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Theorem go2n4 899
Description: 8-variable Godowski equation derived from 4-variable one. The last hypothesis is the 4-variable Godowski equation.
Hypotheses
Ref Expression
go2n4.1 a =< b'
go2n4.2 b =< c'
go2n4.3 c =< d'
go2n4.4 d =< e'
go2n4.5 e =< f'
go2n4.6 f =< g'
go2n4.7 g =< h'
go2n4.8 h =< a'
go2n4.9 (((c ->2 a) ^ (a ->2 g)) ^ ((g ->2 e) ^ (e ->2 c))) =< (a ->2 c)
Assertion
Ref Expression
go2n4 (((a v b) ^ (c v d)) ^ ((e v f) ^ (g v h))) =< (b v c)
Proof of Theorem go2n4
StepHypRef Expression
1 anass 76 . . 3 (((a v b) ^ (c v d)) ^ ((e v f) ^ (g v h))) = ((a v b) ^ ((c v d) ^ ((e v f) ^ (g v h))))
2 ancom 74 . . . 4 ((c v d) ^ ((e v f) ^ (g v h))) = (((e v f) ^ (g v h)) ^ (c v d))
32lan 77 . . 3 ((a v b) ^ ((c v d) ^ ((e v f) ^ (g v h)))) = ((a v b) ^ (((e v f) ^ (g v h)) ^ (c v d)))
41, 3ax-r2 36 . 2 (((a v b) ^ (c v d)) ^ ((e v f) ^ (g v h))) = ((a v b) ^ (((e v f) ^ (g v h)) ^ (c v d)))
5 go2n4.1 . . 3 a =< b'
6 go2n4.2 . . 3 b =< c'
7 anass 76 . . . . . 6 (((c ->2 a) ^ (a ->2 g)) ^ ((g ->2 e) ^ (e ->2 c))) = ((c ->2 a) ^ ((a ->2 g) ^ ((g ->2 e) ^ (e ->2 c))))
8 ancom 74 . . . . . . . 8 ((a ->2 g) ^ ((g ->2 e) ^ (e ->2 c))) = (((g ->2 e) ^ (e ->2 c)) ^ (a ->2 g))
9 an32 83 . . . . . . . 8 (((g ->2 e) ^ (e ->2 c)) ^ (a ->2 g)) = (((g ->2 e) ^ (a ->2 g)) ^ (e ->2 c))
108, 9ax-r2 36 . . . . . . 7 ((a ->2 g) ^ ((g ->2 e) ^ (e ->2 c))) = (((g ->2 e) ^ (a ->2 g)) ^ (e ->2 c))
1110lan 77 . . . . . 6 ((c ->2 a) ^ ((a ->2 g) ^ ((g ->2 e) ^ (e ->2 c)))) = ((c ->2 a) ^ (((g ->2 e) ^ (a ->2 g)) ^ (e ->2 c)))
127, 11ax-r2 36 . . . . 5 (((c ->2 a) ^ (a ->2 g)) ^ ((g ->2 e) ^ (e ->2 c))) = ((c ->2 a) ^ (((g ->2 e) ^ (a ->2 g)) ^ (e ->2 c)))
1312ax-r1 35 . . . 4 ((c ->2 a) ^ (((g ->2 e) ^ (a ->2 g)) ^ (e ->2 c))) = (((c ->2 a) ^ (a ->2 g)) ^ ((g ->2 e) ^ (e ->2 c)))
14 go2n4.9 . . . 4 (((c ->2 a) ^ (a ->2 g)) ^ ((g ->2 e) ^ (e ->2 c))) =< (a ->2 c)
1513, 14bltr 138 . . 3 ((c ->2 a) ^ (((g ->2 e) ^ (a ->2 g)) ^ (e ->2 c))) =< (a ->2 c)
16 go2n4.5 . . . . . 6 e =< f'
17 go2n4.6 . . . . . 6 f =< g'
1816, 17govar2 897 . . . . 5 (e v f) =< (g ->2 e)
19 go2n4.7 . . . . . 6 g =< h'
20 go2n4.8 . . . . . 6 h =< a'
2119, 20govar2 897 . . . . 5 (g v h) =< (a ->2 g)
2218, 21le2an 169 . . . 4 ((e v f) ^ (g v h)) =< ((g ->2 e) ^ (a ->2 g))
23 go2n4.3 . . . . 5 c =< d'
24 go2n4.4 . . . . 5 d =< e'
2523, 24govar2 897 . . . 4 (c v d) =< (e ->2 c)
2622, 25le2an 169 . . 3 (((e v f) ^ (g v h)) ^ (c v d)) =< (((g ->2 e) ^ (a ->2 g)) ^ (e ->2 c))
275, 6, 15, 26gon2n 898 . 2 ((a v b) ^ (((e v f) ^ (g v h)) ^ (c v d))) =< (b v c)
284, 27bltr 138 1 (((a v b) ^ (c v d)) ^ ((e v f) ^ (g v h))) =< (b v c)
Colors of variables: term
Syntax hints:   =< wle 2  'wn 4   v wo 6   ^ wa 7   ->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-i2 45  df-le1 130  df-le2 131  df-c1 132  df-c2 133
This theorem is referenced by:  gomaex4  900
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