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Theorem ud1lem2 561
 Description: Lemma for unified disjunction.
Assertion
Ref Expression
ud1lem2 ((a ∪ (ab )) →1 a) = (ab)

Proof of Theorem ud1lem2
StepHypRef Expression
1 df-i1 44 . 2 ((a ∪ (ab )) →1 a) = ((a ∪ (ab )) ∪ ((a ∪ (ab )) ∩ a))
2 comid 187 . . . 4 (a ∪ (ab )) C (a ∪ (ab ))
32comcom3 454 . . 3 (a ∪ (ab )) C (a ∪ (ab ))
4 comor1 461 . . . 4 (a ∪ (ab )) C a
54comcom3 454 . . 3 (a ∪ (ab )) C a
63, 5fh3 471 . 2 ((a ∪ (ab )) ∪ ((a ∪ (ab )) ∩ a)) = (((a ∪ (ab )) ∪ (a ∪ (ab ))) ∩ ((a ∪ (ab ))a))
7 ancom 74 . . 3 (((a ∪ (ab )) ∪ (a ∪ (ab ))) ∩ ((a ∪ (ab ))a)) = (((a ∪ (ab ))a) ∩ ((a ∪ (ab )) ∪ (a ∪ (ab ))))
8 ax-a2 31 . . . . 5 ((a ∪ (ab )) ∪ (a ∪ (ab ))) = ((a ∪ (ab )) ∪ (a ∪ (ab )) )
9 df-t 41 . . . . . 6 1 = ((a ∪ (ab )) ∪ (a ∪ (ab )) )
109ax-r1 35 . . . . 5 ((a ∪ (ab )) ∪ (a ∪ (ab )) ) = 1
118, 10ax-r2 36 . . . 4 ((a ∪ (ab )) ∪ (a ∪ (ab ))) = 1
1211lan 77 . . 3 (((a ∪ (ab ))a) ∩ ((a ∪ (ab )) ∪ (a ∪ (ab )))) = (((a ∪ (ab ))a) ∩ 1)
13 an1 106 . . . 4 (((a ∪ (ab ))a) ∩ 1) = ((a ∪ (ab ))a)
14 oran 87 . . . . . . 7 (a ∪ (ab )) = (a ∩ (ab ) )
15 oran 87 . . . . . . . . . 10 (ab) = (ab )
1615ax-r1 35 . . . . . . . . 9 (ab ) = (ab)
1716lan 77 . . . . . . . 8 (a ∩ (ab ) ) = (a ∩ (ab))
1817ax-r4 37 . . . . . . 7 (a ∩ (ab ) ) = (a ∩ (ab))
1914, 18ax-r2 36 . . . . . 6 (a ∪ (ab )) = (a ∩ (ab))
2019con2 67 . . . . 5 (a ∪ (ab )) = (a ∩ (ab))
2120ax-r5 38 . . . 4 ((a ∪ (ab ))a) = ((a ∩ (ab)) ∪ a)
22 ax-a2 31 . . . . 5 ((a ∩ (ab)) ∪ a) = (a ∪ (a ∩ (ab)))
23 oml 445 . . . . 5 (a ∪ (a ∩ (ab))) = (ab)
2422, 23ax-r2 36 . . . 4 ((a ∩ (ab)) ∪ a) = (ab)
2513, 21, 243tr 65 . . 3 (((a ∪ (ab ))a) ∩ 1) = (ab)
267, 12, 253tr 65 . 2 (((a ∪ (ab )) ∪ (a ∪ (ab ))) ∩ ((a ∪ (ab ))a)) = (ab)
271, 6, 263tr 65 1 ((a ∪ (ab )) →1 a) = (ab)
 Colors of variables: term Syntax hints:   = wb 1  ⊥ 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-a4 33  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  df-c1 132  df-c2 133 This theorem is referenced by:  ud1  595
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