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Theorem k1-6 353
 Description: Statement (6) in proof of Theorem 1 of Kalmbach, Orthomodular Lattices, p. 21.
Hypothesis
Ref Expression
k1-6.1 x = ((xc) ∪ (xc ))
Assertion
Ref Expression
k1-6 (xc) = ((xc ) ∩ c)

Proof of Theorem k1-6
StepHypRef Expression
1 anor3 90 . . . . 5 ((xc) ∩ (xc ) ) = ((xc) ∪ (xc ))
21cm 61 . . . 4 ((xc) ∪ (xc )) = ((xc) ∩ (xc ) )
3 k1-6.1 . . . . 5 x = ((xc) ∪ (xc ))
43con4 69 . . . 4 x = ((xc) ∪ (xc ))
5 oran3 93 . . . . 5 (xc ) = (xc)
6 oran2 92 . . . . 5 (xc) = (xc )
75, 62an 79 . . . 4 ((xc ) ∩ (xc)) = ((xc) ∩ (xc ) )
82, 4, 73tr1 63 . . 3 x = ((xc ) ∩ (xc))
98ran 78 . 2 (xc) = (((xc ) ∩ (xc)) ∩ c)
10 anass 76 . 2 (((xc ) ∩ (xc)) ∩ c) = ((xc ) ∩ ((xc) ∩ c))
11 ancom 74 . . . 4 ((xc) ∩ c) = (c ∩ (xc))
12 ax-a2 31 . . . . 5 (xc) = (cx )
1312lan 77 . . . 4 (c ∩ (xc)) = (c ∩ (cx ))
14 anabs 121 . . . 4 (c ∩ (cx )) = c
1511, 13, 143tr 65 . . 3 ((xc) ∩ c) = c
1615lan 77 . 2 ((xc ) ∩ ((xc) ∩ c)) = ((xc ) ∩ c)
179, 10, 163tr 65 1 (xc) = ((xc ) ∩ c)
 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 This theorem is referenced by:  k1-8a  355
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