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Theorem k1-2 357
 Description: Statement (2) in proof of Theorem 1 of Kalmbach, Orthomodular Lattices, p. 21. (Contributed by NM, 27-May-2008.)
Hypotheses
Ref Expression
k1-2.1 x = ((xc) ∪ (xc ))
k1-2.2 y = ((yc) ∪ (yc ))
k1-2.3 ((xc) ∪ (yc)) = ((((xc) ∪ (yc))c) ∪ (((xc) ∪ (yc))c ))
Assertion
Ref Expression
k1-2 ((xy) ∩ c) = ((xc) ∪ (yc))

Proof of Theorem k1-2
StepHypRef Expression
1 k1-2.1 . . . . 5 x = ((xc) ∪ (xc ))
2 k1-2.2 . . . . 5 y = ((yc) ∪ (yc ))
31, 22or 72 . . . 4 (xy) = (((xc) ∪ (xc )) ∪ ((yc) ∪ (yc )))
4 or4 84 . . . 4 (((xc) ∪ (xc )) ∪ ((yc) ∪ (yc ))) = (((xc) ∪ (yc)) ∪ ((xc ) ∪ (yc )))
53, 4ax-r2 36 . . 3 (xy) = (((xc) ∪ (yc)) ∪ ((xc ) ∪ (yc )))
65ran 78 . 2 ((xy) ∩ c) = ((((xc) ∪ (yc)) ∪ ((xc ) ∪ (yc ))) ∩ c)
7 k1-2.3 . . . 4 ((xc) ∪ (yc)) = ((((xc) ∪ (yc))c) ∪ (((xc) ∪ (yc))c ))
8 lear 161 . . . . 5 (xc) ≤ c
9 lear 161 . . . . 5 (yc) ≤ c
108, 9lel2or 170 . . . 4 ((xc) ∪ (yc)) ≤ c
11 lear 161 . . . . 5 (xc ) ≤ c
12 lear 161 . . . . 5 (yc ) ≤ c
1311, 12lel2or 170 . . . 4 ((xc ) ∪ (yc )) ≤ c
147, 10, 13k1-8a 355 . . 3 ((xc) ∪ (yc)) = ((((xc) ∪ (yc)) ∪ ((xc ) ∪ (yc ))) ∩ c)
1514ax-r1 35 . 2 ((((xc) ∪ (yc)) ∪ ((xc ) ∪ (yc ))) ∩ c) = ((xc) ∪ (yc))
166, 15tr 62 1 ((xy) ∩ c) = ((xc) ∪ (yc))
 Colors of variables: term Syntax hints:   = wb 1  ⊥ 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 This theorem depends on definitions:  df-a 40  df-t 41  df-f 42  df-le1 130  df-le2 131 This theorem is referenced by: (None)
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