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Theorem u3lemaa 602
 Description: Lemma for Kalmbach implication study.
Assertion
Ref Expression
u3lemaa ((a3 b) ∩ a) = (a ∩ (ab))

Proof of Theorem u3lemaa
StepHypRef Expression
1 df-i3 46 . . 3 (a3 b) = (((ab) ∪ (ab )) ∪ (a ∩ (ab)))
21ran 78 . 2 ((a3 b) ∩ a) = ((((ab) ∪ (ab )) ∪ (a ∩ (ab))) ∩ a)
3 comanr1 464 . . . . . 6 a C (ab)
43comcom6 459 . . . . 5 a C (ab)
5 comanr1 464 . . . . . 6 a C (ab )
65comcom6 459 . . . . 5 a C (ab )
74, 6com2or 483 . . . 4 a C ((ab) ∪ (ab ))
8 comid 187 . . . . 5 a C a
9 comorr 184 . . . . . 6 a C (ab)
109comcom6 459 . . . . 5 a C (ab)
118, 10com2an 484 . . . 4 a C (a ∩ (ab))
127, 11fh1r 473 . . 3 ((((ab) ∪ (ab )) ∪ (a ∩ (ab))) ∩ a) = ((((ab) ∪ (ab )) ∩ a) ∪ ((a ∩ (ab)) ∩ a))
134, 6fh1r 473 . . . . . 6 (((ab) ∪ (ab )) ∩ a) = (((ab) ∩ a) ∪ ((ab ) ∩ a))
14 ancom 74 . . . . . . . . 9 ((ab) ∩ a) = (a ∩ (ab))
15 anass 76 . . . . . . . . . . 11 ((aa ) ∩ b) = (a ∩ (ab))
1615ax-r1 35 . . . . . . . . . 10 (a ∩ (ab)) = ((aa ) ∩ b)
17 ancom 74 . . . . . . . . . . 11 ((aa ) ∩ b) = (b ∩ (aa ))
18 dff 101 . . . . . . . . . . . . . 14 0 = (aa )
1918ax-r1 35 . . . . . . . . . . . . 13 (aa ) = 0
2019lan 77 . . . . . . . . . . . 12 (b ∩ (aa )) = (b ∩ 0)
21 an0 108 . . . . . . . . . . . 12 (b ∩ 0) = 0
2220, 21ax-r2 36 . . . . . . . . . . 11 (b ∩ (aa )) = 0
2317, 22ax-r2 36 . . . . . . . . . 10 ((aa ) ∩ b) = 0
2416, 23ax-r2 36 . . . . . . . . 9 (a ∩ (ab)) = 0
2514, 24ax-r2 36 . . . . . . . 8 ((ab) ∩ a) = 0
26 ancom 74 . . . . . . . . 9 ((ab ) ∩ a) = (a ∩ (ab ))
27 anass 76 . . . . . . . . . . 11 ((aa ) ∩ b ) = (a ∩ (ab ))
2827ax-r1 35 . . . . . . . . . 10 (a ∩ (ab )) = ((aa ) ∩ b )
29 ancom 74 . . . . . . . . . . 11 ((aa ) ∩ b ) = (b ∩ (aa ))
3019lan 77 . . . . . . . . . . . 12 (b ∩ (aa )) = (b ∩ 0)
31 an0 108 . . . . . . . . . . . 12 (b ∩ 0) = 0
3230, 31ax-r2 36 . . . . . . . . . . 11 (b ∩ (aa )) = 0
3329, 32ax-r2 36 . . . . . . . . . 10 ((aa ) ∩ b ) = 0
3428, 33ax-r2 36 . . . . . . . . 9 (a ∩ (ab )) = 0
3526, 34ax-r2 36 . . . . . . . 8 ((ab ) ∩ a) = 0
3625, 352or 72 . . . . . . 7 (((ab) ∩ a) ∪ ((ab ) ∩ a)) = (0 ∪ 0)
37 or0 102 . . . . . . 7 (0 ∪ 0) = 0
3836, 37ax-r2 36 . . . . . 6 (((ab) ∩ a) ∪ ((ab ) ∩ a)) = 0
3913, 38ax-r2 36 . . . . 5 (((ab) ∪ (ab )) ∩ a) = 0
40 an32 83 . . . . . 6 ((a ∩ (ab)) ∩ a) = ((aa) ∩ (ab))
41 anidm 111 . . . . . . 7 (aa) = a
4241ran 78 . . . . . 6 ((aa) ∩ (ab)) = (a ∩ (ab))
4340, 42ax-r2 36 . . . . 5 ((a ∩ (ab)) ∩ a) = (a ∩ (ab))
4439, 432or 72 . . . 4 ((((ab) ∪ (ab )) ∩ a) ∪ ((a ∩ (ab)) ∩ a)) = (0 ∪ (a ∩ (ab)))
45 ax-a2 31 . . . . 5 (0 ∪ (a ∩ (ab))) = ((a ∩ (ab)) ∪ 0)
46 or0 102 . . . . 5 ((a ∩ (ab)) ∪ 0) = (a ∩ (ab))
4745, 46ax-r2 36 . . . 4 (0 ∪ (a ∩ (ab))) = (a ∩ (ab))
4844, 47ax-r2 36 . . 3 ((((ab) ∪ (ab )) ∩ a) ∪ ((a ∩ (ab)) ∩ a)) = (a ∩ (ab))
4912, 48ax-r2 36 . 2 ((((ab) ∪ (ab )) ∪ (a ∩ (ab))) ∩ a) = (a ∩ (ab))
502, 49ax-r2 36 1 ((a3 b) ∩ a) = (a ∩ (ab))
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7  0wf 9   →3 wi3 14 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-i3 46  df-le1 130  df-le2 131  df-c1 132  df-c2 133 This theorem is referenced by:  u3lemnona  667  u3lem13b  790
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