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Theorem gomaex3 924
Description: Proof of Mayet Example 3 from 6-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
gomaex3.1 ab
gomaex3.2 bc
gomaex3.3 cd
gomaex3.5 ef
gomaex3.6 fa
gomaex3.8 (((i2 g) ∩ (g2 y)) ∩ (((y2 w) ∩ (w2 n)) ∩ ((n2 k) ∩ (k2 i)))) ≤ (g2 i)
gomaex3.9 p = ((ab) →1 (de) )
gomaex3.10 q = ((ef) →1 (bc) )
gomaex3.11 r = ((p1 q) ∩ (cd))
gomaex3.12 g = a
gomaex3.14 i = c
gomaex3.16 k = r
gomaex3.18 n = (p1 q)
gomaex3.20 w = q
gomaex3.22 y = (ef)
Assertion
Ref Expression
gomaex3 (((ab) ∩ (de) ) ∩ ((((ab) →1 (de) ) →1 ((ef) →1 (bc) ) )1 (cd))) ≤ ((bc) ∪ (ef) )

Proof of Theorem gomaex3
StepHypRef Expression
1 df-i1 44 . . . 4 ((((ab) →1 (de) ) →1 ((ef) →1 (bc) ) )1 (cd)) = ((((ab) →1 (de) ) →1 ((ef) →1 (bc) ) ) ∪ ((((ab) →1 (de) ) →1 ((ef) →1 (bc) ) ) ∩ (cd)))
2 ax-a2 31 . . . . . 6 (r ∪ (p1 q)) = ((p1 q) ∪ r)
3 gomaex3.9 . . . . . . . . . 10 p = ((ab) →1 (de) )
43con2 67 . . . . . . . . 9 p = ((ab) →1 (de) )
5 gomaex3.10 . . . . . . . . 9 q = ((ef) →1 (bc) )
64, 5ud1lem0ab 257 . . . . . . . 8 (p1 q) = (((ab) →1 (de) ) →1 ((ef) →1 (bc) ) )
7 ax-a1 30 . . . . . . . 8 (((ab) →1 (de) ) →1 ((ef) →1 (bc) ) ) = (((ab) →1 (de) ) →1 ((ef) →1 (bc) ) )
86, 7ax-r2 36 . . . . . . 7 (p1 q) = (((ab) →1 (de) ) →1 ((ef) →1 (bc) ) )
9 gomaex3.11 . . . . . . . 8 r = ((p1 q) ∩ (cd))
106ax-r4 37 . . . . . . . . 9 (p1 q) = (((ab) →1 (de) ) →1 ((ef) →1 (bc) ) )
1110ran 78 . . . . . . . 8 ((p1 q) ∩ (cd)) = ((((ab) →1 (de) ) →1 ((ef) →1 (bc) ) ) ∩ (cd))
129, 11ax-r2 36 . . . . . . 7 r = ((((ab) →1 (de) ) →1 ((ef) →1 (bc) ) ) ∩ (cd))
138, 122or 72 . . . . . 6 ((p1 q) ∪ r) = ((((ab) →1 (de) ) →1 ((ef) →1 (bc) ) ) ∪ ((((ab) →1 (de) ) →1 ((ef) →1 (bc) ) ) ∩ (cd)))
142, 13ax-r2 36 . . . . 5 (r ∪ (p1 q)) = ((((ab) →1 (de) ) →1 ((ef) →1 (bc) ) ) ∪ ((((ab) →1 (de) ) →1 ((ef) →1 (bc) ) ) ∩ (cd)))
1514ax-r1 35 . . . 4 ((((ab) →1 (de) ) →1 ((ef) →1 (bc) ) ) ∪ ((((ab) →1 (de) ) →1 ((ef) →1 (bc) ) ) ∩ (cd))) = (r ∪ (p1 q))
161, 15ax-r2 36 . . 3 ((((ab) →1 (de) ) →1 ((ef) →1 (bc) ) )1 (cd)) = (r ∪ (p1 q))
1716lan 77 . 2 (((ab) ∩ (de) ) ∩ ((((ab) →1 (de) ) →1 ((ef) →1 (bc) ) )1 (cd))) = (((ab) ∩ (de) ) ∩ (r ∪ (p1 q)))
18 gomaex3.1 . . 3 ab
19 gomaex3.2 . . 3 bc
20 gomaex3.3 . . 3 cd
21 gomaex3.5 . . 3 ef
22 gomaex3.6 . . 3 fa
23 gomaex3.8 . . 3 (((i2 g) ∩ (g2 y)) ∩ (((y2 w) ∩ (w2 n)) ∩ ((n2 k) ∩ (k2 i)))) ≤ (g2 i)
24 gomaex3.12 . . 3 g = a
25 id 59 . . 3 b = b
26 gomaex3.14 . . 3 i = c
27 id 59 . . 3 (cd) = (cd)
28 gomaex3.16 . . 3 k = r
29 id 59 . . 3 (p1 q) = (p1 q)
30 gomaex3.18 . . 3 n = (p1 q)
31 id 59 . . 3 (pq) = (pq)
32 gomaex3.20 . . 3 w = q
33 id 59 . . 3 q = q
34 gomaex3.22 . . 3 y = (ef)
35 id 59 . . 3 f = f
3618, 19, 20, 21, 22, 23, 3, 5, 9, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35gomaex3lem10 923 . 2 (((ab) ∩ (de) ) ∩ (r ∪ (p1 q))) ≤ ((bc) ∪ (ef) )
3717, 36bltr 138 1 (((ab) ∩ (de) ) ∩ ((((ab) →1 (de) ) →1 ((ef) →1 (bc) ) )1 (cd))) ≤ ((bc) ∪ (ef) )
Colors of variables: term
Syntax hints:   = wb 1  wle 2   wn 4  wo 6  wa 7  1 wi1 12  2 wi2 13
This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a4 33  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-r3 439
This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i1 44  df-i2 45  df-le1 130  df-le2 131  df-c1 132  df-c2 133
This theorem is referenced by: (None)
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