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Theorem marsdenlem1 880
 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
marsdenlem1 ((ab) ∩ (ad )) = ((a ∩ (ab)) ∪ (d ∩ (ab)))

Proof of Theorem marsdenlem1
StepHypRef Expression
1 ancom 74 . 2 ((ab) ∩ (ad )) = ((ad ) ∩ (ab))
2 comorr 184 . . . 4 a C (ab)
32comcom3 454 . . 3 a C (ab)
4 marsden.4 . . . . 5 d C a
54comcom4 455 . . . 4 d C a
65comcom 453 . . 3 a C d
73, 6fh2r 474 . 2 ((ad ) ∩ (ab)) = ((a ∩ (ab)) ∪ (d ∩ (ab)))
81, 7ax-r2 36 1 ((ab) ∩ (ad )) = ((a ∩ (ab)) ∪ (d ∩ (ab)))
 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: (None)
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