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