Theorem oago3.29 871

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

Step	Hyp	Ref	Expression
1		anass 69	. . . . 5 (((a ->1 b) ^ (b ->2 c)) ^ (c ->1 a)) = ((a ->1 b) ^ ((b ->2 c) ^ (c ->1 a)))
2		i2id 268	. . . . 5 (a ->2 a) = 1
3	1, 2	2an 72	. . . 4 ((((a ->1 b) ^ (b ->2 c)) ^ (c ->1 a)) ^ (a ->2 a)) = (((a ->1 b) ^ ((b ->2 c) ^ (c ->1 a))) ^ 1)
4	3	ax-r1 34	. . 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 98	. . 3 (((a ->1 b) ^ ((b ->2 c) ^ (c ->1 a))) ^ 1) = ((a ->1 b) ^ ((b ->2 c) ^ (c ->1 a)))
6		mhcor1 870	. . 3 ((((a ->1 b) ^ (b ->2 c)) ^ (c ->1 a)) ^ (a ->2 a)) = (((a == b) ^ (b == c)) ^ (c == a))
7	4, 5, 6	3tr2 61	. 2 ((a ->1 b) ^ ((b ->2 c) ^ (c ->1 a))) = (((a == b) ^ (b == c)) ^ (c == a))
8		lear 153	. . 3 (((a == b) ^ (b == c)) ^ (c == a)) =< (c == a)
9		bicom 88	. . 3 (c == a) = (a == c)
10	8, 9	lbtr 131	. 2 (((a == b) ^ (b == c)) ^ (c == a)) =< (a == c)
11	7, 10	bltr 130	1 ((a ->1 b) ^ ((b ->2 c) ^ (c ->1 a))) =< (a == c)

Syntax hints:   =< wle 2   == tb 5   ^ wa 7  1wt 9   ->1 wi1 13   ->2 wi2 14

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  ax-a4 32  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37  ax-r3 421

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41  df-i1 43  df-i2 44  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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