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Theorem k1-8a 355
 Description: First part of statement (8) in proof of Theorem 1 of Kalmbach, Orthomodular Lattices, p. 21.
Hypotheses
Ref Expression
k1-8a.1 x = ((xc) ∪ (xc ))
k1-8a.2 xc
k1-8a.3 yc
Assertion
Ref Expression
k1-8a x = ((xy) ∩ c)

Proof of Theorem k1-8a
StepHypRef Expression
1 leo 158 . . 3 x ≤ (xy)
2 k1-8a.2 . . 3 xc
31, 2ler2an 173 . 2 x ≤ ((xy) ∩ c)
4 k1-8a.3 . . . . 5 yc
54lelor 166 . . . 4 (xy) ≤ (xc )
65leran 153 . . 3 ((xy) ∩ c) ≤ ((xc ) ∩ c)
7 ax-a1 30 . . . . . 6 x = x
87ror 71 . . . . 5 (xc ) = (x c )
98ran 78 . . . 4 ((xc ) ∩ c) = ((x c ) ∩ c)
107ran 78 . . . . . 6 (xc) = (x c)
11 k1-8a.1 . . . . . . 7 x = ((xc) ∪ (xc ))
1211k1-6 353 . . . . . 6 (x c) = ((x c ) ∩ c)
1310, 12tr 62 . . . . 5 (xc) = ((x c ) ∩ c)
1413cm 61 . . . 4 ((x c ) ∩ c) = (xc)
152df2le2 136 . . . 4 (xc) = x
169, 14, 153tr 65 . . 3 ((xc ) ∩ c) = x
176, 16lbtr 139 . 2 ((xy) ∩ c) ≤ x
183, 17lebi 145 1 x = ((xy) ∩ 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:  k1-8b  356  k1-2  357
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