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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 a =< b'
gomaex3.2 b =< c'
gomaex3.3 c =< d'
gomaex3.5 e =< f'
gomaex3.6 f =< a'
gomaex3.8 (((i ->2 g) ^ (g ->2 y)) ^ (((y ->2 w) ^ (w ->2 n)) ^ ((n ->2 k) ^ (k ->2 i)))) =< (g ->2 i)
gomaex3.9 p = ((a v b) ->1 (d v e)')'
gomaex3.10 q = ((e v f) ->1 (b v c)')'
gomaex3.11 r = ((p' ->1 q)' ^ (c v d))
gomaex3.12 g = a
gomaex3.14 i = c
gomaex3.16 k = r
gomaex3.18 n = (p' ->1 q)'
gomaex3.20 w = q'
gomaex3.22 y = (e v f)'
Assertion
Ref Expression
gomaex3 (((a v b) ^ (d v e)') ^ ((((a v b) ->1 (d v e)') ->1 ((e v f) ->1 (b v c)')')' ->1 (c v d))) =< ((b v c) v (e v f)')
Proof of Theorem gomaex3
StepHypRef Expression
1 df-i1 44 . . . 4 ((((a v b) ->1 (d v e)') ->1 ((e v f) ->1 (b v c)')')' ->1 (c v d)) = ((((a v b) ->1 (d v e)') ->1 ((e v f) ->1 (b v c)')')'' v ((((a v b) ->1 (d v e)') ->1 ((e v f) ->1 (b v c)')')' ^ (c v d)))
2 ax-a2 31 . . . . . 6 (r v (p' ->1 q)) = ((p' ->1 q) v r)
3 gomaex3.9 . . . . . . . . . 10 p = ((a v b) ->1 (d v e)')'
43con2 67 . . . . . . . . 9 p' = ((a v b) ->1 (d v e)')
5 gomaex3.10 . . . . . . . . 9 q = ((e v f) ->1 (b v c)')'
64, 5ud1lem0ab 257 . . . . . . . 8 (p' ->1 q) = (((a v b) ->1 (d v e)') ->1 ((e v f) ->1 (b v c)')')
7 ax-a1 30 . . . . . . . 8 (((a v b) ->1 (d v e)') ->1 ((e v f) ->1 (b v c)')') = (((a v b) ->1 (d v e)') ->1 ((e v f) ->1 (b v c)')')''
86, 7ax-r2 36 . . . . . . 7 (p' ->1 q) = (((a v b) ->1 (d v e)') ->1 ((e v f) ->1 (b v c)')')''
9 gomaex3.11 . . . . . . . 8 r = ((p' ->1 q)' ^ (c v d))
106ax-r4 37 . . . . . . . . 9 (p' ->1 q)' = (((a v b) ->1 (d v e)') ->1 ((e v f) ->1 (b v c)')')'
1110ran 78 . . . . . . . 8 ((p' ->1 q)' ^ (c v d)) = ((((a v b) ->1 (d v e)') ->1 ((e v f) ->1 (b v c)')')' ^ (c v d))
129, 11ax-r2 36 . . . . . . 7 r = ((((a v b) ->1 (d v e)') ->1 ((e v f) ->1 (b v c)')')' ^ (c v d))
138, 122or 72 . . . . . 6 ((p' ->1 q) v r) = ((((a v b) ->1 (d v e)') ->1 ((e v f) ->1 (b v c)')')'' v ((((a v b) ->1 (d v e)') ->1 ((e v f) ->1 (b v c)')')' ^ (c v d)))
142, 13ax-r2 36 . . . . 5 (r v (p' ->1 q)) = ((((a v b) ->1 (d v e)') ->1 ((e v f) ->1 (b v c)')')'' v ((((a v b) ->1 (d v e)') ->1 ((e v f) ->1 (b v c)')')' ^ (c v d)))
1514ax-r1 35 . . . 4 ((((a v b) ->1 (d v e)') ->1 ((e v f) ->1 (b v c)')')'' v ((((a v b) ->1 (d v e)') ->1 ((e v f) ->1 (b v c)')')' ^ (c v d))) = (r v (p' ->1 q))
161, 15ax-r2 36 . . 3 ((((a v b) ->1 (d v e)') ->1 ((e v f) ->1 (b v c)')')' ->1 (c v d)) = (r v (p' ->1 q))
1716lan 77 . 2 (((a v b) ^ (d v e)') ^ ((((a v b) ->1 (d v e)') ->1 ((e v f) ->1 (b v c)')')' ->1 (c v d))) = (((a v b) ^ (d v e)') ^ (r v (p' ->1 q)))
18 gomaex3.1 . . 3 a =< b'
19 gomaex3.2 . . 3 b =< c'
20 gomaex3.3 . . 3 c =< d'
21 gomaex3.5 . . 3 e =< f'
22 gomaex3.6 . . 3 f =< a'
23 gomaex3.8 . . 3 (((i ->2 g) ^ (g ->2 y)) ^ (((y ->2 w) ^ (w ->2 n)) ^ ((n ->2 k) ^ (k ->2 i)))) =< (g ->2 i)
24 gomaex3.12 . . 3 g = a
25 id 59 . . 3 b = b
26 gomaex3.14 . . 3 i = c
27 id 59 . . 3 (c v d)' = (c v d)'
28 gomaex3.16 . . 3 k = r
29 id 59 . . 3 (p' ->1 q) = (p' ->1 q)
30 gomaex3.18 . . 3 n = (p' ->1 q)'
31 id 59 . . 3 (p' ^ q) = (p' ^ q)
32 gomaex3.20 . . 3 w = q'
33 id 59 . . 3 q = q
34 gomaex3.22 . . 3 y = (e v f)'
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 (((a v b) ^ (d v e)') ^ (r v (p' ->1 q))) =< ((b v c) v (e v f)')
3717, 36bltr 138 1 (((a v b) ^ (d v e)') ^ ((((a v b) ->1 (d v e)') ->1 ((e v f) ->1 (b v c)')')' ->1 (c v d))) =< ((b v c) v (e v f)')
Colors of variables: term
Syntax hints:   = wb 1   =< wle 2  'wn 4   v 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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