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Theorem k1-5 360
 Description: Statement (5) in proof of Theorem 1 of Kalmbach, Orthomodular Lattices, p. 21. (Contributed by NM, 27-May-2008.)
Hypotheses
Ref Expression
k1-5.1 (x ∩ (xc)) = (((x ∩ (xc)) ∩ c) ∪ ((x ∩ (xc)) ∩ c ))
k1-5.2 xc
Assertion
Ref Expression
k1-5 (x ∪ (xc )) = c

Proof of Theorem k1-5
StepHypRef Expression
1 k1-5.1 . . 3 (x ∩ (xc)) = (((x ∩ (xc)) ∩ c) ∪ ((x ∩ (xc)) ∩ c ))
2 ax-a1 30 . . . . 5 c = c
32lor 70 . . . 4 (xc) = (xc )
43lan 77 . . 3 (x ∩ (xc)) = (x ∩ (xc ))
5 orcom 73 . . . 4 (((x ∩ (xc)) ∩ c) ∪ ((x ∩ (xc)) ∩ c )) = (((x ∩ (xc)) ∩ c ) ∪ ((x ∩ (xc)) ∩ c))
64ran 78 . . . . 5 ((x ∩ (xc)) ∩ c ) = ((x ∩ (xc )) ∩ c )
74, 22an 79 . . . . 5 ((x ∩ (xc)) ∩ c) = ((x ∩ (xc )) ∩ c )
86, 72or 72 . . . 4 (((x ∩ (xc)) ∩ c ) ∪ ((x ∩ (xc)) ∩ c)) = (((x ∩ (xc )) ∩ c ) ∪ ((x ∩ (xc )) ∩ c ))
95, 8tr 62 . . 3 (((x ∩ (xc)) ∩ c) ∪ ((x ∩ (xc)) ∩ c )) = (((x ∩ (xc )) ∩ c ) ∪ ((x ∩ (xc )) ∩ c ))
101, 4, 93tr2 64 . 2 (x ∩ (xc )) = (((x ∩ (xc )) ∩ c ) ∪ ((x ∩ (xc )) ∩ c ))
11 k1-5.2 . 2 xc
1210, 11k1-4 359 1 (x ∪ (xc )) = c
 Colors of variables: term Syntax hints:   = wb 1   ≤ wle 2  ⊥ 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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