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. (Contributed by NM, 22-Jun-2003.)

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 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
3	1, 2	2an 79	. . . 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 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))
7	4, 5, 6	3tr2 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)
10	8, 9	lbtr 139	. 2 (((a ≡ b) ∩ (b ≡ c)) ∩ (c ≡ a)) ≤ (a ≡ c)
11	7, 10	bltr 138	1 ((a →1 b) ∩ ((b →2 c) ∩ (c →1 a))) ≤ (a ≡ c)

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)