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Theorem 4oa 1039
Description: Variant of proper 4-OA.
Hypotheses
Ref Expression
4oa.1 e = (((a ^ c) v ((a ->1 d) ^ (c ->1 d))) ^ ((b ^ c) v ((b ->1 d) ^ (c ->1 d))))
4oa.2 f = (((a ^ b) v ((a ->1 d) ^ (b ->1 d))) v e)
Assertion
Ref Expression
4oa ((a ->1 d) ^ f) =< (b ->1 d)
Proof of Theorem 4oa
StepHypRef Expression
1 4oa.2 . . 3 f = (((a ^ b) v ((a ->1 d) ^ (b ->1 d))) v e)
21lan 77 . 2 ((a ->1 d) ^ f) = ((a ->1 d) ^ (((a ^ b) v ((a ->1 d) ^ (b ->1 d))) v e))
3 axoa4a 1037 . . . 4 (((b'' ->1 d) ->1 d) ^ ((b'' ->1 d) v ((a'' ->1 d) ^ ((((b'' ->1 d) ^ (a'' ->1 d)) v (((b'' ->1 d) ->1 d) ^ ((a'' ->1 d) ->1 d))) v ((((b'' ->1 d) ^ (c'' ->1 d)) v (((b'' ->1 d) ->1 d) ^ ((c'' ->1 d) ->1 d))) ^ (((a'' ->1 d) ^ (c'' ->1 d)) v (((a'' ->1 d) ->1 d) ^ ((c'' ->1 d) ->1 d)))))))) =< ((((b'' ->1 d) ^ d) v ((a'' ->1 d) ^ d)) v ((c'' ->1 d) ^ d))
4 id 59 . . . 4 (b'' ->1 d) = (b'' ->1 d)
5 id 59 . . . 4 (a'' ->1 d) = (a'' ->1 d)
6 id 59 . . . 4 (c'' ->1 d) = (c'' ->1 d)
73, 4, 5, 6oa4to4u2 974 . . 3 ((b' ->1 d) ^ ((b'' ->1 d) v ((a'' ->1 d) ^ ((((b' ->1 d) ^ (a' ->1 d)) v ((b'' ->1 d) ^ (a'' ->1 d))) v ((((b' ->1 d) ^ (c' ->1 d)) v ((b'' ->1 d) ^ (c'' ->1 d))) ^ (((a' ->1 d) ^ (c' ->1 d)) v ((a'' ->1 d) ^ (c'' ->1 d)))))))) =< d
8 4oa.1 . . 3 e = (((a ^ c) v ((a ->1 d) ^ (c ->1 d))) ^ ((b ^ c) v ((b ->1 d) ^ (c ->1 d))))
97, 8oa4uto4g 975 . 2 ((a ->1 d) ^ (((a ^ b) v ((a ->1 d) ^ (b ->1 d))) v e)) =< (b ->1 d)
102, 9bltr 138 1 ((a ->1 d) ^ f) =< (b ->1 d)
Colors of variables: term
Syntax hints:   = wb 1   =< wle 2  'wn 4   v wo 6   ^ wa 7   ->1 wi1 12
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  ax-4oa 1033
This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i1 44  df-le1 130  df-le2 131  df-c1 132  df-c2 133
This theorem is referenced by:  4oaiii  1040
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