Theorem comm0 170

Description: Commutation with 0. Kalmbach 83 p. 20.

Assertion

Ref	Expression
comm0	a C 0

Proof of Theorem comm0

Step	Hyp	Ref	Expression
1		ax-a2 30	. . . . 5 (0 ∪ a) = (a ∪ 0)
2		or0 94	. . . . 5 (a ∪ 0) = a
3	1, 2	ax-r2 35	. . . 4 (0 ∪ a) = a
4	3	ax-r1 34	. . 3 a = (0 ∪ a)
5		an0 100	. . . . 5 (a ∩ 0) = 0
6		df-f 41	. . . . . . . 8 0 = 1⊥
7	6	con2 64	. . . . . . 7 0⊥ = 1
8	7	lan 70	. . . . . 6 (a ∩ 0⊥ ) = (a ∩ 1)
9		an1 98	. . . . . 6 (a ∩ 1) = a
10	8, 9	ax-r2 35	. . . . 5 (a ∩ 0⊥ ) = a
11	5, 10	2or 67	. . . 4 ((a ∩ 0) ∪ (a ∩ 0⊥ )) = (0 ∪ a)
12	11	ax-r1 34	. . 3 (0 ∪ a) = ((a ∩ 0) ∪ (a ∩ 0⊥ ))
13	4, 12	ax-r2 35	. 2 a = ((a ∩ 0) ∪ (a ∩ 0⊥ ))
14	13	df-c1 124	1 a C 0

Syntax hints:   C wc 3  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 9  0wf 10

This theorem is referenced by:  wcom0 389

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a4 32  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37

This theorem depends on definitions:  df-a 39  df-t 40  df-f 41  df-c1 124

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