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Theorem 4oaiii 1040
 Description: Proper OA analog to Godowski/Greechie, Eq. III. (Contributed by NM, 29-Dec-1998.)
Hypotheses
Ref Expression
4oa.1 e = (((ac) ∪ ((a1 d) ∩ (c1 d))) ∩ ((bc) ∪ ((b1 d) ∩ (c1 d))))
4oa.2 f = (((ab) ∪ ((a1 d) ∩ (b1 d))) ∪ e)
Assertion
Ref Expression
4oaiii ((a1 d) ∩ f) = ((b1 d) ∩ f)

Proof of Theorem 4oaiii
StepHypRef Expression
1 4oa.1 . . . 4 e = (((ac) ∪ ((a1 d) ∩ (c1 d))) ∩ ((bc) ∪ ((b1 d) ∩ (c1 d))))
2 4oa.2 . . . 4 f = (((ab) ∪ ((a1 d) ∩ (b1 d))) ∪ e)
31, 24oa 1039 . . 3 ((a1 d) ∩ f) ≤ (b1 d)
4 lear 161 . . 3 ((a1 d) ∩ f) ≤ f
53, 4ler2an 173 . 2 ((a1 d) ∩ f) ≤ ((b1 d) ∩ f)
6 ancom 74 . . . . 5 (((ac) ∪ ((a1 d) ∩ (c1 d))) ∩ ((bc) ∪ ((b1 d) ∩ (c1 d)))) = (((bc) ∪ ((b1 d) ∩ (c1 d))) ∩ ((ac) ∪ ((a1 d) ∩ (c1 d))))
71, 6ax-r2 36 . . . 4 e = (((bc) ∪ ((b1 d) ∩ (c1 d))) ∩ ((ac) ∪ ((a1 d) ∩ (c1 d))))
8 ancom 74 . . . . . . 7 (ab) = (ba)
9 ancom 74 . . . . . . 7 ((a1 d) ∩ (b1 d)) = ((b1 d) ∩ (a1 d))
108, 92or 72 . . . . . 6 ((ab) ∪ ((a1 d) ∩ (b1 d))) = ((ba) ∪ ((b1 d) ∩ (a1 d)))
1110ax-r5 38 . . . . 5 (((ab) ∪ ((a1 d) ∩ (b1 d))) ∪ e) = (((ba) ∪ ((b1 d) ∩ (a1 d))) ∪ e)
122, 11ax-r2 36 . . . 4 f = (((ba) ∪ ((b1 d) ∩ (a1 d))) ∪ e)
137, 124oa 1039 . . 3 ((b1 d) ∩ f) ≤ (a1 d)
14 lear 161 . . 3 ((b1 d) ∩ f) ≤ f
1513, 14ler2an 173 . 2 ((b1 d) ∩ f) ≤ ((a1 d) ∩ f)
165, 15lebi 145 1 ((a1 d) ∩ f) = ((b1 d) ∩ f)
 Colors of variables: term Syntax hints:   = wb 1   ∪ wo 6   ∩ wa 7   →1 wi1 12 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  ax-4oa 1033 This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i1 44  df-le1 130  df-le2 131  df-c1 132  df-c2 133 This theorem is referenced by:  4oath1  1041
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