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Theorem u3lemana 607
 Description: Lemma for Kalmbach implication study.
Assertion
Ref Expression
u3lemana ((a3 b) ∩ a ) = ((ab) ∪ (ab ))

Proof of Theorem u3lemana
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 . . . . 5 a C (ab)
4 comanr1 464 . . . . 5 a C (ab )
53, 4com2or 483 . . . 4 a C ((ab) ∪ (ab ))
6 comid 187 . . . . . 6 a C a
76comcom3 454 . . . . 5 a C a
8 comorr 184 . . . . 5 a C (ab)
97, 8com2an 484 . . . 4 a C (a ∩ (ab))
105, 9fh1r 473 . . 3 ((((ab) ∪ (ab )) ∪ (a ∩ (ab))) ∩ a ) = ((((ab) ∪ (ab )) ∩ a ) ∪ ((a ∩ (ab)) ∩ a ))
11 lea 160 . . . . . . 7 (ab) ≤ a
12 lea 160 . . . . . . 7 (ab ) ≤ a
1311, 12lel2or 170 . . . . . 6 ((ab) ∪ (ab )) ≤ a
1413df2le2 136 . . . . 5 (((ab) ∪ (ab )) ∩ a ) = ((ab) ∪ (ab ))
15 an32 83 . . . . . 6 ((a ∩ (ab)) ∩ a ) = ((aa ) ∩ (ab))
16 ancom 74 . . . . . . 7 ((aa ) ∩ (ab)) = ((ab) ∩ (aa ))
17 dff 101 . . . . . . . . . 10 0 = (aa )
1817ax-r1 35 . . . . . . . . 9 (aa ) = 0
1918lan 77 . . . . . . . 8 ((ab) ∩ (aa )) = ((ab) ∩ 0)
20 an0 108 . . . . . . . 8 ((ab) ∩ 0) = 0
2119, 20ax-r2 36 . . . . . . 7 ((ab) ∩ (aa )) = 0
2216, 21ax-r2 36 . . . . . 6 ((aa ) ∩ (ab)) = 0
2315, 22ax-r2 36 . . . . 5 ((a ∩ (ab)) ∩ a ) = 0
2414, 232or 72 . . . 4 ((((ab) ∪ (ab )) ∩ a ) ∪ ((a ∩ (ab)) ∩ a )) = (((ab) ∪ (ab )) ∪ 0)
25 or0 102 . . . 4 (((ab) ∪ (ab )) ∪ 0) = ((ab) ∪ (ab ))
2624, 25ax-r2 36 . . 3 ((((ab) ∪ (ab )) ∩ a ) ∪ ((a ∩ (ab)) ∩ a )) = ((ab) ∪ (ab ))
2710, 26ax-r2 36 . 2 ((((ab) ∪ (ab )) ∪ (a ∩ (ab))) ∩ a ) = ((ab) ∪ (ab ))
282, 27ax-r2 36 1 ((a3 b) ∩ a ) = ((ab) ∪ (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:  u3lemnoa  662  u3lem13a  789  u3lem13b  790
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