Theorem marsdenlem3 864

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
marsdenlem3	(((b⊥ ∩ c) ∪ (c⊥ ∩ d)) ∩ (b ∩ d⊥ )) = 0

Proof of Theorem marsdenlem3

Step	Hyp	Ref	Expression
1		lea 152	. . . . . . . 8 (b ∩ d⊥ ) ≤ b
2	1	lecon 146	. . . . . . 7 b⊥ ≤ (b ∩ d⊥ )⊥
3	2	lel 143	. . . . . 6 (b⊥ ∩ c) ≤ (b ∩ d⊥ )⊥
4	3	lecom 172	. . . . 5 (b⊥ ∩ c) C (b ∩ d⊥ )⊥
5	4	comcom7 442	. . . 4 (b⊥ ∩ c) C (b ∩ d⊥ )
6	5	comcom 435	. . 3 (b ∩ d⊥ ) C (b⊥ ∩ c)
7		lear 153	. . . . . . . 8 (c⊥ ∩ d) ≤ d
8	7	lerr 142	. . . . . . 7 (c⊥ ∩ d) ≤ (b⊥ ∪ d)
9		oran2 84	. . . . . . 7 (b⊥ ∪ d) = (b ∩ d⊥ )⊥
10	8, 9	lbtr 131	. . . . . 6 (c⊥ ∩ d) ≤ (b ∩ d⊥ )⊥
11	10	lecom 172	. . . . 5 (c⊥ ∩ d) C (b ∩ d⊥ )⊥
12	11	comcom7 442	. . . 4 (c⊥ ∩ d) C (b ∩ d⊥ )
13	12	comcom 435	. . 3 (b ∩ d⊥ ) C (c⊥ ∩ d)
14	6, 13	fh1r 455	. 2 (((b⊥ ∩ c) ∪ (c⊥ ∩ d)) ∩ (b ∩ d⊥ )) = (((b⊥ ∩ c) ∩ (b ∩ d⊥ )) ∪ ((c⊥ ∩ d) ∩ (b ∩ d⊥ )))
15		an4 78	. . . 4 ((b⊥ ∩ c) ∩ (b ∩ d⊥ )) = ((b⊥ ∩ b) ∩ (c ∩ d⊥ ))
16		ancom 68	. . . . . 6 (b⊥ ∩ b) = (b ∩ b⊥ )
17		dff 93	. . . . . . 7 0 = (b ∩ b⊥ )
18	17	ax-r1 34	. . . . . 6 (b ∩ b⊥ ) = 0
19	16, 18	ax-r2 35	. . . . 5 (b⊥ ∩ b) = 0
20	19	ran 71	. . . 4 ((b⊥ ∩ b) ∩ (c ∩ d⊥ )) = (0 ∩ (c ∩ d⊥ ))
21		an0r 101	. . . 4 (0 ∩ (c ∩ d⊥ )) = 0
22	15, 20, 21	3tr 62	. . 3 ((b⊥ ∩ c) ∩ (b ∩ d⊥ )) = 0
23		an4 78	. . . 4 ((c⊥ ∩ d) ∩ (b ∩ d⊥ )) = ((c⊥ ∩ b) ∩ (d ∩ d⊥ ))
24		dff 93	. . . . . 6 0 = (d ∩ d⊥ )
25	24	ax-r1 34	. . . . 5 (d ∩ d⊥ ) = 0
26	25	lan 70	. . . 4 ((c⊥ ∩ b) ∩ (d ∩ d⊥ )) = ((c⊥ ∩ b) ∩ 0)
27		an0 100	. . . 4 ((c⊥ ∩ b) ∩ 0) = 0
28	23, 26, 27	3tr 62	. . 3 ((c⊥ ∩ d) ∩ (b ∩ d⊥ )) = 0
29	22, 28	2or 67	. 2 (((b⊥ ∩ c) ∩ (b ∩ d⊥ )) ∪ ((c⊥ ∩ d) ∩ (b ∩ d⊥ ))) = (0 ∪ 0)
30		or0 94	. 2 (0 ∪ 0) = 0
31	14, 29, 30	3tr 62	1 (((b⊥ ∩ c) ∪ (c⊥ ∩ 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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