QLE Home Quantum Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  QLE Home  >  Th. List  >  oa4to6lem3 Unicode version
Theorem oa4to6lem3 962
Description: Lemma for orthoarguesian law (4-variable to 6-variable proof).
Hypotheses
Ref Expression
oa4to6lem.1 a' =< b
oa4to6lem.2 c' =< d
oa4to6lem.3 e' =< f
oa4to6lem.4 g = (((a ^ b) v (c ^ d)) v (e ^ f))
Assertion
Ref Expression
oa4to6lem3 f =< (e ->1 g)
Proof of Theorem oa4to6lem3
StepHypRef Expression
1 leor 159 . . . 4 f =< (e' v f)
2 comid 187 . . . . . . . . 9 e C e
32comcom3 454 . . . . . . . 8 e' C e
4 oa4to6lem.3 . . . . . . . . 9 e' =< f
54lecom 180 . . . . . . . 8 e' C f
63, 5fh3 471 . . . . . . 7 (e' v (e ^ f)) = ((e' v e) ^ (e' v f))
7 ancom 74 . . . . . . . 8 (1 ^ (e' v f)) = ((e' v f) ^ 1)
8 df-t 41 . . . . . . . . . 10 1 = (e v e')
9 ax-a2 31 . . . . . . . . . 10 (e v e') = (e' v e)
108, 9ax-r2 36 . . . . . . . . 9 1 = (e' v e)
1110ran 78 . . . . . . . 8 (1 ^ (e' v f)) = ((e' v e) ^ (e' v f))
12 an1 106 . . . . . . . 8 ((e' v f) ^ 1) = (e' v f)
137, 11, 123tr2 64 . . . . . . 7 ((e' v e) ^ (e' v f)) = (e' v f)
146, 13ax-r2 36 . . . . . 6 (e' v (e ^ f)) = (e' v f)
1514ax-r1 35 . . . . 5 (e' v f) = (e' v (e ^ f))
16 anidm 111 . . . . . . . . 9 (e ^ e) = e
1716ran 78 . . . . . . . 8 ((e ^ e) ^ f) = (e ^ f)
1817ax-r1 35 . . . . . . 7 (e ^ f) = ((e ^ e) ^ f)
19 anass 76 . . . . . . 7 ((e ^ e) ^ f) = (e ^ (e ^ f))
2018, 19ax-r2 36 . . . . . 6 (e ^ f) = (e ^ (e ^ f))
2120lor 70 . . . . 5 (e' v (e ^ f)) = (e' v (e ^ (e ^ f)))
2215, 21ax-r2 36 . . . 4 (e' v f) = (e' v (e ^ (e ^ f)))
231, 22lbtr 139 . . 3 f =< (e' v (e ^ (e ^ f)))
24 leor 159 . . . . 5 (e ^ f) =< (((a ^ b) v (c ^ d)) v (e ^ f))
2524lelan 167 . . . 4 (e ^ (e ^ f)) =< (e ^ (((a ^ b) v (c ^ d)) v (e ^ f)))
2625lelor 166 . . 3 (e' v (e ^ (e ^ f))) =< (e' v (e ^ (((a ^ b) v (c ^ d)) v (e ^ f))))
2723, 26letr 137 . 2 f =< (e' v (e ^ (((a ^ b) v (c ^ d)) v (e ^ f))))
28 oa4to6lem.4 . . . . 5 g = (((a ^ b) v (c ^ d)) v (e ^ f))
2928ud1lem0a 255 . . . 4 (e ->1 g) = (e ->1 (((a ^ b) v (c ^ d)) v (e ^ f)))
30 df-i1 44 . . . 4 (e ->1 (((a ^ b) v (c ^ d)) v (e ^ f))) = (e' v (e ^ (((a ^ b) v (c ^ d)) v (e ^ f))))
3129, 30ax-r2 36 . . 3 (e ->1 g) = (e' v (e ^ (((a ^ b) v (c ^ d)) v (e ^ f))))
3231ax-r1 35 . 2 (e' v (e ^ (((a ^ b) v (c ^ d)) v (e ^ f)))) = (e ->1 g)
3327, 32lbtr 139 1 f =< (e ->1 g)
Colors of variables: term
Syntax hints:   = wb 1   =< wle 2  'wn 4   v 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:  oa4to6lem4  963
  Copyright terms: Public domain W3C validator