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Theorem testmod2 1213
Description: A modular law experiment.
Assertion
Ref Expression
testmod2 ((a v b) ^ (a v (c v d))) = (a v (b ^ (((a v c) ^ (b v d)) v (d ^ ((a v c) v ((b v c) ^ (d v a)))))))
Proof of Theorem testmod2
StepHypRef Expression
1 orass 75 . . . . 5 ((a v c) v d) = (a v (c v d))
21lan 77 . . . 4 ((a v b) ^ ((a v c) v d)) = ((a v b) ^ (a v (c v d)))
32cm 61 . . 3 ((a v b) ^ (a v (c v d))) = ((a v b) ^ ((a v c) v d))
4 leo 158 . . . . 5 a =< (a v c)
54ler 149 . . . 4 a =< ((a v c) v d)
65mlduali 1126 . . 3 ((a v b) ^ ((a v c) v d)) = (a v (b ^ ((a v c) v d)))
73, 6tr 62 . 2 ((a v b) ^ (a v (c v d))) = (a v (b ^ ((a v c) v d)))
8 leo 158 . . . . . . . . 9 b =< (b v d)
9 leor 159 . . . . . . . . 9 b =< ((a v c) v b)
108, 9ler2an 173 . . . . . . . 8 b =< ((b v d) ^ ((a v c) v b))
1110df2le2 136 . . . . . . 7 (b ^ ((b v d) ^ ((a v c) v b))) = b
1211ran 78 . . . . . 6 ((b ^ ((b v d) ^ ((a v c) v b))) ^ ((a v c) v d)) = (b ^ ((a v c) v d))
1312cm 61 . . . . 5 (b ^ ((a v c) v d)) = ((b ^ ((b v d) ^ ((a v c) v b))) ^ ((a v c) v d))
14 anass 76 . . . . 5 ((b ^ ((b v d) ^ ((a v c) v b))) ^ ((a v c) v d)) = (b ^ (((b v d) ^ ((a v c) v b)) ^ ((a v c) v d)))
1513, 14tr 62 . . . 4 (b ^ ((a v c) v d)) = (b ^ (((b v d) ^ ((a v c) v b)) ^ ((a v c) v d)))
16 an32 83 . . . . . . . . . 10 (((b v d) ^ ((a v c) v b)) ^ ((a v c) v d)) = (((b v d) ^ ((a v c) v d)) ^ ((a v c) v b))
17 leor 159 . . . . . . . . . . . . 13 d =< (b v d)
1817mldual2i 1125 . . . . . . . . . . . 12 ((b v d) ^ ((a v c) v d)) = (((b v d) ^ (a v c)) v d)
19 ancom 74 . . . . . . . . . . . . 13 ((b v d) ^ (a v c)) = ((a v c) ^ (b v d))
2019ror 71 . . . . . . . . . . . 12 (((b v d) ^ (a v c)) v d) = (((a v c) ^ (b v d)) v d)
2118, 20tr 62 . . . . . . . . . . 11 ((b v d) ^ ((a v c) v d)) = (((a v c) ^ (b v d)) v d)
2221ran 78 . . . . . . . . . 10 (((b v d) ^ ((a v c) v d)) ^ ((a v c) v b)) = ((((a v c) ^ (b v d)) v d) ^ ((a v c) v b))
2316, 22tr 62 . . . . . . . . 9 (((b v d) ^ ((a v c) v b)) ^ ((a v c) v d)) = ((((a v c) ^ (b v d)) v d) ^ ((a v c) v b))
24 lea 160 . . . . . . . . . . . . 13 ((a v c) ^ (b v d)) =< (a v c)
2524leror 152 . . . . . . . . . . . 12 (((a v c) ^ (b v d)) v d) =< ((a v c) v d)
2625df2le2 136 . . . . . . . . . . 11 ((((a v c) ^ (b v d)) v d) ^ ((a v c) v d)) = (((a v c) ^ (b v d)) v d)
2726ran 78 . . . . . . . . . 10 (((((a v c) ^ (b v d)) v d) ^ ((a v c) v d)) ^ ((a v c) v b)) = ((((a v c) ^ (b v d)) v d) ^ ((a v c) v b))
2827cm 61 . . . . . . . . 9 ((((a v c) ^ (b v d)) v d) ^ ((a v c) v b)) = (((((a v c) ^ (b v d)) v d) ^ ((a v c) v d)) ^ ((a v c) v b))
2923, 28tr 62 . . . . . . . 8 (((b v d) ^ ((a v c) v b)) ^ ((a v c) v d)) = (((((a v c) ^ (b v d)) v d) ^ ((a v c) v d)) ^ ((a v c) v b))
30 anass 76 . . . . . . . 8 (((((a v c) ^ (b v d)) v d) ^ ((a v c) v d)) ^ ((a v c) v b)) = ((((a v c) ^ (b v d)) v d) ^ (((a v c) v d) ^ ((a v c) v b)))
3129, 30tr 62 . . . . . . 7 (((b v d) ^ ((a v c) v b)) ^ ((a v c) v d)) = ((((a v c) ^ (b v d)) v d) ^ (((a v c) v d) ^ ((a v c) v b)))
32 l42modlem1 1147 . . . . . . . . 9 (((a v c) v d) ^ ((a v c) v b)) = ((a v c) v ((a v d) ^ (c v b)))
33 orcom 73 . . . . . . . . . . . 12 (a v d) = (d v a)
34 orcom 73 . . . . . . . . . . . 12 (c v b) = (b v c)
3533, 342an 79 . . . . . . . . . . 11 ((a v d) ^ (c v b)) = ((d v a) ^ (b v c))
36 ancom 74 . . . . . . . . . . 11 ((d v a) ^ (b v c)) = ((b v c) ^ (d v a))
3735, 36tr 62 . . . . . . . . . 10 ((a v d) ^ (c v b)) = ((b v c) ^ (d v a))
3837lor 70 . . . . . . . . 9 ((a v c) v ((a v d) ^ (c v b))) = ((a v c) v ((b v c) ^ (d v a)))
3932, 38tr 62 . . . . . . . 8 (((a v c) v d) ^ ((a v c) v b)) = ((a v c) v ((b v c) ^ (d v a)))
4039lan 77 . . . . . . 7 ((((a v c) ^ (b v d)) v d) ^ (((a v c) v d) ^ ((a v c) v b))) = ((((a v c) ^ (b v d)) v d) ^ ((a v c) v ((b v c) ^ (d v a))))
4131, 40tr 62 . . . . . 6 (((b v d) ^ ((a v c) v b)) ^ ((a v c) v d)) = ((((a v c) ^ (b v d)) v d) ^ ((a v c) v ((b v c) ^ (d v a))))
42 leao1 162 . . . . . . 7 ((a v c) ^ (b v d)) =< ((a v c) v ((b v c) ^ (d v a)))
4342mlduali 1126 . . . . . 6 ((((a v c) ^ (b v d)) v d) ^ ((a v c) v ((b v c) ^ (d v a)))) = (((a v c) ^ (b v d)) v (d ^ ((a v c) v ((b v c) ^ (d v a)))))
4441, 43tr 62 . . . . 5 (((b v d) ^ ((a v c) v b)) ^ ((a v c) v d)) = (((a v c) ^ (b v d)) v (d ^ ((a v c) v ((b v c) ^ (d v a)))))
4544lan 77 . . . 4 (b ^ (((b v d) ^ ((a v c) v b)) ^ ((a v c) v d))) = (b ^ (((a v c) ^ (b v d)) v (d ^ ((a v c) v ((b v c) ^ (d v a))))))
4615, 45tr 62 . . 3 (b ^ ((a v c) v d)) = (b ^ (((a v c) ^ (b v d)) v (d ^ ((a v c) v ((b v c) ^ (d v a))))))
4746lor 70 . 2 (a v (b ^ ((a v c) v d))) = (a v (b ^ (((a v c) ^ (b v d)) v (d ^ ((a v c) v ((b v c) ^ (d v a)))))))
487, 47tr 62 1 ((a v b) ^ (a v (c v d))) = (a v (b ^ (((a v c) ^ (b v d)) v (d ^ ((a v c) v ((b v c) ^ (d v a)))))))
Colors of variables: term
Syntax hints:   = wb 1   v wo 6   ^ wa 7
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-ml 1120
This theorem depends on definitions:  df-a 40  df-t 41  df-f 42  df-le1 130  df-le2 131
This theorem is referenced by: (None)
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