Theorem com3ii 457

Description: Lemma 3(ii) of Kalmbach 83 p. 23. (Contributed by NM, 27-Aug-1997.)

Hypothesis

Ref	Expression
comcom.1	a C b

Assertion

Ref	Expression
com3ii	(a ∩ (a⊥ ∪ b)) = (a ∩ b)

Proof of Theorem com3ii

Step	Hyp	Ref	Expression
1		comcom.1	. . . . . 6 a C b
2	1	comcom 453	. . . . 5 b C a
3	2	comd 456	. . . 4 b = ((b ∪ a) ∩ (b ∪ a⊥ ))
4	3	lan 77	. . 3 (a ∩ b) = (a ∩ ((b ∪ a) ∩ (b ∪ a⊥ )))
5		anass 76	. . . . 5 ((a ∩ (b ∪ a)) ∩ (b ∪ a⊥ )) = (a ∩ ((b ∪ a) ∩ (b ∪ a⊥ )))
6	5	ax-r1 35	. . . 4 (a ∩ ((b ∪ a) ∩ (b ∪ a⊥ ))) = ((a ∩ (b ∪ a)) ∩ (b ∪ a⊥ ))
7		ax-a2 31	. . . . . . 7 (b ∪ a) = (a ∪ b)
8	7	lan 77	. . . . . 6 (a ∩ (b ∪ a)) = (a ∩ (a ∪ b))
9		anabs 121	. . . . . 6 (a ∩ (a ∪ b)) = a
10	8, 9	ax-r2 36	. . . . 5 (a ∩ (b ∪ a)) = a
11		ax-a2 31	. . . . 5 (b ∪ a⊥ ) = (a⊥ ∪ b)
12	10, 11	2an 79	. . . 4 ((a ∩ (b ∪ a)) ∩ (b ∪ a⊥ )) = (a ∩ (a⊥ ∪ b))
13	6, 12	ax-r2 36	. . 3 (a ∩ ((b ∪ a) ∩ (b ∪ a⊥ ))) = (a ∩ (a⊥ ∪ b))
14	4, 13	ax-r2 36	. 2 (a ∩ b) = (a ∩ (a⊥ ∪ b))
15	14	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  ax-r3 439

This proof depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is used by:  fh1  469  fh2  470