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Theorem mhlemlem2 875
 Description: Lemma for Lemma 7.1 of Kalmbach, p. 91.
Hypothesis
Ref Expression
mhlem.1 (ab) ≤ (cd)
Assertion
Ref Expression
mhlemlem2 (((ab) ∪ d) ∩ (b ∪ (cd))) = (bd)

Proof of Theorem mhlemlem2
StepHypRef Expression
1 ax-a2 31 . . . 4 (ab) = (ba)
21ax-r5 38 . . 3 ((ab) ∪ d) = ((ba) ∪ d)
3 ax-a2 31 . . . 4 (cd) = (dc)
43lor 70 . . 3 (b ∪ (cd)) = (b ∪ (dc))
52, 42an 79 . 2 (((ab) ∪ d) ∩ (b ∪ (cd))) = (((ba) ∪ d) ∩ (b ∪ (dc)))
6 mhlem.1 . . . 4 (ab) ≤ (cd)
7 ax-a2 31 . . . 4 (ba) = (ab)
8 ax-a2 31 . . . . 5 (dc) = (cd)
98ax-r4 37 . . . 4 (dc) = (cd)
106, 7, 9le3tr1 140 . . 3 (ba) ≤ (dc)
1110mhlemlem1 874 . 2 (((ba) ∪ d) ∩ (b ∪ (dc))) = (bd)
125, 11ax-r2 36 1 (((ab) ∪ d) ∩ (b ∪ (cd))) = (bd)
 Colors of variables: term Syntax hints:   = wb 1   ≤ wle 2  ⊥ 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:  mhlem  876
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