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Theorem testmod2expanded 1214
Description: A modular law experiment.
Assertion
Ref Expression
testmod2expanded ((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 testmod2expanded
StepHypRef Expression
1 orass 75 . . . . . . . . . . . . 13 ((a v c) v d) = (a v (c v d))
21lan 77 . . . . . . . . . . . 12 ((a v b) ^ ((a v c) v d)) = ((a v b) ^ (a v (c v d)))
32cm 61 . . . . . . . . . . 11 ((a v b) ^ (a v (c v d))) = ((a v b) ^ ((a v c) v d))
4 leo 158 . . . . . . . . . . . . 13 a =< (a v c)
54ler 149 . . . . . . . . . . . 12 a =< ((a v c) v d)
65mlduali 1126 . . . . . . . . . . 11 ((a v b) ^ ((a v c) v d)) = (a v (b ^ ((a v c) v d)))
73, 6tr 62 . . . . . . . . . 10 ((a v b) ^ (a v (c v d))) = (a v (b ^ ((a v c) v d)))
8 leo 158 . . . . . . . . . . . . . . 15 b =< (b v d)
9 leor 159 . . . . . . . . . . . . . . 15 b =< ((a v c) v b)
108, 9ler2an 173 . . . . . . . . . . . . . 14 b =< ((b v d) ^ ((a v c) v b))
1110df2le2 136 . . . . . . . . . . . . 13 (b ^ ((b v d) ^ ((a v c) v b))) = b
1211ran 78 . . . . . . . . . . . 12 ((b ^ ((b v d) ^ ((a v c) v b))) ^ ((a v c) v d)) = (b ^ ((a v c) v d))
1312cm 61 . . . . . . . . . . 11 (b ^ ((a v c) v d)) = ((b ^ ((b v d) ^ ((a v c) v b))) ^ ((a v c) v d))
1413lor 70 . . . . . . . . . 10 (a v (b ^ ((a v c) v d))) = (a v ((b ^ ((b v d) ^ ((a v c) v b))) ^ ((a v c) v d)))
157, 14tr 62 . . . . . . . . 9 ((a v b) ^ (a v (c v d))) = (a v ((b ^ ((b v d) ^ ((a v c) v b))) ^ ((a v c) v d)))
16 anass 76 . . . . . . . . . 10 ((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)))
1716lor 70 . . . . . . . . 9 (a v ((b ^ ((b v d) ^ ((a v c) v b))) ^ ((a v c) v d))) = (a v (b ^ (((b v d) ^ ((a v c) v b)) ^ ((a v c) v d))))
1815, 17tr 62 . . . . . . . 8 ((a v b) ^ (a v (c v d))) = (a v (b ^ (((b v d) ^ ((a v c) v b)) ^ ((a v c) v d))))
19 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))
2019lan 77 . . . . . . . . 9 (b ^ (((b v d) ^ ((a v c) v b)) ^ ((a v c) v d))) = (b ^ (((b v d) ^ ((a v c) v d)) ^ ((a v c) v b)))
2120lor 70 . . . . . . . 8 (a v (b ^ (((b v d) ^ ((a v c) v b)) ^ ((a v c) v d)))) = (a v (b ^ (((b v d) ^ ((a v c) v d)) ^ ((a v c) v b))))
2218, 21tr 62 . . . . . . 7 ((a v b) ^ (a v (c v d))) = (a v (b ^ (((b v d) ^ ((a v c) v d)) ^ ((a v c) v b))))
23 leor 159 . . . . . . . . . . 11 d =< (b v d)
2423mldual2i 1125 . . . . . . . . . 10 ((b v d) ^ ((a v c) v d)) = (((b v d) ^ (a v c)) v d)
2524ran 78 . . . . . . . . 9 (((b v d) ^ ((a v c) v d)) ^ ((a v c) v b)) = ((((b v d) ^ (a v c)) v d) ^ ((a v c) v b))
2625lan 77 . . . . . . . 8 (b ^ (((b v d) ^ ((a v c) v d)) ^ ((a v c) v b))) = (b ^ ((((b v d) ^ (a v c)) v d) ^ ((a v c) v b)))
2726lor 70 . . . . . . 7 (a v (b ^ (((b v d) ^ ((a v c) v d)) ^ ((a v c) v b)))) = (a v (b ^ ((((b v d) ^ (a v c)) v d) ^ ((a v c) v b))))
2822, 27tr 62 . . . . . 6 ((a v b) ^ (a v (c v d))) = (a v (b ^ ((((b v d) ^ (a v c)) v d) ^ ((a v c) v b))))
29 ancom 74 . . . . . . . . . 10 ((b v d) ^ (a v c)) = ((a v c) ^ (b v d))
3029ror 71 . . . . . . . . 9 (((b v d) ^ (a v c)) v d) = (((a v c) ^ (b v d)) v d)
3130ran 78 . . . . . . . 8 ((((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))
3231lan 77 . . . . . . 7 (b ^ ((((b v d) ^ (a v c)) v d) ^ ((a v c) v b))) = (b ^ ((((a v c) ^ (b v d)) v d) ^ ((a v c) v b)))
3332lor 70 . . . . . 6 (a v (b ^ ((((b v d) ^ (a v c)) v d) ^ ((a v c) v b)))) = (a v (b ^ ((((a v c) ^ (b v d)) v d) ^ ((a v c) v b))))
3428, 33tr 62 . . . . 5 ((a v b) ^ (a v (c v d))) = (a v (b ^ ((((a v c) ^ (b v d)) v d) ^ ((a v c) v b))))
35 lea 160 . . . . . . . . . . . 12 ((a v c) ^ (b v d)) =< (a v c)
3635leror 152 . . . . . . . . . . 11 (((a v c) ^ (b v d)) v d) =< ((a v c) v d)
3736df2le2 136 . . . . . . . . . 10 ((((a v c) ^ (b v d)) v d) ^ ((a v c) v d)) = (((a v c) ^ (b v d)) v d)
3837ran 78 . . . . . . . . 9 (((((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))
3938cm 61 . . . . . . . 8 ((((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))
40 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)))
4139, 40tr 62 . . . . . . 7 ((((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)))
4241lan 77 . . . . . 6 (b ^ ((((a v c) ^ (b v d)) v d) ^ ((a v c) v b))) = (b ^ ((((a v c) ^ (b v d)) v d) ^ (((a v c) v d) ^ ((a v c) v b))))
4342lor 70 . . . . 5 (a v (b ^ ((((a v c) ^ (b v d)) v d) ^ ((a v c) v b)))) = (a v (b ^ ((((a v c) ^ (b v d)) v d) ^ (((a v c) v d) ^ ((a v c) v b)))))
4434, 43tr 62 . . . 4 ((a v b) ^ (a v (c v d))) = (a v (b ^ ((((a v c) ^ (b v d)) v d) ^ (((a v c) v d) ^ ((a v c) v b)))))
45 l42modlem1 1147 . . . . . . 7 (((a v c) v d) ^ ((a v c) v b)) = ((a v c) v ((a v d) ^ (c v b)))
4645lan 77 . . . . . 6 ((((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 ((a v d) ^ (c v b))))
4746lan 77 . . . . 5 (b ^ ((((a v c) ^ (b v d)) v d) ^ (((a v c) v d) ^ ((a v c) v b)))) = (b ^ ((((a v c) ^ (b v d)) v d) ^ ((a v c) v ((a v d) ^ (c v b)))))
4847lor 70 . . . 4 (a v (b ^ ((((a v c) ^ (b v d)) v d) ^ (((a v c) v d) ^ ((a v c) v b))))) = (a v (b ^ ((((a v c) ^ (b v d)) v d) ^ ((a v c) v ((a v d) ^ (c v b))))))
4944, 48tr 62 . . 3 ((a v b) ^ (a v (c v d))) = (a v (b ^ ((((a v c) ^ (b v d)) v d) ^ ((a v c) v ((a v d) ^ (c v b))))))
50 orcom 73 . . . . . . . . 9 (a v d) = (d v a)
51 orcom 73 . . . . . . . . 9 (c v b) = (b v c)
5250, 512an 79 . . . . . . . 8 ((a v d) ^ (c v b)) = ((d v a) ^ (b v c))
53 ancom 74 . . . . . . . 8 ((d v a) ^ (b v c)) = ((b v c) ^ (d v a))
5452, 53tr 62 . . . . . . 7 ((a v d) ^ (c v b)) = ((b v c) ^ (d v a))
5554lor 70 . . . . . 6 ((a v c) v ((a v d) ^ (c v b))) = ((a v c) v ((b v c) ^ (d v a)))
5655lan 77 . . . . 5 ((((a v c) ^ (b v d)) v d) ^ ((a v c) v ((a v d) ^ (c v b)))) = ((((a v c) ^ (b v d)) v d) ^ ((a v c) v ((b v c) ^ (d v a))))
5756lan 77 . . . 4 (b ^ ((((a v c) ^ (b v d)) v d) ^ ((a v c) v ((a v d) ^ (c v b))))) = (b ^ ((((a v c) ^ (b v d)) v d) ^ ((a v c) v ((b v c) ^ (d v a)))))
5857lor 70 . . 3 (a v (b ^ ((((a v c) ^ (b v d)) v d) ^ ((a v c) v ((a v d) ^ (c v b)))))) = (a v (b ^ ((((a v c) ^ (b v d)) v d) ^ ((a v c) v ((b v c) ^ (d v a))))))
5949, 58tr 62 . 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))))))
60 leao1 162 . . . . 5 ((a v c) ^ (b v d)) =< ((a v c) v ((b v c) ^ (d v a)))
6160mlduali 1126 . . . 4 ((((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)))))
6261lan 77 . . 3 (b ^ ((((a v c) ^ (b v d)) v d) ^ ((a v c) v ((b v c) ^ (d v a))))) = (b ^ (((a v c) ^ (b v d)) v (d ^ ((a v c) v ((b v c) ^ (d v a))))))
6362lor 70 . 2 (a v (b ^ ((((a v c) ^ (b v d)) v d) ^ ((a v c) v ((b v c) ^ (d v a)))))) = (a v (b ^ (((a v c) ^ (b v d)) v (d ^ ((a v c) v ((b v c) ^ (d v a)))))))
6459, 63tr 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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