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Theorem nom15 312
Description: Part of Lemma 3.3(14) from "Non-Orthomodular Models..." paper.
Assertion
Ref Expression
nom15 (a ->5 (a ^ b)) = (a ->1 b)
Proof of Theorem nom15
StepHypRef Expression
1 anass 76 . . . . . . . 8 ((a ^ a) ^ b) = (a ^ (a ^ b))
21ax-r1 35 . . . . . . 7 (a ^ (a ^ b)) = ((a ^ a) ^ b)
3 anidm 111 . . . . . . . 8 (a ^ a) = a
43ran 78 . . . . . . 7 ((a ^ a) ^ b) = (a ^ b)
52, 4ax-r2 36 . . . . . 6 (a ^ (a ^ b)) = (a ^ b)
65ax-r5 38 . . . . 5 ((a ^ (a ^ b)) v (a' ^ (a ^ b))) = ((a ^ b) v (a' ^ (a ^ b)))
7 ax-a2 31 . . . . 5 ((a ^ b) v (a' ^ (a ^ b))) = ((a' ^ (a ^ b)) v (a ^ b))
8 lear 161 . . . . . 6 (a' ^ (a ^ b)) =< (a ^ b)
98df-le2 131 . . . . 5 ((a' ^ (a ^ b)) v (a ^ b)) = (a ^ b)
106, 7, 93tr 65 . . . 4 ((a ^ (a ^ b)) v (a' ^ (a ^ b))) = (a ^ b)
11 oran3 93 . . . . . . 7 (a' v b') = (a ^ b)'
1211lan 77 . . . . . 6 (a' ^ (a' v b')) = (a' ^ (a ^ b)')
1312ax-r1 35 . . . . 5 (a' ^ (a ^ b)') = (a' ^ (a' v b'))
14 anabs 121 . . . . 5 (a' ^ (a' v b')) = a'
1513, 14ax-r2 36 . . . 4 (a' ^ (a ^ b)') = a'
1610, 152or 72 . . 3 (((a ^ (a ^ b)) v (a' ^ (a ^ b))) v (a' ^ (a ^ b)')) = ((a ^ b) v a')
17 ax-a2 31 . . 3 ((a ^ b) v a') = (a' v (a ^ b))
1816, 17ax-r2 36 . 2 (((a ^ (a ^ b)) v (a' ^ (a ^ b))) v (a' ^ (a ^ b)')) = (a' v (a ^ b))
19 df-i5 48 . 2 (a ->5 (a ^ b)) = (((a ^ (a ^ b)) v (a' ^ (a ^ b))) v (a' ^ (a ^ b)'))
20 df-i1 44 . 2 (a ->1 b) = (a' v (a ^ b))
2118, 19, 203tr1 63 1 (a ->5 (a ^ b)) = (a ->1 b)
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v wo 6   ^ wa 7   ->1 wi1 12   ->5 wi5 16
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-i5 48  df-le1 130  df-le2 131
This theorem is referenced by:  nom45  330  lem3.3.7i5e3  1074
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