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Theorem k1-8b 356
Description: Second part of statement (8) in proof of Theorem 1 of Kalmbach, Orthomodular Lattices, p. 21.
Hypotheses
Ref Expression
k1-8b.1 y' = ((y' ^ c) v (y' ^ c'))
k1-8b.2 x =< c
k1-8b.3 y =< c'
Assertion
Ref Expression
k1-8b y = ((x v y) ^ c')
Proof of Theorem k1-8b
StepHypRef Expression
1 k1-8b.1 . . . 4 y' = ((y' ^ c) v (y' ^ c'))
2 ax-a1 30 . . . . . 6 c = c''
32lan 77 . . . . 5 (y' ^ c) = (y' ^ c'')
43ror 71 . . . 4 ((y' ^ c) v (y' ^ c')) = ((y' ^ c'') v (y' ^ c'))
5 orcom 73 . . . 4 ((y' ^ c'') v (y' ^ c')) = ((y' ^ c') v (y' ^ c''))
61, 4, 53tr 65 . . 3 y' = ((y' ^ c') v (y' ^ c''))
7 k1-8b.3 . . 3 y =< c'
8 k1-8b.2 . . . 4 x =< c
98, 2lbtr 139 . . 3 x =< c''
106, 7, 9k1-8a 355 . 2 y = ((y v x) ^ c')
11 orcom 73 . . 3 (y v x) = (x v y)
1211ran 78 . 2 ((y v x) ^ c') = ((x v y) ^ c')
1310, 12tr 62 1 y = ((x v y) ^ c')
Colors of variables: term
Syntax hints:   = wb 1   =< wle 2  '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:  k1-3  358
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