Theorem distoah4 923

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
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)) ∪ ((b ∪ 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 ∪ c) →2 ((a →2 b) ∩ (a →2 c)))
5		df-i2 44	. . 3 ((b ∪ c) →2 ((a →2 b) ∩ (a →2 c))) = (((a →2 b) ∩ (a →2 c)) ∪ ((b ∪ c)⊥ ∩ ((a →2 b) ∩ (a →2 c))⊥ ))
6	4, 5	ax-r2 35	. 2 f = (((a →2 b) ∩ (a →2 c)) ∪ ((b ∪ 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   ∪ 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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