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Theorem oaidlem2 931
 Description: Lemma for identity-like OA law. (Contributed by NM, 22-Jan-1999.)
Hypothesis
Ref Expression
oaidlem2.1 ((d ∪ ((a1 c) ∩ (b1 c))) ∪ ((a1 c) →1 (b1 c))) = 1
Assertion
Ref Expression
oaidlem2 ((a1 c) ∩ (d ∪ ((a1 c) ∩ (b1 c)))) ≤ (b1 c)

Proof of Theorem oaidlem2
StepHypRef Expression
1 anidm 111 . . . . . . . . . 10 ((a1 c) ∩ (a1 c)) = (a1 c)
21ax-r1 35 . . . . . . . . 9 (a1 c) = ((a1 c) ∩ (a1 c))
32ran 78 . . . . . . . 8 ((a1 c) ∩ (b1 c)) = (((a1 c) ∩ (a1 c)) ∩ (b1 c))
4 anass 76 . . . . . . . 8 (((a1 c) ∩ (a1 c)) ∩ (b1 c)) = ((a1 c) ∩ ((a1 c) ∩ (b1 c)))
53, 4ax-r2 36 . . . . . . 7 ((a1 c) ∩ (b1 c)) = ((a1 c) ∩ ((a1 c) ∩ (b1 c)))
6 leor 159 . . . . . . . 8 ((a1 c) ∩ (b1 c)) ≤ (d ∪ ((a1 c) ∩ (b1 c)))
76lelan 167 . . . . . . 7 ((a1 c) ∩ ((a1 c) ∩ (b1 c))) ≤ ((a1 c) ∩ (d ∪ ((a1 c) ∩ (b1 c))))
85, 7bltr 138 . . . . . 6 ((a1 c) ∩ (b1 c)) ≤ ((a1 c) ∩ (d ∪ ((a1 c) ∩ (b1 c))))
98df-le2 131 . . . . 5 (((a1 c) ∩ (b1 c)) ∪ ((a1 c) ∩ (d ∪ ((a1 c) ∩ (b1 c))))) = ((a1 c) ∩ (d ∪ ((a1 c) ∩ (b1 c))))
10 ax-a3 32 . . . . . 6 (((d ∪ ((a1 c) ∩ (b1 c))) ∪ (a1 c) ) ∪ ((a1 c) ∩ (b1 c))) = ((d ∪ ((a1 c) ∩ (b1 c))) ∪ ((a1 c) ∪ ((a1 c) ∩ (b1 c))))
11 ax-a2 31 . . . . . . . 8 ((d ∪ ((a1 c) ∩ (b1 c))) ∪ (a1 c) ) = ((a1 c) ∪ (d ∪ ((a1 c) ∩ (b1 c))) )
12 oran3 93 . . . . . . . 8 ((a1 c) ∪ (d ∪ ((a1 c) ∩ (b1 c))) ) = ((a1 c) ∩ (d ∪ ((a1 c) ∩ (b1 c))))
1311, 12ax-r2 36 . . . . . . 7 ((d ∪ ((a1 c) ∩ (b1 c))) ∪ (a1 c) ) = ((a1 c) ∩ (d ∪ ((a1 c) ∩ (b1 c))))
1413ax-r5 38 . . . . . 6 (((d ∪ ((a1 c) ∩ (b1 c))) ∪ (a1 c) ) ∪ ((a1 c) ∩ (b1 c))) = (((a1 c) ∩ (d ∪ ((a1 c) ∩ (b1 c)))) ∪ ((a1 c) ∩ (b1 c)))
15 df-i1 44 . . . . . . . . 9 ((a1 c) →1 (b1 c)) = ((a1 c) ∪ ((a1 c) ∩ (b1 c)))
1615lor 70 . . . . . . . 8 ((d ∪ ((a1 c) ∩ (b1 c))) ∪ ((a1 c) →1 (b1 c))) = ((d ∪ ((a1 c) ∩ (b1 c))) ∪ ((a1 c) ∪ ((a1 c) ∩ (b1 c))))
1716ax-r1 35 . . . . . . 7 ((d ∪ ((a1 c) ∩ (b1 c))) ∪ ((a1 c) ∪ ((a1 c) ∩ (b1 c)))) = ((d ∪ ((a1 c) ∩ (b1 c))) ∪ ((a1 c) →1 (b1 c)))
18 oaidlem2.1 . . . . . . 7 ((d ∪ ((a1 c) ∩ (b1 c))) ∪ ((a1 c) →1 (b1 c))) = 1
1917, 18ax-r2 36 . . . . . 6 ((d ∪ ((a1 c) ∩ (b1 c))) ∪ ((a1 c) ∪ ((a1 c) ∩ (b1 c)))) = 1
2010, 14, 193tr2 64 . . . . 5 (((a1 c) ∩ (d ∪ ((a1 c) ∩ (b1 c)))) ∪ ((a1 c) ∩ (b1 c))) = 1
219, 20lem3.1 443 . . . 4 ((a1 c) ∩ (b1 c)) = ((a1 c) ∩ (d ∪ ((a1 c) ∩ (b1 c))))
2221ax-r1 35 . . 3 ((a1 c) ∩ (d ∪ ((a1 c) ∩ (b1 c)))) = ((a1 c) ∩ (b1 c))
2322bile 142 . 2 ((a1 c) ∩ (d ∪ ((a1 c) ∩ (b1 c)))) ≤ ((a1 c) ∩ (b1 c))
24 lear 161 . 2 ((a1 c) ∩ (b1 c)) ≤ (b1 c)
2523, 24letr 137 1 ((a1 c) ∩ (d ∪ ((a1 c) ∩ (b1 c)))) ≤ (b1 c)
 Colors of variables: term Syntax hints:   = wb 1   ≤ wle 2  ⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8   →1 wi1 12 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-r3 439 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 This theorem is referenced by: (None)
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