QLE Home Quantum Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  QLE Home  >  Th. List  >  oa4to6lem4 Unicode version
Theorem oa4to6lem4 963
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
oa4to6lem4 (b ^ (a v (c ^ (((a ^ c) v (b ^ d)) v (((a ^ e) v (b ^ f)) ^ ((c ^ e) v (d ^ f))))))) =< ((a ->1 g) ^ (a v (c ^ (((a ^ c) v ((a ->1 g) ^ (c ->1 g))) v (((a ^ e) v ((a ->1 g) ^ (e ->1 g))) ^ ((c ^ e) v ((c ->1 g) ^ (e ->1 g))))))))
Proof of Theorem oa4to6lem4
StepHypRef Expression
1 oa4to6lem.1 . . 3 a' =< b
2 oa4to6lem.2 . . 3 c' =< d
3 oa4to6lem.3 . . 3 e' =< f
4 oa4to6lem.4 . . 3 g = (((a ^ b) v (c ^ d)) v (e ^ f))
51, 2, 3, 4oa4to6lem1 960 . 2 b =< (a ->1 g)
61, 2, 3, 4oa4to6lem2 961 . . . . . . 7 d =< (c ->1 g)
75, 6le2an 169 . . . . . 6 (b ^ d) =< ((a ->1 g) ^ (c ->1 g))
87lelor 166 . . . . 5 ((a ^ c) v (b ^ d)) =< ((a ^ c) v ((a ->1 g) ^ (c ->1 g)))
91, 2, 3, 4oa4to6lem3 962 . . . . . . . 8 f =< (e ->1 g)
105, 9le2an 169 . . . . . . 7 (b ^ f) =< ((a ->1 g) ^ (e ->1 g))
1110lelor 166 . . . . . 6 ((a ^ e) v (b ^ f)) =< ((a ^ e) v ((a ->1 g) ^ (e ->1 g)))
126, 9le2an 169 . . . . . . 7 (d ^ f) =< ((c ->1 g) ^ (e ->1 g))
1312lelor 166 . . . . . 6 ((c ^ e) v (d ^ f)) =< ((c ^ e) v ((c ->1 g) ^ (e ->1 g)))
1411, 13le2an 169 . . . . 5 (((a ^ e) v (b ^ f)) ^ ((c ^ e) v (d ^ f))) =< (((a ^ e) v ((a ->1 g) ^ (e ->1 g))) ^ ((c ^ e) v ((c ->1 g) ^ (e ->1 g))))
158, 14le2or 168 . . . 4 (((a ^ c) v (b ^ d)) v (((a ^ e) v (b ^ f)) ^ ((c ^ e) v (d ^ f)))) =< (((a ^ c) v ((a ->1 g) ^ (c ->1 g))) v (((a ^ e) v ((a ->1 g) ^ (e ->1 g))) ^ ((c ^ e) v ((c ->1 g) ^ (e ->1 g)))))
1615lelan 167 . . 3 (c ^ (((a ^ c) v (b ^ d)) v (((a ^ e) v (b ^ f)) ^ ((c ^ e) v (d ^ f))))) =< (c ^ (((a ^ c) v ((a ->1 g) ^ (c ->1 g))) v (((a ^ e) v ((a ->1 g) ^ (e ->1 g))) ^ ((c ^ e) v ((c ->1 g) ^ (e ->1 g))))))
1716lelor 166 . 2 (a v (c ^ (((a ^ c) v (b ^ d)) v (((a ^ e) v (b ^ f)) ^ ((c ^ e) v (d ^ f)))))) =< (a v (c ^ (((a ^ c) v ((a ->1 g) ^ (c ->1 g))) v (((a ^ e) v ((a ->1 g) ^ (e ->1 g))) ^ ((c ^ e) v ((c ->1 g) ^ (e ->1 g)))))))
185, 17le2an 169 1 (b ^ (a v (c ^ (((a ^ c) v (b ^ d)) v (((a ^ e) v (b ^ f)) ^ ((c ^ e) v (d ^ f))))))) =< ((a ->1 g) ^ (a v (c ^ (((a ^ c) v ((a ->1 g) ^ (c ->1 g))) v (((a ^ e) v ((a ->1 g) ^ (e ->1 g))) ^ ((c ^ e) v ((c ->1 g) ^ (e ->1 g))))))))
Colors of variables: term
Syntax hints:   = wb 1   =< wle 2  'wn 4   v wo 6   ^ wa 7   ->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:  oa4to6dual  964
  Copyright terms: Public domain W3C validator