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Theorem oaliv 1003
Description: Orthoarguesian law. Godowski/Greechie, Eq. IV.
Assertion
Ref Expression
oaliv (b' ^ ((a ->2 b) v ((a ->2 c) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c)))))) =< ((b' ^ (a ->2 b)) v (c' ^ (a ->2 c)))
Proof of Theorem oaliv
StepHypRef Expression
1 lea 160 . . . 4 (b' ^ ((a ->2 b) v ((a ->2 c) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c)))))) =< b'
2 oalii 1002 . . . 4 (b' ^ ((a ->2 b) v ((a ->2 c) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c)))))) =< a'
31, 2ler2an 173 . . 3 (b' ^ ((a ->2 b) v ((a ->2 c) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c)))))) =< (b' ^ a')
4 df-i2 45 . . . . . . 7 (a ->2 b) = (b v (a' ^ b'))
5 ancom 74 . . . . . . . 8 (a' ^ b') = (b' ^ a')
65lor 70 . . . . . . 7 (b v (a' ^ b')) = (b v (b' ^ a'))
74, 6ax-r2 36 . . . . . 6 (a ->2 b) = (b v (b' ^ a'))
87lan 77 . . . . 5 (b' ^ (a ->2 b)) = (b' ^ (b v (b' ^ a')))
9 omlan 448 . . . . 5 (b' ^ (b v (b' ^ a'))) = (b' ^ a')
108, 9ax-r2 36 . . . 4 (b' ^ (a ->2 b)) = (b' ^ a')
1110ax-r1 35 . . 3 (b' ^ a') = (b' ^ (a ->2 b))
123, 11lbtr 139 . 2 (b' ^ ((a ->2 b) v ((a ->2 c) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c)))))) =< (b' ^ (a ->2 b))
13 leo 158 . 2 (b' ^ (a ->2 b)) =< ((b' ^ (a ->2 b)) v (c' ^ (a ->2 c)))
1412, 13letr 137 1 (b' ^ ((a ->2 b) v ((a ->2 c) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c)))))) =< ((b' ^ (a ->2 b)) v (c' ^ (a ->2 c)))
Colors of variables: term
Syntax hints:   =< wle 2  'wn 4   v wo 6   ^ wa 7   ->2 wi2 13
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  ax-r3 439  ax-3oa 998
This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i1 44  df-i2 45  df-le1 130  df-le2 131
This theorem is referenced by: (None)
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