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Theorem k1-2 357
Description: Statement (2) in proof of Theorem 1 of Kalmbach, Orthomodular Lattices, p. 21.
Hypotheses
Ref Expression
k1-2.1 x = ((x ^ c) v (x ^ c'))
k1-2.2 y = ((y ^ c) v (y ^ c'))
k1-2.3 ((x ^ c) v (y ^ c))' = ((((x ^ c) v (y ^ c))' ^ c) v (((x ^ c) v (y ^ c))' ^ c'))
Assertion
Ref Expression
k1-2 ((x v y) ^ c) = ((x ^ c) v (y ^ c))
Proof of Theorem k1-2
StepHypRef Expression
1 k1-2.1 . . . . 5 x = ((x ^ c) v (x ^ c'))
2 k1-2.2 . . . . 5 y = ((y ^ c) v (y ^ c'))
31, 22or 72 . . . 4 (x v y) = (((x ^ c) v (x ^ c')) v ((y ^ c) v (y ^ c')))
4 or4 84 . . . 4 (((x ^ c) v (x ^ c')) v ((y ^ c) v (y ^ c'))) = (((x ^ c) v (y ^ c)) v ((x ^ c') v (y ^ c')))
53, 4ax-r2 36 . . 3 (x v y) = (((x ^ c) v (y ^ c)) v ((x ^ c') v (y ^ c')))
65ran 78 . 2 ((x v y) ^ c) = ((((x ^ c) v (y ^ c)) v ((x ^ c') v (y ^ c'))) ^ c)
7 k1-2.3 . . . 4 ((x ^ c) v (y ^ c))' = ((((x ^ c) v (y ^ c))' ^ c) v (((x ^ c) v (y ^ c))' ^ c'))
8 lear 161 . . . . 5 (x ^ c) =< c
9 lear 161 . . . . 5 (y ^ c) =< c
108, 9lel2or 170 . . . 4 ((x ^ c) v (y ^ c)) =< c
11 lear 161 . . . . 5 (x ^ c') =< c'
12 lear 161 . . . . 5 (y ^ c') =< c'
1311, 12lel2or 170 . . . 4 ((x ^ c') v (y ^ c')) =< c'
147, 10, 13k1-8a 355 . . 3 ((x ^ c) v (y ^ c)) = ((((x ^ c) v (y ^ c)) v ((x ^ c') v (y ^ c'))) ^ c)
1514ax-r1 35 . 2 ((((x ^ c) v (y ^ c)) v ((x ^ c') v (y ^ c'))) ^ c) = ((x ^ c) v (y ^ c))
166, 15tr 62 1 ((x v y) ^ c) = ((x ^ c) v (y ^ c))
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v 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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