Theorem i1abs 801

Description: An absorption law for →₁ . (Contributed by NM, 21-Feb-2002.)

Assertion

Ref	Expression
i1abs	((a →1 b)⊥ ∪ (a ∩ b)) = a

Proof of Theorem i1abs

Step	Hyp	Ref	Expression
1		ud1lem0c 277	. . 3 (a →1 b)⊥ = (a ∩ (a⊥ ∪ b⊥ ))
2	1	ax-r5 38	. 2 ((a →1 b)⊥ ∪ (a ∩ b)) = ((a ∩ (a⊥ ∪ b⊥ )) ∪ (a ∩ b))
3		comanr1 464	. . 3 a C (a ∩ b)
4		comorr 184	. . . 4 a⊥ C (a⊥ ∪ b⊥ )
5	4	comcom6 459	. . 3 a C (a⊥ ∪ b⊥ )
6	3, 5	fh4r 476	. 2 ((a ∩ (a⊥ ∪ b⊥ )) ∪ (a ∩ b)) = ((a ∪ (a ∩ b)) ∩ ((a⊥ ∪ b⊥ ) ∪ (a ∩ b)))
7		orabs 120	. . . 4 (a ∪ (a ∩ b)) = a
8		df-a 40	. . . . . 6 (a ∩ b) = (a⊥ ∪ b⊥ )⊥
9	8	lor 70	. . . . 5 ((a⊥ ∪ b⊥ ) ∪ (a ∩ b)) = ((a⊥ ∪ b⊥ ) ∪ (a⊥ ∪ b⊥ )⊥ )
10		df-t 41	. . . . . 6 1 = ((a⊥ ∪ b⊥ ) ∪ (a⊥ ∪ b⊥ )⊥ )
11	10	ax-r1 35	. . . . 5 ((a⊥ ∪ b⊥ ) ∪ (a⊥ ∪ b⊥ )⊥ ) = 1
12	9, 11	ax-r2 36	. . . 4 ((a⊥ ∪ b⊥ ) ∪ (a ∩ b)) = 1
13	7, 12	2an 79	. . 3 ((a ∪ (a ∩ b)) ∩ ((a⊥ ∪ b⊥ ) ∪ (a ∩ b))) = (a ∩ 1)
14		an1 106	. . 3 (a ∩ 1) = a
15	13, 14	ax-r2 36	. 2 ((a ∪ (a ∩ b)) ∩ ((a⊥ ∪ b⊥ ) ∪ (a ∩ b))) = a
16	2, 6, 15	3tr 65	1 ((a →1 b)⊥ ∪ (a ∩ b)) = a

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8   →₁ wi1 12

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-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is referenced by:  cancellem  891