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Theorem oagen1 1014
 Description: "Generalized" OA.
Hypothesis
Ref Expression
oagen1.1 d ≤ ((bc) →0 ((a2 b) ∩ (a2 c)))
Assertion
Ref Expression
oagen1 ((a2 b) ∩ (d ∪ ((a2 b) ∩ (a2 c)))) = ((a2 b) ∩ (a2 c))

Proof of Theorem oagen1
StepHypRef Expression
1 oagen1.1 . . . . . . 7 d ≤ ((bc) →0 ((a2 b) ∩ (a2 c)))
2 df-i0 43 . . . . . . 7 ((bc) →0 ((a2 b) ∩ (a2 c))) = ((bc) ∪ ((a2 b) ∩ (a2 c)))
31, 2lbtr 139 . . . . . 6 d ≤ ((bc) ∪ ((a2 b) ∩ (a2 c)))
43leror 152 . . . . 5 (d ∪ ((a2 b) ∩ (a2 c))) ≤ (((bc) ∪ ((a2 b) ∩ (a2 c))) ∪ ((a2 b) ∩ (a2 c)))
5 ax-a3 32 . . . . . 6 (((bc) ∪ ((a2 b) ∩ (a2 c))) ∪ ((a2 b) ∩ (a2 c))) = ((bc) ∪ (((a2 b) ∩ (a2 c)) ∪ ((a2 b) ∩ (a2 c))))
6 oridm 110 . . . . . . 7 (((a2 b) ∩ (a2 c)) ∪ ((a2 b) ∩ (a2 c))) = ((a2 b) ∩ (a2 c))
76lor 70 . . . . . 6 ((bc) ∪ (((a2 b) ∩ (a2 c)) ∪ ((a2 b) ∩ (a2 c)))) = ((bc) ∪ ((a2 b) ∩ (a2 c)))
85, 7ax-r2 36 . . . . 5 (((bc) ∪ ((a2 b) ∩ (a2 c))) ∪ ((a2 b) ∩ (a2 c))) = ((bc) ∪ ((a2 b) ∩ (a2 c)))
94, 8lbtr 139 . . . 4 (d ∪ ((a2 b) ∩ (a2 c))) ≤ ((bc) ∪ ((a2 b) ∩ (a2 c)))
109lelan 167 . . 3 ((a2 b) ∩ (d ∪ ((a2 b) ∩ (a2 c)))) ≤ ((a2 b) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c))))
11 oath1 1004 . . 3 ((a2 b) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c)))) = ((a2 b) ∩ (a2 c))
1210, 11lbtr 139 . 2 ((a2 b) ∩ (d ∪ ((a2 b) ∩ (a2 c)))) ≤ ((a2 b) ∩ (a2 c))
13 lea 160 . . 3 ((a2 b) ∩ (a2 c)) ≤ (a2 b)
14 leor 159 . . 3 ((a2 b) ∩ (a2 c)) ≤ (d ∪ ((a2 b) ∩ (a2 c)))
1513, 14ler2an 173 . 2 ((a2 b) ∩ (a2 c)) ≤ ((a2 b) ∩ (d ∪ ((a2 b) ∩ (a2 c))))
1612, 15lebi 145 1 ((a2 b) ∩ (d ∪ ((a2 b) ∩ (a2 c)))) = ((a2 b) ∩ (a2 c))
 Colors of variables: term Syntax hints:   = wb 1   ≤ wle 2  ⊥ wn 4   ∪ wo 6   ∩ wa 7   →0 wi0 11   →2 wi2 13 This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-3oa 998 This theorem depends on definitions:  df-a 40  df-t 41  df-f 42  df-i0 43  df-i1 44  df-i2 45  df-le1 130  df-le2 131 This theorem is referenced by:  oagen1b  1015  oadist  1019  oadistb  1020
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