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Theorem distoah3 942
 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
distoah3 f ≤ ((bc) →0 ((a2 b) ∩ (a2 c)))

Proof of Theorem distoah3
StepHypRef Expression
1 leor 159 . 2 ((bc) →2 ((a2 b) ∩ (a2 c))) ≤ (((bc) →1 ((a2 b) ∩ (a2 c))) ∪ ((bc) →2 ((a2 b) ∩ (a2 c))))
2 distoa.3 . . 3 f = ((bc) →2 ((a2 b) ∩ (a2 c)))
32ax-r1 35 . 2 ((bc) →2 ((a2 b) ∩ (a2 c))) = f
4 u12lem 771 . 2 (((bc) →1 ((a2 b) ∩ (a2 c))) ∪ ((bc) →2 ((a2 b) ∩ (a2 c)))) = ((bc) →0 ((a2 b) ∩ (a2 c)))
51, 3, 4le3tr2 141 1 f ≤ ((bc) →0 ((a2 b) ∩ (a2 c)))
 Colors of variables: term 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)
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