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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