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Theorem oadist2a 1007
Description: Distributive inference derived from OA.
Hypothesis
Ref Expression
oadist2a.1 (d v ((b v c) ->2 ((a ->2 b) ^ (a ->2 c)))) =< ((b v c) ->0 ((a ->2 b) ^ (a ->2 c)))
Assertion
Ref Expression
oadist2a ((a ->2 b) ^ (d v ((b v c) ->2 ((a ->2 b) ^ (a ->2 c))))) = (((a ->2 b) ^ d) v ((a ->2 b) ^ ((b v c) ->2 ((a ->2 b) ^ (a ->2 c)))))
Proof of Theorem oadist2a
StepHypRef Expression
1 ax-a2 31 . . 3 (d v ((b v c) ->2 ((a ->2 b) ^ (a ->2 c)))) = (((b v c) ->2 ((a ->2 b) ^ (a ->2 c))) v d)
21lan 77 . 2 ((a ->2 b) ^ (d v ((b v c) ->2 ((a ->2 b) ^ (a ->2 c))))) = ((a ->2 b) ^ (((b v c) ->2 ((a ->2 b) ^ (a ->2 c))) v d))
3 ax-a2 31 . . . . . . 7 (((b v c) ->2 ((a ->2 b) ^ (a ->2 c))) v d) = (d v ((b v c) ->2 ((a ->2 b) ^ (a ->2 c))))
4 oadist2a.1 . . . . . . 7 (d v ((b v c) ->2 ((a ->2 b) ^ (a ->2 c)))) =< ((b v c) ->0 ((a ->2 b) ^ (a ->2 c)))
53, 4bltr 138 . . . . . 6 (((b v c) ->2 ((a ->2 b) ^ (a ->2 c))) v d) =< ((b v c) ->0 ((a ->2 b) ^ (a ->2 c)))
65lelan 167 . . . . 5 ((a ->2 b) ^ (((b v c) ->2 ((a ->2 b) ^ (a ->2 c))) v d)) =< ((a ->2 b) ^ ((b v c) ->0 ((a ->2 b) ^ (a ->2 c))))
7 df-i0 43 . . . . . . . 8 ((b v c) ->0 ((a ->2 b) ^ (a ->2 c))) = ((b v c)' v ((a ->2 b) ^ (a ->2 c)))
87lan 77 . . . . . . 7 ((a ->2 b) ^ ((b v c) ->0 ((a ->2 b) ^ (a ->2 c)))) = ((a ->2 b) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c))))
9 oath1 1004 . . . . . . 7 ((a ->2 b) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c)))) = ((a ->2 b) ^ (a ->2 c))
108, 9ax-r2 36 . . . . . 6 ((a ->2 b) ^ ((b v c) ->0 ((a ->2 b) ^ (a ->2 c)))) = ((a ->2 b) ^ (a ->2 c))
11 leo 158 . . . . . . 7 ((a ->2 b) ^ (a ->2 c)) =< (((a ->2 b) ^ (a ->2 c)) v ((b v c)' ^ ((a ->2 b) ^ (a ->2 c))'))
12 df-i2 45 . . . . . . . 8 ((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))'))
1312ax-r1 35 . . . . . . 7 (((a ->2 b) ^ (a ->2 c)) v ((b v c)' ^ ((a ->2 b) ^ (a ->2 c))')) = ((b v c) ->2 ((a ->2 b) ^ (a ->2 c)))
1411, 13lbtr 139 . . . . . 6 ((a ->2 b) ^ (a ->2 c)) =< ((b v c) ->2 ((a ->2 b) ^ (a ->2 c)))
1510, 14bltr 138 . . . . 5 ((a ->2 b) ^ ((b v c) ->0 ((a ->2 b) ^ (a ->2 c)))) =< ((b v c) ->2 ((a ->2 b) ^ (a ->2 c)))
166, 15letr 137 . . . 4 ((a ->2 b) ^ (((b v c) ->2 ((a ->2 b) ^ (a ->2 c))) v d)) =< ((b v c) ->2 ((a ->2 b) ^ (a ->2 c)))
1716distlem 188 . . 3 ((a ->2 b) ^ (((b v c) ->2 ((a ->2 b) ^ (a ->2 c))) v d)) = (((a ->2 b) ^ ((b v c) ->2 ((a ->2 b) ^ (a ->2 c)))) v ((a ->2 b) ^ d))
18 ax-a2 31 . . 3 (((a ->2 b) ^ ((b v c) ->2 ((a ->2 b) ^ (a ->2 c)))) v ((a ->2 b) ^ d)) = (((a ->2 b) ^ d) v ((a ->2 b) ^ ((b v c) ->2 ((a ->2 b) ^ (a ->2 c)))))
1917, 18ax-r2 36 . 2 ((a ->2 b) ^ (((b v c) ->2 ((a ->2 b) ^ (a ->2 c))) v d)) = (((a ->2 b) ^ d) v ((a ->2 b) ^ ((b v c) ->2 ((a ->2 b) ^ (a ->2 c)))))
202, 19ax-r2 36 1 ((a ->2 b) ^ (d v ((b v c) ->2 ((a ->2 b) ^ (a ->2 c))))) = (((a ->2 b) ^ d) v ((a ->2 b) ^ ((b v c) ->2 ((a ->2 b) ^ (a ->2 c)))))
Colors of variables: term
Syntax hints:   = wb 1   =< wle 2  'wn 4   v wo 6   ^ wa 7   ->0 wi0 11   ->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-3oa 998
This theorem depends on definitions:  df-a 40  df-t 41  df-f 42  df-i0 43  df-i1 44  df-i2 45  df-le1 130  df-le2 131
This theorem is referenced by:  oadist2b  1008  oadist2  1009
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