Quantum Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  QLE Home  >  Th. List  >  u1lem4 GIF version

Theorem u1lem4 757
 Description: Lemma for unified implication study.
Assertion
Ref Expression
u1lem4 (a1 (a1 (b1 a))) = (a1 (b1 a))

Proof of Theorem u1lem4
StepHypRef Expression
1 df-i1 44 . 2 (a1 (a1 (b1 a))) = (a ∪ (a ∩ (a1 (b1 a))))
2 comid 187 . . . . 5 a C a
32comcom2 183 . . . 4 a C a
4 u1lemc1 680 . . . 4 a C (a1 (b1 a))
53, 4fh4 472 . . 3 (a ∪ (a ∩ (a1 (b1 a)))) = ((aa) ∩ (a ∪ (a1 (b1 a))))
6 ax-a2 31 . . . . . 6 (aa) = (aa )
7 df-t 41 . . . . . . 7 1 = (aa )
87ax-r1 35 . . . . . 6 (aa ) = 1
96, 8ax-r2 36 . . . . 5 (aa) = 1
10 ax-a2 31 . . . . . 6 (a ∪ (a1 (b1 a))) = ((a1 (b1 a)) ∪ a )
11 u1lemona 625 . . . . . . 7 ((a1 (b1 a)) ∪ a ) = (a ∪ (a ∩ (b1 a)))
12 df-i1 44 . . . . . . . . . . 11 (b1 a) = (b ∪ (ba))
13 ancom 74 . . . . . . . . . . . 12 (ba) = (ab)
1413lor 70 . . . . . . . . . . 11 (b ∪ (ba)) = (b ∪ (ab))
1512, 14ax-r2 36 . . . . . . . . . 10 (b1 a) = (b ∪ (ab))
1615lan 77 . . . . . . . . 9 (a ∩ (b1 a)) = (a ∩ (b ∪ (ab)))
1716lor 70 . . . . . . . 8 (a ∪ (a ∩ (b1 a))) = (a ∪ (a ∩ (b ∪ (ab))))
18 u1lem3 749 . . . . . . . . . 10 (a1 (b1 a)) = (a ∪ ((ab) ∪ (ab )))
19 ax-a2 31 . . . . . . . . . . . . . 14 (b ∪ (ab)) = ((ab) ∪ b )
2019lan 77 . . . . . . . . . . . . 13 (a ∩ (b ∪ (ab))) = (a ∩ ((ab) ∪ b ))
21 coman1 185 . . . . . . . . . . . . . . 15 (ab) C a
22 coman2 186 . . . . . . . . . . . . . . . 16 (ab) C b
2322comcom2 183 . . . . . . . . . . . . . . 15 (ab) C b
2421, 23fh2 470 . . . . . . . . . . . . . 14 (a ∩ ((ab) ∪ b )) = ((a ∩ (ab)) ∪ (ab ))
25 anass 76 . . . . . . . . . . . . . . . . 17 ((aa) ∩ b) = (a ∩ (ab))
2625ax-r1 35 . . . . . . . . . . . . . . . 16 (a ∩ (ab)) = ((aa) ∩ b)
27 anidm 111 . . . . . . . . . . . . . . . . 17 (aa) = a
2827ran 78 . . . . . . . . . . . . . . . 16 ((aa) ∩ b) = (ab)
2926, 28ax-r2 36 . . . . . . . . . . . . . . 15 (a ∩ (ab)) = (ab)
3029ax-r5 38 . . . . . . . . . . . . . 14 ((a ∩ (ab)) ∪ (ab )) = ((ab) ∪ (ab ))
3124, 30ax-r2 36 . . . . . . . . . . . . 13 (a ∩ ((ab) ∪ b )) = ((ab) ∪ (ab ))
3220, 31ax-r2 36 . . . . . . . . . . . 12 (a ∩ (b ∪ (ab))) = ((ab) ∪ (ab ))
3332ax-r1 35 . . . . . . . . . . 11 ((ab) ∪ (ab )) = (a ∩ (b ∪ (ab)))
3433lor 70 . . . . . . . . . 10 (a ∪ ((ab) ∪ (ab ))) = (a ∪ (a ∩ (b ∪ (ab))))
3518, 34ax-r2 36 . . . . . . . . 9 (a1 (b1 a)) = (a ∪ (a ∩ (b ∪ (ab))))
3635ax-r1 35 . . . . . . . 8 (a ∪ (a ∩ (b ∪ (ab)))) = (a1 (b1 a))
3717, 36ax-r2 36 . . . . . . 7 (a ∪ (a ∩ (b1 a))) = (a1 (b1 a))
3811, 37ax-r2 36 . . . . . 6 ((a1 (b1 a)) ∪ a ) = (a1 (b1 a))
3910, 38ax-r2 36 . . . . 5 (a ∪ (a1 (b1 a))) = (a1 (b1 a))
409, 392an 79 . . . 4 ((aa) ∩ (a ∪ (a1 (b1 a)))) = (1 ∩ (a1 (b1 a)))
41 ancom 74 . . . . 5 (1 ∩ (a1 (b1 a))) = ((a1 (b1 a)) ∩ 1)
42 an1 106 . . . . 5 ((a1 (b1 a)) ∩ 1) = (a1 (b1 a))
4341, 42ax-r2 36 . . . 4 (1 ∩ (a1 (b1 a))) = (a1 (b1 a))
4440, 43ax-r2 36 . . 3 ((aa) ∩ (a ∪ (a1 (b1 a)))) = (a1 (b1 a))
455, 44ax-r2 36 . 2 (a ∪ (a ∩ (a1 (b1 a)))) = (a1 (b1 a))
461, 45ax-r2 36 1 (a1 (a1 (b1 a))) = (a1 (b1 a))
 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: (None)
 Copyright terms: Public domain W3C validator