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Theorem k1-7 354
 Description: Statement (7) in proof of Theorem 1 of Kalmbach, Orthomodular Lattices, p. 21.
Hypothesis
Ref Expression
k1-7.1 x = ((xc) ∪ (xc ))
Assertion
Ref Expression
k1-7 (xc ) = ((xc) ∩ c )

Proof of Theorem k1-7
StepHypRef Expression
1 anor3 90 . . . . 5 ((xc ) ∩ (xc ) ) = ((xc ) ∪ (xc ))
21cm 61 . . . 4 ((xc ) ∪ (xc )) = ((xc ) ∩ (xc ) )
3 k1-7.1 . . . . . 6 x = ((xc) ∪ (xc ))
4 ax-a1 30 . . . . . . . 8 c = c
54lan 77 . . . . . . 7 (xc) = (xc )
65ror 71 . . . . . 6 ((xc) ∪ (xc )) = ((xc ) ∪ (xc ))
7 orcom 73 . . . . . 6 ((xc ) ∪ (xc )) = ((xc ) ∪ (xc ))
83, 6, 73tr 65 . . . . 5 x = ((xc ) ∪ (xc ))
98con4 69 . . . 4 x = ((xc ) ∪ (xc ))
10 oran3 93 . . . . 5 (xc ) = (xc )
11 oran2 92 . . . . 5 (xc ) = (xc )
1210, 112an 79 . . . 4 ((xc ) ∩ (xc )) = ((xc ) ∩ (xc ) )
132, 9, 123tr1 63 . . 3 x = ((xc ) ∩ (xc ))
1413ran 78 . 2 (xc ) = (((xc ) ∩ (xc )) ∩ c )
154lor 70 . . . . . 6 (xc) = (xc )
1615ran 78 . . . . 5 ((xc) ∩ (xc )) = ((xc ) ∩ (xc ))
1716ran 78 . . . 4 (((xc) ∩ (xc )) ∩ c ) = (((xc ) ∩ (xc )) ∩ c )
1817cm 61 . . 3 (((xc ) ∩ (xc )) ∩ c ) = (((xc) ∩ (xc )) ∩ c )
19 anass 76 . . 3 (((xc) ∩ (xc )) ∩ c ) = ((xc) ∩ ((xc ) ∩ c ))
2018, 19tr 62 . 2 (((xc ) ∩ (xc )) ∩ c ) = ((xc) ∩ ((xc ) ∩ c ))
21 ancom 74 . . . 4 ((xc ) ∩ c ) = (c ∩ (xc ))
22 ax-a2 31 . . . . 5 (xc ) = (cx )
2322lan 77 . . . 4 (c ∩ (xc )) = (c ∩ (cx ))
24 anabs 121 . . . 4 (c ∩ (cx )) = c
2521, 23, 243tr 65 . . 3 ((xc ) ∩ c ) = c
2625lan 77 . 2 ((xc) ∩ ((xc ) ∩ c )) = ((xc) ∩ c )
2714, 20, 263tr 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: (None)
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