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Theorem distoah4 943
 Description: Satisfaction of distributive law hypothesis.
Hypotheses
Ref Expression
distoa.1 d = (a2 b)
distoa.2 e = ((bc) →1 ((a2 b) ∩ (a2 c)))
distoa.3 f = ((bc) →2 ((a2 b) ∩ (a2 c)))
Assertion
Ref Expression
distoah4 (d ∩ (a2 c)) ≤ f

Proof of Theorem distoah4
StepHypRef Expression
1 leo 158 . 2 ((a2 b) ∩ (a2 c)) ≤ (((a2 b) ∩ (a2 c)) ∪ ((bc) ∩ ((a2 b) ∩ (a2 c)) ))
2 distoa.1 . . 3 d = (a2 b)
32ran 78 . 2 (d ∩ (a2 c)) = ((a2 b) ∩ (a2 c))
4 distoa.3 . . 3 f = ((bc) →2 ((a2 b) ∩ (a2 c)))
5 df-i2 45 . . 3 ((bc) →2 ((a2 b) ∩ (a2 c))) = (((a2 b) ∩ (a2 c)) ∪ ((bc) ∩ ((a2 b) ∩ (a2 c)) ))
64, 5ax-r2 36 . 2 f = (((a2 b) ∩ (a2 c)) ∪ ((bc) ∩ ((a2 b) ∩ (a2 c)) ))
71, 3, 6le3tr1 140 1 (d ∩ (a2 c)) ≤ f
 Colors of variables: term Syntax hints:   = wb 1   ≤ wle 2  ⊥ wn 4   ∪ wo 6   ∩ wa 7   →1 wi1 12   →2 wi2 13 This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38 This theorem depends on definitions:  df-a 40  df-i2 45  df-le1 130  df-le2 131 This theorem is referenced by: (None)
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