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Theorem oa4to4u2 974
Description: A weaker-looking "universal" proper 4-OA.
Hypotheses
Ref Expression
oa4to4u.1 ((e ->1 d) ^ (e v (f ^ (((e ^ f) v ((e ->1 d) ^ (f ->1 d))) v (((e ^ g) v ((e ->1 d) ^ (g ->1 d))) ^ ((f ^ g) v ((f ->1 d) ^ (g ->1 d)))))))) =< (((e ^ d) v (f ^ d)) v (g ^ d))
oa4to4u.2 e = (a' ->1 d)
oa4to4u3 f = (b' ->1 d)
oa4to4u.4 g = (c' ->1 d)
Assertion
Ref Expression
oa4to4u2 ((a ->1 d) ^ ((a' ->1 d) v ((b' ->1 d) ^ ((((a ->1 d) ^ (b ->1 d)) v ((a' ->1 d) ^ (b' ->1 d))) v ((((a ->1 d) ^ (c ->1 d)) v ((a' ->1 d) ^ (c' ->1 d))) ^ (((b ->1 d) ^ (c ->1 d)) v ((b' ->1 d) ^ (c' ->1 d)))))))) =< d
Proof of Theorem oa4to4u2
StepHypRef Expression
1 oa4to4u.1 . . 3 ((e ->1 d) ^ (e v (f ^ (((e ^ f) v ((e ->1 d) ^ (f ->1 d))) v (((e ^ g) v ((e ->1 d) ^ (g ->1 d))) ^ ((f ^ g) v ((f ->1 d) ^ (g ->1 d)))))))) =< (((e ^ d) v (f ^ d)) v (g ^ d))
2 oa4to4u.2 . . 3 e = (a' ->1 d)
3 oa4to4u3 . . 3 f = (b' ->1 d)
4 oa4to4u.4 . . 3 g = (c' ->1 d)
51, 2, 3, 4oa4to4u 973 . 2 ((a ->1 d) ^ ((a' ->1 d) v ((b' ->1 d) ^ ((((a ->1 d) ^ (b ->1 d)) v ((a' ->1 d) ^ (b' ->1 d))) v ((((a ->1 d) ^ (c ->1 d)) v ((a' ->1 d) ^ (c' ->1 d))) ^ (((b ->1 d) ^ (c ->1 d)) v ((b' ->1 d) ^ (c' ->1 d)))))))) =< ((((a ->1 d) ^ (a' ->1 d)) v ((b ->1 d) ^ (b' ->1 d))) v ((c ->1 d) ^ (c' ->1 d)))
6 u1lem8 776 . . . . 5 ((a ->1 d) ^ (a' ->1 d)) = ((a ^ d) v (a' ^ d))
7 lear 161 . . . . . 6 (a ^ d) =< d
8 lear 161 . . . . . 6 (a' ^ d) =< d
97, 8lel2or 170 . . . . 5 ((a ^ d) v (a' ^ d)) =< d
106, 9bltr 138 . . . 4 ((a ->1 d) ^ (a' ->1 d)) =< d
11 u1lem8 776 . . . . 5 ((b ->1 d) ^ (b' ->1 d)) = ((b ^ d) v (b' ^ d))
12 lear 161 . . . . . 6 (b ^ d) =< d
13 lear 161 . . . . . 6 (b' ^ d) =< d
1412, 13lel2or 170 . . . . 5 ((b ^ d) v (b' ^ d)) =< d
1511, 14bltr 138 . . . 4 ((b ->1 d) ^ (b' ->1 d)) =< d
1610, 15lel2or 170 . . 3 (((a ->1 d) ^ (a' ->1 d)) v ((b ->1 d) ^ (b' ->1 d))) =< d
17 u1lem8 776 . . . 4 ((c ->1 d) ^ (c' ->1 d)) = ((c ^ d) v (c' ^ d))
18 lear 161 . . . . 5 (c ^ d) =< d
19 lear 161 . . . . 5 (c' ^ d) =< d
2018, 19lel2or 170 . . . 4 ((c ^ d) v (c' ^ d)) =< d
2117, 20bltr 138 . . 3 ((c ->1 d) ^ (c' ->1 d)) =< d
2216, 21lel2or 170 . 2 ((((a ->1 d) ^ (a' ->1 d)) v ((b ->1 d) ^ (b' ->1 d))) v ((c ->1 d) ^ (c' ->1 d))) =< d
235, 22letr 137 1 ((a ->1 d) ^ ((a' ->1 d) v ((b' ->1 d) ^ ((((a ->1 d) ^ (b ->1 d)) v ((a' ->1 d) ^ (b' ->1 d))) v ((((a ->1 d) ^ (c ->1 d)) v ((a' ->1 d) ^ (c' ->1 d))) ^ (((b ->1 d) ^ (c ->1 d)) v ((b' ->1 d) ^ (c' ->1 d)))))))) =< d
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:  4oa  1039
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