Theorem com3i 449

Description: Lemma 3(i) of Kalmbach 83 p. 23.

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 80	. . . . . . . 8 (a ∩ b⊥ ) = (a⊥ ∪ b)⊥
2	1	con2 64	. . . . . . 7 (a ∩ b⊥ )⊥ = (a⊥ ∪ b)
3	2	ran 71	. . . . . 6 ((a ∩ b⊥ )⊥ ∩ a) = ((a⊥ ∪ b) ∩ a)
4		ancom 68	. . . . . 6 ((a⊥ ∪ b) ∩ a) = (a ∩ (a⊥ ∪ b))
5	3, 4	ax-r2 35	. . . . 5 ((a ∩ b⊥ )⊥ ∩ a) = (a ∩ (a⊥ ∪ b))
6		com3i.1	. . . . 5 (a ∩ (a⊥ ∪ b)) = (a ∩ b)
7	5, 6	ax-r2 35	. . . 4 ((a ∩ b⊥ )⊥ ∩ a) = (a ∩ b)
8	7	lor 66	. . 3 ((a ∩ b⊥ ) ∪ ((a ∩ b⊥ )⊥ ∩ a)) = ((a ∩ b⊥ ) ∪ (a ∩ b))
9		lea 152	. . . 4 (a ∩ b⊥ ) ≤ a
10	9	oml2 433	. . 3 ((a ∩ b⊥ ) ∪ ((a ∩ b⊥ )⊥ ∩ a)) = a
11		ax-a2 30	. . 3 ((a ∩ b⊥ ) ∪ (a ∩ b)) = ((a ∩ b) ∪ (a ∩ b⊥ ))
12	8, 10, 11	3tr2 61	. 2 a = ((a ∩ b) ∪ (a ∩ b⊥ ))
13	12	df-c1 124	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 29  ax-a2 30  ax-a3 31  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37  ax-r3 421

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41  df-le1 122  df-le2 123  df-c1 124

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