Theorem marsdenlem2 863

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
marsdenlem2	((c v d) ^ (b_|_ v c_|_)) = (((b_|_ ^ c) v (c_|_ ^ d)) v (b_|_ ^ d))

Proof of Theorem marsdenlem2

Step	Hyp	Ref	Expression
1		ancom 68	. 2 ((c v d) ^ (b_|_ v c_|_)) = ((b_|_ v c_|_) ^ (c v d))
2		comorr 176	. . . 4 c C (c v d)
3	2	comcom3 436	. . 3 c_|_ C (c v d)
4		marsden.2	. . . . 5 b C c
5	4	comcom4 437	. . . 4 b_|_ C c_|_
6	5	comcom 435	. . 3 c_|_ C b_|_
7	3, 6	fh2rc 462	. 2 ((b_|_ v c_|_) ^ (c v d)) = ((b_|_ ^ (c v d)) v (c_|_ ^ (c v d)))
8	6	comcom6 441	. . . . 5 c C b_|_
9		marsden.3	. . . . 5 c C d
10	8, 9	fh2 452	. . . 4 (b_|_ ^ (c v d)) = ((b_|_ ^ c) v (b_|_ ^ d))
11		comid 179	. . . . . . 7 c C c
12	11	comcom2 175	. . . . . 6 c C c_|_
13	12, 9	fh2 452	. . . . 5 (c_|_ ^ (c v d)) = ((c_|_ ^ c) v (c_|_ ^ d))
14		dff 93	. . . . . . . 8 0 = (c ^ c_|_)
15		ancom 68	. . . . . . . 8 (c ^ c_|_) = (c_|_ ^ c)
16	14, 15	ax-r2 35	. . . . . . 7 0 = (c_|_ ^ c)
17	16	ax-r5 37	. . . . . 6 (0 v (c_|_ ^ d)) = ((c_|_ ^ c) v (c_|_ ^ d))
18	17	ax-r1 34	. . . . 5 ((c_|_ ^ c) v (c_|_ ^ d)) = (0 v (c_|_ ^ d))
19		or0r 95	. . . . 5 (0 v (c_|_ ^ d)) = (c_|_ ^ d)
20	13, 18, 19	3tr 62	. . . 4 (c_|_ ^ (c v d)) = (c_|_ ^ d)
21	10, 20	2or 67	. . 3 ((b_|_ ^ (c v d)) v (c_|_ ^ (c v d))) = (((b_|_ ^ c) v (b_|_ ^ d)) v (c_|_ ^ d))
22		or32 75	. . 3 (((b_|_ ^ c) v (b_|_ ^ d)) v (c_|_ ^ d)) = (((b_|_ ^ c) v (c_|_ ^ d)) v (b_|_ ^ d))
23	21, 22	ax-r2 35	. 2 ((b_|_ ^ (c v d)) v (c_|_ ^ (c v d))) = (((b_|_ ^ c) v (c_|_ ^ d)) v (b_|_ ^ d))
24	1, 7, 23	3tr 62	1 ((c v d) ^ (b_|_ v c_|_)) = (((b_|_ ^ c) v (c_|_ ^ d)) v (b_|_ ^ d))

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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