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Theorem oa4b 959
Description: Derivation of 4-OA law variant.
Hypothesis
Ref Expression
oa4b.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))
Assertion
Ref Expression
oa4b ((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)))))))) =< g
Proof of Theorem oa4b
StepHypRef Expression
1 oa4b.1 . 2 ((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))
2 lear 161 . . . 4 (a ^ g) =< g
3 lear 161 . . . 4 (c ^ g) =< g
42, 3lel2or 170 . . 3 ((a ^ g) v (c ^ g)) =< g
5 lear 161 . . 3 (e ^ g) =< g
64, 5lel2or 170 . 2 (((a ^ g) v (c ^ g)) v (e ^ g)) =< g
71, 6letr 137 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)))))))) =< g
Colors of variables: term
Syntax hints:   =< wle 2   v wo 6   ^ wa 7   ->1 wi1 12
This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38
This theorem depends on definitions:  df-a 40  df-t 41  df-f 42  df-le1 130  df-le2 131
This theorem is referenced by: (None)
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