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Theorem wom2 434
Description: Weak orthomodular law for study of weakly orthomodular lattices.
Assertion
Ref Expression
wom2 a =< ((a == b)' v ((a v c) == (b v c)))
Proof of Theorem wom2
StepHypRef Expression
1 le1 146 . 2 a =< 1
2 conb 122 . . . . . 6 (a == b) = (a' == b')
32ax-r4 37 . . . . 5 (a == b)' = (a' == b')'
4 oran 87 . . . . . . 7 (a v c) = (a' ^ c')'
5 oran 87 . . . . . . 7 (b v c) = (b' ^ c')'
64, 52bi 99 . . . . . 6 ((a v c) == (b v c)) = ((a' ^ c')' == (b' ^ c')')
7 conb 122 . . . . . . 7 ((a' ^ c') == (b' ^ c')) = ((a' ^ c')' == (b' ^ c')')
87ax-r1 35 . . . . . 6 ((a' ^ c')' == (b' ^ c')') = ((a' ^ c') == (b' ^ c'))
96, 8ax-r2 36 . . . . 5 ((a v c) == (b v c)) = ((a' ^ c') == (b' ^ c'))
103, 92or 72 . . . 4 ((a == b)' v ((a v c) == (b v c))) = ((a' == b')' v ((a' ^ c') == (b' ^ c')))
11 ska4 433 . . . 4 ((a' == b')' v ((a' ^ c') == (b' ^ c'))) = 1
1210, 11ax-r2 36 . . 3 ((a == b)' v ((a v c) == (b v c))) = 1
1312ax-r1 35 . 2 1 = ((a == b)' v ((a v c) == (b v c)))
141, 13lbtr 139 1 a =< ((a == b)' v ((a v c) == (b v c)))
Colors of variables: term
Syntax hints:   =< wle 2  'wn 4   == tb 5   v wo 6   ^ wa 7  1wt 8
This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a4 33  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-wom 361
This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i1 44  df-i2 45  df-le 129  df-le1 130  df-le2 131  df-cmtr 134
This theorem is referenced by:  ka4ot  435
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