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Theorem i3lem1 504
 Description: Lemma for Kalmbach implication.
Hypothesis
Ref Expression
i3lem.1 (a3 b) = 1
Assertion
Ref Expression
i3lem1 ((ab) ∪ (ab )) = a

Proof of Theorem i3lem1
StepHypRef Expression
1 coman1 185 . . . . . . 7 (ab ) C a
21comcom 453 . . . . . 6 a C (ab )
3 comorr 184 . . . . . . 7 a C (ab)
4 comorr 184 . . . . . . . 8 a C (a ∪ (ab))
54comcom3 454 . . . . . . 7 a C (a ∪ (ab))
63, 5com2an 484 . . . . . 6 a C ((ab) ∩ (a ∪ (ab)))
72, 6fh1 469 . . . . 5 (a ∩ ((ab ) ∪ ((ab) ∩ (a ∪ (ab))))) = ((a ∩ (ab )) ∪ (a ∩ ((ab) ∩ (a ∪ (ab)))))
8 anass 76 . . . . . . . . 9 ((aa ) ∩ b ) = (a ∩ (ab ))
98ax-r1 35 . . . . . . . 8 (a ∩ (ab )) = ((aa ) ∩ b )
10 anidm 111 . . . . . . . . 9 (aa ) = a
1110ran 78 . . . . . . . 8 ((aa ) ∩ b ) = (ab )
129, 11ax-r2 36 . . . . . . 7 (a ∩ (ab )) = (ab )
13 anass 76 . . . . . . . . 9 ((a ∩ (ab)) ∩ (a ∪ (ab))) = (a ∩ ((ab) ∩ (a ∪ (ab))))
1413ax-r1 35 . . . . . . . 8 (a ∩ ((ab) ∩ (a ∪ (ab)))) = ((a ∩ (ab)) ∩ (a ∪ (ab)))
15 anabs 121 . . . . . . . . . 10 (a ∩ (ab)) = a
1615ran 78 . . . . . . . . 9 ((a ∩ (ab)) ∩ (a ∪ (ab))) = (a ∩ (a ∪ (ab)))
17 omlan 448 . . . . . . . . 9 (a ∩ (a ∪ (ab))) = (ab)
1816, 17ax-r2 36 . . . . . . . 8 ((a ∩ (ab)) ∩ (a ∪ (ab))) = (ab)
1914, 18ax-r2 36 . . . . . . 7 (a ∩ ((ab) ∩ (a ∪ (ab)))) = (ab)
2012, 192or 72 . . . . . 6 ((a ∩ (ab )) ∪ (a ∩ ((ab) ∩ (a ∪ (ab))))) = ((ab ) ∪ (ab))
21 ax-a2 31 . . . . . 6 ((ab ) ∪ (ab)) = ((ab) ∪ (ab ))
2220, 21ax-r2 36 . . . . 5 ((a ∩ (ab )) ∪ (a ∩ ((ab) ∩ (a ∪ (ab))))) = ((ab) ∪ (ab ))
237, 22ax-r2 36 . . . 4 (a ∩ ((ab ) ∪ ((ab) ∩ (a ∪ (ab))))) = ((ab) ∪ (ab ))
2423ax-r1 35 . . 3 ((ab) ∪ (ab )) = (a ∩ ((ab ) ∪ ((ab) ∩ (a ∪ (ab)))))
25 df2i3 498 . . . . . 6 (a3 b) = ((ab ) ∪ ((ab) ∩ (a ∪ (ab))))
2625ax-r1 35 . . . . 5 ((ab ) ∪ ((ab) ∩ (a ∪ (ab)))) = (a3 b)
27 i3lem.1 . . . . 5 (a3 b) = 1
2826, 27ax-r2 36 . . . 4 ((ab ) ∪ ((ab) ∩ (a ∪ (ab)))) = 1
2928lan 77 . . 3 (a ∩ ((ab ) ∪ ((ab) ∩ (a ∪ (ab))))) = (a ∩ 1)
3024, 29ax-r2 36 . 2 ((ab) ∪ (ab )) = (a ∩ 1)
31 an1 106 . 2 (a ∩ 1) = a
3230, 31ax-r2 36 1 ((ab) ∪ (ab )) = a
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8   →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:  i3lem2  505  i3lem3  506  i3lem4  507
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