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Theorem 4oagen1b 1043
 Description: "Generalized" OA.
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)
4oagen1b.1 gf
4oagen1b.2 h ≤ (a1 d)
Assertion
Ref Expression
4oagen1b (h ∩ (g ∪ ((a1 d) ∩ (b1 d)))) = (h ∩ (b1 d))

Proof of Theorem 4oagen1b
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)
3 4oagen1b.1 . . . 4 gf
41, 2, 34oagen1 1042 . . 3 ((a1 d) ∩ (g ∪ ((a1 d) ∩ (b1 d)))) = ((a1 d) ∩ (b1 d))
54lan 77 . 2 (h ∩ ((a1 d) ∩ (g ∪ ((a1 d) ∩ (b1 d))))) = (h ∩ ((a1 d) ∩ (b1 d)))
6 anass 76 . . . 4 ((h ∩ (a1 d)) ∩ (g ∪ ((a1 d) ∩ (b1 d)))) = (h ∩ ((a1 d) ∩ (g ∪ ((a1 d) ∩ (b1 d)))))
76ax-r1 35 . . 3 (h ∩ ((a1 d) ∩ (g ∪ ((a1 d) ∩ (b1 d))))) = ((h ∩ (a1 d)) ∩ (g ∪ ((a1 d) ∩ (b1 d))))
8 4oagen1b.2 . . . . 5 h ≤ (a1 d)
98df2le2 136 . . . 4 (h ∩ (a1 d)) = h
109ran 78 . . 3 ((h ∩ (a1 d)) ∩ (g ∪ ((a1 d) ∩ (b1 d)))) = (h ∩ (g ∪ ((a1 d) ∩ (b1 d))))
117, 10ax-r2 36 . 2 (h ∩ ((a1 d) ∩ (g ∪ ((a1 d) ∩ (b1 d))))) = (h ∩ (g ∪ ((a1 d) ∩ (b1 d))))
12 anass 76 . . . 4 ((h ∩ (a1 d)) ∩ (b1 d)) = (h ∩ ((a1 d) ∩ (b1 d)))
1312ax-r1 35 . . 3 (h ∩ ((a1 d) ∩ (b1 d))) = ((h ∩ (a1 d)) ∩ (b1 d))
149ran 78 . . 3 ((h ∩ (a1 d)) ∩ (b1 d)) = (h ∩ (b1 d))
1513, 14ax-r2 36 . 2 (h ∩ ((a1 d) ∩ (b1 d))) = (h ∩ (b1 d))
165, 11, 153tr2 64 1 (h ∩ (g ∪ ((a1 d) ∩ (b1 d)))) = (h ∩ (b1 d))
 Colors of variables: term Syntax hints:   = wb 1   ≤ wle 2   ∪ 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:  4oadist  1044
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