Theorem gomaex3lem1 894

Description: Lemma for Godowski 6-var -> Mayet Example 3.

Hypothesis

Ref	Expression
gomaex3lem1.3	c =< d_|_

Assertion

Ref	Expression
gomaex3lem1	(c v (c v d)_|_) = d_|_

Proof of Theorem gomaex3lem1

Step	Hyp	Ref	Expression
1		comid 179	. . . 4 c C c
2	1	comcom2 175	. . 3 c C c_|_
3		gomaex3lem1.3	. . . 4 c =< d_|_
4	3	lecom 172	. . 3 c C d_|_
5	2, 4	fh3 453	. 2 (c v (c_|_ ^ d_|_)) = ((c v c_|_) ^ (c v d_|_))
6		anor3 82	. . 3 (c_|_ ^ d_|_) = (c v d)_|_
7	6	lor 66	. 2 (c v (c_|_ ^ d_|_)) = (c v (c v d)_|_)
8		ancom 68	. . 3 ((c v c_|_) ^ (c v d_|_)) = ((c v d_|_) ^ (c v c_|_))
9	3	df-le2 123	. . . . . 6 (c v d_|_) = d_|_
10	9	ax-r1 34	. . . . 5 d_|_ = (c v d_|_)
11		df-t 40	. . . . 5 1 = (c v c_|_)
12	10, 11	2an 72	. . . 4 (d_|_ ^ 1) = ((c v d_|_) ^ (c v c_|_))
13	12	ax-r1 34	. . 3 ((c v d_|_) ^ (c v c_|_)) = (d_|_ ^ 1)
14		an1 98	. . 3 (d_|_ ^ 1) = d_|_
15	8, 13, 14	3tr 62	. 2 ((c v c_|_) ^ (c v d_|_)) = d_|_
16	5, 7, 15	3tr2 61	1 (c v (c v d)_|_) = d_|_

Syntax hints:   = wb 1   =< wle 2  _|_wn 4   v wo 6   ^ wa 7  1wt 9

This theorem is referenced by:  gomaex3lem7 900

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

metamath.org