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Theorem 2oai1u 822
Description: Orthoarguesian-like OM law.
Assertion
Ref Expression
2oai1u ((a ->1 c) ^ (((a ->1 c) ^ (b ->1 c))' ->2 ((a' ->1 c) ^ (b' ->1 c)))) =< (b ->1 c)
Proof of Theorem 2oai1u
StepHypRef Expression
1 1oai1 821 . 2 (((a' ->1 c) ->1 c) ^ (((a' ->1 c) ^ (b' ->1 c))' ->1 (((a' ->1 c) ->1 c) ^ ((b' ->1 c) ->1 c)))) =< ((b' ->1 c) ->1 c)
2 u1lem11 780 . . 3 ((a' ->1 c) ->1 c) = (a ->1 c)
3 u1lem11 780 . . . . . . . 8 ((b' ->1 c) ->1 c) = (b ->1 c)
42, 32an 79 . . . . . . 7 (((a' ->1 c) ->1 c) ^ ((b' ->1 c) ->1 c)) = ((a ->1 c) ^ (b ->1 c))
54ax-r1 35 . . . . . 6 ((a ->1 c) ^ (b ->1 c)) = (((a' ->1 c) ->1 c) ^ ((b' ->1 c) ->1 c))
65ud1lem0a 255 . . . . 5 (((a' ->1 c) ^ (b' ->1 c))' ->1 ((a ->1 c) ^ (b ->1 c))) = (((a' ->1 c) ^ (b' ->1 c))' ->1 (((a' ->1 c) ->1 c) ^ ((b' ->1 c) ->1 c)))
76ax-r1 35 . . . 4 (((a' ->1 c) ^ (b' ->1 c))' ->1 (((a' ->1 c) ->1 c) ^ ((b' ->1 c) ->1 c))) = (((a' ->1 c) ^ (b' ->1 c))' ->1 ((a ->1 c) ^ (b ->1 c)))
8 i1i2con2 269 . . . 4 (((a' ->1 c) ^ (b' ->1 c))' ->1 ((a ->1 c) ^ (b ->1 c))) = (((a ->1 c) ^ (b ->1 c))' ->2 ((a' ->1 c) ^ (b' ->1 c)))
97, 8ax-r2 36 . . 3 (((a' ->1 c) ^ (b' ->1 c))' ->1 (((a' ->1 c) ->1 c) ^ ((b' ->1 c) ->1 c))) = (((a ->1 c) ^ (b ->1 c))' ->2 ((a' ->1 c) ^ (b' ->1 c)))
102, 92an 79 . 2 (((a' ->1 c) ->1 c) ^ (((a' ->1 c) ^ (b' ->1 c))' ->1 (((a' ->1 c) ->1 c) ^ ((b' ->1 c) ->1 c)))) = ((a ->1 c) ^ (((a ->1 c) ^ (b ->1 c))' ->2 ((a' ->1 c) ^ (b' ->1 c))))
111, 10, 3le3tr2 141 1 ((a ->1 c) ^ (((a ->1 c) ^ (b ->1 c))' ->2 ((a' ->1 c) ^ (b' ->1 c)))) =< (b ->1 c)
Colors of variables: term
Syntax hints:   =< wle 2  'wn 4   ^ wa 7   ->1 wi1 12   ->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-i1 44  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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