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Theorem lem4.6.6i3j0 1096
Description: Equation 4.14 of [MegPav2000] p. 23. The variable i in the paper is set to 3, and j is set to 0. (Contributed by Roy F. Longton, 3-Jul-05.)
Assertion
Ref Expression
lem4.6.6i3j0 ((a ->3 b) v (a ->0 b)) = (a ->0 b)
Proof of Theorem lem4.6.6i3j0
StepHypRef Expression
1 ax-a3 32 . . 3 ((((a' ^ b) v (a' ^ b')) v (a ^ (a' v b))) v (a' v b)) = (((a' ^ b) v (a' ^ b')) v ((a ^ (a' v b)) v (a' v b)))
2 ax-a3 32 . . . . 5 (((a ^ (a' v b)) v a') v b) = ((a ^ (a' v b)) v (a' v b))
32ax-r1 35 . . . 4 ((a ^ (a' v b)) v (a' v b)) = (((a ^ (a' v b)) v a') v b)
43lor 70 . . 3 (((a' ^ b) v (a' ^ b')) v ((a ^ (a' v b)) v (a' v b))) = (((a' ^ b) v (a' ^ b')) v (((a ^ (a' v b)) v a') v b))
5 ax-a2 31 . . . . . . 7 ((a ^ (a' v b)) v a') = (a' v (a ^ (a' v b)))
6 omln 446 . . . . . . 7 (a' v (a ^ (a' v b))) = (a' v b)
75, 6ax-r2 36 . . . . . 6 ((a ^ (a' v b)) v a') = (a' v b)
87ax-r5 38 . . . . 5 (((a ^ (a' v b)) v a') v b) = ((a' v b) v b)
98lor 70 . . . 4 (((a' ^ b) v (a' ^ b')) v (((a ^ (a' v b)) v a') v b)) = (((a' ^ b) v (a' ^ b')) v ((a' v b) v b))
10 leid 148 . . . . . . 7 (a' v b) =< (a' v b)
11 leor 159 . . . . . . 7 b =< (a' v b)
1210, 11lel2or 170 . . . . . 6 ((a' v b) v b) =< (a' v b)
13 leo 158 . . . . . 6 (a' v b) =< ((a' v b) v b)
1412, 13lebi 145 . . . . 5 ((a' v b) v b) = (a' v b)
1514lor 70 . . . 4 (((a' ^ b) v (a' ^ b')) v ((a' v b) v b)) = (((a' ^ b) v (a' ^ b')) v (a' v b))
16 leao1 162 . . . . . 6 (a' ^ b) =< (a' v b)
17 leao1 162 . . . . . 6 (a' ^ b') =< (a' v b)
1816, 17lel2or 170 . . . . 5 ((a' ^ b) v (a' ^ b')) =< (a' v b)
1918df-le2 131 . . . 4 (((a' ^ b) v (a' ^ b')) v (a' v b)) = (a' v b)
209, 15, 193tr 65 . . 3 (((a' ^ b) v (a' ^ b')) v (((a ^ (a' v b)) v a') v b)) = (a' v b)
211, 4, 203tr 65 . 2 ((((a' ^ b) v (a' ^ b')) v (a ^ (a' v b))) v (a' v b)) = (a' v b)
22 df-i3 46 . . 3 (a ->3 b) = (((a' ^ b) v (a' ^ b')) v (a ^ (a' v b)))
23 df-i0 43 . . 3 (a ->0 b) = (a' v b)
2422, 232or 72 . 2 ((a ->3 b) v (a ->0 b)) = ((((a' ^ b) v (a' ^ b')) v (a ^ (a' v b))) v (a' v b))
2521, 24, 233tr1 63 1 ((a ->3 b) v (a ->0 b)) = (a ->0 b)
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v wo 6   ^ wa 7   ->0 wi0 11   ->3 wi3 14
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  ax-r3 439
This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i0 43  df-i3 46  df-le1 130  df-le2 131
This theorem is referenced by: (None)
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