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Theorem u3lem15 795
 Description: Lemma for Kalmbach implication. (Contributed by NM, 7-Aug-2001.)
Assertion
Ref Expression
u3lem15 ((a3 b) ∩ (ab)) = ((ab) ∩ (a ∪ (ab)))

Proof of Theorem u3lem15
StepHypRef Expression
1 dfi3b 499 . . 3 (a3 b) = ((ab) ∩ ((a ∪ (ab )) ∪ (ab)))
21ran 78 . 2 ((a3 b) ∩ (ab)) = (((ab) ∩ ((a ∪ (ab )) ∪ (ab))) ∩ (ab))
3 anass 76 . 2 (((ab) ∩ ((a ∪ (ab )) ∪ (ab))) ∩ (ab)) = ((ab) ∩ (((a ∪ (ab )) ∪ (ab)) ∩ (ab)))
4 comor1 461 . . . . . 6 (ab) C a
54comcom2 183 . . . . . . 7 (ab) C a
6 comor2 462 . . . . . . . 8 (ab) C b
76comcom2 183 . . . . . . 7 (ab) C b
85, 7com2an 484 . . . . . 6 (ab) C (ab )
94, 8com2or 483 . . . . 5 (ab) C (a ∪ (ab ))
10 leao4 165 . . . . . . 7 (ab) ≤ (ab)
1110lecom 180 . . . . . 6 (ab) C (ab)
1211comcom 453 . . . . 5 (ab) C (ab)
139, 12fh1r 473 . . . 4 (((a ∪ (ab )) ∪ (ab)) ∩ (ab)) = (((a ∪ (ab )) ∩ (ab)) ∪ ((ab) ∩ (ab)))
144, 8fh1r 473 . . . . . 6 ((a ∪ (ab )) ∩ (ab)) = ((a ∩ (ab)) ∪ ((ab ) ∩ (ab)))
15 anabs 121 . . . . . . 7 (a ∩ (ab)) = a
16 oran 87 . . . . . . . . 9 (ab) = (ab )
1716lan 77 . . . . . . . 8 ((ab ) ∩ (ab)) = ((ab ) ∩ (ab ) )
18 dff 101 . . . . . . . . 9 0 = ((ab ) ∩ (ab ) )
1918ax-r1 35 . . . . . . . 8 ((ab ) ∩ (ab ) ) = 0
2017, 19ax-r2 36 . . . . . . 7 ((ab ) ∩ (ab)) = 0
2115, 202or 72 . . . . . 6 ((a ∩ (ab)) ∪ ((ab ) ∩ (ab))) = (a ∪ 0)
22 or0 102 . . . . . 6 (a ∪ 0) = a
2314, 21, 223tr 65 . . . . 5 ((a ∪ (ab )) ∩ (ab)) = a
2410df2le2 136 . . . . 5 ((ab) ∩ (ab)) = (ab)
2523, 242or 72 . . . 4 (((a ∪ (ab )) ∩ (ab)) ∪ ((ab) ∩ (ab))) = (a ∪ (ab))
2613, 25ax-r2 36 . . 3 (((a ∪ (ab )) ∪ (ab)) ∩ (ab)) = (a ∪ (ab))
2726lan 77 . 2 ((ab) ∩ (((a ∪ (ab )) ∪ (ab)) ∩ (ab))) = ((ab) ∩ (a ∪ (ab)))
282, 3, 273tr 65 1 ((a3 b) ∩ (ab)) = ((ab) ∩ (a ∪ (ab)))
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7  0wf 9   →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:  neg3antlem2  865
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