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Theorem nom14 311
Description: Part of Lemma 3.3(14) from "Non-Orthomodular Models..." paper.
Assertion
Ref Expression
nom14 (a ->4 (a ^ b)) = (a ->1 b)
Proof of Theorem nom14
StepHypRef Expression
1 ax-a2 31 . . . . 5 ((a ^ (a ^ b)) v (a' ^ (a ^ b))) = ((a' ^ (a ^ b)) v (a ^ (a ^ b)))
2 anass 76 . . . . . . . 8 ((a ^ a) ^ b) = (a ^ (a ^ b))
32ax-r1 35 . . . . . . 7 (a ^ (a ^ b)) = ((a ^ a) ^ b)
4 anidm 111 . . . . . . . 8 (a ^ a) = a
54ran 78 . . . . . . 7 ((a ^ a) ^ b) = (a ^ b)
63, 5ax-r2 36 . . . . . 6 (a ^ (a ^ b)) = (a ^ b)
76lor 70 . . . . 5 ((a' ^ (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)
101, 7, 93tr 65 . . . 4 ((a ^ (a ^ b)) v (a' ^ (a ^ b))) = (a ^ b)
1110ax-r5 38 . . 3 (((a ^ (a ^ b)) v (a' ^ (a ^ b))) v ((a' v (a ^ b)) ^ (a ^ b)')) = ((a ^ b) v ((a' v (a ^ b)) ^ (a ^ b)'))
12 leo 158 . . . . 5 (a ^ b) =< ((a ^ b) v a')
13 lea 160 . . . . . 6 ((a' v (a ^ b)) ^ (a ^ b)') =< (a' v (a ^ b))
14 ax-a2 31 . . . . . 6 (a' v (a ^ b)) = ((a ^ b) v a')
1513, 14lbtr 139 . . . . 5 ((a' v (a ^ b)) ^ (a ^ b)') =< ((a ^ b) v a')
1612, 15lel2or 170 . . . 4 ((a ^ b) v ((a' v (a ^ b)) ^ (a ^ b)')) =< ((a ^ b) v a')
17 leo 158 . . . . . 6 a' =< (a' v (a ^ b))
18 lea 160 . . . . . . 7 (a ^ b) =< a
1918lecon 154 . . . . . 6 a' =< (a ^ b)'
2017, 19ler2an 173 . . . . 5 a' =< ((a' v (a ^ b)) ^ (a ^ b)')
2120lelor 166 . . . 4 ((a ^ b) v a') =< ((a ^ b) v ((a' v (a ^ b)) ^ (a ^ b)'))
2216, 21lebi 145 . . 3 ((a ^ b) v ((a' v (a ^ b)) ^ (a ^ b)')) = ((a ^ b) v a')
23 ax-a2 31 . . 3 ((a ^ b) v a') = (a' v (a ^ b))
2411, 22, 233tr 65 . 2 (((a ^ (a ^ b)) v (a' ^ (a ^ b))) v ((a' v (a ^ b)) ^ (a ^ b)')) = (a' v (a ^ b))
25 df-i4 47 . 2 (a ->4 (a ^ b)) = (((a ^ (a ^ b)) v (a' ^ (a ^ b))) v ((a' v (a ^ b)) ^ (a ^ b)'))
26 df-i1 44 . 2 (a ->1 b) = (a' v (a ^ b))
2724, 25, 263tr1 63 1 (a ->4 (a ^ b)) = (a ->1 b)
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v 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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