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Theorem oale 829
Description: Relation for studying OA.
Assertion
Ref Expression
oale ((a ->2 b) ^ ((b v c) v ((a ->2 b) ^ (a ->2 c)))') =< (a ->2 c)
Proof of Theorem oale
StepHypRef Expression
1 df-i2 45 . . . . . . 7 ((b v c) ->2 ((a ->2 b) ^ (a ->2 c))) = (((a ->2 b) ^ (a ->2 c)) v ((b v c)' ^ ((a ->2 b) ^ (a ->2 c))'))
21lan 77 . . . . . 6 ((a ->2 b) ^ ((b v c) ->2 ((a ->2 b) ^ (a ->2 c)))) = ((a ->2 b) ^ (((a ->2 b) ^ (a ->2 c)) v ((b v c)' ^ ((a ->2 b) ^ (a ->2 c))')))
3 coman1 185 . . . . . . 7 ((a ->2 b) ^ (a ->2 c)) C (a ->2 b)
4 comanr2 465 . . . . . . . 8 ((a ->2 b) ^ (a ->2 c))' C ((b v c)' ^ ((a ->2 b) ^ (a ->2 c))')
54comcom6 459 . . . . . . 7 ((a ->2 b) ^ (a ->2 c)) C ((b v c)' ^ ((a ->2 b) ^ (a ->2 c))')
63, 5fh2 470 . . . . . 6 ((a ->2 b) ^ (((a ->2 b) ^ (a ->2 c)) v ((b v c)' ^ ((a ->2 b) ^ (a ->2 c))'))) = (((a ->2 b) ^ ((a ->2 b) ^ (a ->2 c))) v ((a ->2 b) ^ ((b v c)' ^ ((a ->2 b) ^ (a ->2 c))')))
7 anass 76 . . . . . . . . . 10 (((a ->2 b) ^ (a ->2 b)) ^ (a ->2 c)) = ((a ->2 b) ^ ((a ->2 b) ^ (a ->2 c)))
87ax-r1 35 . . . . . . . . 9 ((a ->2 b) ^ ((a ->2 b) ^ (a ->2 c))) = (((a ->2 b) ^ (a ->2 b)) ^ (a ->2 c))
9 anidm 111 . . . . . . . . . 10 ((a ->2 b) ^ (a ->2 b)) = (a ->2 b)
109ran 78 . . . . . . . . 9 (((a ->2 b) ^ (a ->2 b)) ^ (a ->2 c)) = ((a ->2 b) ^ (a ->2 c))
118, 10ax-r2 36 . . . . . . . 8 ((a ->2 b) ^ ((a ->2 b) ^ (a ->2 c))) = ((a ->2 b) ^ (a ->2 c))
12 anor3 90 . . . . . . . . 9 ((b v c)' ^ ((a ->2 b) ^ (a ->2 c))') = ((b v c) v ((a ->2 b) ^ (a ->2 c)))'
1312lan 77 . . . . . . . 8 ((a ->2 b) ^ ((b v c)' ^ ((a ->2 b) ^ (a ->2 c))')) = ((a ->2 b) ^ ((b v c) v ((a ->2 b) ^ (a ->2 c)))')
1411, 132or 72 . . . . . . 7 (((a ->2 b) ^ ((a ->2 b) ^ (a ->2 c))) v ((a ->2 b) ^ ((b v c)' ^ ((a ->2 b) ^ (a ->2 c))'))) = (((a ->2 b) ^ (a ->2 c)) v ((a ->2 b) ^ ((b v c) v ((a ->2 b) ^ (a ->2 c)))'))
15 ax-a2 31 . . . . . . 7 (((a ->2 b) ^ (a ->2 c)) v ((a ->2 b) ^ ((b v c) v ((a ->2 b) ^ (a ->2 c)))')) = (((a ->2 b) ^ ((b v c) v ((a ->2 b) ^ (a ->2 c)))') v ((a ->2 b) ^ (a ->2 c)))
1614, 15ax-r2 36 . . . . . 6 (((a ->2 b) ^ ((a ->2 b) ^ (a ->2 c))) v ((a ->2 b) ^ ((b v c)' ^ ((a ->2 b) ^ (a ->2 c))'))) = (((a ->2 b) ^ ((b v c) v ((a ->2 b) ^ (a ->2 c)))') v ((a ->2 b) ^ (a ->2 c)))
172, 6, 163tr 65 . . . . 5 ((a ->2 b) ^ ((b v c) ->2 ((a ->2 b) ^ (a ->2 c)))) = (((a ->2 b) ^ ((b v c) v ((a ->2 b) ^ (a ->2 c)))') v ((a ->2 b) ^ (a ->2 c)))
1817ax-r1 35 . . . 4 (((a ->2 b) ^ ((b v c) v ((a ->2 b) ^ (a ->2 c)))') v ((a ->2 b) ^ (a ->2 c))) = ((a ->2 b) ^ ((b v c) ->2 ((a ->2 b) ^ (a ->2 c))))
19 2oath1 826 . . . 4 ((a ->2 b) ^ ((b v c) ->2 ((a ->2 b) ^ (a ->2 c)))) = ((a ->2 b) ^ (a ->2 c))
2018, 19ax-r2 36 . . 3 (((a ->2 b) ^ ((b v c) v ((a ->2 b) ^ (a ->2 c)))') v ((a ->2 b) ^ (a ->2 c))) = ((a ->2 b) ^ (a ->2 c))
2120df-le1 130 . 2 ((a ->2 b) ^ ((b v c) v ((a ->2 b) ^ (a ->2 c)))') =< ((a ->2 b) ^ (a ->2 c))
22 lear 161 . 2 ((a ->2 b) ^ (a ->2 c)) =< (a ->2 c)
2321, 22letr 137 1 ((a ->2 b) ^ ((b v c) v ((a ->2 b) ^ (a ->2 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-a4 33  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-r3 439
This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i2 45  df-le1 130  df-le2 131  df-c1 132  df-c2 133
This theorem is referenced by: (None)
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