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Theorem u3lemona 627
 Description: Lemma for Kalmbach implication study.
Assertion
Ref Expression
u3lemona ((a3 b) ∪ a ) = (ab)

Proof of Theorem u3lemona
StepHypRef Expression
1 df-i3 46 . . 3 (a3 b) = (((ab) ∪ (ab )) ∪ (a ∩ (ab)))
21ax-r5 38 . 2 ((a3 b) ∪ a ) = ((((ab) ∪ (ab )) ∪ (a ∩ (ab))) ∪ a )
3 or32 82 . . 3 ((((ab) ∪ (ab )) ∪ (a ∩ (ab))) ∪ a ) = ((((ab) ∪ (ab )) ∪ a ) ∪ (a ∩ (ab)))
4 lea 160 . . . . . . 7 (ab) ≤ a
5 lea 160 . . . . . . 7 (ab ) ≤ a
64, 5lel2or 170 . . . . . 6 ((ab) ∪ (ab )) ≤ a
76df-le2 131 . . . . 5 (((ab) ∪ (ab )) ∪ a ) = a
87ax-r5 38 . . . 4 ((((ab) ∪ (ab )) ∪ a ) ∪ (a ∩ (ab))) = (a ∪ (a ∩ (ab)))
9 omln 446 . . . 4 (a ∪ (a ∩ (ab))) = (ab)
108, 9ax-r2 36 . . 3 ((((ab) ∪ (ab )) ∪ a ) ∪ (a ∩ (ab))) = (ab)
113, 10ax-r2 36 . 2 ((((ab) ∪ (ab )) ∪ (a ∩ (ab))) ∪ a ) = (ab)
122, 11ax-r2 36 1 ((a3 b) ∪ a ) = (ab)
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7   →3 wi3 14 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-i3 46  df-le1 130  df-le2 131 This theorem is referenced by:  u3lemnaa  642  u3lem5  763
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