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Theorem salem1 837
 Description: Lemma for attempt at Sasaki algebra.
Assertion
Ref Expression
salem1 (((a1 b) ∪ b) →1 b) = (a2 b)

Proof of Theorem salem1
StepHypRef Expression
1 u1lemob 630 . . . . . 6 ((a1 b) ∪ b) = (a b)
21ax-r4 37 . . . . 5 ((a1 b) ∪ b) = (a b)
3 anor1 88 . . . . . 6 (ab ) = (a b)
43ax-r1 35 . . . . 5 (a b) = (ab )
52, 4ax-r2 36 . . . 4 ((a1 b) ∪ b) = (ab )
61ran 78 . . . . 5 (((a1 b) ∪ b) ∩ b) = ((a b) ∩ b)
7 ax-a2 31 . . . . . . 7 (a b) = (ba )
87ran 78 . . . . . 6 ((a b) ∩ b) = ((ba ) ∩ b)
9 ancom 74 . . . . . 6 ((ba ) ∩ b) = (b ∩ (ba ))
108, 9ax-r2 36 . . . . 5 ((a b) ∩ b) = (b ∩ (ba ))
11 anabs 121 . . . . 5 (b ∩ (ba )) = b
126, 10, 113tr 65 . . . 4 (((a1 b) ∪ b) ∩ b) = b
135, 122or 72 . . 3 (((a1 b) ∪ b) ∪ (((a1 b) ∪ b) ∩ b)) = ((ab ) ∪ b)
14 ax-a2 31 . . 3 ((ab ) ∪ b) = (b ∪ (ab ))
1513, 14ax-r2 36 . 2 (((a1 b) ∪ b) ∪ (((a1 b) ∪ b) ∩ b)) = (b ∪ (ab ))
16 df-i1 44 . 2 (((a1 b) ∪ b) →1 b) = (((a1 b) ∪ b) ∪ (((a1 b) ∪ b) ∩ b))
17 df-i2 45 . 2 (a2 b) = (b ∪ (ab ))
1815, 16, 173tr1 63 1 (((a1 b) ∪ b) →1 b) = (a2 b)
 Colors of variables: term Syntax hints:   = wb 1  ⊥ 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-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-i1 44  df-i2 45  df-le1 130  df-le2 131 This theorem is referenced by: (None)
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