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Theorem nom14 311
 Description: Part of Lemma 3.3(14) from "Non-Orthomodular Models..." paper.
Assertion
Ref Expression
nom14 (a4 (ab)) = (a1 b)

Proof of Theorem nom14
StepHypRef Expression
1 ax-a2 31 . . . . 5 ((a ∩ (ab)) ∪ (a ∩ (ab))) = ((a ∩ (ab)) ∪ (a ∩ (ab)))
2 anass 76 . . . . . . . 8 ((aa) ∩ b) = (a ∩ (ab))
32ax-r1 35 . . . . . . 7 (a ∩ (ab)) = ((aa) ∩ b)
4 anidm 111 . . . . . . . 8 (aa) = a
54ran 78 . . . . . . 7 ((aa) ∩ b) = (ab)
63, 5ax-r2 36 . . . . . 6 (a ∩ (ab)) = (ab)
76lor 70 . . . . 5 ((a ∩ (ab)) ∪ (a ∩ (ab))) = ((a ∩ (ab)) ∪ (ab))
8 lear 161 . . . . . 6 (a ∩ (ab)) ≤ (ab)
98df-le2 131 . . . . 5 ((a ∩ (ab)) ∪ (ab)) = (ab)
101, 7, 93tr 65 . . . 4 ((a ∩ (ab)) ∪ (a ∩ (ab))) = (ab)
1110ax-r5 38 . . 3 (((a ∩ (ab)) ∪ (a ∩ (ab))) ∪ ((a ∪ (ab)) ∩ (ab) )) = ((ab) ∪ ((a ∪ (ab)) ∩ (ab) ))
12 leo 158 . . . . 5 (ab) ≤ ((ab) ∪ a )
13 lea 160 . . . . . 6 ((a ∪ (ab)) ∩ (ab) ) ≤ (a ∪ (ab))
14 ax-a2 31 . . . . . 6 (a ∪ (ab)) = ((ab) ∪ a )
1513, 14lbtr 139 . . . . 5 ((a ∪ (ab)) ∩ (ab) ) ≤ ((ab) ∪ a )
1612, 15lel2or 170 . . . 4 ((ab) ∪ ((a ∪ (ab)) ∩ (ab) )) ≤ ((ab) ∪ a )
17 leo 158 . . . . . 6 a ≤ (a ∪ (ab))
18 lea 160 . . . . . . 7 (ab) ≤ a
1918lecon 154 . . . . . 6 a ≤ (ab)
2017, 19ler2an 173 . . . . 5 a ≤ ((a ∪ (ab)) ∩ (ab) )
2120lelor 166 . . . 4 ((ab) ∪ a ) ≤ ((ab) ∪ ((a ∪ (ab)) ∩ (ab) ))
2216, 21lebi 145 . . 3 ((ab) ∪ ((a ∪ (ab)) ∩ (ab) )) = ((ab) ∪ a )
23 ax-a2 31 . . 3 ((ab) ∪ a ) = (a ∪ (ab))
2411, 22, 233tr 65 . 2 (((a ∩ (ab)) ∪ (a ∩ (ab))) ∪ ((a ∪ (ab)) ∩ (ab) )) = (a ∪ (ab))
25 df-i4 47 . 2 (a4 (ab)) = (((a ∩ (ab)) ∪ (a ∩ (ab))) ∪ ((a ∪ (ab)) ∩ (ab) ))
26 df-i1 44 . 2 (a1 b) = (a ∪ (ab))
2724, 25, 263tr1 63 1 (a4 (ab)) = (a1 b)
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7   →1 wi1 12   →4 wi4 15 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-i4 47  df-le1 130  df-le2 131 This theorem is referenced by:  nom43  328  lem3.3.7i4e3  1071
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