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Theorem lem4.6.6i2j4 1095
 Description: Equation 4.14 of [MegPav2000] p. 23. The variable i in the paper is set to 2, and j is set to 4. (Contributed by Roy F. Longton, 3-Jul-05.)
Assertion
Ref Expression
lem4.6.6i2j4 ((a2 b) ∪ (a4 b)) = (a0 b)

Proof of Theorem lem4.6.6i2j4
StepHypRef Expression
1 ax-a2 31 . . . 4 (b ∪ (ab )) = ((ab ) ∪ b)
21ax-r5 38 . . 3 ((b ∪ (ab )) ∪ (((ab) ∪ (ab)) ∪ ((ab) ∩ b ))) = (((ab ) ∪ b) ∪ (((ab) ∪ (ab)) ∪ ((ab) ∩ b )))
3 ax-a3 32 . . 3 (((ab ) ∪ b) ∪ (((ab) ∪ (ab)) ∪ ((ab) ∩ b ))) = ((ab ) ∪ (b ∪ (((ab) ∪ (ab)) ∪ ((ab) ∩ b ))))
4 ax-a3 32 . . . . . 6 ((b ∪ ((ab) ∪ (ab))) ∪ ((ab) ∩ b )) = (b ∪ (((ab) ∪ (ab)) ∪ ((ab) ∩ b )))
54ax-r1 35 . . . . 5 (b ∪ (((ab) ∪ (ab)) ∪ ((ab) ∩ b ))) = ((b ∪ ((ab) ∪ (ab))) ∪ ((ab) ∩ b ))
65lor 70 . . . 4 ((ab ) ∪ (b ∪ (((ab) ∪ (ab)) ∪ ((ab) ∩ b )))) = ((ab ) ∪ ((b ∪ ((ab) ∪ (ab))) ∪ ((ab) ∩ b )))
7 ax-a2 31 . . . . . 6 (b ∪ ((ab) ∪ (ab))) = (((ab) ∪ (ab)) ∪ b)
87ax-r5 38 . . . . 5 ((b ∪ ((ab) ∪ (ab))) ∪ ((ab) ∩ b )) = ((((ab) ∪ (ab)) ∪ b) ∪ ((ab) ∩ b ))
98lor 70 . . . 4 ((ab ) ∪ ((b ∪ ((ab) ∪ (ab))) ∪ ((ab) ∩ b ))) = ((ab ) ∪ ((((ab) ∪ (ab)) ∪ b) ∪ ((ab) ∩ b )))
10 ax-a3 32 . . . . . 6 ((((ab) ∪ (ab)) ∪ b) ∪ ((ab) ∩ b )) = (((ab) ∪ (ab)) ∪ (b ∪ ((ab) ∩ b )))
1110lor 70 . . . . 5 ((ab ) ∪ ((((ab) ∪ (ab)) ∪ b) ∪ ((ab) ∩ b ))) = ((ab ) ∪ (((ab) ∪ (ab)) ∪ (b ∪ ((ab) ∩ b ))))
12 ancom 74 . . . . . . . . 9 ((ab) ∩ b ) = (b ∩ (ab))
1312lor 70 . . . . . . . 8 (b ∪ ((ab) ∩ b )) = (b ∪ (b ∩ (ab)))
14 ax-a2 31 . . . . . . . . . 10 (ab) = (ba )
1514lan 77 . . . . . . . . 9 (b ∩ (ab)) = (b ∩ (ba ))
1615lor 70 . . . . . . . 8 (b ∪ (b ∩ (ab))) = (b ∪ (b ∩ (ba )))
17 oml 445 . . . . . . . . 9 (b ∪ (b ∩ (ba ))) = (ba )
18 ax-a2 31 . . . . . . . . 9 (ba ) = (ab)
1917, 18ax-r2 36 . . . . . . . 8 (b ∪ (b ∩ (ba ))) = (ab)
2013, 16, 193tr 65 . . . . . . 7 (b ∪ ((ab) ∩ b )) = (ab)
2120lor 70 . . . . . 6 (((ab) ∪ (ab)) ∪ (b ∪ ((ab) ∩ b ))) = (((ab) ∪ (ab)) ∪ (ab))
2221lor 70 . . . . 5 ((ab ) ∪ (((ab) ∪ (ab)) ∪ (b ∪ ((ab) ∩ b )))) = ((ab ) ∪ (((ab) ∪ (ab)) ∪ (ab)))
23 leao1 162 . . . . . . 7 (ab ) ≤ (ab)
24 leao4 165 . . . . . . . . 9 (ab) ≤ (ab)
25 leao1 162 . . . . . . . . 9 (ab) ≤ (ab)
2624, 25lel2or 170 . . . . . . . 8 ((ab) ∪ (ab)) ≤ (ab)
27 leid 148 . . . . . . . 8 (ab) ≤ (ab)
2826, 27lel2or 170 . . . . . . 7 (((ab) ∪ (ab)) ∪ (ab)) ≤ (ab)
2923, 28lel2or 170 . . . . . 6 ((ab ) ∪ (((ab) ∪ (ab)) ∪ (ab))) ≤ (ab)
30 leor 159 . . . . . . 7 (ab) ≤ (((ab) ∪ (ab)) ∪ (ab))
3130lerr 150 . . . . . 6 (ab) ≤ ((ab ) ∪ (((ab) ∪ (ab)) ∪ (ab)))
3229, 31lebi 145 . . . . 5 ((ab ) ∪ (((ab) ∪ (ab)) ∪ (ab))) = (ab)
3311, 22, 323tr 65 . . . 4 ((ab ) ∪ ((((ab) ∪ (ab)) ∪ b) ∪ ((ab) ∩ b ))) = (ab)
346, 9, 333tr 65 . . 3 ((ab ) ∪ (b ∪ (((ab) ∪ (ab)) ∪ ((ab) ∩ b )))) = (ab)
352, 3, 343tr 65 . 2 ((b ∪ (ab )) ∪ (((ab) ∪ (ab)) ∪ ((ab) ∩ b ))) = (ab)
36 df-i2 45 . . 3 (a2 b) = (b ∪ (ab ))
37 df-i4 47 . . 3 (a4 b) = (((ab) ∪ (ab)) ∪ ((ab) ∩ b ))
3836, 372or 72 . 2 ((a2 b) ∪ (a4 b)) = ((b ∪ (ab )) ∪ (((ab) ∪ (ab)) ∪ ((ab) ∩ b )))
39 df-i0 43 . 2 (a0 b) = (ab)
4035, 38, 393tr1 63 1 ((a2 b) ∪ (a4 b)) = (a0 b)
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7   →0 wi0 11   →2 wi2 13   →4 wi4 15 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-r3 439 This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i0 43  df-i2 45  df-i4 47  df-le1 130  df-le2 131 This theorem is referenced by: (None)
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