Theorem com3i 467

Description: Lemma 3(i) of Kalmbach 83 p. 23. (Contributed by NM, 28-Aug-1997.)

Hypothesis

Ref	Expression
com3i.1	(a ∩ (a⊥ ∪ b)) = (a ∩ b)

Assertion

Ref	Expression
com3i	a C b

Proof of Theorem com3i

Step	Hyp	Ref	Expression
1		anor1 88	. . . . . . . 8 (a ∩ b⊥ ) = (a⊥ ∪ b)⊥
2	1	con2 67	. . . . . . 7 (a ∩ b⊥ )⊥ = (a⊥ ∪ b)
3	2	ran 78	. . . . . 6 ((a ∩ b⊥ )⊥ ∩ a) = ((a⊥ ∪ b) ∩ a)
4		ancom 74	. . . . . 6 ((a⊥ ∪ b) ∩ a) = (a ∩ (a⊥ ∪ b))
5	3, 4	ax-r2 36	. . . . 5 ((a ∩ b⊥ )⊥ ∩ a) = (a ∩ (a⊥ ∪ b))
6		com3i.1	. . . . 5 (a ∩ (a⊥ ∪ b)) = (a ∩ b)
7	5, 6	ax-r2 36	. . . 4 ((a ∩ b⊥ )⊥ ∩ a) = (a ∩ b)
8	7	lor 70	. . 3 ((a ∩ b⊥ ) ∪ ((a ∩ b⊥ )⊥ ∩ a)) = ((a ∩ b⊥ ) ∪ (a ∩ b))
9		lea 160	. . . 4 (a ∩ b⊥ ) ≤ a
10	9	oml2 451	. . 3 ((a ∩ b⊥ ) ∪ ((a ∩ b⊥ )⊥ ∩ a)) = a
11		ax-a2 31	. . 3 ((a ∩ b⊥ ) ∪ (a ∩ b)) = ((a ∩ b) ∪ (a ∩ b⊥ ))
12	8, 10, 11	3tr2 64	. 2 a = ((a ∩ b) ∪ (a ∩ b⊥ ))
13	12	df-c1 132	1 a C 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

This theorem is referenced by: (None)