Theorem oa4lem2 918

Description: Lemma for 3-var to 4-var OA.

Hypotheses

Ref	Expression
oa4lem1.1	a =< b_|_
oa4lem1.2	c =< d_|_

Assertion

Ref	Expression
oa4lem2	(c v d) =< ((a v c)_|_ ->2 d)

Proof of Theorem oa4lem2

Step	Hyp	Ref	Expression
1		leor 151	. . . . 5 c =< (a v c)
2		ax-a1 29	. . . . 5 (a v c) = (a v c)_|__|_
3	1, 2	lbtr 131	. . . 4 c =< (a v c)_|__|_
4		oa4lem1.2	. . . 4 c =< d_|_
5	3, 4	ler2an 165	. . 3 c =< ((a v c)_|__|_ ^ d_|_)
6	5	lelor 158	. 2 (d v c) =< (d v ((a v c)_|__|_ ^ d_|_))
7		ax-a2 30	. 2 (c v d) = (d v c)
8		df-i2 44	. 2 ((a v c)_|_ ->2 d) = (d v ((a v c)_|__|_ ^ d_|_))
9	6, 7, 8	le3tr1 132	1 (c v d) =< ((a v c)_|_ ->2 d)

Syntax hints:   =< wle 2  _|_wn 4   v wo 6   ^ wa 7   ->2 wi2 14

This theorem is referenced by:  oa4lem3 919

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-i2 44  df-le1 122  df-le2 123

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