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Theorem k1-4 359
Description: Statement (4) in proof of Theorem 1 of Kalmbach, Orthomodular Lattices, p. 21.
Hypotheses
Ref Expression
k1-4.1 (x' ^ (x v c')) = (((x' ^ (x v c')) ^ c) v ((x' ^ (x v c')) ^ c'))
k1-4.2 x =< c
Assertion
Ref Expression
k1-4 (x v (x' ^ c)) = c
Proof of Theorem k1-4
StepHypRef Expression
1 oran1 91 . . . . 5 (x v c') = (x' ^ c)'
21lan 77 . . . 4 (x' ^ (x v c')) = (x' ^ (x' ^ c)')
32cm 61 . . 3 (x' ^ (x' ^ c)') = (x' ^ (x v c'))
4 anor3 90 . . 3 (x' ^ (x' ^ c)') = (x v (x' ^ c))'
5 k1-4.1 . . . 4 (x' ^ (x v c')) = (((x' ^ (x v c')) ^ c) v ((x' ^ (x v c')) ^ c'))
61lan 77 . . . . . 6 ((x' ^ c) ^ (x v c')) = ((x' ^ c) ^ (x' ^ c)')
7 an32 83 . . . . . 6 ((x' ^ (x v c')) ^ c) = ((x' ^ c) ^ (x v c'))
8 dff 101 . . . . . 6 0 = ((x' ^ c) ^ (x' ^ c)')
96, 7, 83tr1 63 . . . . 5 ((x' ^ (x v c')) ^ c) = 0
10 an32 83 . . . . . 6 ((x' ^ (x v c')) ^ c') = ((x' ^ c') ^ (x v c'))
11 leao4 165 . . . . . . 7 (x' ^ c') =< (x v c')
1211df2le2 136 . . . . . 6 ((x' ^ c') ^ (x v c')) = (x' ^ c')
13 anor3 90 . . . . . . 7 (x' ^ c') = (x v c)'
14 k1-4.2 . . . . . . . . 9 x =< c
1514df-le2 131 . . . . . . . 8 (x v c) = c
1615ax-r4 37 . . . . . . 7 (x v c)' = c'
1713, 16tr 62 . . . . . 6 (x' ^ c') = c'
1810, 12, 173tr 65 . . . . 5 ((x' ^ (x v c')) ^ c') = c'
199, 182or 72 . . . 4 (((x' ^ (x v c')) ^ c) v ((x' ^ (x v c')) ^ c')) = (0 v c')
20 or0r 103 . . . 4 (0 v c') = c'
215, 19, 203tr 65 . . 3 (x' ^ (x v c')) = c'
223, 4, 213tr2 64 . 2 (x v (x' ^ c))' = c'
2322con1 66 1 (x v (x' ^ c)) = c
Colors of variables: term
Syntax hints:   = wb 1   =< wle 2  'wn 4   v wo 6   ^ wa 7  0wf 9
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:  k1-5  360
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