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Theorem marsdenlem3 882
 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
marsdenlem3 (((bc) ∪ (cd)) ∩ (bd )) = 0

Proof of Theorem marsdenlem3
StepHypRef Expression
1 lea 160 . . . . . . . 8 (bd ) ≤ b
21lecon 154 . . . . . . 7 b ≤ (bd )
32lel 151 . . . . . 6 (bc) ≤ (bd )
43lecom 180 . . . . 5 (bc) C (bd )
54comcom7 460 . . . 4 (bc) C (bd )
65comcom 453 . . 3 (bd ) C (bc)
7 lear 161 . . . . . . . 8 (cd) ≤ d
87lerr 150 . . . . . . 7 (cd) ≤ (bd)
9 oran2 92 . . . . . . 7 (bd) = (bd )
108, 9lbtr 139 . . . . . 6 (cd) ≤ (bd )
1110lecom 180 . . . . 5 (cd) C (bd )
1211comcom7 460 . . . 4 (cd) C (bd )
1312comcom 453 . . 3 (bd ) C (cd)
146, 13fh1r 473 . 2 (((bc) ∪ (cd)) ∩ (bd )) = (((bc) ∩ (bd )) ∪ ((cd) ∩ (bd )))
15 an4 86 . . . 4 ((bc) ∩ (bd )) = ((bb) ∩ (cd ))
16 ancom 74 . . . . . 6 (bb) = (bb )
17 dff 101 . . . . . . 7 0 = (bb )
1817ax-r1 35 . . . . . 6 (bb ) = 0
1916, 18ax-r2 36 . . . . 5 (bb) = 0
2019ran 78 . . . 4 ((bb) ∩ (cd )) = (0 ∩ (cd ))
21 an0r 109 . . . 4 (0 ∩ (cd )) = 0
2215, 20, 213tr 65 . . 3 ((bc) ∩ (bd )) = 0
23 an4 86 . . . 4 ((cd) ∩ (bd )) = ((cb) ∩ (dd ))
24 dff 101 . . . . . 6 0 = (dd )
2524ax-r1 35 . . . . 5 (dd ) = 0
2625lan 77 . . . 4 ((cb) ∩ (dd )) = ((cb) ∩ 0)
27 an0 108 . . . 4 ((cb) ∩ 0) = 0
2823, 26, 273tr 65 . . 3 ((cd) ∩ (bd )) = 0
2922, 282or 72 . 2 (((bc) ∩ (bd )) ∪ ((cd) ∩ (bd ))) = (0 ∪ 0)
30 or0 102 . 2 (0 ∪ 0) = 0
3114, 29, 303tr 65 1 (((bc) ∪ (cd)) ∩ (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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