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Theorem u3lemanb 617
Description: Lemma for Kalmbach implication study.
Assertion
Ref Expression
u3lemanb ((a3 b) ∩ b ) = (ab )

Proof of Theorem u3lemanb
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 . . . . . . 7 b C (ab)
43comcom3 454 . . . . . 6 b C (ab)
5 comanr2 465 . . . . . 6 b C (ab )
64, 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 ))
2410comcom2 183 . . . . . . 7 (ab) C b
258, 24com2an 484 . . . . . . 7 (ab) C (ab )
2624, 25fh2r 474 . . . . . 6 (((ab) ∪ (ab )) ∩ b ) = (((ab) ∩ b ) ∪ ((ab ) ∩ b ))
27 ax-a2 31 . . . . . . 7 (((ab) ∩ b ) ∪ ((ab ) ∩ b )) = (((ab ) ∩ b ) ∪ ((ab) ∩ b ))
28 anass 76 . . . . . . . . . 10 ((ab ) ∩ b ) = (a ∩ (bb ))
29 anidm 111 . . . . . . . . . . 11 (bb ) = b
3029lan 77 . . . . . . . . . 10 (a ∩ (bb )) = (ab )
3128, 30ax-r2 36 . . . . . . . . 9 ((ab ) ∩ b ) = (ab )
32 anass 76 . . . . . . . . . 10 ((ab) ∩ b ) = (a ∩ (bb ))
33 dff 101 . . . . . . . . . . . . 13 0 = (bb )
3433lan 77 . . . . . . . . . . . 12 (a ∩ 0) = (a ∩ (bb ))
3534ax-r1 35 . . . . . . . . . . 11 (a ∩ (bb )) = (a ∩ 0)
36 an0 108 . . . . . . . . . . 11 (a ∩ 0) = 0
3735, 36ax-r2 36 . . . . . . . . . 10 (a ∩ (bb )) = 0
3832, 37ax-r2 36 . . . . . . . . 9 ((ab) ∩ b ) = 0
3931, 382or 72 . . . . . . . 8 (((ab ) ∩ b ) ∪ ((ab) ∩ b )) = ((ab ) ∪ 0)
40 or0 102 . . . . . . . 8 ((ab ) ∪ 0) = (ab )
4139, 40ax-r2 36 . . . . . . 7 (((ab ) ∩ b ) ∪ ((ab) ∩ b )) = (ab )
4227, 41ax-r2 36 . . . . . 6 (((ab) ∩ b ) ∪ ((ab ) ∩ b )) = (ab )
4326, 42ax-r2 36 . . . . 5 (((ab) ∪ (ab )) ∩ b ) = (ab )
44 an32 83 . . . . . 6 ((a ∩ (ab)) ∩ b ) = ((ab ) ∩ (ab))
45 ancom 74 . . . . . . 7 ((ab ) ∩ (ab)) = ((ab) ∩ (ab ))
46 anor1 88 . . . . . . . . 9 (ab ) = (ab)
4746lan 77 . . . . . . . 8 ((ab) ∩ (ab )) = ((ab) ∩ (ab) )
48 dff 101 . . . . . . . . 9 0 = ((ab) ∩ (ab) )
4948ax-r1 35 . . . . . . . 8 ((ab) ∩ (ab) ) = 0
5047, 49ax-r2 36 . . . . . . 7 ((ab) ∩ (ab )) = 0
5145, 50ax-r2 36 . . . . . 6 ((ab ) ∩ (ab)) = 0
5244, 51ax-r2 36 . . . . 5 ((a ∩ (ab)) ∩ b ) = 0
5343, 522or 72 . . . 4 ((((ab) ∪ (ab )) ∩ b ) ∪ ((a ∩ (ab)) ∩ b )) = ((ab ) ∪ 0)
5453, 40ax-r2 36 . . 3 ((((ab) ∪ (ab )) ∩ b ) ∪ ((a ∩ (ab)) ∩ b )) = (ab )
5523, 54ax-r2 36 . 2 ((((ab) ∪ (ab )) ∪ (a ∩ (ab))) ∩ b ) = (ab )
562, 55ax-r2 36 1 ((a3 b) ∩ b ) = (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:  u3lemnob  672  u3lem3  751  u3lem13b  790  neg3antlem2  865
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