Theorem com3iia 1102

Description: The dual of com3ii 457. (Contributed by Roy F. Longton, 2-Jul-2005.)

Hypothesis

Ref	Expression
com3iia.1	a C b

Assertion

Ref	Expression
com3iia	(a ∪ (a⊥ ∩ b)) = (a ∪ b)

Proof of Theorem com3iia

Step	Hyp	Ref	Expression
1		comid 187	. . . 4 a C a
2	1	comcom2 183	. . 3 a C a⊥
3		com3iia.1	. . 3 a C b
4	2, 3	fh3 471	. 2 (a ∪ (a⊥ ∩ b)) = ((a ∪ a⊥ ) ∩ (a ∪ b))
5		lear 161	. . 3 ((a ∪ a⊥ ) ∩ (a ∪ b)) ≤ (a ∪ b)
6		ax-a4 33	. . . . 5 ((a ∪ b) ∪ (a ∪ a⊥ )) = (a ∪ a⊥ )
7	6	df-le1 130	. . . 4 (a ∪ b) ≤ (a ∪ a⊥ )
8		leid 148	. . . 4 (a ∪ b) ≤ (a ∪ b)
9	7, 8	ler2an 173	. . 3 (a ∪ b) ≤ ((a ∪ a⊥ ) ∩ (a ∪ b))
10	5, 9	lebi 145	. 2 ((a ∪ a⊥ ) ∩ (a ∪ b)) = (a ∪ b)
11	4, 10	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-a4 33  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: (None)