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Theorem gomaex3lem6 919
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
gomaex3lem6 (((a v b) ^ (c v (c v d)')) ^ (((r v (p' ->1 q)) ^ ((p' ->1 q)' v (p' ^ q))) ^ ((q' v q) ^ ((e v f)' v f)))) =< (b v c)
Proof of Theorem gomaex3lem6
StepHypRef Expression
1 gomaex3lem5.1 . . 3 a =< b'
2 gomaex3lem5.2 . . 3 b =< c'
3 gomaex3lem5.3 . . 3 c =< d'
4 gomaex3lem5.5 . . 3 e =< f'
5 gomaex3lem5.6 . . 3 f =< a'
6 gomaex3lem5.8 . . 3 (((i ->2 g) ^ (g ->2 y)) ^ (((y ->2 w) ^ (w ->2 n)) ^ ((n ->2 k) ^ (k ->2 i)))) =< (g ->2 i)
7 gomaex3lem5.9 . . 3 p = ((a v b) ->1 (d v e)')'
8 gomaex3lem5.10 . . 3 q = ((e v f) ->1 (b v c)')'
9 gomaex3lem5.11 . . 3 r = ((p' ->1 q)' ^ (c v d))
10 gomaex3lem5.12 . . 3 g = a
11 gomaex3lem5.13 . . 3 h = b
12 gomaex3lem5.14 . . 3 i = c
13 gomaex3lem5.15 . . 3 j = (c v d)'
14 gomaex3lem5.16 . . 3 k = r
15 gomaex3lem5.17 . . 3 m = (p' ->1 q)
16 gomaex3lem5.18 . . 3 n = (p' ->1 q)'
17 gomaex3lem5.19 . . 3 u = (p' ^ q)
18 gomaex3lem5.20 . . 3 w = q'
19 gomaex3lem5.21 . . 3 x = q
20 gomaex3lem5.22 . . 3 y = (e v f)'
21 gomaex3lem5.23 . . 3 z = f
221, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21gomaex3lem5 918 . 2 (((g v h) ^ (i v j)) ^ (((k v m) ^ (n v u)) ^ ((w v x) ^ (y v z)))) =< (h v i)
2310, 112or 72 . . . 4 (g v h) = (a v b)
2412, 132or 72 . . . 4 (i v j) = (c v (c v d)')
2523, 242an 79 . . 3 ((g v h) ^ (i v j)) = ((a v b) ^ (c v (c v d)'))
2614, 152or 72 . . . . 5 (k v m) = (r v (p' ->1 q))
2716, 172or 72 . . . . 5 (n v u) = ((p' ->1 q)' v (p' ^ q))
2826, 272an 79 . . . 4 ((k v m) ^ (n v u)) = ((r v (p' ->1 q)) ^ ((p' ->1 q)' v (p' ^ q)))
2918, 192or 72 . . . . 5 (w v x) = (q' v q)
3020, 212or 72 . . . . 5 (y v z) = ((e v f)' v f)
3129, 302an 79 . . . 4 ((w v x) ^ (y v z)) = ((q' v q) ^ ((e v f)' v f))
3228, 312an 79 . . 3 (((k v m) ^ (n v u)) ^ ((w v x) ^ (y v z))) = (((r v (p' ->1 q)) ^ ((p' ->1 q)' v (p' ^ q))) ^ ((q' v q) ^ ((e v f)' v f)))
3325, 322an 79 . 2 (((g v h) ^ (i v j)) ^ (((k v m) ^ (n v u)) ^ ((w v x) ^ (y v z)))) = (((a v b) ^ (c v (c v d)')) ^ (((r v (p' ->1 q)) ^ ((p' ->1 q)' v (p' ^ q))) ^ ((q' v q) ^ ((e v f)' v f))))
3411, 122or 72 . 2 (h v i) = (b v c)
3522, 33, 34le3tr2 141 1 (((a v b) ^ (c v (c v d)')) ^ (((r v (p' ->1 q)) ^ ((p' ->1 q)' v (p' ^ q))) ^ ((q' v q) ^ ((e v f)' v f)))) =< (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:  gomaex3lem7  920
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