Theorem marsdenlem4 865

Description: Lemma for Marsden-Herman distributive law.

Hypotheses

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

Assertion

Ref	Expression
marsdenlem4	(((a⊥ ∩ b) ∪ (a ∩ d⊥ )) ∩ (b⊥ ∩ d)) = 0

Proof of Theorem marsdenlem4

Step	Hyp	Ref	Expression
1		leao3 156	. . . . . 6 (b⊥ ∩ d) ≤ (a ∪ b⊥ )
2		oran1 83	. . . . . 6 (a ∪ b⊥ ) = (a⊥ ∩ b)⊥
3	1, 2	lbtr 131	. . . . 5 (b⊥ ∩ d) ≤ (a⊥ ∩ b)⊥
4	3	lecom 172	. . . 4 (b⊥ ∩ d) C (a⊥ ∩ b)⊥
5	4	comcom7 442	. . 3 (b⊥ ∩ d) C (a⊥ ∩ b)
6		leao4 157	. . . . . 6 (b⊥ ∩ d) ≤ (a⊥ ∪ d)
7		oran2 84	. . . . . 6 (a⊥ ∪ d) = (a ∩ d⊥ )⊥
8	6, 7	lbtr 131	. . . . 5 (b⊥ ∩ d) ≤ (a ∩ d⊥ )⊥
9	8	lecom 172	. . . 4 (b⊥ ∩ d) C (a ∩ d⊥ )⊥
10	9	comcom7 442	. . 3 (b⊥ ∩ d) C (a ∩ d⊥ )
11	5, 10	fh1r 455	. 2 (((a⊥ ∩ b) ∪ (a ∩ d⊥ )) ∩ (b⊥ ∩ d)) = (((a⊥ ∩ b) ∩ (b⊥ ∩ d)) ∪ ((a ∩ d⊥ ) ∩ (b⊥ ∩ d)))
12		ancom 68	. . . . 5 (b⊥ ∩ d) = (d ∩ b⊥ )
13	12	lan 70	. . . 4 ((a⊥ ∩ b) ∩ (b⊥ ∩ d)) = ((a⊥ ∩ b) ∩ (d ∩ b⊥ ))
14		an4 78	. . . 4 ((a⊥ ∩ b) ∩ (d ∩ b⊥ )) = ((a⊥ ∩ d) ∩ (b ∩ b⊥ ))
15		dff 93	. . . . . . 7 0 = (b ∩ b⊥ )
16	15	lan 70	. . . . . 6 ((a⊥ ∩ d) ∩ 0) = ((a⊥ ∩ d) ∩ (b ∩ b⊥ ))
17	16	ax-r1 34	. . . . 5 ((a⊥ ∩ d) ∩ (b ∩ b⊥ )) = ((a⊥ ∩ d) ∩ 0)
18		an0 100	. . . . 5 ((a⊥ ∩ d) ∩ 0) = 0
19	17, 18	ax-r2 35	. . . 4 ((a⊥ ∩ d) ∩ (b ∩ b⊥ )) = 0
20	13, 14, 19	3tr 62	. . 3 ((a⊥ ∩ b) ∩ (b⊥ ∩ d)) = 0
21		an4 78	. . . 4 ((a ∩ d⊥ ) ∩ (b⊥ ∩ d)) = ((a ∩ b⊥ ) ∩ (d⊥ ∩ d))
22		ancom 68	. . . . . 6 (d⊥ ∩ d) = (d ∩ d⊥ )
23		dff 93	. . . . . . 7 0 = (d ∩ d⊥ )
24	23	ax-r1 34	. . . . . 6 (d ∩ d⊥ ) = 0
25	22, 24	ax-r2 35	. . . . 5 (d⊥ ∩ d) = 0
26	25	lan 70	. . . 4 ((a ∩ b⊥ ) ∩ (d⊥ ∩ d)) = ((a ∩ b⊥ ) ∩ 0)
27		an0 100	. . . 4 ((a ∩ b⊥ ) ∩ 0) = 0
28	21, 26, 27	3tr 62	. . 3 ((a ∩ d⊥ ) ∩ (b⊥ ∩ d)) = 0
29	20, 28	2or 67	. 2 (((a⊥ ∩ b) ∩ (b⊥ ∩ d)) ∪ ((a ∩ d⊥ ) ∩ (b⊥ ∩ d))) = (0 ∪ 0)
30		or0 94	. 2 (0 ∪ 0) = 0
31	11, 29, 30	3tr 62	1 (((a⊥ ∩ b) ∪ (a ∩ d⊥ )) ∩ (b⊥ ∩ d)) = 0

Syntax hints:   = wb 1   C wc 3  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  0wf 10

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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