Theorem comm1 179

Description: Commutation with 1. Kalmbach 83 p. 20. (Contributed by NM, 27-Aug-1997.)

Assertion

Ref	Expression
comm1	1 C a

Proof of Theorem comm1

Step	Hyp	Ref	Expression
1		df-t 41	. . 3 1 = (a ∪ a⊥ )
2		ancom 74	. . . . . 6 (1 ∩ a) = (a ∩ 1)
3		an1 106	. . . . . 6 (a ∩ 1) = a
4	2, 3	ax-r2 36	. . . . 5 (1 ∩ a) = a
5		ancom 74	. . . . . 6 (1 ∩ a⊥ ) = (a⊥ ∩ 1)
6		an1 106	. . . . . 6 (a⊥ ∩ 1) = a⊥
7	5, 6	ax-r2 36	. . . . 5 (1 ∩ a⊥ ) = a⊥
8	4, 7	2or 72	. . . 4 ((1 ∩ a) ∪ (1 ∩ a⊥ )) = (a ∪ a⊥ )
9	8	ax-r1 35	. . 3 (a ∪ a⊥ ) = ((1 ∩ a) ∪ (1 ∩ a⊥ ))
10	1, 9	ax-r2 36	. 2 1 = ((1 ∩ a) ∪ (1 ∩ a⊥ ))
11	10	df-c1 132	1 1 C a

Syntax hints:   C wc 3  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8

This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38

This theorem depends on definitions:  df-a 40  df-t 41  df-f 42  df-c1 132

This theorem is referenced by:  wcom1  408