Theorem mhlemlem2 857

Description: Lemma for Lemma 7.1 of Kalmbach, p. 91.

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 30	. . . 4 (a ∪ b) = (b ∪ a)
2	1	ax-r5 37	. . 3 ((a ∪ b) ∪ d) = ((b ∪ a) ∪ d)
3		ax-a2 30	. . . 4 (c ∪ d) = (d ∪ c)
4	3	lor 66	. . 3 (b ∪ (c ∪ d)) = (b ∪ (d ∪ c))
5	2, 4	2an 72	. 2 (((a ∪ b) ∪ d) ∩ (b ∪ (c ∪ d))) = (((b ∪ a) ∪ d) ∩ (b ∪ (d ∪ c)))
6		mhlem.1	. . . 4 (a ∪ b) ≤ (c ∪ d)⊥
7		ax-a2 30	. . . 4 (b ∪ a) = (a ∪ b)
8		ax-a2 30	. . . . 5 (d ∪ c) = (c ∪ d)
9	8	ax-r4 36	. . . 4 (d ∪ c)⊥ = (c ∪ d)⊥
10	6, 7, 9	le3tr1 132	. . 3 (b ∪ a) ≤ (d ∪ c)⊥
11	10	mhlemlem1 856	. 2 (((b ∪ a) ∪ d) ∩ (b ∪ (d ∪ c))) = (b ∪ d)
12	5, 11	ax-r2 35	1 (((a ∪ b) ∪ d) ∩ (b ∪ (c ∪ d))) = (b ∪ d)

Syntax hints:   = wb 1   ≤ wle 2  ^⊥ wn 4   ∪ wo 6   ∩ wa 7

This theorem is referenced by:  mhlem 858

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  ax-a4 32  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  df-c2 125

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