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Theorem dp41lemc0 1184
Description: Part of proof (4)=>(1) in Day/Pickering 1982. (Contributed by NM, 4-Apr-2012.)
Hypotheses
Ref Expression
dp41lem.1 c0 = ((a1a2) ∩ (b1b2))
dp41lem.2 c1 = ((a0a2) ∩ (b0b2))
dp41lem.3 c2 = ((a0a1) ∩ (b0b1))
dp41lem.4 p = (((a0b0) ∩ (a1b1)) ∩ (a2b2))
dp41lem.5 p2 = ((a0b0) ∩ (a1b1))
dp41lem.6 p2 ≤ (a2b2)
Assertion
Ref Expression
dp41lemc0 (((a0b0) ∪ b1) ∩ ((a0a1) ∪ b1)) = ((a0b1) ∪ ((a0b0) ∩ (a1b1)))

Proof of Theorem dp41lemc0
StepHypRef Expression
1 ax-a2 31 . . . . . . 7 (a0a1) = (a1a0)
21ror 71 . . . . . 6 ((a0a1) ∪ b1) = ((a1a0) ∪ b1)
3 or32 82 . . . . . 6 ((a1a0) ∪ b1) = ((a1b1) ∪ a0)
42, 3tr 62 . . . . 5 ((a0a1) ∪ b1) = ((a1b1) ∪ a0)
54lan 77 . . . 4 (((a0b0) ∪ b1) ∩ ((a0a1) ∪ b1)) = (((a0b0) ∪ b1) ∩ ((a1b1) ∪ a0))
6 ancom 74 . . . 4 (((a0b0) ∪ b1) ∩ ((a1b1) ∪ a0)) = (((a1b1) ∪ a0) ∩ ((a0b0) ∪ b1))
75, 6tr 62 . . 3 (((a0b0) ∪ b1) ∩ ((a0a1) ∪ b1)) = (((a1b1) ∪ a0) ∩ ((a0b0) ∪ b1))
8 leor 159 . . . . 5 b1 ≤ (a1b1)
98ler 149 . . . 4 b1 ≤ ((a1b1) ∪ a0)
109mldual2i 1127 . . 3 (((a1b1) ∪ a0) ∩ ((a0b0) ∪ b1)) = ((((a1b1) ∪ a0) ∩ (a0b0)) ∪ b1)
11 ancom 74 . . . . 5 (((a1b1) ∪ a0) ∩ (a0b0)) = ((a0b0) ∩ ((a1b1) ∪ a0))
12 leo 158 . . . . . 6 a0 ≤ (a0b0)
1312mldual2i 1127 . . . . 5 ((a0b0) ∩ ((a1b1) ∪ a0)) = (((a0b0) ∩ (a1b1)) ∪ a0)
1411, 13tr 62 . . . 4 (((a1b1) ∪ a0) ∩ (a0b0)) = (((a0b0) ∩ (a1b1)) ∪ a0)
1514ror 71 . . 3 ((((a1b1) ∪ a0) ∩ (a0b0)) ∪ b1) = ((((a0b0) ∩ (a1b1)) ∪ a0) ∪ b1)
167, 10, 153tr 65 . 2 (((a0b0) ∪ b1) ∩ ((a0a1) ∪ b1)) = ((((a0b0) ∩ (a1b1)) ∪ a0) ∪ b1)
17 orass 75 . 2 ((((a0b0) ∩ (a1b1)) ∪ a0) ∪ b1) = (((a0b0) ∩ (a1b1)) ∪ (a0b1))
18 orcom 73 . 2 (((a0b0) ∩ (a1b1)) ∪ (a0b1)) = ((a0b1) ∪ ((a0b0) ∩ (a1b1)))
1916, 17, 183tr 65 1 (((a0b0) ∪ b1) ∩ ((a0a1) ∪ b1)) = ((a0b1) ∪ ((a0b0) ∩ (a1b1)))
Colors of variables: term
Syntax hints:   = wb 1  wle 2  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:  dp41lemc  1185
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