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Theorem marsdenlem4 883
 Description: Lemma for Marsden-Herman distributive law.
Hypotheses
Ref Expression
marsden.1 a C b
marsden.2 b C c
marsden.3 c C d
marsden.4 d C a
Assertion
Ref Expression
marsdenlem4 (((ab) ∪ (ad )) ∩ (bd)) = 0

Proof of Theorem marsdenlem4
StepHypRef Expression
1 leao3 164 . . . . . 6 (bd) ≤ (ab )
2 oran1 91 . . . . . 6 (ab ) = (ab)
31, 2lbtr 139 . . . . 5 (bd) ≤ (ab)
43lecom 180 . . . 4 (bd) C (ab)
54comcom7 460 . . 3 (bd) C (ab)
6 leao4 165 . . . . . 6 (bd) ≤ (ad)
7 oran2 92 . . . . . 6 (ad) = (ad )
86, 7lbtr 139 . . . . 5 (bd) ≤ (ad )
98lecom 180 . . . 4 (bd) C (ad )
109comcom7 460 . . 3 (bd) C (ad )
115, 10fh1r 473 . 2 (((ab) ∪ (ad )) ∩ (bd)) = (((ab) ∩ (bd)) ∪ ((ad ) ∩ (bd)))
12 ancom 74 . . . . 5 (bd) = (db )
1312lan 77 . . . 4 ((ab) ∩ (bd)) = ((ab) ∩ (db ))
14 an4 86 . . . 4 ((ab) ∩ (db )) = ((ad) ∩ (bb ))
15 dff 101 . . . . . . 7 0 = (bb )
1615lan 77 . . . . . 6 ((ad) ∩ 0) = ((ad) ∩ (bb ))
1716ax-r1 35 . . . . 5 ((ad) ∩ (bb )) = ((ad) ∩ 0)
18 an0 108 . . . . 5 ((ad) ∩ 0) = 0
1917, 18ax-r2 36 . . . 4 ((ad) ∩ (bb )) = 0
2013, 14, 193tr 65 . . 3 ((ab) ∩ (bd)) = 0
21 an4 86 . . . 4 ((ad ) ∩ (bd)) = ((ab ) ∩ (dd))
22 ancom 74 . . . . . 6 (dd) = (dd )
23 dff 101 . . . . . . 7 0 = (dd )
2423ax-r1 35 . . . . . 6 (dd ) = 0
2522, 24ax-r2 36 . . . . 5 (dd) = 0
2625lan 77 . . . 4 ((ab ) ∩ (dd)) = ((ab ) ∩ 0)
27 an0 108 . . . 4 ((ab ) ∩ 0) = 0
2821, 26, 273tr 65 . . 3 ((ad ) ∩ (bd)) = 0
2920, 282or 72 . 2 (((ab) ∩ (bd)) ∪ ((ad ) ∩ (bd))) = (0 ∪ 0)
30 or0 102 . 2 (0 ∪ 0) = 0
3111, 29, 303tr 65 1 (((ab) ∪ (ad )) ∩ (bd)) = 0
 Colors of variables: term Syntax hints:   = wb 1   C wc 3  ⊥ wn 4   ∪ wo 6   ∩ wa 7  0wf 9 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-le1 130  df-le2 131  df-c1 132  df-c2 133 This theorem is referenced by: (None)
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