Theorem imp3 841

Description: Implicational product with 3 variables. Theorem 3.20 of "Equations, states, and lattices..." paper. (Contributed by NM, 3-Mar-2000.)

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 44	. . 3 (b →1 c) = (b⊥ ∪ (b ∩ c))
2	1	lan 77	. 2 ((a →2 b) ∩ (b →1 c)) = ((a →2 b) ∩ (b⊥ ∪ (b ∩ c)))
3		u2lemc1 681	. . . 4 b C (a →2 b)
4	3	comcom3 454	. . 3 b⊥ C (a →2 b)
5		comanr1 464	. . . 4 b C (b ∩ c)
6	5	comcom3 454	. . 3 b⊥ C (b ∩ c)
7	4, 6	fh2 470	. 2 ((a →2 b) ∩ (b⊥ ∪ (b ∩ c))) = (((a →2 b) ∩ b⊥ ) ∪ ((a →2 b) ∩ (b ∩ c)))
8		u2lemanb 616	. . 3 ((a →2 b) ∩ b⊥ ) = (a⊥ ∩ b⊥ )
9		ancom 74	. . . 4 ((a →2 b) ∩ (b ∩ c)) = ((b ∩ c) ∩ (a →2 b))
10		lea 160	. . . . . 6 (b ∩ c) ≤ b
11		u2lem3 750	. . . . . . 7 (b →2 (a →2 b)) = 1
12	11	u2lemle2 716	. . . . . 6 b ≤ (a →2 b)
13	10, 12	letr 137	. . . . 5 (b ∩ c) ≤ (a →2 b)
14	13	df2le2 136	. . . 4 ((b ∩ c) ∩ (a →2 b)) = (b ∩ c)
15	9, 14	ax-r2 36	. . 3 ((a →2 b) ∩ (b ∩ c)) = (b ∩ c)
16	8, 15	2or 72	. 2 (((a →2 b) ∩ b⊥ ) ∪ ((a →2 b) ∩ (b ∩ c))) = ((a⊥ ∩ b⊥ ) ∪ (b ∩ c))
17	2, 7, 16	3tr 65	1 ((a →2 b) ∩ (b →1 c)) = ((a⊥ ∩ b⊥ ) ∪ (b ∩ c))

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →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:  orbi  842  mlaconj4  844  mhcor1  888