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Theorem nom20 313
Description: Part of Lemma 3.3(14) from "Non-Orthomodular Models..." paper.
Assertion
Ref Expression
nom20 (a ==0 (a ^ b)) = (a ->1 b)
Proof of Theorem nom20
StepHypRef Expression
1 lea 160 . . . . . 6 (a ^ b) =< a
2 leor 159 . . . . . 6 a =< (b' v a)
31, 2letr 137 . . . . 5 (a ^ b) =< (b' v a)
43lelor 166 . . . 4 (a' v (a ^ b)) =< (a' v (b' v a))
5 ax-a3 32 . . . . . 6 ((a' v b') v a) = (a' v (b' v a))
65ax-r1 35 . . . . 5 (a' v (b' v a)) = ((a' v b') v a)
7 oran3 93 . . . . . 6 (a' v b') = (a ^ b)'
87ax-r5 38 . . . . 5 ((a' v b') v a) = ((a ^ b)' v a)
96, 8ax-r2 36 . . . 4 (a' v (b' v a)) = ((a ^ b)' v a)
104, 9lbtr 139 . . 3 (a' v (a ^ b)) =< ((a ^ b)' v a)
1110df2le2 136 . 2 ((a' v (a ^ b)) ^ ((a ^ b)' v a)) = (a' v (a ^ b))
12 df-id0 49 . 2 (a ==0 (a ^ b)) = ((a' v (a ^ b)) ^ ((a ^ b)' v a))
13 df-i1 44 . 2 (a ->1 b) = (a' v (a ^ b))
1411, 12, 133tr1 63 1 (a ==0 (a ^ b)) = (a ->1 b)
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v wo 6   ^ wa 7   ->1 wi1 12   ==0 wid0 17
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-id0 49  df-le1 130  df-le2 131
This theorem is referenced by:  nom30  319  nom50  331
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