Theorem imp3 823

Description: Implicational product with 3 variables. Theorem 3.20 of "Equations, states, and lattices..." paper.

Assertion

Ref	Expression
imp3	((a →2 b) ∩ (b →1 c)) = ((a⊥ ∩ b⊥ ) ∪ (b ∩ c))

Proof of Theorem imp3

Step	Hyp	Ref	Expression
1		df-i1 43	. . 3 (b →1 c) = (b⊥ ∪ (b ∩ c))
2	1	lan 70	. 2 ((a →2 b) ∩ (b →1 c)) = ((a →2 b) ∩ (b⊥ ∪ (b ∩ c)))
3		u2lemc1 663	. . . 4 b C (a →2 b)
4	3	comcom3 436	. . 3 b⊥ C (a →2 b)
5		comanr1 446	. . . 4 b C (b ∩ c)
6	5	comcom3 436	. . 3 b⊥ C (b ∩ c)
7	4, 6	fh2 452	. 2 ((a →2 b) ∩ (b⊥ ∪ (b ∩ c))) = (((a →2 b) ∩ b⊥ ) ∪ ((a →2 b) ∩ (b ∩ c)))
8		u2lemanb 598	. . 3 ((a →2 b) ∩ b⊥ ) = (a⊥ ∩ b⊥ )
9		ancom 68	. . . 4 ((a →2 b) ∩ (b ∩ c)) = ((b ∩ c) ∩ (a →2 b))
10		lea 152	. . . . . 6 (b ∩ c) ≤ b
11		u2lem3 732	. . . . . . 7 (b →2 (a →2 b)) = 1
12	11	u2lemle2 698	. . . . . 6 b ≤ (a →2 b)
13	10, 12	letr 129	. . . . 5 (b ∩ c) ≤ (a →2 b)
14	13	df2le2 128	. . . 4 ((b ∩ c) ∩ (a →2 b)) = (b ∩ c)
15	9, 14	ax-r2 35	. . 3 ((a →2 b) ∩ (b ∩ c)) = (b ∩ c)
16	8, 15	2or 67	. 2 (((a →2 b) ∩ b⊥ ) ∪ ((a →2 b) ∩ (b ∩ c))) = ((a⊥ ∩ b⊥ ) ∪ (b ∩ c))
17	2, 7, 16	3tr 62	1 ((a →2 b) ∩ (b →1 c)) = ((a⊥ ∩ b⊥ ) ∪ (b ∩ c))

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₁ wi1 13   →₂ wi2 14

This theorem is referenced by:  orbi 824  mlaconj4 826  mhcor1 870

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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