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Theorem gomaex3lem9 922
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
gomaex3lem9 (((a v b) ^ (d v e)') ^ (r v (p' ->1 q))) =< (b v c)
Proof of Theorem gomaex3lem9
StepHypRef Expression
1 ancom 74 . . 3 (((a v b) ^ (d v e)') ^ (r v (p' ->1 q))) = ((r v (p' ->1 q)) ^ ((a v b) ^ (d v e)'))
2 gomaex3lem5.9 . . . . . . 7 p = ((a v b) ->1 (d v e)')'
32gomaex3lem4 917 . . . . . 6 ((a v b) ^ (d v e)') =< p'
43df2le2 136 . . . . 5 (((a v b) ^ (d v e)') ^ p') = ((a v b) ^ (d v e)')
54ax-r1 35 . . . 4 ((a v b) ^ (d v e)') = (((a v b) ^ (d v e)') ^ p')
65lan 77 . . 3 ((r v (p' ->1 q)) ^ ((a v b) ^ (d v e)')) = ((r v (p' ->1 q)) ^ (((a v b) ^ (d v e)') ^ p'))
7 an12 81 . . 3 ((r v (p' ->1 q)) ^ (((a v b) ^ (d v e)') ^ p')) = (((a v b) ^ (d v e)') ^ ((r v (p' ->1 q)) ^ p'))
81, 6, 73tr 65 . 2 (((a v b) ^ (d v e)') ^ (r v (p' ->1 q))) = (((a v b) ^ (d v e)') ^ ((r v (p' ->1 q)) ^ p'))
9 gomaex3lem5.1 . . 3 a =< b'
10 gomaex3lem5.2 . . 3 b =< c'
11 gomaex3lem5.3 . . 3 c =< d'
12 gomaex3lem5.5 . . 3 e =< f'
13 gomaex3lem5.6 . . 3 f =< a'
14 gomaex3lem5.8 . . 3 (((i ->2 g) ^ (g ->2 y)) ^ (((y ->2 w) ^ (w ->2 n)) ^ ((n ->2 k) ^ (k ->2 i)))) =< (g ->2 i)
15 gomaex3lem5.10 . . 3 q = ((e v f) ->1 (b v c)')'
16 gomaex3lem5.11 . . 3 r = ((p' ->1 q)' ^ (c v d))
17 gomaex3lem5.12 . . 3 g = a
18 gomaex3lem5.13 . . 3 h = b
19 gomaex3lem5.14 . . 3 i = c
20 gomaex3lem5.15 . . 3 j = (c v d)'
21 gomaex3lem5.16 . . 3 k = r
22 gomaex3lem5.17 . . 3 m = (p' ->1 q)
23 gomaex3lem5.18 . . 3 n = (p' ->1 q)'
24 gomaex3lem5.19 . . 3 u = (p' ^ q)
25 gomaex3lem5.20 . . 3 w = q'
26 gomaex3lem5.21 . . 3 x = q
27 gomaex3lem5.22 . . 3 y = (e v f)'
28 gomaex3lem5.23 . . 3 z = f
299, 10, 11, 12, 13, 14, 2, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28gomaex3lem8 921 . 2 (((a v b) ^ (d v e)') ^ ((r v (p' ->1 q)) ^ p')) =< (b v c)
308, 29bltr 138 1 (((a v b) ^ (d v e)') ^ (r v (p' ->1 q))) =< (b v c)
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:  gomaex3lem10  923
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