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Theorem testmod 1213
 Description: A modular law experiment. (Contributed by NM, 21-Apr-2012.)
Assertion
Ref Expression
testmod (((ca) ∪ ((bc) ∩ (da))) ∩ (a ∪ (b ∩ (d ∪ ((ac) ∩ (bd)))))) = ((b ∩ ((((ac) ∪ ((bc) ∩ (da))) ∩ d) ∪ ((ac) ∩ (bd)))) ∪ a)

Proof of Theorem testmod
StepHypRef Expression
1 leao1 162 . . . . . . . 8 ((ac) ∩ (bd)) ≤ ((ac) ∪ ((bc) ∩ (da)))
21mli 1126 . . . . . . 7 ((((ac) ∪ ((bc) ∩ (da))) ∩ d) ∪ ((ac) ∩ (bd))) = (((ac) ∪ ((bc) ∩ (da))) ∩ (d ∪ ((ac) ∩ (bd))))
3 orass 75 . . . . . . . 8 ((ac) ∪ ((bc) ∩ (da))) = (a ∪ (c ∪ ((bc) ∩ (da))))
43ran 78 . . . . . . 7 (((ac) ∪ ((bc) ∩ (da))) ∩ (d ∪ ((ac) ∩ (bd)))) = ((a ∪ (c ∪ ((bc) ∩ (da)))) ∩ (d ∪ ((ac) ∩ (bd))))
52, 4tr 62 . . . . . 6 ((((ac) ∪ ((bc) ∩ (da))) ∩ d) ∪ ((ac) ∩ (bd))) = ((a ∪ (c ∪ ((bc) ∩ (da)))) ∩ (d ∪ ((ac) ∩ (bd))))
65lan 77 . . . . 5 (b ∩ ((((ac) ∪ ((bc) ∩ (da))) ∩ d) ∪ ((ac) ∩ (bd)))) = (b ∩ ((a ∪ (c ∪ ((bc) ∩ (da)))) ∩ (d ∪ ((ac) ∩ (bd)))))
76ror 71 . . . 4 ((b ∩ ((((ac) ∪ ((bc) ∩ (da))) ∩ d) ∪ ((ac) ∩ (bd)))) ∪ a) = ((b ∩ ((a ∪ (c ∪ ((bc) ∩ (da)))) ∩ (d ∪ ((ac) ∩ (bd))))) ∪ a)
8 an12 81 . . . . 5 (b ∩ ((a ∪ (c ∪ ((bc) ∩ (da)))) ∩ (d ∪ ((ac) ∩ (bd))))) = ((a ∪ (c ∪ ((bc) ∩ (da)))) ∩ (b ∩ (d ∪ ((ac) ∩ (bd)))))
98ror 71 . . . 4 ((b ∩ ((a ∪ (c ∪ ((bc) ∩ (da)))) ∩ (d ∪ ((ac) ∩ (bd))))) ∪ a) = (((a ∪ (c ∪ ((bc) ∩ (da)))) ∩ (b ∩ (d ∪ ((ac) ∩ (bd))))) ∪ a)
107, 9tr 62 . . 3 ((b ∩ ((((ac) ∪ ((bc) ∩ (da))) ∩ d) ∪ ((ac) ∩ (bd)))) ∪ a) = (((a ∪ (c ∪ ((bc) ∩ (da)))) ∩ (b ∩ (d ∪ ((ac) ∩ (bd))))) ∪ a)
11 leo 158 . . . 4 a ≤ (a ∪ (c ∪ ((bc) ∩ (da))))
1211mli 1126 . . 3 (((a ∪ (c ∪ ((bc) ∩ (da)))) ∩ (b ∩ (d ∪ ((ac) ∩ (bd))))) ∪ a) = ((a ∪ (c ∪ ((bc) ∩ (da)))) ∩ ((b ∩ (d ∪ ((ac) ∩ (bd)))) ∪ a))
13 orcom 73 . . . . 5 (a ∪ (c ∪ ((bc) ∩ (da)))) = ((c ∪ ((bc) ∩ (da))) ∪ a)
14 or32 82 . . . . 5 ((c ∪ ((bc) ∩ (da))) ∪ a) = ((ca) ∪ ((bc) ∩ (da)))
1513, 14tr 62 . . . 4 (a ∪ (c ∪ ((bc) ∩ (da)))) = ((ca) ∪ ((bc) ∩ (da)))
16 orcom 73 . . . 4 ((b ∩ (d ∪ ((ac) ∩ (bd)))) ∪ a) = (a ∪ (b ∩ (d ∪ ((ac) ∩ (bd)))))
1715, 162an 79 . . 3 ((a ∪ (c ∪ ((bc) ∩ (da)))) ∩ ((b ∩ (d ∪ ((ac) ∩ (bd)))) ∪ a)) = (((ca) ∪ ((bc) ∩ (da))) ∩ (a ∪ (b ∩ (d ∪ ((ac) ∩ (bd))))))
1810, 12, 173tr 65 . 2 ((b ∩ ((((ac) ∪ ((bc) ∩ (da))) ∩ d) ∪ ((ac) ∩ (bd)))) ∪ a) = (((ca) ∪ ((bc) ∩ (da))) ∩ (a ∪ (b ∩ (d ∪ ((ac) ∩ (bd))))))
1918cm 61 1 (((ca) ∪ ((bc) ∩ (da))) ∩ (a ∪ (b ∩ (d ∪ ((ac) ∩ (bd)))))) = ((b ∩ ((((ac) ∪ ((bc) ∩ (da))) ∩ d) ∪ ((ac) ∩ (bd)))) ∪ a)
 Colors of variables: term Syntax hints:   = wb 1   ∪ 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 1122 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:  testmod1  1214
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