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Theorem k1-5 360
Description: Statement (5) in proof of Theorem 1 of Kalmbach, Orthomodular Lattices, p. 21.
Hypotheses
Ref Expression
k1-5.1 (x' ^ (x v c)) = (((x' ^ (x v c)) ^ c) v ((x' ^ (x v c)) ^ c'))
k1-5.2 x =< c'
Assertion
Ref Expression
k1-5 (x v (x' ^ c')) = c'

Proof of Theorem k1-5
StepHypRef Expression
1 k1-5.1 . . 3 (x' ^ (x v c)) = (((x' ^ (x v c)) ^ c) v ((x' ^ (x v c)) ^ c'))
2 ax-a1 30 . . . . 5 c = c''
32lor 70 . . . 4 (x v c) = (x v c'')
43lan 77 . . 3 (x' ^ (x v c)) = (x' ^ (x v c''))
5 orcom 73 . . . 4 (((x' ^ (x v c)) ^ c) v ((x' ^ (x v c)) ^ c')) = (((x' ^ (x v c)) ^ c') v ((x' ^ (x v c)) ^ c))
64ran 78 . . . . 5 ((x' ^ (x v c)) ^ c') = ((x' ^ (x v c'')) ^ c')
74, 22an 79 . . . . 5 ((x' ^ (x v c)) ^ c) = ((x' ^ (x v c'')) ^ c'')
86, 72or 72 . . . 4 (((x' ^ (x v c)) ^ c') v ((x' ^ (x v c)) ^ c)) = (((x' ^ (x v c'')) ^ c') v ((x' ^ (x v c'')) ^ c''))
95, 8tr 62 . . 3 (((x' ^ (x v c)) ^ c) v ((x' ^ (x v c)) ^ c')) = (((x' ^ (x v c'')) ^ c') v ((x' ^ (x v c'')) ^ c''))
101, 4, 93tr2 64 . 2 (x' ^ (x v c'')) = (((x' ^ (x v c'')) ^ c') v ((x' ^ (x v c'')) ^ c''))
11 k1-5.2 . 2 x =< c'
1210, 11k1-4 359 1 (x v (x' ^ c')) = 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
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