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Theorem 4oagen1 1042
 Description: "Generalized" 4-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)
4oagen1.1 gf
Assertion
Ref Expression
4oagen1 ((a1 d) ∩ (g ∪ ((a1 d) ∩ (b1 d)))) = ((a1 d) ∩ (b1 d))

Proof of Theorem 4oagen1
StepHypRef Expression
1 4oagen1.1 . . . . . . 7 gf
2 4oa.2 . . . . . . . 8 f = (((ab) ∪ ((a1 d) ∩ (b1 d))) ∪ e)
3 or32 82 . . . . . . . 8 (((ab) ∪ ((a1 d) ∩ (b1 d))) ∪ e) = (((ab) ∪ e) ∪ ((a1 d) ∩ (b1 d)))
42, 3ax-r2 36 . . . . . . 7 f = (((ab) ∪ e) ∪ ((a1 d) ∩ (b1 d)))
51, 4lbtr 139 . . . . . 6 g ≤ (((ab) ∪ e) ∪ ((a1 d) ∩ (b1 d)))
65leror 152 . . . . 5 (g ∪ ((a1 d) ∩ (b1 d))) ≤ ((((ab) ∪ e) ∪ ((a1 d) ∩ (b1 d))) ∪ ((a1 d) ∩ (b1 d)))
7 ax-a3 32 . . . . . 6 ((((ab) ∪ e) ∪ ((a1 d) ∩ (b1 d))) ∪ ((a1 d) ∩ (b1 d))) = (((ab) ∪ e) ∪ (((a1 d) ∩ (b1 d)) ∪ ((a1 d) ∩ (b1 d))))
8 oridm 110 . . . . . . . 8 (((a1 d) ∩ (b1 d)) ∪ ((a1 d) ∩ (b1 d))) = ((a1 d) ∩ (b1 d))
98lor 70 . . . . . . 7 (((ab) ∪ e) ∪ (((a1 d) ∩ (b1 d)) ∪ ((a1 d) ∩ (b1 d)))) = (((ab) ∪ e) ∪ ((a1 d) ∩ (b1 d)))
104ax-r1 35 . . . . . . 7 (((ab) ∪ e) ∪ ((a1 d) ∩ (b1 d))) = f
119, 10ax-r2 36 . . . . . 6 (((ab) ∪ e) ∪ (((a1 d) ∩ (b1 d)) ∪ ((a1 d) ∩ (b1 d)))) = f
127, 11ax-r2 36 . . . . 5 ((((ab) ∪ e) ∪ ((a1 d) ∩ (b1 d))) ∪ ((a1 d) ∩ (b1 d))) = f
136, 12lbtr 139 . . . 4 (g ∪ ((a1 d) ∩ (b1 d))) ≤ f
1413lelan 167 . . 3 ((a1 d) ∩ (g ∪ ((a1 d) ∩ (b1 d)))) ≤ ((a1 d) ∩ f)
15 4oa.1 . . . 4 e = (((ac) ∪ ((a1 d) ∩ (c1 d))) ∩ ((bc) ∪ ((b1 d) ∩ (c1 d))))
1615, 24oath1 1041 . . 3 ((a1 d) ∩ f) = ((a1 d) ∩ (b1 d))
1714, 16lbtr 139 . 2 ((a1 d) ∩ (g ∪ ((a1 d) ∩ (b1 d)))) ≤ ((a1 d) ∩ (b1 d))
18 lea 160 . . 3 ((a1 d) ∩ (b1 d)) ≤ (a1 d)
19 leor 159 . . 3 ((a1 d) ∩ (b1 d)) ≤ (g ∪ ((a1 d) ∩ (b1 d)))
2018, 19ler2an 173 . 2 ((a1 d) ∩ (b1 d)) ≤ ((a1 d) ∩ (g ∪ ((a1 d) ∩ (b1 d))))
2117, 20lebi 145 1 ((a1 d) ∩ (g ∪ ((a1 d) ∩ (b1 d)))) = ((a1 d) ∩ (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:  4oagen1b  1043
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