Theorem wwcom3ii 215

Description: Lemma 3(ii) (weak) of Kalmbach 83 p. 23. (Contributed by NM, 2-Sep-1997.)

Hypothesis

Ref	Expression
wwcom3ii.1	b⊥ C a

Assertion

Ref	Expression
wwcom3ii	(a ∩ (a⊥ ∪ b)) = (a ∩ b)

Proof of Theorem wwcom3ii

Step	Hyp	Ref	Expression
1		wwcom3ii.1	. . . . 5 b⊥ C a
2	1	wwcomd 214	. . . 4 b = ((b ∪ a) ∩ (b ∪ a⊥ ))
3	2	lan 77	. . 3 (a ∩ b) = (a ∩ ((b ∪ a) ∩ (b ∪ a⊥ )))
4		anass 76	. . . . 5 ((a ∩ (b ∪ a)) ∩ (b ∪ a⊥ )) = (a ∩ ((b ∪ a) ∩ (b ∪ a⊥ )))
5	4	ax-r1 35	. . . 4 (a ∩ ((b ∪ a) ∩ (b ∪ a⊥ ))) = ((a ∩ (b ∪ a)) ∩ (b ∪ a⊥ ))
6		ax-a2 31	. . . . . . 7 (b ∪ a) = (a ∪ b)
7	6	lan 77	. . . . . 6 (a ∩ (b ∪ a)) = (a ∩ (a ∪ b))
8		anabs 121	. . . . . 6 (a ∩ (a ∪ b)) = a
9	7, 8	ax-r2 36	. . . . 5 (a ∩ (b ∪ a)) = a
10		ax-a2 31	. . . . 5 (b ∪ a⊥ ) = (a⊥ ∪ b)
11	9, 10	2an 79	. . . 4 ((a ∩ (b ∪ a)) ∩ (b ∪ a⊥ )) = (a ∩ (a⊥ ∪ b))
12	5, 11	ax-r2 36	. . 3 (a ∩ ((b ∪ a) ∩ (b ∪ a⊥ ))) = (a ∩ (a⊥ ∪ b))
13	3, 12	ax-r2 36	. 2 (a ∩ b) = (a ∩ (a⊥ ∪ b))
14	13	ax-r1 35	1 (a ∩ (a⊥ ∪ b)) = (a ∩ b)

This proof depends on syntax axioms:   = wb 1   C wc 3  ^⊥ wn 4   ∪ wo 6   ∩ wa 7

This proof depends on axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38

This proof depends on definitions:  df-a 40  df-c2 133

This theorem is used by:  wwfh1  216  wwfh2  217