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Theorem oa6to4 958
Description: Derivation of 4-variable proper OA law, assuming 6-variable OA law.
Hypotheses
Ref Expression
oa6to4.1 b' = (a ->1 g)'
oa6to4.2 d' = (c ->1 g)'
oa6to4.3 f' = (e ->1 g)'
oa6to4.oa6 (((a' v b') ^ (c' v d')) ^ (e' v f')) =< (b' v (a' ^ (c' v (((a' v c') ^ (b' v d')) ^ (((a' v e') ^ (b' v f')) v ((c' v e') ^ (d' v f')))))))
Assertion
Ref Expression
oa6to4 ((a ->1 g) ^ (a v (c ^ (((a ^ c) v ((a ->1 g) ^ (c ->1 g))) v (((a ^ e) v ((a ->1 g) ^ (e ->1 g))) ^ ((c ^ e) v ((c ->1 g) ^ (e ->1 g)))))))) =< (((a ^ g) v (c ^ g)) v (e ^ g))
Proof of Theorem oa6to4
StepHypRef Expression
1 oa6to4.oa6 . . 3 (((a' v b') ^ (c' v d')) ^ (e' v f')) =< (b' v (a' ^ (c' v (((a' v c') ^ (b' v d')) ^ (((a' v e') ^ (b' v f')) v ((c' v e') ^ (d' v f')))))))
21oa6todual 952 . 2 (b ^ (a v (c ^ (((a ^ c) v (b ^ d)) v (((a ^ e) v (b ^ f)) ^ ((c ^ e) v (d ^ f))))))) =< (((a ^ b) v (c ^ d)) v (e ^ f))
3 oa6to4.1 . . . 4 b' = (a ->1 g)'
43con1 66 . . 3 b = (a ->1 g)
5 oa6to4.2 . . . . . . . . 9 d' = (c ->1 g)'
65con1 66 . . . . . . . 8 d = (c ->1 g)
74, 62an 79 . . . . . . 7 (b ^ d) = ((a ->1 g) ^ (c ->1 g))
87lor 70 . . . . . 6 ((a ^ c) v (b ^ d)) = ((a ^ c) v ((a ->1 g) ^ (c ->1 g)))
9 oa6to4.3 . . . . . . . . . 10 f' = (e ->1 g)'
109con1 66 . . . . . . . . 9 f = (e ->1 g)
114, 102an 79 . . . . . . . 8 (b ^ f) = ((a ->1 g) ^ (e ->1 g))
1211lor 70 . . . . . . 7 ((a ^ e) v (b ^ f)) = ((a ^ e) v ((a ->1 g) ^ (e ->1 g)))
136, 102an 79 . . . . . . . 8 (d ^ f) = ((c ->1 g) ^ (e ->1 g))
1413lor 70 . . . . . . 7 ((c ^ e) v (d ^ f)) = ((c ^ e) v ((c ->1 g) ^ (e ->1 g)))
1512, 142an 79 . . . . . 6 (((a ^ e) v (b ^ f)) ^ ((c ^ e) v (d ^ f))) = (((a ^ e) v ((a ->1 g) ^ (e ->1 g))) ^ ((c ^ e) v ((c ->1 g) ^ (e ->1 g))))
168, 152or 72 . . . . 5 (((a ^ c) v (b ^ d)) v (((a ^ e) v (b ^ f)) ^ ((c ^ e) v (d ^ f)))) = (((a ^ c) v ((a ->1 g) ^ (c ->1 g))) v (((a ^ e) v ((a ->1 g) ^ (e ->1 g))) ^ ((c ^ e) v ((c ->1 g) ^ (e ->1 g)))))
1716lan 77 . . . 4 (c ^ (((a ^ c) v (b ^ d)) v (((a ^ e) v (b ^ f)) ^ ((c ^ e) v (d ^ f))))) = (c ^ (((a ^ c) v ((a ->1 g) ^ (c ->1 g))) v (((a ^ e) v ((a ->1 g) ^ (e ->1 g))) ^ ((c ^ e) v ((c ->1 g) ^ (e ->1 g))))))
1817lor 70 . . 3 (a v (c ^ (((a ^ c) v (b ^ d)) v (((a ^ e) v (b ^ f)) ^ ((c ^ e) v (d ^ f)))))) = (a v (c ^ (((a ^ c) v ((a ->1 g) ^ (c ->1 g))) v (((a ^ e) v ((a ->1 g) ^ (e ->1 g))) ^ ((c ^ e) v ((c ->1 g) ^ (e ->1 g)))))))
194, 182an 79 . 2 (b ^ (a v (c ^ (((a ^ c) v (b ^ d)) v (((a ^ e) v (b ^ f)) ^ ((c ^ e) v (d ^ f))))))) = ((a ->1 g) ^ (a v (c ^ (((a ^ c) v ((a ->1 g) ^ (c ->1 g))) v (((a ^ e) v ((a ->1 g) ^ (e ->1 g))) ^ ((c ^ e) v ((c ->1 g) ^ (e ->1 g))))))))
204lan 77 . . . . 5 (a ^ b) = (a ^ (a ->1 g))
21 ancom 74 . . . . 5 (a ^ (a ->1 g)) = ((a ->1 g) ^ a)
22 u1lemaa 600 . . . . 5 ((a ->1 g) ^ a) = (a ^ g)
2320, 21, 223tr 65 . . . 4 (a ^ b) = (a ^ g)
246lan 77 . . . . 5 (c ^ d) = (c ^ (c ->1 g))
25 ancom 74 . . . . 5 (c ^ (c ->1 g)) = ((c ->1 g) ^ c)
26 u1lemaa 600 . . . . 5 ((c ->1 g) ^ c) = (c ^ g)
2724, 25, 263tr 65 . . . 4 (c ^ d) = (c ^ g)
2823, 272or 72 . . 3 ((a ^ b) v (c ^ d)) = ((a ^ g) v (c ^ g))
2910lan 77 . . . 4 (e ^ f) = (e ^ (e ->1 g))
30 ancom 74 . . . 4 (e ^ (e ->1 g)) = ((e ->1 g) ^ e)
31 u1lemaa 600 . . . 4 ((e ->1 g) ^ e) = (e ^ g)
3229, 30, 313tr 65 . . 3 (e ^ f) = (e ^ g)
3328, 322or 72 . 2 (((a ^ b) v (c ^ d)) v (e ^ f)) = (((a ^ g) v (c ^ g)) v (e ^ g))
342, 19, 33le3tr2 141 1 ((a ->1 g) ^ (a v (c ^ (((a ^ c) v ((a ->1 g) ^ (c ->1 g))) v (((a ^ e) v ((a ->1 g) ^ (e ->1 g))) ^ ((c ^ e) v ((c ->1 g) ^ (e ->1 g)))))))) =< (((a ^ g) v (c ^ g)) v (e ^ g))
Colors of variables: term
Syntax hints:   = wb 1   =< wle 2  'wn 4   v wo 6   ^ wa 7   ->1 wi1 12
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-le1 130  df-le2 131  df-c1 132  df-c2 133
This theorem is referenced by:  oa3-2to2s  990  axoa4a  1037
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