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Theorem gomaex3lem7 920
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
gomaex3lem7 (((a v b) ^ d') ^ (((r v (p' ->1 q)) ^ p') ^ e')) =< (b v c)
Proof of Theorem gomaex3lem7
StepHypRef Expression
1 gomaex3lem5.3 . . . . . 6 c =< d'
21gomaex3lem1 914 . . . . 5 (c v (c v d)') = d'
32lan 77 . . . 4 ((a v b) ^ (c v (c v d)')) = ((a v b) ^ d')
4 gomaex3lem3 916 . . . . . 6 ((p' ->1 q)' v (p' ^ q)) = p'
54lan 77 . . . . 5 ((r v (p' ->1 q)) ^ ((p' ->1 q)' v (p' ^ q))) = ((r v (p' ->1 q)) ^ p')
6 ancom 74 . . . . . 6 ((q' v q) ^ ((e v f)' v f)) = (((e v f)' v f) ^ (q' v q))
7 gomaex3lem5.5 . . . . . . . 8 e =< f'
87gomaex3lem2 915 . . . . . . 7 ((e v f)' v f) = e'
9 ax-a2 31 . . . . . . . 8 (q' v q) = (q v q')
10 df-t 41 . . . . . . . . 9 1 = (q v q')
1110ax-r1 35 . . . . . . . 8 (q v q') = 1
129, 11ax-r2 36 . . . . . . 7 (q' v q) = 1
138, 122an 79 . . . . . 6 (((e v f)' v f) ^ (q' v q)) = (e' ^ 1)
14 an1 106 . . . . . 6 (e' ^ 1) = e'
156, 13, 143tr 65 . . . . 5 ((q' v q) ^ ((e v f)' v f)) = e'
165, 152an 79 . . . 4 (((r v (p' ->1 q)) ^ ((p' ->1 q)' v (p' ^ q))) ^ ((q' v q) ^ ((e v f)' v f))) = (((r v (p' ->1 q)) ^ p') ^ e')
173, 162an 79 . . 3 (((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)))) = (((a v b) ^ d') ^ (((r v (p' ->1 q)) ^ p') ^ e'))
1817ax-r1 35 . 2 (((a v b) ^ d') ^ (((r v (p' ->1 q)) ^ p') ^ e')) = (((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))))
19 gomaex3lem5.1 . . 3 a =< b'
20 gomaex3lem5.2 . . 3 b =< c'
21 gomaex3lem5.6 . . 3 f =< a'
22 gomaex3lem5.8 . . 3 (((i ->2 g) ^ (g ->2 y)) ^ (((y ->2 w) ^ (w ->2 n)) ^ ((n ->2 k) ^ (k ->2 i)))) =< (g ->2 i)
23 gomaex3lem5.9 . . 3 p = ((a v b) ->1 (d v e)')'
24 gomaex3lem5.10 . . 3 q = ((e v f) ->1 (b v c)')'
25 gomaex3lem5.11 . . 3 r = ((p' ->1 q)' ^ (c v d))
26 gomaex3lem5.12 . . 3 g = a
27 gomaex3lem5.13 . . 3 h = b
28 gomaex3lem5.14 . . 3 i = c
29 gomaex3lem5.15 . . 3 j = (c v d)'
30 gomaex3lem5.16 . . 3 k = r
31 gomaex3lem5.17 . . 3 m = (p' ->1 q)
32 gomaex3lem5.18 . . 3 n = (p' ->1 q)'
33 gomaex3lem5.19 . . 3 u = (p' ^ q)
34 gomaex3lem5.20 . . 3 w = q'
35 gomaex3lem5.21 . . 3 x = q
36 gomaex3lem5.22 . . 3 y = (e v f)'
37 gomaex3lem5.23 . . 3 z = f
3819, 20, 1, 7, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37gomaex3lem6 919 . 2 (((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)
3918, 38bltr 138 1 (((a v b) ^ d') ^ (((r v (p' ->1 q)) ^ p') ^ e')) =< (b v c)
Colors of variables: term
Syntax hints:   = wb 1   =< wle 2  'wn 4   v wo 6   ^ wa 7  1wt 8   ->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:  gomaex3lem8  921
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