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Theorem u1lem11 780
 Description: Lemma used in study of orthoarguesian law. (Contributed by NM, 28-Dec-1998.)
Assertion
Ref Expression
u1lem11 ((a1 b) →1 b) = (a1 b)

Proof of Theorem u1lem11
StepHypRef Expression
1 ud1lem0c 277 . . . . 5 (a1 b) = (a ∩ (a b ))
2 ax-a1 30 . . . . . . . 8 a = a
32ax-r1 35 . . . . . . 7 a = a
43ax-r5 38 . . . . . 6 (a b ) = (ab )
54lan 77 . . . . 5 (a ∩ (a b )) = (a ∩ (ab ))
61, 5ax-r2 36 . . . 4 (a1 b) = (a ∩ (ab ))
7 u1lemab 610 . . . . 5 ((a1 b) ∩ b) = ((ab) ∪ (a b))
8 ax-a2 31 . . . . 5 ((ab) ∪ (a b)) = ((a b) ∪ (ab))
92ran 78 . . . . . . 7 (ab) = (a b)
109ax-r5 38 . . . . . 6 ((ab) ∪ (ab)) = ((a b) ∪ (ab))
1110ax-r1 35 . . . . 5 ((a b) ∪ (ab)) = ((ab) ∪ (ab))
127, 8, 113tr 65 . . . 4 ((a1 b) ∩ b) = ((ab) ∪ (ab))
136, 122or 72 . . 3 ((a1 b) ∪ ((a1 b) ∩ b)) = ((a ∩ (ab )) ∪ ((ab) ∪ (ab)))
14 comanr1 464 . . . . . . 7 a C (ab)
1514comcom3 454 . . . . . 6 a C (ab)
16 comanr1 464 . . . . . 6 a C (ab)
1715, 16com2or 483 . . . . 5 a C ((ab) ∪ (ab))
1817comcom 453 . . . 4 ((ab) ∪ (ab)) C a
19 comor1 461 . . . . . . 7 (ab ) C a
20 comor2 462 . . . . . . . 8 (ab ) C b
2120comcom7 460 . . . . . . 7 (ab ) C b
2219, 21com2an 484 . . . . . 6 (ab ) C (ab)
2319comcom2 183 . . . . . . 7 (ab ) C a
2423, 21com2an 484 . . . . . 6 (ab ) C (ab)
2522, 24com2or 483 . . . . 5 (ab ) C ((ab) ∪ (ab))
2625comcom 453 . . . 4 ((ab) ∪ (ab)) C (ab )
2718, 26fh3r 475 . . 3 ((a ∩ (ab )) ∪ ((ab) ∪ (ab))) = ((a ∪ ((ab) ∪ (ab))) ∩ ((ab ) ∪ ((ab) ∪ (ab))))
28 or32 82 . . . . . 6 ((a ∪ (ab)) ∪ (ab)) = ((a ∪ (ab)) ∪ (ab))
29 ax-a3 32 . . . . . 6 ((a ∪ (ab)) ∪ (ab)) = (a ∪ ((ab) ∪ (ab)))
30 orabs 120 . . . . . . 7 (a ∪ (ab)) = a
3130ax-r5 38 . . . . . 6 ((a ∪ (ab)) ∪ (ab)) = (a ∪ (ab))
3228, 29, 313tr2 64 . . . . 5 (a ∪ ((ab) ∪ (ab))) = (a ∪ (ab))
33 or12 80 . . . . . 6 ((ab ) ∪ ((ab) ∪ (ab))) = ((ab) ∪ ((ab ) ∪ (ab)))
34 anor2 89 . . . . . . . . 9 (ab) = (ab )
3534lor 70 . . . . . . . 8 ((ab ) ∪ (ab)) = ((ab ) ∪ (ab ) )
36 df-t 41 . . . . . . . . 9 1 = ((ab ) ∪ (ab ) )
3736ax-r1 35 . . . . . . . 8 ((ab ) ∪ (ab ) ) = 1
3835, 37ax-r2 36 . . . . . . 7 ((ab ) ∪ (ab)) = 1
3938lor 70 . . . . . 6 ((ab) ∪ ((ab ) ∪ (ab))) = ((ab) ∪ 1)
40 or1 104 . . . . . 6 ((ab) ∪ 1) = 1
4133, 39, 403tr 65 . . . . 5 ((ab ) ∪ ((ab) ∪ (ab))) = 1
4232, 412an 79 . . . 4 ((a ∪ ((ab) ∪ (ab))) ∩ ((ab ) ∪ ((ab) ∪ (ab)))) = ((a ∪ (ab)) ∩ 1)
43 an1 106 . . . 4 ((a ∪ (ab)) ∩ 1) = (a ∪ (ab))
4442, 43ax-r2 36 . . 3 ((a ∪ ((ab) ∪ (ab))) ∩ ((ab ) ∪ ((ab) ∪ (ab)))) = (a ∪ (ab))
4513, 27, 443tr 65 . 2 ((a1 b) ∪ ((a1 b) ∩ b)) = (a ∪ (ab))
46 df-i1 44 . 2 ((a1 b) →1 b) = ((a1 b) ∪ ((a1 b) ∩ b))
47 df-i1 44 . 2 (a1 b) = (a ∪ (ab))
4845, 46, 473tr1 63 1 ((a1 b) →1 b) = (a1 b)
 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:  u1lem12  781  2oai1u  822  1oath1i1u  828  oa4to4u  973  3oa2  1024
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