Theorem distoah3 922

Description: Satisfaction of distributive law hypothesis.

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 151	. 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 34	. 2 ((b ∪ c) →2 ((a →2 b) ∩ (a →2 c))) = f
4		u12lem 753	. 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 133	1 f ≤ ((b ∪ c) →0 ((a →2 b) ∩ (a →2 c)))

Syntax hints:   = wb 1   ≤ wle 2   ∪ wo 6   ∩ wa 7   →₀ wi0 12   →₁ wi1 13   →₂ wi2 14

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-i0 42  df-i1 43  df-i2 44  df-le1 122  df-le2 123

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