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Theorem oago3.29 889
Description: Equation (3.29) of "Equations, states, and lattices..." paper. This shows that it holds in all OMLs, not just 4GO.
Assertion
Ref Expression
oago3.29 ((a ->1 b) ^ ((b ->2 c) ^ (c ->1 a))) =< (a == c)
Proof of Theorem oago3.29
StepHypRef Expression
1 anass 76 . . . . 5 (((a ->1 b) ^ (b ->2 c)) ^ (c ->1 a)) = ((a ->1 b) ^ ((b ->2 c) ^ (c ->1 a)))
2 i2id 276 . . . . 5 (a ->2 a) = 1
31, 22an 79 . . . 4 ((((a ->1 b) ^ (b ->2 c)) ^ (c ->1 a)) ^ (a ->2 a)) = (((a ->1 b) ^ ((b ->2 c) ^ (c ->1 a))) ^ 1)
43ax-r1 35 . . 3 (((a ->1 b) ^ ((b ->2 c) ^ (c ->1 a))) ^ 1) = ((((a ->1 b) ^ (b ->2 c)) ^ (c ->1 a)) ^ (a ->2 a))
5 an1 106 . . 3 (((a ->1 b) ^ ((b ->2 c) ^ (c ->1 a))) ^ 1) = ((a ->1 b) ^ ((b ->2 c) ^ (c ->1 a)))
6 mhcor1 888 . . 3 ((((a ->1 b) ^ (b ->2 c)) ^ (c ->1 a)) ^ (a ->2 a)) = (((a == b) ^ (b == c)) ^ (c == a))
74, 5, 63tr2 64 . 2 ((a ->1 b) ^ ((b ->2 c) ^ (c ->1 a))) = (((a == b) ^ (b == c)) ^ (c == a))
8 lear 161 . . 3 (((a == b) ^ (b == c)) ^ (c == a)) =< (c == a)
9 bicom 96 . . 3 (c == a) = (a == c)
108, 9lbtr 139 . 2 (((a == b) ^ (b == c)) ^ (c == a)) =< (a == c)
117, 10bltr 138 1 ((a ->1 b) ^ ((b ->2 c) ^ (c ->1 a))) =< (a == c)
Colors of variables: term
Syntax hints:   =< wle 2   == tb 5   ^ wa 7  1wt 8   ->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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