3(i) of Kalmbach 83 p. 23. (Contributed by NM, 28-Aug-1997.)
Hypothesis
Ref Expression
com3i.1 (a ∩ (ab)) = (ab)
Assertion
Ref Expression
com3i a C b

Proof of Theorem com3i
StepHypRef Expression
1 anor1 88 . . . . . . . 8 (ab ) = (ab)
21con2 67 . . . . . . 7 (ab ) = (ab)
32ran 78 . . . . . 6 ((ab )a) = ((ab) ∩ a)
4 ancom 74 . . . . . 6 ((ab) ∩ a) = (a ∩ (ab))
53, 4ax-r2 36 . . . . 5 ((ab )a) = (a ∩ (ab))
6 com3i.1 . . . . 5 (a ∩ (ab)) = (ab)
75, 6ax-r2 36 . . . 4 ((ab )a) = (ab)
87lor 70 . . 3 ((ab ) ∪ ((ab )a)) = ((ab ) ∪ (ab))
9 lea 160 . . . 4 (ab ) ≤ a
109oml2 451 . . 3 ((ab ) ∪ ((ab )a)) = a
11 ax-a2 31 . . 3 ((ab ) ∪ (ab)) = ((ab) ∪ (ab ))
128, 10, 113tr2 64 . 2 a = ((ab) ∪ (ab ))
1312df-c1 132 1 a C b
 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 This theorem is referenced by: (None)
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