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Theorem oaur 930
Description: Transformation lemma for studying the orthoarguesian law.
Hypothesis
Ref Expression
oaur.1 b =< (a ->1 c)
Assertion
Ref Expression
oaur (a ^ ((a ->1 c) v b)) =< c
Proof of Theorem oaur
StepHypRef Expression
1 leid 148 . . . . 5 (a ->1 c) =< (a ->1 c)
2 oaur.1 . . . . 5 b =< (a ->1 c)
31, 2lel2or 170 . . . 4 ((a ->1 c) v b) =< (a ->1 c)
43lelan 167 . . 3 (a ^ ((a ->1 c) v b)) =< (a ^ (a ->1 c))
5 ancom 74 . . . 4 (a ^ (a ->1 c)) = ((a ->1 c) ^ a)
6 u1lemaa 600 . . . 4 ((a ->1 c) ^ a) = (a ^ c)
75, 6ax-r2 36 . . 3 (a ^ (a ->1 c)) = (a ^ c)
84, 7lbtr 139 . 2 (a ^ ((a ->1 c) v b)) =< (a ^ c)
9 lear 161 . 2 (a ^ c) =< c
108, 9letr 137 1 (a ^ ((a ->1 c) v b)) =< c
Colors of variables: term
Syntax hints:   =< wle 2   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
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:  oa4gto4u  976
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