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Theorem u3lemab 612
Description: Lemma for Kalmbach implication study. (Contributed by NM, 14-Dec-1997.)
Assertion
Ref Expression
u3lemab ((a3 b) ∩ b) = ((ab) ∪ (ab))

Proof of Theorem u3lemab
StepHypRef Expression
1 df-i3 46 . . 3 (a3 b) = (((ab) ∪ (ab )) ∪ (a ∩ (ab)))
21ran 78 . 2 ((a3 b) ∩ b) = ((((ab) ∪ (ab )) ∪ (a ∩ (ab))) ∩ b)
3 comanr2 465 . . . . . 6 b C (ab)
4 comanr2 465 . . . . . . 7 b C (ab )
54comcom6 459 . . . . . 6 b C (ab )
63, 5com2or 483 . . . . 5 b C ((ab) ∪ (ab ))
76comcom 453 . . . 4 ((ab) ∪ (ab )) C b
8 coman1 185 . . . . . . . . 9 (ab) C a
98comcom7 460 . . . . . . . 8 (ab) C a
10 coman2 186 . . . . . . . . 9 (ab) C b
118, 10com2or 483 . . . . . . . 8 (ab) C (ab)
129, 11com2an 484 . . . . . . 7 (ab) C (a ∩ (ab))
1312comcom 453 . . . . . 6 (a ∩ (ab)) C (ab)
14 coman1 185 . . . . . . . . 9 (ab ) C a
1514comcom7 460 . . . . . . . 8 (ab ) C a
16 coman2 186 . . . . . . . . . 10 (ab ) C b
1716comcom7 460 . . . . . . . . 9 (ab ) C b
1814, 17com2or 483 . . . . . . . 8 (ab ) C (ab)
1915, 18com2an 484 . . . . . . 7 (ab ) C (a ∩ (ab))
2019comcom 453 . . . . . 6 (a ∩ (ab)) C (ab )
2113, 20com2or 483 . . . . 5 (a ∩ (ab)) C ((ab) ∪ (ab ))
2221comcom 453 . . . 4 ((ab) ∪ (ab )) C (a ∩ (ab))
237, 22fh2r 474 . . 3 ((((ab) ∪ (ab )) ∪ (a ∩ (ab))) ∩ b) = ((((ab) ∪ (ab )) ∩ b) ∪ ((a ∩ (ab)) ∩ b))
243, 5fh1r 473 . . . . . 6 (((ab) ∪ (ab )) ∩ b) = (((ab) ∩ b) ∪ ((ab ) ∩ b))
25 anass 76 . . . . . . . . 9 ((ab) ∩ b) = (a ∩ (bb))
26 anidm 111 . . . . . . . . . 10 (bb) = b
2726lan 77 . . . . . . . . 9 (a ∩ (bb)) = (ab)
2825, 27ax-r2 36 . . . . . . . 8 ((ab) ∩ b) = (ab)
29 an32 83 . . . . . . . . 9 ((ab ) ∩ b) = ((ab) ∩ b )
30 anass 76 . . . . . . . . . 10 ((ab) ∩ b ) = (a ∩ (bb ))
31 dff 101 . . . . . . . . . . . . 13 0 = (bb )
3231ax-r1 35 . . . . . . . . . . . 12 (bb ) = 0
3332lan 77 . . . . . . . . . . 11 (a ∩ (bb )) = (a ∩ 0)
34 an0 108 . . . . . . . . . . 11 (a ∩ 0) = 0
3533, 34ax-r2 36 . . . . . . . . . 10 (a ∩ (bb )) = 0
3630, 35ax-r2 36 . . . . . . . . 9 ((ab) ∩ b ) = 0
3729, 36ax-r2 36 . . . . . . . 8 ((ab ) ∩ b) = 0
3828, 372or 72 . . . . . . 7 (((ab) ∩ b) ∪ ((ab ) ∩ b)) = ((ab) ∪ 0)
39 or0 102 . . . . . . 7 ((ab) ∪ 0) = (ab)
4038, 39ax-r2 36 . . . . . 6 (((ab) ∩ b) ∪ ((ab ) ∩ b)) = (ab)
4124, 40ax-r2 36 . . . . 5 (((ab) ∪ (ab )) ∩ b) = (ab)
42 anass 76 . . . . . 6 ((a ∩ (ab)) ∩ b) = (a ∩ ((ab) ∩ b))
43 ancom 74 . . . . . . . 8 ((ab) ∩ b) = (b ∩ (ab))
44 ax-a2 31 . . . . . . . . . 10 (ab) = (ba )
4544lan 77 . . . . . . . . 9 (b ∩ (ab)) = (b ∩ (ba ))
46 anabs 121 . . . . . . . . 9 (b ∩ (ba )) = b
4745, 46ax-r2 36 . . . . . . . 8 (b ∩ (ab)) = b
4843, 47ax-r2 36 . . . . . . 7 ((ab) ∩ b) = b
4948lan 77 . . . . . 6 (a ∩ ((ab) ∩ b)) = (ab)
5042, 49ax-r2 36 . . . . 5 ((a ∩ (ab)) ∩ b) = (ab)
5141, 502or 72 . . . 4 ((((ab) ∪ (ab )) ∩ b) ∪ ((a ∩ (ab)) ∩ b)) = ((ab) ∪ (ab))
52 ax-a2 31 . . . 4 ((ab) ∪ (ab)) = ((ab) ∪ (ab))
5351, 52ax-r2 36 . . 3 ((((ab) ∪ (ab )) ∩ b) ∪ ((a ∩ (ab)) ∩ b)) = ((ab) ∪ (ab))
5423, 53ax-r2 36 . 2 ((((ab) ∪ (ab )) ∪ (a ∩ (ab))) ∩ b) = ((ab) ∪ (ab))
552, 54ax-r2 36 1 ((a3 b) ∩ b) = ((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:  u3lemnonb  677  neg3antlem1  864  neg3antlem2  865
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