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Theorem testmod3 1217
 Description: A modular law experiment. (Contributed by NM, 21-Apr-2012.)
Assertion
Ref Expression
testmod3 (((ca) ∪ ((bc) ∩ (da))) ∩ (a ∪ (b ∩ (d ∪ ((ac) ∩ (bd)))))) = (a ∪ (((ca) ∪ ((bc) ∩ (da))) ∩ (b ∩ (d ∪ ((ac) ∩ (bd))))))

Proof of Theorem testmod3
StepHypRef Expression
1 orcom 73 . . . 4 (a ∪ (((ca) ∪ ((bc) ∩ (da))) ∩ (b ∩ (d ∪ ((ac) ∩ (bd)))))) = ((((ca) ∪ ((bc) ∩ (da))) ∩ (b ∩ (d ∪ ((ac) ∩ (bd))))) ∪ a)
2 leor 159 . . . . . 6 a ≤ (ca)
32ler 149 . . . . 5 a ≤ ((ca) ∪ ((bc) ∩ (da)))
43mli 1126 . . . 4 ((((ca) ∪ ((bc) ∩ (da))) ∩ (b ∩ (d ∪ ((ac) ∩ (bd))))) ∪ a) = (((ca) ∪ ((bc) ∩ (da))) ∩ ((b ∩ (d ∪ ((ac) ∩ (bd)))) ∪ a))
51, 4tr 62 . . 3 (a ∪ (((ca) ∪ ((bc) ∩ (da))) ∩ (b ∩ (d ∪ ((ac) ∩ (bd)))))) = (((ca) ∪ ((bc) ∩ (da))) ∩ ((b ∩ (d ∪ ((ac) ∩ (bd)))) ∪ a))
6 orcom 73 . . . 4 ((b ∩ (d ∪ ((ac) ∩ (bd)))) ∪ a) = (a ∪ (b ∩ (d ∪ ((ac) ∩ (bd)))))
76lan 77 . . 3 (((ca) ∪ ((bc) ∩ (da))) ∩ ((b ∩ (d ∪ ((ac) ∩ (bd)))) ∪ a)) = (((ca) ∪ ((bc) ∩ (da))) ∩ (a ∪ (b ∩ (d ∪ ((ac) ∩ (bd))))))
85, 7tr 62 . 2 (a ∪ (((ca) ∪ ((bc) ∩ (da))) ∩ (b ∩ (d ∪ ((ac) ∩ (bd)))))) = (((ca) ∪ ((bc) ∩ (da))) ∩ (a ∪ (b ∩ (d ∪ ((ac) ∩ (bd))))))
98cm 61 1 (((ca) ∪ ((bc) ∩ (da))) ∩ (a ∪ (b ∩ (d ∪ ((ac) ∩ (bd)))))) = (a ∪ (((ca) ∪ ((bc) ∩ (da))) ∩ (b ∩ (d ∪ ((ac) ∩ (bd))))))
 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: (None)
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