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Theorem oa3-u2lem 986
Description: Lemma for a "universal" 3-OA. Equivalence with substitution into 6-OA dual.
Assertion
Ref Expression
oa3-u2lem ((a ->1 c) ^ ((a' ->1 c) v (c ^ ((((a' ->1 c) ^ c) v ((a ->1 c) ^ 1)) v ((((a' ->1 c) ^ (b' ->1 c)) v ((a ->1 c) ^ (b ->1 c))) ^ ((c ^ (b' ->1 c)) v (1 ^ (b ->1 c)))))))) = ((a ->1 c) ^ ((a' ->1 c) v (c ^ ((a ->1 c) v ((b ->1 c) ^ (((a ->1 c) ^ (b ->1 c)) v ((a' ->1 c) ^ (b' ->1 c))))))))
Proof of Theorem oa3-u2lem
StepHypRef Expression
1 u1lemab 610 . . . . . . 7 ((a' ->1 c) ^ c) = ((a' ^ c) v (a'' ^ c))
2 an1 106 . . . . . . 7 ((a ->1 c) ^ 1) = (a ->1 c)
31, 22or 72 . . . . . 6 (((a' ->1 c) ^ c) v ((a ->1 c) ^ 1)) = (((a' ^ c) v (a'' ^ c)) v (a ->1 c))
4 lea 160 . . . . . . . . 9 (a' ^ c) =< a'
5 ax-a1 30 . . . . . . . . . . . 12 a = a''
65ax-r1 35 . . . . . . . . . . 11 a'' = a
7 leid 148 . . . . . . . . . . 11 a =< a
86, 7bltr 138 . . . . . . . . . 10 a'' =< a
98leran 153 . . . . . . . . 9 (a'' ^ c) =< (a ^ c)
104, 9le2or 168 . . . . . . . 8 ((a' ^ c) v (a'' ^ c)) =< (a' v (a ^ c))
11 df-i1 44 . . . . . . . . 9 (a ->1 c) = (a' v (a ^ c))
1211ax-r1 35 . . . . . . . 8 (a' v (a ^ c)) = (a ->1 c)
1310, 12lbtr 139 . . . . . . 7 ((a' ^ c) v (a'' ^ c)) =< (a ->1 c)
1413df-le2 131 . . . . . 6 (((a' ^ c) v (a'' ^ c)) v (a ->1 c)) = (a ->1 c)
153, 14ax-r2 36 . . . . 5 (((a' ->1 c) ^ c) v ((a ->1 c) ^ 1)) = (a ->1 c)
16 ancom 74 . . . . . 6 ((((a' ->1 c) ^ (b' ->1 c)) v ((a ->1 c) ^ (b ->1 c))) ^ ((c ^ (b' ->1 c)) v (1 ^ (b ->1 c)))) = (((c ^ (b' ->1 c)) v (1 ^ (b ->1 c))) ^ (((a' ->1 c) ^ (b' ->1 c)) v ((a ->1 c) ^ (b ->1 c))))
17 ancom 74 . . . . . . . . . 10 (c ^ (b' ->1 c)) = ((b' ->1 c) ^ c)
18 u1lemab 610 . . . . . . . . . 10 ((b' ->1 c) ^ c) = ((b' ^ c) v (b'' ^ c))
1917, 18ax-r2 36 . . . . . . . . 9 (c ^ (b' ->1 c)) = ((b' ^ c) v (b'' ^ c))
20 ancom 74 . . . . . . . . . 10 (1 ^ (b ->1 c)) = ((b ->1 c) ^ 1)
21 an1 106 . . . . . . . . . 10 ((b ->1 c) ^ 1) = (b ->1 c)
2220, 21ax-r2 36 . . . . . . . . 9 (1 ^ (b ->1 c)) = (b ->1 c)
2319, 222or 72 . . . . . . . 8 ((c ^ (b' ->1 c)) v (1 ^ (b ->1 c))) = (((b' ^ c) v (b'' ^ c)) v (b ->1 c))
24 lea 160 . . . . . . . . . . 11 (b' ^ c) =< b'
25 ax-a1 30 . . . . . . . . . . . . . 14 b = b''
2625ax-r1 35 . . . . . . . . . . . . 13 b'' = b
27 leid 148 . . . . . . . . . . . . 13 b =< b
2826, 27bltr 138 . . . . . . . . . . . 12 b'' =< b
2928leran 153 . . . . . . . . . . 11 (b'' ^ c) =< (b ^ c)
3024, 29le2or 168 . . . . . . . . . 10 ((b' ^ c) v (b'' ^ c)) =< (b' v (b ^ c))
31 df-i1 44 . . . . . . . . . . 11 (b ->1 c) = (b' v (b ^ c))
3231ax-r1 35 . . . . . . . . . 10 (b' v (b ^ c)) = (b ->1 c)
3330, 32lbtr 139 . . . . . . . . 9 ((b' ^ c) v (b'' ^ c)) =< (b ->1 c)
3433df-le2 131 . . . . . . . 8 (((b' ^ c) v (b'' ^ c)) v (b ->1 c)) = (b ->1 c)
3523, 34ax-r2 36 . . . . . . 7 ((c ^ (b' ->1 c)) v (1 ^ (b ->1 c))) = (b ->1 c)
36 ax-a2 31 . . . . . . 7 (((a' ->1 c) ^ (b' ->1 c)) v ((a ->1 c) ^ (b ->1 c))) = (((a ->1 c) ^ (b ->1 c)) v ((a' ->1 c) ^ (b' ->1 c)))
3735, 362an 79 . . . . . 6 (((c ^ (b' ->1 c)) v (1 ^ (b ->1 c))) ^ (((a' ->1 c) ^ (b' ->1 c)) v ((a ->1 c) ^ (b ->1 c)))) = ((b ->1 c) ^ (((a ->1 c) ^ (b ->1 c)) v ((a' ->1 c) ^ (b' ->1 c))))
3816, 37ax-r2 36 . . . . 5 ((((a' ->1 c) ^ (b' ->1 c)) v ((a ->1 c) ^ (b ->1 c))) ^ ((c ^ (b' ->1 c)) v (1 ^ (b ->1 c)))) = ((b ->1 c) ^ (((a ->1 c) ^ (b ->1 c)) v ((a' ->1 c) ^ (b' ->1 c))))
3915, 382or 72 . . . 4 ((((a' ->1 c) ^ c) v ((a ->1 c) ^ 1)) v ((((a' ->1 c) ^ (b' ->1 c)) v ((a ->1 c) ^ (b ->1 c))) ^ ((c ^ (b' ->1 c)) v (1 ^ (b ->1 c))))) = ((a ->1 c) v ((b ->1 c) ^ (((a ->1 c) ^ (b ->1 c)) v ((a' ->1 c) ^ (b' ->1 c)))))
4039lan 77 . . 3 (c ^ ((((a' ->1 c) ^ c) v ((a ->1 c) ^ 1)) v ((((a' ->1 c) ^ (b' ->1 c)) v ((a ->1 c) ^ (b ->1 c))) ^ ((c ^ (b' ->1 c)) v (1 ^ (b ->1 c)))))) = (c ^ ((a ->1 c) v ((b ->1 c) ^ (((a ->1 c) ^ (b ->1 c)) v ((a' ->1 c) ^ (b' ->1 c))))))
4140lor 70 . 2 ((a' ->1 c) v (c ^ ((((a' ->1 c) ^ c) v ((a ->1 c) ^ 1)) v ((((a' ->1 c) ^ (b' ->1 c)) v ((a ->1 c) ^ (b ->1 c))) ^ ((c ^ (b' ->1 c)) v (1 ^ (b ->1 c))))))) = ((a' ->1 c) v (c ^ ((a ->1 c) v ((b ->1 c) ^ (((a ->1 c) ^ (b ->1 c)) v ((a' ->1 c) ^ (b' ->1 c)))))))
4241lan 77 1 ((a ->1 c) ^ ((a' ->1 c) v (c ^ ((((a' ->1 c) ^ c) v ((a ->1 c) ^ 1)) v ((((a' ->1 c) ^ (b' ->1 c)) v ((a ->1 c) ^ (b ->1 c))) ^ ((c ^ (b' ->1 c)) v (1 ^ (b ->1 c)))))))) = ((a ->1 c) ^ ((a' ->1 c) v (c ^ ((a ->1 c) v ((b ->1 c) ^ (((a ->1 c) ^ (b ->1 c)) v ((a' ->1 c) ^ (b' ->1 c))))))))
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v wo 6   ^ wa 7  1wt 8   ->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-u2  992
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