Theorem mhlemlem2 875

Description: Lemma for Lemma 7.1 of Kalmbach, p. 91. (Contributed by NM, 10-Mar-2002.)

Hypothesis

Ref	Expression
mhlem.1	(a ∪ b) ≤ (c ∪ d)⊥

Assertion

Ref	Expression
mhlemlem2	(((a ∪ b) ∪ d) ∩ (b ∪ (c ∪ d))) = (b ∪ d)

Proof of Theorem mhlemlem2

Step	Hyp	Ref	Expression
1		ax-a2 31	. . . 4 (a ∪ b) = (b ∪ a)
2	1	ax-r5 38	. . 3 ((a ∪ b) ∪ d) = ((b ∪ a) ∪ d)
3		ax-a2 31	. . . 4 (c ∪ d) = (d ∪ c)
4	3	lor 70	. . 3 (b ∪ (c ∪ d)) = (b ∪ (d ∪ c))
5	2, 4	2an 79	. 2 (((a ∪ b) ∪ d) ∩ (b ∪ (c ∪ d))) = (((b ∪ a) ∪ d) ∩ (b ∪ (d ∪ c)))
6		mhlem.1	. . . 4 (a ∪ b) ≤ (c ∪ d)⊥
7		ax-a2 31	. . . 4 (b ∪ a) = (a ∪ b)
8		ax-a2 31	. . . . 5 (d ∪ c) = (c ∪ d)
9	8	ax-r4 37	. . . 4 (d ∪ c)⊥ = (c ∪ d)⊥
10	6, 7, 9	le3tr1 140	. . 3 (b ∪ a) ≤ (d ∪ c)⊥
11	10	mhlemlem1 874	. 2 (((b ∪ a) ∪ d) ∩ (b ∪ (d ∪ c))) = (b ∪ d)
12	5, 11	ax-r2 36	1 (((a ∪ b) ∪ d) ∩ (b ∪ (c ∪ d))) = (b ∪ d)

Syntax hints:   = wb 1   ≤ wle 2  ^⊥ 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:  mhlem  876