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Theorem gomaex4 900
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
StepHypRef 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)
101, 2, 3, 4, 5, 6, 7, 8, 9go2n4 899 . . . . . 6 (((g v h) ^ (a v b)) ^ ((c v d) ^ (e v f))) =< (h v a)
11 an4 86 . . . . . . 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 74 . . . . . . . 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 74 . . . . . . . . 9 ((c v d) ^ (g v h)) = ((g v h) ^ (c v d))
1413ran 78 . . . . . . . 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)))
1512, 14ax-r2 36 . . . . . . 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 86 . . . . . . 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)))
1711, 15, 163tr 65 . . . . . 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 31 . . . . . 6 (a v h) = (h v a)
1910, 17, 18le3tr1 140 . . . . 5 (((a v b) ^ (c v d)) ^ ((e v f) ^ (g v h))) =< (a v h)
20 ancom 74 . . . . . . . . 9 ((e v f) ^ (g v h)) = ((g v h) ^ (e v f))
2120lan 77 . . . . . . . 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 86 . . . . . . . 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 74 . . . . . . . . 9 ((c v d) ^ (e v f)) = ((e v f) ^ (c v d))
2423lan 77 . . . . . . . 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)))
2521, 22, 243tr 65 . . . . . . 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 74 . . . . . . . 8 ((a v b) ^ (g v h)) = ((g v h) ^ (a v b))
27 ancom 74 . . . . . . . 8 ((e v f) ^ (c v d)) = ((c v d) ^ (e v f))
2826, 272an 79 . . . . . . 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 74 . . . . . . 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)))
3025, 28, 293tr 65 . . . . . 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)
325, 6, 7, 8, 1, 2, 3, 4, 31go2n4 899 . . . . . 6 (((c v d) ^ (e v f)) ^ ((g v h) ^ (a v b))) =< (d v e)
3330, 32bltr 138 . . . . 5 (((a v b) ^ (c v d)) ^ ((e v f) ^ (g v h))) =< (d v e)
3419, 33ler2an 173 . . . 4 (((a v b) ^ (c v d)) ^ ((e v f) ^ (g v h))) =< ((a v h) ^ (d v e))
3534leran 153 . . 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 343 . . 3 (((a v h) ^ (d v e)) ^ ((a v h) ->1 (d v e)')) = 0
3735, 36lbtr 139 . 2 ((((a v b) ^ (c v d)) ^ ((e v f) ^ (g v h))) ^ ((a v h) ->1 (d v e)')) =< 0
38 le0 147 . 2 0 =< ((((a v b) ^ (c v d)) ^ ((e v f) ^ (g v h))) ^ ((a v h) ->1 (d v e)'))
3937, 38lebi 145 1 ((((a v b) ^ (c v d)) ^ ((e v f) ^ (g v h))) ^ ((a v h) ->1 (d v e)')) = 0
Colors of variables: term
Syntax hints:   = wb 1   =< wle 2  'wn 4   v wo 6   ^ wa 7  0wf 9   ->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
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