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Theorem mhlem2 878
 Description: Lemma for Marsden-Herman distributive law. (Contributed by NM, 10-Mar-2002.)
Hypotheses
Ref Expression
mh.1 a C c
mh.2 a C d
mh.3 b C c
mh.4 b C d
Assertion
Ref Expression
mhlem2 (((ac) ∩ (cb )) ∩ ((bd) ∩ (ad ))) = (((ac ) ∩ (bd )) ∪ ((cb ) ∩ (da )))

Proof of Theorem mhlem2
StepHypRef Expression
1 mh.1 . . . 4 a C c
2 mh.3 . . . . 5 b C c
32comcom3 454 . . . 4 b C c
41, 3mhlem1 877 . . 3 ((ac) ∩ (cb )) = ((ac ) ∪ (cb ))
5 ax-a2 31 . . . . 5 (ad ) = (da )
65lan 77 . . . 4 ((bd) ∩ (ad )) = ((bd) ∩ (da ))
7 mh.4 . . . . 5 b C d
8 mh.2 . . . . . 6 a C d
98comcom3 454 . . . . 5 a C d
107, 9mhlem1 877 . . . 4 ((bd) ∩ (da )) = ((bd ) ∪ (da ))
116, 10ax-r2 36 . . 3 ((bd) ∩ (ad )) = ((bd ) ∪ (da ))
124, 112an 79 . 2 (((ac) ∩ (cb )) ∩ ((bd) ∩ (ad ))) = (((ac ) ∪ (cb )) ∩ ((bd ) ∪ (da )))
13 leao2 163 . . . . . 6 (ac ) ≤ (cb)
14 leao3 164 . . . . . 6 (ac ) ≤ (da)
1513, 14ler2an 173 . . . . 5 (ac ) ≤ ((cb) ∩ (da))
16 leao3 164 . . . . . 6 (bd ) ≤ (cb)
17 leao2 163 . . . . . 6 (bd ) ≤ (da)
1816, 17ler2an 173 . . . . 5 (bd ) ≤ ((cb) ∩ (da))
1915, 18lel2or 170 . . . 4 ((ac ) ∪ (bd )) ≤ ((cb) ∩ (da))
20 oran2 92 . . . . . 6 (cb) = (cb )
21 oran2 92 . . . . . 6 (da) = (da )
2220, 212an 79 . . . . 5 ((cb) ∩ (da)) = ((cb ) ∩ (da ) )
23 anor3 90 . . . . 5 ((cb ) ∩ (da ) ) = ((cb ) ∪ (da ))
2422, 23ax-r2 36 . . . 4 ((cb) ∩ (da)) = ((cb ) ∪ (da ))
2519, 24lbtr 139 . . 3 ((ac ) ∪ (bd )) ≤ ((cb ) ∪ (da ))
2625mhlem 876 . 2 (((ac ) ∪ (cb )) ∩ ((bd ) ∪ (da ))) = (((ac ) ∩ (bd )) ∪ ((cb ) ∩ (da )))
2712, 26ax-r2 36 1 (((ac) ∩ (cb )) ∩ ((bd) ∩ (ad ))) = (((ac ) ∩ (bd )) ∪ ((cb ) ∩ (da )))
 Colors of variables: term Syntax hints:   = wb 1   C wc 3  ⊥ wn 4   ∪ wo 6   ∩ wa 7 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:  mh  879
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