Theorem oa4b 939

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

Step	Hyp	Ref	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 153	. . . 4 (a ^ g) =< g
3		lear 153	. . . 4 (c ^ g) =< g
4	2, 3	lel2or 162	. . 3 ((a ^ g) v (c ^ g)) =< g
5		lear 153	. . 3 (e ^ g) =< g
6	4, 5	lel2or 162	. 2 (((a ^ g) v (c ^ g)) v (e ^ g)) =< g
7	1, 6	letr 129	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

Syntax hints:   =< wle 2   v wo 6   ^ wa 7   ->1 wi1 13

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37

This theorem depends on definitions:  df-a 39  df-t 40  df-f 41  df-le1 122  df-le2 123

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