ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ecopovtrng Unicode version

Theorem ecopovtrng 6536
Description: Assuming that operation  F is commutative (second hypothesis), closed (third hypothesis), associative (fourth hypothesis), and has the cancellation property (fifth hypothesis), show that the relation  .~, specified by the first hypothesis, is transitive. (Contributed by Jim Kingdon, 1-Sep-2019.)
Hypotheses
Ref Expression
ecopopr.1  |-  .~  =  { <. x ,  y
>.  |  ( (
x  e.  ( S  X.  S )  /\  y  e.  ( S  X.  S ) )  /\  E. z E. w E. v E. u ( ( x  =  <. z ,  w >.  /\  y  =  <. v ,  u >. )  /\  ( z 
.+  u )  =  ( w  .+  v
) ) ) }
ecopoprg.com  |-  ( ( x  e.  S  /\  y  e.  S )  ->  ( x  .+  y
)  =  ( y 
.+  x ) )
ecopoprg.cl  |-  ( ( x  e.  S  /\  y  e.  S )  ->  ( x  .+  y
)  e.  S )
ecopoprg.ass  |-  ( ( x  e.  S  /\  y  e.  S  /\  z  e.  S )  ->  ( ( x  .+  y )  .+  z
)  =  ( x 
.+  ( y  .+  z ) ) )
ecopoprg.can  |-  ( ( x  e.  S  /\  y  e.  S  /\  z  e.  S )  ->  ( ( x  .+  y )  =  ( x  .+  z )  ->  y  =  z ) )
Assertion
Ref Expression
ecopovtrng  |-  ( ( A  .~  B  /\  B  .~  C )  ->  A  .~  C )
Distinct variable groups:    x, y, z, w, v, u,  .+    x, S, y, z, w, v, u
Allowed substitution hints:    A( x, y, z, w, v, u)    B( x, y, z, w, v, u)    C( x, y, z, w, v, u)    .~ ( x, y, z, w, v, u)

Proof of Theorem ecopovtrng
Dummy variables  f  g  h  t  s  r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ecopopr.1 . . . . . . 7  |-  .~  =  { <. x ,  y
>.  |  ( (
x  e.  ( S  X.  S )  /\  y  e.  ( S  X.  S ) )  /\  E. z E. w E. v E. u ( ( x  =  <. z ,  w >.  /\  y  =  <. v ,  u >. )  /\  ( z 
.+  u )  =  ( w  .+  v
) ) ) }
2 opabssxp 4620 . . . . . . 7  |-  { <. x ,  y >.  |  ( ( x  e.  ( S  X.  S )  /\  y  e.  ( S  X.  S ) )  /\  E. z E. w E. v E. u ( ( x  =  <. z ,  w >.  /\  y  =  <. v ,  u >. )  /\  ( z  .+  u
)  =  ( w 
.+  v ) ) ) }  C_  (
( S  X.  S
)  X.  ( S  X.  S ) )
31, 2eqsstri 3133 . . . . . 6  |-  .~  C_  (
( S  X.  S
)  X.  ( S  X.  S ) )
43brel 4598 . . . . 5  |-  ( A  .~  B  ->  ( A  e.  ( S  X.  S )  /\  B  e.  ( S  X.  S
) ) )
54simpld 111 . . . 4  |-  ( A  .~  B  ->  A  e.  ( S  X.  S
) )
63brel 4598 . . . 4  |-  ( B  .~  C  ->  ( B  e.  ( S  X.  S )  /\  C  e.  ( S  X.  S
) ) )
75, 6anim12i 336 . . 3  |-  ( ( A  .~  B  /\  B  .~  C )  -> 
( A  e.  ( S  X.  S )  /\  ( B  e.  ( S  X.  S
)  /\  C  e.  ( S  X.  S
) ) ) )
8 3anass 967 . . 3  |-  ( ( A  e.  ( S  X.  S )  /\  B  e.  ( S  X.  S )  /\  C  e.  ( S  X.  S
) )  <->  ( A  e.  ( S  X.  S
)  /\  ( B  e.  ( S  X.  S
)  /\  C  e.  ( S  X.  S
) ) ) )
97, 8sylibr 133 . 2  |-  ( ( A  .~  B  /\  B  .~  C )  -> 
( A  e.  ( S  X.  S )  /\  B  e.  ( S  X.  S )  /\  C  e.  ( S  X.  S ) ) )
10 eqid 2140 . . 3  |-  ( S  X.  S )  =  ( S  X.  S
)
11 breq1 3939 . . . . 5  |-  ( <.
f ,  g >.  =  A  ->  ( <.
f ,  g >.  .~  <. h ,  t
>. 
<->  A  .~  <. h ,  t >. )
)
1211anbi1d 461 . . . 4  |-  ( <.
f ,  g >.  =  A  ->  ( (
<. f ,  g >.  .~  <. h ,  t
>.  /\  <. h ,  t
>.  .~  <. s ,  r
>. )  <->  ( A  .~  <.
h ,  t >.  /\  <. h ,  t
>.  .~  <. s ,  r
>. ) ) )
13 breq1 3939 . . . 4  |-  ( <.
f ,  g >.  =  A  ->  ( <.
f ,  g >.  .~  <. s ,  r
>. 
<->  A  .~  <. s ,  r >. )
)
1412, 13imbi12d 233 . . 3  |-  ( <.
f ,  g >.  =  A  ->  ( ( ( <. f ,  g
>.  .~  <. h ,  t
>.  /\  <. h ,  t
>.  .~  <. s ,  r
>. )  ->  <. f ,  g >.  .~  <. s ,  r >. )  <->  ( ( A  .~  <. h ,  t >.  /\  <. h ,  t >.  .~  <. s ,  r >. )  ->  A  .~  <. s ,  r >. )
) )
15 breq2 3940 . . . . 5  |-  ( <.
h ,  t >.  =  B  ->  ( A  .~  <. h ,  t
>. 
<->  A  .~  B ) )
16 breq1 3939 . . . . 5  |-  ( <.
h ,  t >.  =  B  ->  ( <.
h ,  t >.  .~  <. s ,  r
>. 
<->  B  .~  <. s ,  r >. )
)
1715, 16anbi12d 465 . . . 4  |-  ( <.
h ,  t >.  =  B  ->  ( ( A  .~  <. h ,  t >.  /\  <. h ,  t >.  .~  <. s ,  r >. )  <->  ( A  .~  B  /\  B  .~  <. s ,  r
>. ) ) )
1817imbi1d 230 . . 3  |-  ( <.
h ,  t >.  =  B  ->  ( ( ( A  .~  <. h ,  t >.  /\  <. h ,  t >.  .~  <. s ,  r >. )  ->  A  .~  <. s ,  r >. )  <->  ( ( A  .~  B  /\  B  .~  <. s ,  r >. )  ->  A  .~  <. s ,  r >. )
) )
19 breq2 3940 . . . . 5  |-  ( <.
s ,  r >.  =  C  ->  ( B  .~  <. s ,  r
>. 
<->  B  .~  C ) )
2019anbi2d 460 . . . 4  |-  ( <.
s ,  r >.  =  C  ->  ( ( A  .~  B  /\  B  .~  <. s ,  r
>. )  <->  ( A  .~  B  /\  B  .~  C
) ) )
21 breq2 3940 . . . 4  |-  ( <.
s ,  r >.  =  C  ->  ( A  .~  <. s ,  r
>. 
<->  A  .~  C ) )
2220, 21imbi12d 233 . . 3  |-  ( <.
s ,  r >.  =  C  ->  ( ( ( A  .~  B  /\  B  .~  <. s ,  r >. )  ->  A  .~  <. s ,  r >. )  <->  ( ( A  .~  B  /\  B  .~  C )  ->  A  .~  C
) ) )
231ecopoveq 6531 . . . . . . . 8  |-  ( ( ( f  e.  S  /\  g  e.  S
)  /\  ( h  e.  S  /\  t  e.  S ) )  -> 
( <. f ,  g
>.  .~  <. h ,  t
>. 
<->  ( f  .+  t
)  =  ( g 
.+  h ) ) )
24233adant3 1002 . . . . . . 7  |-  ( ( ( f  e.  S  /\  g  e.  S
)  /\  ( h  e.  S  /\  t  e.  S )  /\  (
s  e.  S  /\  r  e.  S )
)  ->  ( <. f ,  g >.  .~  <. h ,  t >.  <->  ( f  .+  t )  =  ( g  .+  h ) ) )
251ecopoveq 6531 . . . . . . . 8  |-  ( ( ( h  e.  S  /\  t  e.  S
)  /\  ( s  e.  S  /\  r  e.  S ) )  -> 
( <. h ,  t
>.  .~  <. s ,  r
>. 
<->  ( h  .+  r
)  =  ( t 
.+  s ) ) )
26253adant1 1000 . . . . . . 7  |-  ( ( ( f  e.  S  /\  g  e.  S
)  /\  ( h  e.  S  /\  t  e.  S )  /\  (
s  e.  S  /\  r  e.  S )
)  ->  ( <. h ,  t >.  .~  <. s ,  r >.  <->  ( h  .+  r )  =  ( t  .+  s ) ) )
2724, 26anbi12d 465 . . . . . 6  |-  ( ( ( f  e.  S  /\  g  e.  S
)  /\  ( h  e.  S  /\  t  e.  S )  /\  (
s  e.  S  /\  r  e.  S )
)  ->  ( ( <. f ,  g >.  .~  <. h ,  t
>.  /\  <. h ,  t
>.  .~  <. s ,  r
>. )  <->  ( ( f 
.+  t )  =  ( g  .+  h
)  /\  ( h  .+  r )  =  ( t  .+  s ) ) ) )
28 oveq12 5790 . . . . . . 7  |-  ( ( ( f  .+  t
)  =  ( g 
.+  h )  /\  ( h  .+  r )  =  ( t  .+  s ) )  -> 
( ( f  .+  t )  .+  (
h  .+  r )
)  =  ( ( g  .+  h ) 
.+  ( t  .+  s ) ) )
29 simp2l 1008 . . . . . . . . 9  |-  ( ( ( f  e.  S  /\  g  e.  S
)  /\  ( h  e.  S  /\  t  e.  S )  /\  (
s  e.  S  /\  r  e.  S )
)  ->  h  e.  S )
30 simp2r 1009 . . . . . . . . 9  |-  ( ( ( f  e.  S  /\  g  e.  S
)  /\  ( h  e.  S  /\  t  e.  S )  /\  (
s  e.  S  /\  r  e.  S )
)  ->  t  e.  S )
31 simp1l 1006 . . . . . . . . 9  |-  ( ( ( f  e.  S  /\  g  e.  S
)  /\  ( h  e.  S  /\  t  e.  S )  /\  (
s  e.  S  /\  r  e.  S )
)  ->  f  e.  S )
32 ecopoprg.com . . . . . . . . . 10  |-  ( ( x  e.  S  /\  y  e.  S )  ->  ( x  .+  y
)  =  ( y 
.+  x ) )
3332adantl 275 . . . . . . . . 9  |-  ( ( ( ( f  e.  S  /\  g  e.  S )  /\  (
h  e.  S  /\  t  e.  S )  /\  ( s  e.  S  /\  r  e.  S
) )  /\  (
x  e.  S  /\  y  e.  S )
)  ->  ( x  .+  y )  =  ( y  .+  x ) )
34 ecopoprg.ass . . . . . . . . . 10  |-  ( ( x  e.  S  /\  y  e.  S  /\  z  e.  S )  ->  ( ( x  .+  y )  .+  z
)  =  ( x 
.+  ( y  .+  z ) ) )
3534adantl 275 . . . . . . . . 9  |-  ( ( ( ( f  e.  S  /\  g  e.  S )  /\  (
h  e.  S  /\  t  e.  S )  /\  ( s  e.  S  /\  r  e.  S
) )  /\  (
x  e.  S  /\  y  e.  S  /\  z  e.  S )
)  ->  ( (
x  .+  y )  .+  z )  =  ( x  .+  ( y 
.+  z ) ) )
36 simp3r 1011 . . . . . . . . 9  |-  ( ( ( f  e.  S  /\  g  e.  S
)  /\  ( h  e.  S  /\  t  e.  S )  /\  (
s  e.  S  /\  r  e.  S )
)  ->  r  e.  S )
37 ecopoprg.cl . . . . . . . . . 10  |-  ( ( x  e.  S  /\  y  e.  S )  ->  ( x  .+  y
)  e.  S )
3837adantl 275 . . . . . . . . 9  |-  ( ( ( ( f  e.  S  /\  g  e.  S )  /\  (
h  e.  S  /\  t  e.  S )  /\  ( s  e.  S  /\  r  e.  S
) )  /\  (
x  e.  S  /\  y  e.  S )
)  ->  ( x  .+  y )  e.  S
)
3929, 30, 31, 33, 35, 36, 38caov411d 5963 . . . . . . . 8  |-  ( ( ( f  e.  S  /\  g  e.  S
)  /\  ( h  e.  S  /\  t  e.  S )  /\  (
s  e.  S  /\  r  e.  S )
)  ->  ( (
h  .+  t )  .+  ( f  .+  r
) )  =  ( ( f  .+  t
)  .+  ( h  .+  r ) ) )
40 simp1r 1007 . . . . . . . . . 10  |-  ( ( ( f  e.  S  /\  g  e.  S
)  /\  ( h  e.  S  /\  t  e.  S )  /\  (
s  e.  S  /\  r  e.  S )
)  ->  g  e.  S )
41 simp3l 1010 . . . . . . . . . 10  |-  ( ( ( f  e.  S  /\  g  e.  S
)  /\  ( h  e.  S  /\  t  e.  S )  /\  (
s  e.  S  /\  r  e.  S )
)  ->  s  e.  S )
4240, 30, 29, 33, 35, 41, 38caov411d 5963 . . . . . . . . 9  |-  ( ( ( f  e.  S  /\  g  e.  S
)  /\  ( h  e.  S  /\  t  e.  S )  /\  (
s  e.  S  /\  r  e.  S )
)  ->  ( (
g  .+  t )  .+  ( h  .+  s
) )  =  ( ( h  .+  t
)  .+  ( g  .+  s ) ) )
4340, 30, 29, 33, 35, 41, 38caov4d 5962 . . . . . . . . 9  |-  ( ( ( f  e.  S  /\  g  e.  S
)  /\  ( h  e.  S  /\  t  e.  S )  /\  (
s  e.  S  /\  r  e.  S )
)  ->  ( (
g  .+  t )  .+  ( h  .+  s
) )  =  ( ( g  .+  h
)  .+  ( t  .+  s ) ) )
4442, 43eqtr3d 2175 . . . . . . . 8  |-  ( ( ( f  e.  S  /\  g  e.  S
)  /\  ( h  e.  S  /\  t  e.  S )  /\  (
s  e.  S  /\  r  e.  S )
)  ->  ( (
h  .+  t )  .+  ( g  .+  s
) )  =  ( ( g  .+  h
)  .+  ( t  .+  s ) ) )
4539, 44eqeq12d 2155 . . . . . . 7  |-  ( ( ( f  e.  S  /\  g  e.  S
)  /\  ( h  e.  S  /\  t  e.  S )  /\  (
s  e.  S  /\  r  e.  S )
)  ->  ( (
( h  .+  t
)  .+  ( f  .+  r ) )  =  ( ( h  .+  t )  .+  (
g  .+  s )
)  <->  ( ( f 
.+  t )  .+  ( h  .+  r ) )  =  ( ( g  .+  h ) 
.+  ( t  .+  s ) ) ) )
4628, 45syl5ibr 155 . . . . . 6  |-  ( ( ( f  e.  S  /\  g  e.  S
)  /\  ( h  e.  S  /\  t  e.  S )  /\  (
s  e.  S  /\  r  e.  S )
)  ->  ( (
( f  .+  t
)  =  ( g 
.+  h )  /\  ( h  .+  r )  =  ( t  .+  s ) )  -> 
( ( h  .+  t )  .+  (
f  .+  r )
)  =  ( ( h  .+  t ) 
.+  ( g  .+  s ) ) ) )
4727, 46sylbid 149 . . . . 5  |-  ( ( ( f  e.  S  /\  g  e.  S
)  /\  ( h  e.  S  /\  t  e.  S )  /\  (
s  e.  S  /\  r  e.  S )
)  ->  ( ( <. f ,  g >.  .~  <. h ,  t
>.  /\  <. h ,  t
>.  .~  <. s ,  r
>. )  ->  ( ( h  .+  t ) 
.+  ( f  .+  r ) )  =  ( ( h  .+  t )  .+  (
g  .+  s )
) ) )
48 ecopoprg.can . . . . . . . 8  |-  ( ( x  e.  S  /\  y  e.  S  /\  z  e.  S )  ->  ( ( x  .+  y )  =  ( x  .+  z )  ->  y  =  z ) )
49 oveq2 5789 . . . . . . . 8  |-  ( y  =  z  ->  (
x  .+  y )  =  ( x  .+  z ) )
5048, 49impbid1 141 . . . . . . 7  |-  ( ( x  e.  S  /\  y  e.  S  /\  z  e.  S )  ->  ( ( x  .+  y )  =  ( x  .+  z )  <-> 
y  =  z ) )
5150adantl 275 . . . . . 6  |-  ( ( ( ( f  e.  S  /\  g  e.  S )  /\  (
h  e.  S  /\  t  e.  S )  /\  ( s  e.  S  /\  r  e.  S
) )  /\  (
x  e.  S  /\  y  e.  S  /\  z  e.  S )
)  ->  ( (
x  .+  y )  =  ( x  .+  z )  <->  y  =  z ) )
5237caovcl 5932 . . . . . . 7  |-  ( ( h  e.  S  /\  t  e.  S )  ->  ( h  .+  t
)  e.  S )
5329, 30, 52syl2anc 409 . . . . . 6  |-  ( ( ( f  e.  S  /\  g  e.  S
)  /\  ( h  e.  S  /\  t  e.  S )  /\  (
s  e.  S  /\  r  e.  S )
)  ->  ( h  .+  t )  e.  S
)
5437caovcl 5932 . . . . . . 7  |-  ( ( f  e.  S  /\  r  e.  S )  ->  ( f  .+  r
)  e.  S )
5531, 36, 54syl2anc 409 . . . . . 6  |-  ( ( ( f  e.  S  /\  g  e.  S
)  /\  ( h  e.  S  /\  t  e.  S )  /\  (
s  e.  S  /\  r  e.  S )
)  ->  ( f  .+  r )  e.  S
)
5638, 40, 41caovcld 5931 . . . . . 6  |-  ( ( ( f  e.  S  /\  g  e.  S
)  /\  ( h  e.  S  /\  t  e.  S )  /\  (
s  e.  S  /\  r  e.  S )
)  ->  ( g  .+  s )  e.  S
)
5751, 53, 55, 56caovcand 5940 . . . . 5  |-  ( ( ( f  e.  S  /\  g  e.  S
)  /\  ( h  e.  S  /\  t  e.  S )  /\  (
s  e.  S  /\  r  e.  S )
)  ->  ( (
( h  .+  t
)  .+  ( f  .+  r ) )  =  ( ( h  .+  t )  .+  (
g  .+  s )
)  <->  ( f  .+  r )  =  ( g  .+  s ) ) )
5847, 57sylibd 148 . . . 4  |-  ( ( ( f  e.  S  /\  g  e.  S
)  /\  ( h  e.  S  /\  t  e.  S )  /\  (
s  e.  S  /\  r  e.  S )
)  ->  ( ( <. f ,  g >.  .~  <. h ,  t
>.  /\  <. h ,  t
>.  .~  <. s ,  r
>. )  ->  ( f 
.+  r )  =  ( g  .+  s
) ) )
591ecopoveq 6531 . . . . 5  |-  ( ( ( f  e.  S  /\  g  e.  S
)  /\  ( s  e.  S  /\  r  e.  S ) )  -> 
( <. f ,  g
>.  .~  <. s ,  r
>. 
<->  ( f  .+  r
)  =  ( g 
.+  s ) ) )
60593adant2 1001 . . . 4  |-  ( ( ( f  e.  S  /\  g  e.  S
)  /\  ( h  e.  S  /\  t  e.  S )  /\  (
s  e.  S  /\  r  e.  S )
)  ->  ( <. f ,  g >.  .~  <. s ,  r >.  <->  ( f  .+  r )  =  ( g  .+  s ) ) )
6158, 60sylibrd 168 . . 3  |-  ( ( ( f  e.  S  /\  g  e.  S
)  /\  ( h  e.  S  /\  t  e.  S )  /\  (
s  e.  S  /\  r  e.  S )
)  ->  ( ( <. f ,  g >.  .~  <. h ,  t
>.  /\  <. h ,  t
>.  .~  <. s ,  r
>. )  ->  <. f ,  g >.  .~  <. s ,  r >. )
)
6210, 14, 18, 22, 613optocl 4624 . 2  |-  ( ( A  e.  ( S  X.  S )  /\  B  e.  ( S  X.  S )  /\  C  e.  ( S  X.  S
) )  ->  (
( A  .~  B  /\  B  .~  C )  ->  A  .~  C
) )
639, 62mpcom 36 1  |-  ( ( A  .~  B  /\  B  .~  C )  ->  A  .~  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 963    = wceq 1332   E.wex 1469    e. wcel 1481   <.cop 3534   class class class wbr 3936   {copab 3995    X. cxp 4544  (class class class)co 5781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-br 3937  df-opab 3997  df-xp 4552  df-iota 5095  df-fv 5138  df-ov 5784
This theorem is referenced by:  ecopoverg  6537
  Copyright terms: Public domain W3C validator