Theorem comd 456

Description: Commutation dual. Kalmbach 83 p. 23.

Hypothesis

Ref	Expression
comcom.1	a C b

Assertion

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

Proof of Theorem comd

Step	Hyp	Ref	Expression
1		comcom.1	. . . . 5 a C b
2	1	comcom4 455	. . . 4 a⊥ C b⊥
3	2	df-c2 133	. . 3 a⊥ = ((a⊥ ∩ b⊥ ) ∪ (a⊥ ∩ b⊥ ⊥ ))
4	3	con3 68	. 2 a = ((a⊥ ∩ b⊥ ) ∪ (a⊥ ∩ b⊥ ⊥ ))⊥
5		oran 87	. . . 4 ((a⊥ ∩ b⊥ ) ∪ (a⊥ ∩ b⊥ ⊥ )) = ((a⊥ ∩ b⊥ )⊥ ∩ (a⊥ ∩ b⊥ ⊥ )⊥ )⊥
6	5	con2 67	. . 3 ((a⊥ ∩ b⊥ ) ∪ (a⊥ ∩ b⊥ ⊥ ))⊥ = ((a⊥ ∩ b⊥ )⊥ ∩ (a⊥ ∩ b⊥ ⊥ )⊥ )
7		oran 87	. . . . 5 (a ∪ b) = (a⊥ ∩ b⊥ )⊥
8		oran 87	. . . . 5 (a ∪ b⊥ ) = (a⊥ ∩ b⊥ ⊥ )⊥
9	7, 8	2an 79	. . . 4 ((a ∪ b) ∩ (a ∪ b⊥ )) = ((a⊥ ∩ b⊥ )⊥ ∩ (a⊥ ∩ b⊥ ⊥ )⊥ )
10	9	ax-r1 35	. . 3 ((a⊥ ∩ b⊥ )⊥ ∩ (a⊥ ∩ b⊥ ⊥ )⊥ ) = ((a ∪ b) ∩ (a ∪ b⊥ ))
11	6, 10	ax-r2 36	. 2 ((a⊥ ∩ b⊥ ) ∪ (a⊥ ∩ b⊥ ⊥ ))⊥ = ((a ∪ b) ∩ (a ∪ b⊥ ))
12	4, 11	ax-r2 36	1 a = ((a ∪ b) ∩ (a ∪ b⊥ ))

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

This theorem was proved from 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 theorem 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 referenced by:  com3ii  457  gsth  489