Theorem distoah3 942

Description: Satisfaction of distributive law hypothesis. (Contributed by NM, 29-Nov-1998.)

Hypotheses

Ref	Expression
distoa.1	d = (a →2 b)
distoa.2	e = ((b ∪ c) →1 ((a →2 b) ∩ (a →2 c)))
distoa.3	f = ((b ∪ c) →2 ((a →2 b) ∩ (a →2 c)))

Assertion

Ref	Expression
distoah3	f ≤ ((b ∪ c) →0 ((a →2 b) ∩ (a →2 c)))

Proof of Theorem distoah3

Step	Hyp	Ref	Expression
1		leor 159	. 2 ((b ∪ c) →2 ((a →2 b) ∩ (a →2 c))) ≤ (((b ∪ c) →1 ((a →2 b) ∩ (a →2 c))) ∪ ((b ∪ c) →2 ((a →2 b) ∩ (a →2 c))))
2		distoa.3	. . 3 f = ((b ∪ c) →2 ((a →2 b) ∩ (a →2 c)))
3	2	ax-r1 35	. 2 ((b ∪ c) →2 ((a →2 b) ∩ (a →2 c))) = f
4		u12lem 771	. 2 (((b ∪ c) →1 ((a →2 b) ∩ (a →2 c))) ∪ ((b ∪ c) →2 ((a →2 b) ∩ (a →2 c)))) = ((b ∪ c) →0 ((a →2 b) ∩ (a →2 c)))
5	1, 3, 4	le3tr2 141	1 f ≤ ((b ∪ c) →0 ((a →2 b) ∩ (a →2 c)))

Syntax hints:   = wb 1   ≤ wle 2   ∪ wo 6   ∩ wa 7   →₀ wi0 11   →₁ wi1 12   →₂ wi2 13

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-i0 43  df-i1 44  df-i2 45  df-le1 130  df-le2 131

This theorem is referenced by: (None)