Theorem comm0 178

Description: Commutation with 0. Kalmbach 83 p. 20. (Contributed by NM, 27-Aug-1997.)

Assertion

Ref	Expression
comm0	a C 0

Proof of Theorem comm0

Step	Hyp	Ref	Expression
1		ax-a2 31	. . . . 5 (0 ∪ a) = (a ∪ 0)
2		or0 102	. . . . 5 (a ∪ 0) = a
3	1, 2	ax-r2 36	. . . 4 (0 ∪ a) = a
4	3	ax-r1 35	. . 3 a = (0 ∪ a)
5		an0 108	. . . . 5 (a ∩ 0) = 0
6		df-f 42	. . . . . . . 8 0 = 1⊥
7	6	con2 67	. . . . . . 7 0⊥ = 1
8	7	lan 77	. . . . . 6 (a ∩ 0⊥ ) = (a ∩ 1)
9		an1 106	. . . . . 6 (a ∩ 1) = a
10	8, 9	ax-r2 36	. . . . 5 (a ∩ 0⊥ ) = a
11	5, 10	2or 72	. . . 4 ((a ∩ 0) ∪ (a ∩ 0⊥ )) = (0 ∪ a)
12	11	ax-r1 35	. . 3 (0 ∪ a) = ((a ∩ 0) ∪ (a ∩ 0⊥ ))
13	4, 12	ax-r2 36	. 2 a = ((a ∩ 0) ∪ (a ∩ 0⊥ ))
14	13	df-c1 132	1 a C 0

Syntax hints:   C wc 3  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8  0wf 9

This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a4 33  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:  wcom0  407