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Theorem k1-4 359
 Description: Statement (4) in proof of Theorem 1 of Kalmbach, Orthomodular Lattices, p. 21.
Hypotheses
Ref Expression
k1-4.1 (x ∩ (xc )) = (((x ∩ (xc )) ∩ c) ∪ ((x ∩ (xc )) ∩ c ))
k1-4.2 xc
Assertion
Ref Expression
k1-4 (x ∪ (xc)) = c

Proof of Theorem k1-4
StepHypRef Expression
1 oran1 91 . . . . 5 (xc ) = (xc)
21lan 77 . . . 4 (x ∩ (xc )) = (x ∩ (xc) )
32cm 61 . . 3 (x ∩ (xc) ) = (x ∩ (xc ))
4 anor3 90 . . 3 (x ∩ (xc) ) = (x ∪ (xc))
5 k1-4.1 . . . 4 (x ∩ (xc )) = (((x ∩ (xc )) ∩ c) ∪ ((x ∩ (xc )) ∩ c ))
61lan 77 . . . . . 6 ((xc) ∩ (xc )) = ((xc) ∩ (xc) )
7 an32 83 . . . . . 6 ((x ∩ (xc )) ∩ c) = ((xc) ∩ (xc ))
8 dff 101 . . . . . 6 0 = ((xc) ∩ (xc) )
96, 7, 83tr1 63 . . . . 5 ((x ∩ (xc )) ∩ c) = 0
10 an32 83 . . . . . 6 ((x ∩ (xc )) ∩ c ) = ((xc ) ∩ (xc ))
11 leao4 165 . . . . . . 7 (xc ) ≤ (xc )
1211df2le2 136 . . . . . 6 ((xc ) ∩ (xc )) = (xc )
13 anor3 90 . . . . . . 7 (xc ) = (xc)
14 k1-4.2 . . . . . . . . 9 xc
1514df-le2 131 . . . . . . . 8 (xc) = c
1615ax-r4 37 . . . . . . 7 (xc) = c
1713, 16tr 62 . . . . . 6 (xc ) = c
1810, 12, 173tr 65 . . . . 5 ((x ∩ (xc )) ∩ c ) = c
199, 182or 72 . . . 4 (((x ∩ (xc )) ∩ c) ∪ ((x ∩ (xc )) ∩ c )) = (0 ∪ c )
20 or0r 103 . . . 4 (0 ∪ c ) = c
215, 19, 203tr 65 . . 3 (x ∩ (xc )) = c
223, 4, 213tr2 64 . 2 (x ∪ (xc)) = c
2322con1 66 1 (x ∪ (xc)) = c
 Colors of variables: term Syntax hints:   = wb 1   ≤ wle 2  ⊥ wn 4   ∪ wo 6   ∩ wa 7  0wf 9 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:  k1-5  360
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