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Theorem u5lemaa 604
Description: Lemma for relevance implication study. (Contributed by NM, 14-Dec-1997.)
Assertion
Ref Expression
u5lemaa ((a5 b) ∩ a) = (ab)

Proof of Theorem u5lemaa
StepHypRef Expression
1 df-i5 48 . . 3 (a5 b) = (((ab) ∪ (ab)) ∪ (ab ))
21ran 78 . 2 ((a5 b) ∩ a) = ((((ab) ∪ (ab)) ∪ (ab )) ∩ a)
3 comanr1 464 . . . . 5 a C (ab)
4 comanr1 464 . . . . . 6 a C (ab)
54comcom6 459 . . . . 5 a C (ab)
63, 5com2or 483 . . . 4 a C ((ab) ∪ (ab))
7 comanr1 464 . . . . 5 a C (ab )
87comcom6 459 . . . 4 a C (ab )
96, 8fh1r 473 . . 3 ((((ab) ∪ (ab)) ∪ (ab )) ∩ a) = ((((ab) ∪ (ab)) ∩ a) ∪ ((ab ) ∩ a))
103, 5fh1r 473 . . . . . 6 (((ab) ∪ (ab)) ∩ a) = (((ab) ∩ a) ∪ ((ab) ∩ a))
11 an32 83 . . . . . . . . 9 ((ab) ∩ a) = ((aa) ∩ b)
12 anidm 111 . . . . . . . . . 10 (aa) = a
1312ran 78 . . . . . . . . 9 ((aa) ∩ b) = (ab)
1411, 13ax-r2 36 . . . . . . . 8 ((ab) ∩ a) = (ab)
15 an32 83 . . . . . . . . 9 ((ab) ∩ a) = ((aa) ∩ b)
16 ancom 74 . . . . . . . . . 10 ((aa) ∩ b) = (b ∩ (aa))
17 ancom 74 . . . . . . . . . . . . . 14 (aa ) = (aa)
1817ax-r1 35 . . . . . . . . . . . . 13 (aa) = (aa )
19 dff 101 . . . . . . . . . . . . . 14 0 = (aa )
2019ax-r1 35 . . . . . . . . . . . . 13 (aa ) = 0
2118, 20ax-r2 36 . . . . . . . . . . . 12 (aa) = 0
2221lan 77 . . . . . . . . . . 11 (b ∩ (aa)) = (b ∩ 0)
23 an0 108 . . . . . . . . . . 11 (b ∩ 0) = 0
2422, 23ax-r2 36 . . . . . . . . . 10 (b ∩ (aa)) = 0
2516, 24ax-r2 36 . . . . . . . . 9 ((aa) ∩ b) = 0
2615, 25ax-r2 36 . . . . . . . 8 ((ab) ∩ a) = 0
2714, 262or 72 . . . . . . 7 (((ab) ∩ a) ∪ ((ab) ∩ a)) = ((ab) ∪ 0)
28 or0 102 . . . . . . 7 ((ab) ∪ 0) = (ab)
2927, 28ax-r2 36 . . . . . 6 (((ab) ∩ a) ∪ ((ab) ∩ a)) = (ab)
3010, 29ax-r2 36 . . . . 5 (((ab) ∪ (ab)) ∩ a) = (ab)
31 ancom 74 . . . . 5 ((ab ) ∩ a) = (a ∩ (ab ))
3230, 312or 72 . . . 4 ((((ab) ∪ (ab)) ∩ a) ∪ ((ab ) ∩ a)) = ((ab) ∪ (a ∩ (ab )))
333, 8fh4 472 . . . . 5 ((ab) ∪ (a ∩ (ab ))) = (((ab) ∪ a) ∩ ((ab) ∪ (ab )))
34 ax-a2 31 . . . . . . . 8 ((ab) ∪ a) = (a ∪ (ab))
35 orabs 120 . . . . . . . 8 (a ∪ (ab)) = a
3634, 35ax-r2 36 . . . . . . 7 ((ab) ∪ a) = a
3736ran 78 . . . . . 6 (((ab) ∪ a) ∩ ((ab) ∪ (ab ))) = (a ∩ ((ab) ∪ (ab )))
383, 8fh1 469 . . . . . . 7 (a ∩ ((ab) ∪ (ab ))) = ((a ∩ (ab)) ∪ (a ∩ (ab )))
39 anass 76 . . . . . . . . . . 11 ((aa) ∩ b) = (a ∩ (ab))
4039ax-r1 35 . . . . . . . . . 10 (a ∩ (ab)) = ((aa) ∩ b)
4140, 13ax-r2 36 . . . . . . . . 9 (a ∩ (ab)) = (ab)
42 anass 76 . . . . . . . . . . 11 ((aa ) ∩ b ) = (a ∩ (ab ))
4342ax-r1 35 . . . . . . . . . 10 (a ∩ (ab )) = ((aa ) ∩ b )
44 ancom 74 . . . . . . . . . . 11 ((aa ) ∩ b ) = (b ∩ (aa ))
4519lan 77 . . . . . . . . . . . . 13 (b ∩ 0) = (b ∩ (aa ))
4645ax-r1 35 . . . . . . . . . . . 12 (b ∩ (aa )) = (b ∩ 0)
47 an0 108 . . . . . . . . . . . 12 (b ∩ 0) = 0
4846, 47ax-r2 36 . . . . . . . . . . 11 (b ∩ (aa )) = 0
4944, 48ax-r2 36 . . . . . . . . . 10 ((aa ) ∩ b ) = 0
5043, 49ax-r2 36 . . . . . . . . 9 (a ∩ (ab )) = 0
5141, 502or 72 . . . . . . . 8 ((a ∩ (ab)) ∪ (a ∩ (ab ))) = ((ab) ∪ 0)
5251, 28ax-r2 36 . . . . . . 7 ((a ∩ (ab)) ∪ (a ∩ (ab ))) = (ab)
5338, 52ax-r2 36 . . . . . 6 (a ∩ ((ab) ∪ (ab ))) = (ab)
5437, 53ax-r2 36 . . . . 5 (((ab) ∪ a) ∩ ((ab) ∪ (ab ))) = (ab)
5533, 54ax-r2 36 . . . 4 ((ab) ∪ (a ∩ (ab ))) = (ab)
5632, 55ax-r2 36 . . 3 ((((ab) ∪ (ab)) ∩ a) ∪ ((ab ) ∩ a)) = (ab)
579, 56ax-r2 36 . 2 ((((ab) ∪ (ab)) ∪ (ab )) ∩ a) = (ab)
582, 57ax-r2 36 1 ((a5 b) ∩ a) = (ab)
Colors of variables: term
Syntax hints:   = wb 1   wn 4  wo 6  wa 7  0wf 9  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-i5 48  df-le1 130  df-le2 131  df-c1 132  df-c2 133
This theorem is referenced by:  u5lemnona  669  u5lembi  725
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