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Theorem oaidlem2g 932
 Description: Lemma for identity-like OA law (generalized).
Hypothesis
Ref Expression
oaidlem2g.1 ((c ∪ (ab)) ∪ (a1 b)) = 1
Assertion
Ref Expression
oaidlem2g (a ∩ (c ∪ (ab))) ≤ b

Proof of Theorem oaidlem2g
StepHypRef Expression
1 anidm 111 . . . . . . . . . 10 (aa) = a
21ax-r1 35 . . . . . . . . 9 a = (aa)
32ran 78 . . . . . . . 8 (ab) = ((aa) ∩ b)
4 anass 76 . . . . . . . 8 ((aa) ∩ b) = (a ∩ (ab))
53, 4ax-r2 36 . . . . . . 7 (ab) = (a ∩ (ab))
6 leor 159 . . . . . . . 8 (ab) ≤ (c ∪ (ab))
76lelan 167 . . . . . . 7 (a ∩ (ab)) ≤ (a ∩ (c ∪ (ab)))
85, 7bltr 138 . . . . . 6 (ab) ≤ (a ∩ (c ∪ (ab)))
98df-le2 131 . . . . 5 ((ab) ∪ (a ∩ (c ∪ (ab)))) = (a ∩ (c ∪ (ab)))
10 ax-a3 32 . . . . . 6 (((c ∪ (ab))a ) ∪ (ab)) = ((c ∪ (ab)) ∪ (a ∪ (ab)))
11 ax-a2 31 . . . . . . . 8 ((c ∪ (ab))a ) = (a ∪ (c ∪ (ab)) )
12 oran3 93 . . . . . . . 8 (a ∪ (c ∪ (ab)) ) = (a ∩ (c ∪ (ab)))
1311, 12ax-r2 36 . . . . . . 7 ((c ∪ (ab))a ) = (a ∩ (c ∪ (ab)))
1413ax-r5 38 . . . . . 6 (((c ∪ (ab))a ) ∪ (ab)) = ((a ∩ (c ∪ (ab))) ∪ (ab))
15 df-i1 44 . . . . . . . . 9 (a1 b) = (a ∪ (ab))
1615lor 70 . . . . . . . 8 ((c ∪ (ab)) ∪ (a1 b)) = ((c ∪ (ab)) ∪ (a ∪ (ab)))
1716ax-r1 35 . . . . . . 7 ((c ∪ (ab)) ∪ (a ∪ (ab))) = ((c ∪ (ab)) ∪ (a1 b))
18 oaidlem2g.1 . . . . . . 7 ((c ∪ (ab)) ∪ (a1 b)) = 1
1917, 18ax-r2 36 . . . . . 6 ((c ∪ (ab)) ∪ (a ∪ (ab))) = 1
2010, 14, 193tr2 64 . . . . 5 ((a ∩ (c ∪ (ab))) ∪ (ab)) = 1
219, 20lem3.1 443 . . . 4 (ab) = (a ∩ (c ∪ (ab)))
2221ax-r1 35 . . 3 (a ∩ (c ∪ (ab))) = (ab)
2322bile 142 . 2 (a ∩ (c ∪ (ab))) ≤ (ab)
24 lear 161 . 2 (ab) ≤ b
2523, 24letr 137 1 (a ∩ (c ∪ (ab))) ≤ b
 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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