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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 ((a1 b) ∩ ((b2 c) ∩ (c1 a))) ≤ (ac)

Proof of Theorem oago3.29
StepHypRef Expression
1 anass 76 . . . . 5 (((a1 b) ∩ (b2 c)) ∩ (c1 a)) = ((a1 b) ∩ ((b2 c) ∩ (c1 a)))
2 i2id 276 . . . . 5 (a2 a) = 1
31, 22an 79 . . . 4 ((((a1 b) ∩ (b2 c)) ∩ (c1 a)) ∩ (a2 a)) = (((a1 b) ∩ ((b2 c) ∩ (c1 a))) ∩ 1)
43ax-r1 35 . . 3 (((a1 b) ∩ ((b2 c) ∩ (c1 a))) ∩ 1) = ((((a1 b) ∩ (b2 c)) ∩ (c1 a)) ∩ (a2 a))
5 an1 106 . . 3 (((a1 b) ∩ ((b2 c) ∩ (c1 a))) ∩ 1) = ((a1 b) ∩ ((b2 c) ∩ (c1 a)))
6 mhcor1 888 . . 3 ((((a1 b) ∩ (b2 c)) ∩ (c1 a)) ∩ (a2 a)) = (((ab) ∩ (bc)) ∩ (ca))
74, 5, 63tr2 64 . 2 ((a1 b) ∩ ((b2 c) ∩ (c1 a))) = (((ab) ∩ (bc)) ∩ (ca))
8 lear 161 . . 3 (((ab) ∩ (bc)) ∩ (ca)) ≤ (ca)
9 bicom 96 . . 3 (ca) = (ac)
108, 9lbtr 139 . 2 (((ab) ∩ (bc)) ∩ (ca)) ≤ (ac)
117, 10bltr 138 1 ((a1 b) ∩ ((b2 c) ∩ (c1 a))) ≤ (ac)
 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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