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) v (a ^ d_|_)) ^ (b_|_ ^ d)) = 0

Proof of Theorem marsdenlem4

Step	Hyp	Ref	Expression
1		leao3 156	. . . . . 6 (b_|_ ^ d) =< (a v b_|_)
2		oran1 83	. . . . . 6 (a v 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_|_ v d)
7		oran2 84	. . . . . 6 (a_|_ v 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) v (a ^ d_|_)) ^ (b_|_ ^ d)) = (((a_|_ ^ b) ^ (b_|_ ^ d)) v ((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)) v ((a ^ d_|_) ^ (b_|_ ^ d))) = (0 v 0)
30		or0 94	. 2 (0 v 0) = 0
31	11, 29, 30	3tr 62	1 (((a_|_ ^ b) v (a ^ d_|_)) ^ (b_|_ ^ d)) = 0

Syntax hints:   = wb 1   C wc 3  _|_wn 4   v 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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