Theorem distoah4 923

Description: Satisfaction of distributive law hypothesis.

Hypotheses

Ref	Expression
distoa.1	d = (a ->2 b)
distoa.2	e = ((b v c) ->1 ((a ->2 b) ^ (a ->2 c)))
distoa.3	f = ((b v c) ->2 ((a ->2 b) ^ (a ->2 c)))

Assertion

Ref	Expression
distoah4	(d ^ (a ->2 c)) =< f

Proof of Theorem distoah4

Step	Hyp	Ref	Expression
1		leo 150	. 2 ((a ->2 b) ^ (a ->2 c)) =< (((a ->2 b) ^ (a ->2 c)) v ((b v c)_|_ ^ ((a ->2 b) ^ (a ->2 c))_|_))
2		distoa.1	. . 3 d = (a ->2 b)
3	2	ran 71	. 2 (d ^ (a ->2 c)) = ((a ->2 b) ^ (a ->2 c))
4		distoa.3	. . 3 f = ((b v c) ->2 ((a ->2 b) ^ (a ->2 c)))
5		df-i2 44	. . 3 ((b v c) ->2 ((a ->2 b) ^ (a ->2 c))) = (((a ->2 b) ^ (a ->2 c)) v ((b v c)_|_ ^ ((a ->2 b) ^ (a ->2 c))_|_))
6	4, 5	ax-r2 35	. 2 f = (((a ->2 b) ^ (a ->2 c)) v ((b v c)_|_ ^ ((a ->2 b) ^ (a ->2 c))_|_))
7	1, 3, 6	le3tr1 132	1 (d ^ (a ->2 c)) =< f

Syntax hints:   = wb 1   =< wle 2  _|_wn 4   v wo 6   ^ wa 7   ->1 wi1 13   ->2 wi2 14

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

This theorem depends on definitions:  df-a 39  df-i2 44  df-le1 122  df-le2 123

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