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Theorem u24lem 770
 Description: Lemma for unified implication study.
Assertion
Ref Expression
u24lem ((a2 b) ∩ (a4 b)) = (a5 b)

Proof of Theorem u24lem
StepHypRef Expression
1 df-i2 45 . . 3 (a2 b) = (b ∪ (ab ))
21ran 78 . 2 ((a2 b) ∩ (a4 b)) = ((b ∪ (ab )) ∩ (a4 b))
3 u4lemc1 683 . . . 4 b C (a4 b)
4 comanr2 465 . . . . 5 b C (ab )
54comcom6 459 . . . 4 b C (ab )
63, 5fh2r 474 . . 3 ((b ∪ (ab )) ∩ (a4 b)) = ((b ∩ (a4 b)) ∪ ((ab ) ∩ (a4 b)))
7 ancom 74 . . . . . 6 (b ∩ (a4 b)) = ((a4 b) ∩ b)
8 ancom 74 . . . . . 6 ((a4 b) ∩ b) = (b ∩ (a4 b))
97, 8ax-r2 36 . . . . 5 (b ∩ (a4 b)) = (b ∩ (a4 b))
10 anass 76 . . . . . 6 ((ab ) ∩ (a4 b)) = (a ∩ (b ∩ (a4 b)))
11 ancom 74 . . . . . . . . 9 (b ∩ (a4 b)) = ((a4 b) ∩ b )
12 u4lemanb 618 . . . . . . . . 9 ((a4 b) ∩ b ) = ((ab) ∩ b )
1311, 12ax-r2 36 . . . . . . . 8 (b ∩ (a4 b)) = ((ab) ∩ b )
1413lan 77 . . . . . . 7 (a ∩ (b ∩ (a4 b))) = (a ∩ ((ab) ∩ b ))
15 anass 76 . . . . . . . . 9 ((a ∩ (ab)) ∩ b ) = (a ∩ ((ab) ∩ b ))
1615ax-r1 35 . . . . . . . 8 (a ∩ ((ab) ∩ b )) = ((a ∩ (ab)) ∩ b )
17 anabs 121 . . . . . . . . . 10 (a ∩ (ab)) = a
1817ran 78 . . . . . . . . 9 ((a ∩ (ab)) ∩ b ) = (ab )
19 ancom 74 . . . . . . . . 9 (ab ) = (ba )
2018, 19ax-r2 36 . . . . . . . 8 ((a ∩ (ab)) ∩ b ) = (ba )
2116, 20ax-r2 36 . . . . . . 7 (a ∩ ((ab) ∩ b )) = (ba )
2214, 21ax-r2 36 . . . . . 6 (a ∩ (b ∩ (a4 b))) = (ba )
2310, 22ax-r2 36 . . . . 5 ((ab ) ∩ (a4 b)) = (ba )
249, 232or 72 . . . 4 ((b ∩ (a4 b)) ∪ ((ab ) ∩ (a4 b))) = ((b ∩ (a4 b)) ∪ (ba ))
25 comanr1 464 . . . . . . 7 b C (ba )
2625comcom6 459 . . . . . 6 b C (ba )
2726, 3fh4r 476 . . . . 5 ((b ∩ (a4 b)) ∪ (ba )) = ((b ∪ (ba )) ∩ ((a4 b) ∪ (ba )))
283, 26com2or 483 . . . . . . 7 b C ((a4 b) ∪ (ba ))
2928, 26fh2r 474 . . . . . 6 ((b ∪ (ba )) ∩ ((a4 b) ∪ (ba ))) = ((b ∩ ((a4 b) ∪ (ba ))) ∪ ((ba ) ∩ ((a4 b) ∪ (ba ))))
303, 26fh1 469 . . . . . . . . 9 (b ∩ ((a4 b) ∪ (ba ))) = ((b ∩ (a4 b)) ∪ (b ∩ (ba )))
31 u4lemab 613 . . . . . . . . . . . 12 ((a4 b) ∩ b) = ((ab) ∪ (ab))
327, 31ax-r2 36 . . . . . . . . . . 11 (b ∩ (a4 b)) = ((ab) ∪ (ab))
3332ax-r5 38 . . . . . . . . . 10 ((b ∩ (a4 b)) ∪ (b ∩ (ba ))) = (((ab) ∪ (ab)) ∪ (b ∩ (ba )))
34 id 59 . . . . . . . . . 10 (((ab) ∪ (ab)) ∪ (b ∩ (ba ))) = (((ab) ∪ (ab)) ∪ (b ∩ (ba )))
3533, 34ax-r2 36 . . . . . . . . 9 ((b ∩ (a4 b)) ∪ (b ∩ (ba ))) = (((ab) ∪ (ab)) ∪ (b ∩ (ba )))
3630, 35ax-r2 36 . . . . . . . 8 (b ∩ ((a4 b) ∪ (ba ))) = (((ab) ∪ (ab)) ∪ (b ∩ (ba )))
37 leor 159 . . . . . . . . 9 (ba ) ≤ ((a4 b) ∪ (ba ))
3837df2le2 136 . . . . . . . 8 ((ba ) ∩ ((a4 b) ∪ (ba ))) = (ba )
3936, 382or 72 . . . . . . 7 ((b ∩ ((a4 b) ∪ (ba ))) ∪ ((ba ) ∩ ((a4 b) ∪ (ba )))) = ((((ab) ∪ (ab)) ∪ (b ∩ (ba ))) ∪ (ba ))
40 ax-a3 32 . . . . . . . 8 ((((ab) ∪ (ab)) ∪ (b ∩ (ba ))) ∪ (ba )) = (((ab) ∪ (ab)) ∪ ((b ∩ (ba )) ∪ (ba )))
41 lear 161 . . . . . . . . . . . 12 (b ∩ (ba )) ≤ (ba )
4241df-le2 131 . . . . . . . . . . 11 ((b ∩ (ba )) ∪ (ba )) = (ba )
43 ancom 74 . . . . . . . . . . 11 (ba ) = (ab )
4442, 43ax-r2 36 . . . . . . . . . 10 ((b ∩ (ba )) ∪ (ba )) = (ab )
4544lor 70 . . . . . . . . 9 (((ab) ∪ (ab)) ∪ ((b ∩ (ba )) ∪ (ba ))) = (((ab) ∪ (ab)) ∪ (ab ))
46 df-i5 48 . . . . . . . . . . 11 (a5 b) = (((ab) ∪ (ab)) ∪ (ab ))
4746ax-r1 35 . . . . . . . . . 10 (((ab) ∪ (ab)) ∪ (ab )) = (a5 b)
48 id 59 . . . . . . . . . 10 (a5 b) = (a5 b)
4947, 48ax-r2 36 . . . . . . . . 9 (((ab) ∪ (ab)) ∪ (ab )) = (a5 b)
5045, 49ax-r2 36 . . . . . . . 8 (((ab) ∪ (ab)) ∪ ((b ∩ (ba )) ∪ (ba ))) = (a5 b)
5140, 50ax-r2 36 . . . . . . 7 ((((ab) ∪ (ab)) ∪ (b ∩ (ba ))) ∪ (ba )) = (a5 b)
5239, 51ax-r2 36 . . . . . 6 ((b ∩ ((a4 b) ∪ (ba ))) ∪ ((ba ) ∩ ((a4 b) ∪ (ba )))) = (a5 b)
5329, 52ax-r2 36 . . . . 5 ((b ∪ (ba )) ∩ ((a4 b) ∪ (ba ))) = (a5 b)
5427, 53ax-r2 36 . . . 4 ((b ∩ (a4 b)) ∪ (ba )) = (a5 b)
5524, 54ax-r2 36 . . 3 ((b ∩ (a4 b)) ∪ ((ab ) ∩ (a4 b))) = (a5 b)
566, 55ax-r2 36 . 2 ((b ∪ (ab )) ∩ (a4 b)) = (a5 b)
572, 56ax-r2 36 1 ((a2 b) ∩ (a4 b)) = (a5 b)
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7   →2 wi2 13   →4 wi4 15   →5 wi5 16 This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a4 33  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-i2 45  df-i4 47  df-i5 48  df-le1 130  df-le2 131  df-c1 132  df-c2 133 This theorem is referenced by:  negant5  863
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