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Theorem gomaex3lem5 918
Description: Lemma for Godowski 6-var -> Mayet Example 3.
Hypotheses
Ref Expression
gomaex3lem5.1 a =< b'
gomaex3lem5.2 b =< c'
gomaex3lem5.3 c =< d'
gomaex3lem5.5 e =< f'
gomaex3lem5.6 f =< a'
gomaex3lem5.8 (((i ->2 g) ^ (g ->2 y)) ^ (((y ->2 w) ^ (w ->2 n)) ^ ((n ->2 k) ^ (k ->2 i)))) =< (g ->2 i)
gomaex3lem5.9 p = ((a v b) ->1 (d v e)')'
gomaex3lem5.10 q = ((e v f) ->1 (b v c)')'
gomaex3lem5.11 r = ((p' ->1 q)' ^ (c v d))
gomaex3lem5.12 g = a
gomaex3lem5.13 h = b
gomaex3lem5.14 i = c
gomaex3lem5.15 j = (c v d)'
gomaex3lem5.16 k = r
gomaex3lem5.17 m = (p' ->1 q)
gomaex3lem5.18 n = (p' ->1 q)'
gomaex3lem5.19 u = (p' ^ q)
gomaex3lem5.20 w = q'
gomaex3lem5.21 x = q
gomaex3lem5.22 y = (e v f)'
gomaex3lem5.23 z = f
Assertion
Ref Expression
gomaex3lem5 (((g v h) ^ (i v j)) ^ (((k v m) ^ (n v u)) ^ ((w v x) ^ (y v z)))) =< (h v i)
Proof of Theorem gomaex3lem5
StepHypRef Expression
1 gomaex3lem5.1 . . 3 a =< b'
2 gomaex3lem5.12 . . 3 g = a
3 gomaex3lem5.13 . . 3 h = b
41, 2, 3gomaex3h1 902 . 2 g =< h'
5 gomaex3lem5.2 . . 3 b =< c'
6 gomaex3lem5.14 . . 3 i = c
75, 3, 6gomaex3h2 903 . 2 h =< i'
8 gomaex3lem5.15 . . 3 j = (c v d)'
96, 8gomaex3h3 904 . 2 i =< j'
10 gomaex3lem5.11 . . 3 r = ((p' ->1 q)' ^ (c v d))
11 gomaex3lem5.16 . . 3 k = r
1210, 8, 11gomaex3h4 905 . 2 j =< k'
13 gomaex3lem5.17 . . 3 m = (p' ->1 q)
1410, 11, 13gomaex3h5 906 . 2 k =< m'
15 gomaex3lem5.18 . . 3 n = (p' ->1 q)'
1613, 15gomaex3h6 907 . 2 m =< n'
17 gomaex3lem5.19 . . 3 u = (p' ^ q)
1815, 17gomaex3h7 908 . 2 n =< u'
19 gomaex3lem5.20 . . 3 w = q'
2017, 19gomaex3h8 909 . 2 u =< w'
21 gomaex3lem5.21 . . 3 x = q
2219, 21gomaex3h9 910 . 2 w =< x'
23 gomaex3lem5.10 . . 3 q = ((e v f) ->1 (b v c)')'
24 gomaex3lem5.22 . . 3 y = (e v f)'
2523, 21, 24gomaex3h10 911 . 2 x =< y'
26 gomaex3lem5.23 . . 3 z = f
2724, 26gomaex3h11 912 . 2 y =< z'
28 gomaex3lem5.6 . . 3 f =< a'
2928, 2, 26gomaex3h12 913 . 2 z =< g'
30 gomaex3lem5.8 . 2 (((i ->2 g) ^ (g ->2 y)) ^ (((y ->2 w) ^ (w ->2 n)) ^ ((n ->2 k) ^ (k ->2 i)))) =< (g ->2 i)
314, 7, 9, 12, 14, 16, 18, 20, 22, 25, 27, 29, 30go2n6 901 1 (((g v h) ^ (i v j)) ^ (((k v m) ^ (n v u)) ^ ((w v x) ^ (y v z)))) =< (h v i)
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:  gomaex3lem6  919
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