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 ∪ b) ∩ (c ∪ d)) ∩ ((e ∪ f) ∩ (g ∪ h))) ∩ ((a ∪ h) →1 (d ∪ 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 ∪ h) ∩ (a ∪ b)) ∩ ((c ∪ d) ∩ (e ∪ f))) ≤ (h ∪ a)
11		an4 78	. . . . . . 7 (((a ∪ b) ∩ (c ∪ d)) ∩ ((e ∪ f) ∩ (g ∪ h))) = (((a ∪ b) ∩ (e ∪ f)) ∩ ((c ∪ d) ∩ (g ∪ h)))
12		ancom 68	. . . . . . . 8 (((a ∪ b) ∩ (e ∪ f)) ∩ ((c ∪ d) ∩ (g ∪ h))) = (((c ∪ d) ∩ (g ∪ h)) ∩ ((a ∪ b) ∩ (e ∪ f)))
13		ancom 68	. . . . . . . . 9 ((c ∪ d) ∩ (g ∪ h)) = ((g ∪ h) ∩ (c ∪ d))
14	13	ran 71	. . . . . . . 8 (((c ∪ d) ∩ (g ∪ h)) ∩ ((a ∪ b) ∩ (e ∪ f))) = (((g ∪ h) ∩ (c ∪ d)) ∩ ((a ∪ b) ∩ (e ∪ f)))
15	12, 14	ax-r2 35	. . . . . . 7 (((a ∪ b) ∩ (e ∪ f)) ∩ ((c ∪ d) ∩ (g ∪ h))) = (((g ∪ h) ∩ (c ∪ d)) ∩ ((a ∪ b) ∩ (e ∪ f)))
16		an4 78	. . . . . . 7 (((g ∪ h) ∩ (c ∪ d)) ∩ ((a ∪ b) ∩ (e ∪ f))) = (((g ∪ h) ∩ (a ∪ b)) ∩ ((c ∪ d) ∩ (e ∪ f)))
17	11, 15, 16	3tr 62	. . . . . 6 (((a ∪ b) ∩ (c ∪ d)) ∩ ((e ∪ f) ∩ (g ∪ h))) = (((g ∪ h) ∩ (a ∪ b)) ∩ ((c ∪ d) ∩ (e ∪ f)))
18		ax-a2 30	. . . . . 6 (a ∪ h) = (h ∪ a)
19	10, 17, 18	le3tr1 132	. . . . 5 (((a ∪ b) ∩ (c ∪ d)) ∩ ((e ∪ f) ∩ (g ∪ h))) ≤ (a ∪ h)
20		ancom 68	. . . . . . . . 9 ((e ∪ f) ∩ (g ∪ h)) = ((g ∪ h) ∩ (e ∪ f))
21	20	lan 70	. . . . . . . 8 (((a ∪ b) ∩ (c ∪ d)) ∩ ((e ∪ f) ∩ (g ∪ h))) = (((a ∪ b) ∩ (c ∪ d)) ∩ ((g ∪ h) ∩ (e ∪ f)))
22		an4 78	. . . . . . . 8 (((a ∪ b) ∩ (c ∪ d)) ∩ ((g ∪ h) ∩ (e ∪ f))) = (((a ∪ b) ∩ (g ∪ h)) ∩ ((c ∪ d) ∩ (e ∪ f)))
23		ancom 68	. . . . . . . . 9 ((c ∪ d) ∩ (e ∪ f)) = ((e ∪ f) ∩ (c ∪ d))
24	23	lan 70	. . . . . . . 8 (((a ∪ b) ∩ (g ∪ h)) ∩ ((c ∪ d) ∩ (e ∪ f))) = (((a ∪ b) ∩ (g ∪ h)) ∩ ((e ∪ f) ∩ (c ∪ d)))
25	21, 22, 24	3tr 62	. . . . . . 7 (((a ∪ b) ∩ (c ∪ d)) ∩ ((e ∪ f) ∩ (g ∪ h))) = (((a ∪ b) ∩ (g ∪ h)) ∩ ((e ∪ f) ∩ (c ∪ d)))
26		ancom 68	. . . . . . . 8 ((a ∪ b) ∩ (g ∪ h)) = ((g ∪ h) ∩ (a ∪ b))
27		ancom 68	. . . . . . . 8 ((e ∪ f) ∩ (c ∪ d)) = ((c ∪ d) ∩ (e ∪ f))
28	26, 27	2an 72	. . . . . . 7 (((a ∪ b) ∩ (g ∪ h)) ∩ ((e ∪ f) ∩ (c ∪ d))) = (((g ∪ h) ∩ (a ∪ b)) ∩ ((c ∪ d) ∩ (e ∪ f)))
29		ancom 68	. . . . . . 7 (((g ∪ h) ∩ (a ∪ b)) ∩ ((c ∪ d) ∩ (e ∪ f))) = (((c ∪ d) ∩ (e ∪ f)) ∩ ((g ∪ h) ∩ (a ∪ b)))
30	25, 28, 29	3tr 62	. . . . . 6 (((a ∪ b) ∩ (c ∪ d)) ∩ ((e ∪ f) ∩ (g ∪ h))) = (((c ∪ d) ∩ (e ∪ f)) ∩ ((g ∪ h) ∩ (a ∪ 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 ∪ d) ∩ (e ∪ f)) ∩ ((g ∪ h) ∩ (a ∪ b))) ≤ (d ∪ e)
33	30, 32	bltr 130	. . . . 5 (((a ∪ b) ∩ (c ∪ d)) ∩ ((e ∪ f) ∩ (g ∪ h))) ≤ (d ∪ e)
34	19, 33	ler2an 165	. . . 4 (((a ∪ b) ∩ (c ∪ d)) ∩ ((e ∪ f) ∩ (g ∪ h))) ≤ ((a ∪ h) ∩ (d ∪ e))
35	34	leran 145	. . 3 ((((a ∪ b) ∩ (c ∪ d)) ∩ ((e ∪ f) ∩ (g ∪ h))) ∩ ((a ∪ h) →1 (d ∪ e)⊥ )) ≤ (((a ∪ h) ∩ (d ∪ e)) ∩ ((a ∪ h) →1 (d ∪ e)⊥ ))
36		go1 335	. . 3 (((a ∪ h) ∩ (d ∪ e)) ∩ ((a ∪ h) →1 (d ∪ e)⊥ )) = 0
37	35, 36	lbtr 131	. 2 ((((a ∪ b) ∩ (c ∪ d)) ∩ ((e ∪ f) ∩ (g ∪ h))) ∩ ((a ∪ h) →1 (d ∪ e)⊥ )) ≤ 0
38		le0 139	. 2 0 ≤ ((((a ∪ b) ∩ (c ∪ d)) ∩ ((e ∪ f) ∩ (g ∪ h))) ∩ ((a ∪ h) →1 (d ∪ e)⊥ ))
39	37, 38	lebi 137	1 ((((a ∪ b) ∩ (c ∪ d)) ∩ ((e ∪ f) ∩ (g ∪ h))) ∩ ((a ∪ h) →1 (d ∪ e)⊥ )) = 0

Syntax hints:   = wb 1   ≤ wle 2  ^⊥ wn 4   ∪ 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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