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Theorem nom51 332
Description: Part of Lemma 3.3(15) from "Non-Orthomodular Models..." paper. (Contributed by NM, 7-Feb-1999.)
Assertion
Ref Expression
nom51 ((ab) ≡1 b) = (a2 b)

Proof of Theorem nom51
StepHypRef Expression
1 ancom 74 . . . . . . . . 9 (ba ) = (ab )
2 anor3 90 . . . . . . . . 9 (ab ) = (ab)
31, 2ax-r2 36 . . . . . . . 8 (ba ) = (ab)
43ax-r1 35 . . . . . . 7 (ab) = (ba )
54ax-r4 37 . . . . . 6 (ab) = (ba )
65lor 70 . . . . 5 (b ∪ (ab) ) = (b ∪ (ba ) )
72ax-r1 35 . . . . . . . . 9 (ab) = (ab )
8 ancom 74 . . . . . . . . 9 (ab ) = (ba )
97, 8ax-r2 36 . . . . . . . 8 (ab) = (ba )
109ax-r4 37 . . . . . . 7 (ab) = (ba )
1110lan 77 . . . . . 6 (b ∩ (ab) ) = (b ∩ (ba ) )
124, 112or 72 . . . . 5 ((ab) ∪ (b ∩ (ab) )) = ((ba ) ∪ (b ∩ (ba ) ))
136, 122an 79 . . . 4 ((b ∪ (ab) ) ∩ ((ab) ∪ (b ∩ (ab) ))) = ((b ∪ (ba ) ) ∩ ((ba ) ∪ (b ∩ (ba ) )))
14 df-id2 51 . . . 4 (b2 (ab) ) = ((b ∪ (ab) ) ∩ ((ab) ∪ (b ∩ (ab) )))
15 df-id2 51 . . . 4 (b2 (ba )) = ((b ∪ (ba ) ) ∩ ((ba ) ∪ (b ∩ (ba ) )))
1613, 14, 153tr1 63 . . 3 (b2 (ab) ) = (b2 (ba ))
17 nom22 315 . . 3 (b2 (ba )) = (b1 a )
1816, 17ax-r2 36 . 2 (b2 (ab) ) = (b1 a )
19 nomcon1 302 . 2 ((ab) ≡1 b) = (b2 (ab) )
20 i2i1 267 . 2 (a2 b) = (b1 a )
2118, 19, 203tr1 63 1 ((ab) ≡1 b) = (a2 b)
Colors of variables: term
Syntax hints:   = wb 1   wn 4  wo 6  wa 7  1 wi1 12  2 wi2 13  1 wid1 18  2 wid2 19
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-i1 44  df-i2 45  df-id1 50  df-id2 51  df-le1 130  df-le2 131
This theorem is referenced by:  nom64  341
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