Theorem mhlem2 878

Description: Lemma for Marsden-Herman distributive law. (Contributed by NM, 10-Mar-2002.)

Hypotheses

Ref	Expression
mh.1	a C c
mh.2	a C d
mh.3	b C c
mh.4	b C d

Assertion

Ref	Expression
mhlem2	(((a ∪ c) ∩ (c⊥ ∪ b⊥ )) ∩ ((b ∪ d) ∩ (a⊥ ∪ d⊥ ))) = (((a ∩ c⊥ ) ∩ (b ∩ d⊥ )) ∪ ((c ∩ b⊥ ) ∩ (d ∩ a⊥ )))

Proof of Theorem mhlem2

Step	Hyp	Ref	Expression
1		mh.1	. . . 4 a C c
2		mh.3	. . . . 5 b C c
3	2	comcom3 454	. . . 4 b⊥ C c
4	1, 3	mhlem1 877	. . 3 ((a ∪ c) ∩ (c⊥ ∪ b⊥ )) = ((a ∩ c⊥ ) ∪ (c ∩ b⊥ ))
5		ax-a2 31	. . . . 5 (a⊥ ∪ d⊥ ) = (d⊥ ∪ a⊥ )
6	5	lan 77	. . . 4 ((b ∪ d) ∩ (a⊥ ∪ d⊥ )) = ((b ∪ d) ∩ (d⊥ ∪ a⊥ ))
7		mh.4	. . . . 5 b C d
8		mh.2	. . . . . 6 a C d
9	8	comcom3 454	. . . . 5 a⊥ C d
10	7, 9	mhlem1 877	. . . 4 ((b ∪ d) ∩ (d⊥ ∪ a⊥ )) = ((b ∩ d⊥ ) ∪ (d ∩ a⊥ ))
11	6, 10	ax-r2 36	. . 3 ((b ∪ d) ∩ (a⊥ ∪ d⊥ )) = ((b ∩ d⊥ ) ∪ (d ∩ a⊥ ))
12	4, 11	2an 79	. 2 (((a ∪ c) ∩ (c⊥ ∪ b⊥ )) ∩ ((b ∪ d) ∩ (a⊥ ∪ d⊥ ))) = (((a ∩ c⊥ ) ∪ (c ∩ b⊥ )) ∩ ((b ∩ d⊥ ) ∪ (d ∩ a⊥ )))
13		leao2 163	. . . . . 6 (a ∩ c⊥ ) ≤ (c⊥ ∪ b)
14		leao3 164	. . . . . 6 (a ∩ c⊥ ) ≤ (d⊥ ∪ a)
15	13, 14	ler2an 173	. . . . 5 (a ∩ c⊥ ) ≤ ((c⊥ ∪ b) ∩ (d⊥ ∪ a))
16		leao3 164	. . . . . 6 (b ∩ d⊥ ) ≤ (c⊥ ∪ b)
17		leao2 163	. . . . . 6 (b ∩ d⊥ ) ≤ (d⊥ ∪ a)
18	16, 17	ler2an 173	. . . . 5 (b ∩ d⊥ ) ≤ ((c⊥ ∪ b) ∩ (d⊥ ∪ a))
19	15, 18	lel2or 170	. . . 4 ((a ∩ c⊥ ) ∪ (b ∩ d⊥ )) ≤ ((c⊥ ∪ b) ∩ (d⊥ ∪ a))
20		oran2 92	. . . . . 6 (c⊥ ∪ b) = (c ∩ b⊥ )⊥
21		oran2 92	. . . . . 6 (d⊥ ∪ a) = (d ∩ a⊥ )⊥
22	20, 21	2an 79	. . . . 5 ((c⊥ ∪ b) ∩ (d⊥ ∪ a)) = ((c ∩ b⊥ )⊥ ∩ (d ∩ a⊥ )⊥ )
23		anor3 90	. . . . 5 ((c ∩ b⊥ )⊥ ∩ (d ∩ a⊥ )⊥ ) = ((c ∩ b⊥ ) ∪ (d ∩ a⊥ ))⊥
24	22, 23	ax-r2 36	. . . 4 ((c⊥ ∪ b) ∩ (d⊥ ∪ a)) = ((c ∩ b⊥ ) ∪ (d ∩ a⊥ ))⊥
25	19, 24	lbtr 139	. . 3 ((a ∩ c⊥ ) ∪ (b ∩ d⊥ )) ≤ ((c ∩ b⊥ ) ∪ (d ∩ a⊥ ))⊥
26	25	mhlem 876	. 2 (((a ∩ c⊥ ) ∪ (c ∩ b⊥ )) ∩ ((b ∩ d⊥ ) ∪ (d ∩ a⊥ ))) = (((a ∩ c⊥ ) ∩ (b ∩ d⊥ )) ∪ ((c ∩ b⊥ ) ∩ (d ∩ a⊥ )))
27	12, 26	ax-r2 36	1 (((a ∪ c) ∩ (c⊥ ∪ b⊥ )) ∩ ((b ∪ d) ∩ (a⊥ ∪ d⊥ ))) = (((a ∩ c⊥ ) ∩ (b ∩ d⊥ )) ∪ ((c ∩ b⊥ ) ∩ (d ∩ a⊥ )))

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:  mh  879