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Theorem lem4.6.6i3j1 1097
 Description: Equation 4.14 of [MegPav2000] p. 23. The variable i in the paper is set to 3, and j is set to 1. (Contributed by Roy F. Longton, 3-Jul-05.)
Assertion
Ref Expression
lem4.6.6i3j1 ((a3 b) ∪ (a1 b)) = (a0 b)

Proof of Theorem lem4.6.6i3j1
StepHypRef Expression
1 ax-a3 32 . . 3 ((((ab) ∪ (ab )) ∪ (a ∩ (ab))) ∪ (a ∪ (ab))) = (((ab) ∪ (ab )) ∪ ((a ∩ (ab)) ∪ (a ∪ (ab))))
2 ax-a3 32 . . . . 5 (((a ∩ (ab)) ∪ a ) ∪ (ab)) = ((a ∩ (ab)) ∪ (a ∪ (ab)))
32ax-r1 35 . . . 4 ((a ∩ (ab)) ∪ (a ∪ (ab))) = (((a ∩ (ab)) ∪ a ) ∪ (ab))
43lor 70 . . 3 (((ab) ∪ (ab )) ∪ ((a ∩ (ab)) ∪ (a ∪ (ab)))) = (((ab) ∪ (ab )) ∪ (((a ∩ (ab)) ∪ a ) ∪ (ab)))
5 ax-a2 31 . . . . . . 7 ((a ∩ (ab)) ∪ a ) = (a ∪ (a ∩ (ab)))
6 omln 446 . . . . . . 7 (a ∪ (a ∩ (ab))) = (ab)
75, 6ax-r2 36 . . . . . 6 ((a ∩ (ab)) ∪ a ) = (ab)
87ax-r5 38 . . . . 5 (((a ∩ (ab)) ∪ a ) ∪ (ab)) = ((ab) ∪ (ab))
98lor 70 . . . 4 (((ab) ∪ (ab )) ∪ (((a ∩ (ab)) ∪ a ) ∪ (ab))) = (((ab) ∪ (ab )) ∪ ((ab) ∪ (ab)))
10 leao1 162 . . . . . . 7 (ab) ≤ (ab)
11 leao1 162 . . . . . . 7 (ab ) ≤ (ab)
1210, 11lel2or 170 . . . . . 6 ((ab) ∪ (ab )) ≤ (ab)
13 leid 148 . . . . . . 7 (ab) ≤ (ab)
14 leao4 165 . . . . . . 7 (ab) ≤ (ab)
1513, 14lel2or 170 . . . . . 6 ((ab) ∪ (ab)) ≤ (ab)
1612, 15lel2or 170 . . . . 5 (((ab) ∪ (ab )) ∪ ((ab) ∪ (ab))) ≤ (ab)
17 leo 158 . . . . . 6 (ab) ≤ ((ab) ∪ (ab))
1817lerr 150 . . . . 5 (ab) ≤ (((ab) ∪ (ab )) ∪ ((ab) ∪ (ab)))
1916, 18lebi 145 . . . 4 (((ab) ∪ (ab )) ∪ ((ab) ∪ (ab))) = (ab)
209, 19ax-r2 36 . . 3 (((ab) ∪ (ab )) ∪ (((a ∩ (ab)) ∪ a ) ∪ (ab))) = (ab)
211, 4, 203tr 65 . 2 ((((ab) ∪ (ab )) ∪ (a ∩ (ab))) ∪ (a ∪ (ab))) = (ab)
22 df-i3 46 . . 3 (a3 b) = (((ab) ∪ (ab )) ∪ (a ∩ (ab)))
23 df-i1 44 . . 3 (a1 b) = (a ∪ (ab))
2422, 232or 72 . 2 ((a3 b) ∪ (a1 b)) = ((((ab) ∪ (ab )) ∪ (a ∩ (ab))) ∪ (a ∪ (ab)))
25 df-i0 43 . 2 (a0 b) = (ab)
2621, 24, 253tr1 63 1 ((a3 b) ∪ (a1 b)) = (a0 b)
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7   →0 wi0 11   →1 wi1 12   →3 wi3 14 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-i1 44  df-i3 46  df-le1 130  df-le2 131 This theorem is referenced by: (None)
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