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Theorem lem4 511
 Description: Lemma 4 of Kalmbach p. 240. (Contributed by NM, 5-Nov-1997.)
Assertion
Ref Expression
lem4 (a3 (a3 b)) = (ab)

Proof of Theorem lem4
StepHypRef Expression
1 df-i3 46 . 2 (a3 (a3 b)) = (((a ∩ (a3 b)) ∪ (a ∩ (a3 b) )) ∪ (a ∩ (a ∪ (a3 b))))
2 df-i3 46 . . . . . . . 8 (a3 b) = (((ab) ∪ (ab )) ∪ (a ∩ (ab)))
32lan 77 . . . . . . 7 (a ∩ (a3 b)) = (a ∩ (((ab) ∪ (ab )) ∪ (a ∩ (ab))))
4 lea 160 . . . . . . . . . . . . 13 (ab) ≤ a
5 lea 160 . . . . . . . . . . . . 13 (ab ) ≤ a
64, 5le2or 168 . . . . . . . . . . . 12 ((ab) ∪ (ab )) ≤ (aa )
7 oridm 110 . . . . . . . . . . . 12 (aa ) = a
86, 7lbtr 139 . . . . . . . . . . 11 ((ab) ∪ (ab )) ≤ a
98lecom 180 . . . . . . . . . 10 ((ab) ∪ (ab )) C a
109comcom 453 . . . . . . . . 9 a C ((ab) ∪ (ab ))
11 lea 160 . . . . . . . . . . . 12 (a ∩ (ab)) ≤ a
1211lecom 180 . . . . . . . . . . 11 (a ∩ (ab)) C a
1312comcom 453 . . . . . . . . . 10 a C (a ∩ (ab))
1413comcom3 454 . . . . . . . . 9 a C (a ∩ (ab))
1510, 14fh1 469 . . . . . . . 8 (a ∩ (((ab) ∪ (ab )) ∪ (a ∩ (ab)))) = ((a ∩ ((ab) ∪ (ab ))) ∪ (a ∩ (a ∩ (ab))))
16 ancom 74 . . . . . . . . . . 11 ((aa) ∩ (ab)) = ((ab) ∩ (aa))
17 anass 76 . . . . . . . . . . 11 ((aa) ∩ (ab)) = (a ∩ (a ∩ (ab)))
18 dff 101 . . . . . . . . . . . . . . 15 0 = (aa )
19 ancom 74 . . . . . . . . . . . . . . 15 (aa ) = (aa)
2018, 19ax-r2 36 . . . . . . . . . . . . . 14 0 = (aa)
2120lan 77 . . . . . . . . . . . . 13 ((ab) ∩ 0) = ((ab) ∩ (aa))
2221ax-r1 35 . . . . . . . . . . . 12 ((ab) ∩ (aa)) = ((ab) ∩ 0)
23 an0 108 . . . . . . . . . . . 12 ((ab) ∩ 0) = 0
2422, 23ax-r2 36 . . . . . . . . . . 11 ((ab) ∩ (aa)) = 0
2516, 17, 243tr2 64 . . . . . . . . . 10 (a ∩ (a ∩ (ab))) = 0
2625lor 70 . . . . . . . . 9 ((a ∩ ((ab) ∪ (ab ))) ∪ (a ∩ (a ∩ (ab)))) = ((a ∩ ((ab) ∪ (ab ))) ∪ 0)
27 or0 102 . . . . . . . . . 10 ((a ∩ ((ab) ∪ (ab ))) ∪ 0) = (a ∩ ((ab) ∪ (ab )))
28 ancom 74 . . . . . . . . . . 11 (a ∩ ((ab) ∪ (ab ))) = (((ab) ∪ (ab )) ∩ a )
298df2le2 136 . . . . . . . . . . 11 (((ab) ∪ (ab )) ∩ a ) = ((ab) ∪ (ab ))
3028, 29ax-r2 36 . . . . . . . . . 10 (a ∩ ((ab) ∪ (ab ))) = ((ab) ∪ (ab ))
3127, 30ax-r2 36 . . . . . . . . 9 ((a ∩ ((ab) ∪ (ab ))) ∪ 0) = ((ab) ∪ (ab ))
3226, 31ax-r2 36 . . . . . . . 8 ((a ∩ ((ab) ∪ (ab ))) ∪ (a ∩ (a ∩ (ab)))) = ((ab) ∪ (ab ))
3315, 32ax-r2 36 . . . . . . 7 (a ∩ (((ab) ∪ (ab )) ∪ (a ∩ (ab)))) = ((ab) ∪ (ab ))
343, 33ax-r2 36 . . . . . 6 (a ∩ (a3 b)) = ((ab) ∪ (ab ))
352lor 70 . . . . . . . . 9 (a ∪ (a3 b)) = (a ∪ (((ab) ∪ (ab )) ∪ (a ∩ (ab))))
36 orordi 112 . . . . . . . . . 10 (a ∪ (((ab) ∪ (ab )) ∪ (a ∩ (ab)))) = ((a ∪ ((ab) ∪ (ab ))) ∪ (a ∪ (a ∩ (ab))))
37 orabs 120 . . . . . . . . . . . 12 (a ∪ (a ∩ (ab))) = a
3837lor 70 . . . . . . . . . . 11 ((a ∪ ((ab) ∪ (ab ))) ∪ (a ∪ (a ∩ (ab)))) = ((a ∪ ((ab) ∪ (ab ))) ∪ a)
39 or32 82 . . . . . . . . . . . 12 ((a ∪ ((ab) ∪ (ab ))) ∪ a) = ((aa) ∪ ((ab) ∪ (ab )))
40 oridm 110 . . . . . . . . . . . . 13 (aa) = a
4140ax-r5 38 . . . . . . . . . . . 12 ((aa) ∪ ((ab) ∪ (ab ))) = (a ∪ ((ab) ∪ (ab )))
4239, 41ax-r2 36 . . . . . . . . . . 11 ((a ∪ ((ab) ∪ (ab ))) ∪ a) = (a ∪ ((ab) ∪ (ab )))
4338, 42ax-r2 36 . . . . . . . . . 10 ((a ∪ ((ab) ∪ (ab ))) ∪ (a ∪ (a ∩ (ab)))) = (a ∪ ((ab) ∪ (ab )))
4436, 43ax-r2 36 . . . . . . . . 9 (a ∪ (((ab) ∪ (ab )) ∪ (a ∩ (ab)))) = (a ∪ ((ab) ∪ (ab )))
4535, 44ax-r2 36 . . . . . . . 8 (a ∪ (a3 b)) = (a ∪ ((ab) ∪ (ab )))
4645ax-r4 37 . . . . . . 7 (a ∪ (a3 b)) = (a ∪ ((ab) ∪ (ab )))
47 oran 87 . . . . . . . 8 (a ∪ (a3 b)) = (a ∩ (a3 b) )
4847con2 67 . . . . . . 7 (a ∪ (a3 b)) = (a ∩ (a3 b) )
49 oran 87 . . . . . . . . 9 (a ∪ ((ab) ∪ (ab ))) = (a ∩ ((ab) ∪ (ab )) )
5049con2 67 . . . . . . . 8 (a ∪ ((ab) ∪ (ab ))) = (a ∩ ((ab) ∪ (ab )) )
51 ancom 74 . . . . . . . 8 (a ∩ ((ab) ∪ (ab )) ) = (((ab) ∪ (ab ))a )
5250, 51ax-r2 36 . . . . . . 7 (a ∪ ((ab) ∪ (ab ))) = (((ab) ∪ (ab ))a )
5346, 48, 523tr2 64 . . . . . 6 (a ∩ (a3 b) ) = (((ab) ∪ (ab ))a )
5434, 532or 72 . . . . 5 ((a ∩ (a3 b)) ∪ (a ∩ (a3 b) )) = (((ab) ∪ (ab )) ∪ (((ab) ∪ (ab ))a ))
558oml2 451 . . . . 5 (((ab) ∪ (ab )) ∪ (((ab) ∪ (ab ))a )) = a
5654, 55ax-r2 36 . . . 4 ((a ∩ (a3 b)) ∪ (a ∩ (a3 b) )) = a
572lor 70 . . . . . . 7 (a ∪ (a3 b)) = (a ∪ (((ab) ∪ (ab )) ∪ (a ∩ (ab))))
58 ax-a3 32 . . . . . . . . 9 ((a ∪ ((ab) ∪ (ab ))) ∪ (a ∩ (ab))) = (a ∪ (((ab) ∪ (ab )) ∪ (a ∩ (ab))))
5958ax-r1 35 . . . . . . . 8 (a ∪ (((ab) ∪ (ab )) ∪ (a ∩ (ab)))) = ((a ∪ ((ab) ∪ (ab ))) ∪ (a ∩ (ab)))
60 orordi 112 . . . . . . . . . 10 (a ∪ ((ab) ∪ (ab ))) = ((a ∪ (ab)) ∪ (a ∪ (ab )))
61 orabs 120 . . . . . . . . . . . 12 (a ∪ (ab)) = a
62 orabs 120 . . . . . . . . . . . 12 (a ∪ (ab )) = a
6361, 622or 72 . . . . . . . . . . 11 ((a ∪ (ab)) ∪ (a ∪ (ab ))) = (aa )
6463, 7ax-r2 36 . . . . . . . . . 10 ((a ∪ (ab)) ∪ (a ∪ (ab ))) = a
6560, 64ax-r2 36 . . . . . . . . 9 (a ∪ ((ab) ∪ (ab ))) = a
6665ax-r5 38 . . . . . . . 8 ((a ∪ ((ab) ∪ (ab ))) ∪ (a ∩ (ab))) = (a ∪ (a ∩ (ab)))
6759, 66ax-r2 36 . . . . . . 7 (a ∪ (((ab) ∪ (ab )) ∪ (a ∩ (ab)))) = (a ∪ (a ∩ (ab)))
6857, 67ax-r2 36 . . . . . 6 (a ∪ (a3 b)) = (a ∪ (a ∩ (ab)))
69 omln 446 . . . . . 6 (a ∪ (a ∩ (ab))) = (ab)
7068, 69ax-r2 36 . . . . 5 (a ∪ (a3 b)) = (ab)
7170lan 77 . . . 4 (a ∩ (a ∪ (a3 b))) = (a ∩ (ab))
7256, 712or 72 . . 3 (((a ∩ (a3 b)) ∪ (a ∩ (a3 b) )) ∪ (a ∩ (a ∪ (a3 b)))) = (a ∪ (a ∩ (ab)))
7372, 69ax-r2 36 . 2 (((a ∩ (a3 b)) ∪ (a ∩ (a3 b) )) ∪ (a ∩ (a ∪ (a3 b)))) = (ab)
741, 73ax-r2 36 1 (a3 (a3 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:  i0i3  512  i3i0  513  ska14  514  i3abs1  522  i3abs3  524  i3th1  543  i3th2  544  i3th3  545  i3th5  547  i3th7  549  i3th8  550  u3lem4  758  u3lem12  788  u3lemax4  796  u3lemax5  797
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