Theorem gomaex4 880

Description: Proof of Mayet Example 4 from 4-variable Godowski equation. R. Mayet, "Equational bases for some varieties of orthomodular lattices related to states," Algebra Universalis 23 (1986), 167-195.

Hypotheses

Ref	Expression
go2n4.1	a =< b_|_
go2n4.2	b =< c_|_
go2n4.3	c =< d_|_
go2n4.4	d =< e_|_
go2n4.5	e =< f_|_
go2n4.6	f =< g_|_
go2n4.7	g =< h_|_
go2n4.8	h =< a_|_
gomaex4.9	(((a ->2 g) ^ (g ->2 e)) ^ ((e ->2 c) ^ (c ->2 a))) =< (g ->2 a)
gomaex4.10	(((e ->2 c) ^ (c ->2 a)) ^ ((a ->2 g) ^ (g ->2 e))) =< (c ->2 e)

Assertion

Ref	Expression
gomaex4	((((a v b) ^ (c v d)) ^ ((e v f) ^ (g v h))) ^ ((a v h) ->1 (d v e)_|_)) = 0

Proof of Theorem gomaex4

Step	Hyp	Ref	Expression
1		go2n4.7	. . . . . . 7 g =< h_|_
2		go2n4.8	. . . . . . 7 h =< a_|_
3		go2n4.1	. . . . . . 7 a =< b_|_
4		go2n4.2	. . . . . . 7 b =< c_|_
5		go2n4.3	. . . . . . 7 c =< d_|_
6		go2n4.4	. . . . . . 7 d =< e_|_
7		go2n4.5	. . . . . . 7 e =< f_|_
8		go2n4.6	. . . . . . 7 f =< g_|_
9		gomaex4.9	. . . . . . 7 (((a ->2 g) ^ (g ->2 e)) ^ ((e ->2 c) ^ (c ->2 a))) =< (g ->2 a)
10	1, 2, 3, 4, 5, 6, 7, 8, 9	go2n4 879	. . . . . 6 (((g v h) ^ (a v b)) ^ ((c v d) ^ (e v f))) =< (h v a)
11		an4 78	. . . . . . 7 (((a v b) ^ (c v d)) ^ ((e v f) ^ (g v h))) = (((a v b) ^ (e v f)) ^ ((c v d) ^ (g v h)))
12		ancom 68	. . . . . . . 8 (((a v b) ^ (e v f)) ^ ((c v d) ^ (g v h))) = (((c v d) ^ (g v h)) ^ ((a v b) ^ (e v f)))
13		ancom 68	. . . . . . . . 9 ((c v d) ^ (g v h)) = ((g v h) ^ (c v d))
14	13	ran 71	. . . . . . . 8 (((c v d) ^ (g v h)) ^ ((a v b) ^ (e v f))) = (((g v h) ^ (c v d)) ^ ((a v b) ^ (e v f)))
15	12, 14	ax-r2 35	. . . . . . 7 (((a v b) ^ (e v f)) ^ ((c v d) ^ (g v h))) = (((g v h) ^ (c v d)) ^ ((a v b) ^ (e v f)))
16		an4 78	. . . . . . 7 (((g v h) ^ (c v d)) ^ ((a v b) ^ (e v f))) = (((g v h) ^ (a v b)) ^ ((c v d) ^ (e v f)))
17	11, 15, 16	3tr 62	. . . . . 6 (((a v b) ^ (c v d)) ^ ((e v f) ^ (g v h))) = (((g v h) ^ (a v b)) ^ ((c v d) ^ (e v f)))
18		ax-a2 30	. . . . . 6 (a v h) = (h v a)
19	10, 17, 18	le3tr1 132	. . . . 5 (((a v b) ^ (c v d)) ^ ((e v f) ^ (g v h))) =< (a v h)
20		ancom 68	. . . . . . . . 9 ((e v f) ^ (g v h)) = ((g v h) ^ (e v f))
21	20	lan 70	. . . . . . . 8 (((a v b) ^ (c v d)) ^ ((e v f) ^ (g v h))) = (((a v b) ^ (c v d)) ^ ((g v h) ^ (e v f)))
22		an4 78	. . . . . . . 8 (((a v b) ^ (c v d)) ^ ((g v h) ^ (e v f))) = (((a v b) ^ (g v h)) ^ ((c v d) ^ (e v f)))
23		ancom 68	. . . . . . . . 9 ((c v d) ^ (e v f)) = ((e v f) ^ (c v d))
24	23	lan 70	. . . . . . . 8 (((a v b) ^ (g v h)) ^ ((c v d) ^ (e v f))) = (((a v b) ^ (g v h)) ^ ((e v f) ^ (c v d)))
25	21, 22, 24	3tr 62	. . . . . . 7 (((a v b) ^ (c v d)) ^ ((e v f) ^ (g v h))) = (((a v b) ^ (g v h)) ^ ((e v f) ^ (c v d)))
26		ancom 68	. . . . . . . 8 ((a v b) ^ (g v h)) = ((g v h) ^ (a v b))
27		ancom 68	. . . . . . . 8 ((e v f) ^ (c v d)) = ((c v d) ^ (e v f))
28	26, 27	2an 72	. . . . . . 7 (((a v b) ^ (g v h)) ^ ((e v f) ^ (c v d))) = (((g v h) ^ (a v b)) ^ ((c v d) ^ (e v f)))
29		ancom 68	. . . . . . 7 (((g v h) ^ (a v b)) ^ ((c v d) ^ (e v f))) = (((c v d) ^ (e v f)) ^ ((g v h) ^ (a v b)))
30	25, 28, 29	3tr 62	. . . . . 6 (((a v b) ^ (c v d)) ^ ((e v f) ^ (g v h))) = (((c v d) ^ (e v f)) ^ ((g v h) ^ (a v b)))
31		gomaex4.10	. . . . . . 7 (((e ->2 c) ^ (c ->2 a)) ^ ((a ->2 g) ^ (g ->2 e))) =< (c ->2 e)
32	5, 6, 7, 8, 1, 2, 3, 4, 31	go2n4 879	. . . . . 6 (((c v d) ^ (e v f)) ^ ((g v h) ^ (a v b))) =< (d v e)
33	30, 32	bltr 130	. . . . 5 (((a v b) ^ (c v d)) ^ ((e v f) ^ (g v h))) =< (d v e)
34	19, 33	ler2an 165	. . . 4 (((a v b) ^ (c v d)) ^ ((e v f) ^ (g v h))) =< ((a v h) ^ (d v e))
35	34	leran 145	. . 3 ((((a v b) ^ (c v d)) ^ ((e v f) ^ (g v h))) ^ ((a v h) ->1 (d v e)_|_)) =< (((a v h) ^ (d v e)) ^ ((a v h) ->1 (d v e)_|_))
36		go1 335	. . 3 (((a v h) ^ (d v e)) ^ ((a v h) ->1 (d v e)_|_)) = 0
37	35, 36	lbtr 131	. 2 ((((a v b) ^ (c v d)) ^ ((e v f) ^ (g v h))) ^ ((a v h) ->1 (d v e)_|_)) =< 0
38		le0 139	. 2 0 =< ((((a v b) ^ (c v d)) ^ ((e v f) ^ (g v h))) ^ ((a v h) ->1 (d v e)_|_))
39	37, 38	lebi 137	1 ((((a v b) ^ (c v d)) ^ ((e v f) ^ (g v h))) ^ ((a v h) ->1 (d v e)_|_)) = 0

Syntax hints:   = wb 1   =< wle 2  _|_wn 4   v wo 6   ^ wa 7  0wf 10   ->1 wi1 13   ->2 wi2 14

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-i1 43  df-i2 44  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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