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Theorem ecopovtrn 6389
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 NM, 11-Feb-1996.) (Revised by Mario Carneiro, 26-Apr-2015.)
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
) ) ) }
ecopopr.com  |-  ( x 
.+  y )  =  ( y  .+  x
)
ecopopr.cl  |-  ( ( x  e.  S  /\  y  e.  S )  ->  ( x  .+  y
)  e.  S )
ecopopr.ass  |-  ( ( x  .+  y ) 
.+  z )  =  ( x  .+  (
y  .+  z )
)
ecopopr.can  |-  ( ( x  e.  S  /\  y  e.  S )  ->  ( ( x  .+  y )  =  ( x  .+  z )  ->  y  =  z ) )
Assertion
Ref Expression
ecopovtrn  |-  ( ( 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 ecopovtrn
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 4512 . . . . . . 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 3056 . . . . . 6  |-  .~  C_  (
( S  X.  S
)  X.  ( S  X.  S ) )
43brel 4490 . . . . 5  |-  ( A  .~  B  ->  ( A  e.  ( S  X.  S )  /\  B  e.  ( S  X.  S
) ) )
54simpld 110 . . . 4  |-  ( A  .~  B  ->  A  e.  ( S  X.  S
) )
63brel 4490 . . . 4  |-  ( B  .~  C  ->  ( B  e.  ( S  X.  S )  /\  C  e.  ( S  X.  S
) ) )
75, 6anim12i 331 . . 3  |-  ( ( A  .~  B  /\  B  .~  C )  -> 
( A  e.  ( S  X.  S )  /\  ( B  e.  ( S  X.  S
)  /\  C  e.  ( S  X.  S
) ) ) )
8 3anass 928 . . 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 132 . 2  |-  ( ( A  .~  B  /\  B  .~  C )  -> 
( A  e.  ( S  X.  S )  /\  B  e.  ( S  X.  S )  /\  C  e.  ( S  X.  S ) ) )
10 eqid 2088 . . 3  |-  ( S  X.  S )  =  ( S  X.  S
)
11 breq1 3848 . . . . 5  |-  ( <.
f ,  g >.  =  A  ->  ( <.
f ,  g >.  .~  <. h ,  t
>. 
<->  A  .~  <. h ,  t >. )
)
1211anbi1d 453 . . . 4  |-  ( <.
f ,  g >.  =  A  ->  ( (
<. f ,  g >.  .~  <. h ,  t
>.  /\  <. h ,  t
>.  .~  <. s ,  r
>. )  <->  ( A  .~  <.
h ,  t >.  /\  <. h ,  t
>.  .~  <. s ,  r
>. ) ) )
13 breq1 3848 . . . 4  |-  ( <.
f ,  g >.  =  A  ->  ( <.
f ,  g >.  .~  <. s ,  r
>. 
<->  A  .~  <. s ,  r >. )
)
1412, 13imbi12d 232 . . 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 3849 . . . . 5  |-  ( <.
h ,  t >.  =  B  ->  ( A  .~  <. h ,  t
>. 
<->  A  .~  B ) )
16 breq1 3848 . . . . 5  |-  ( <.
h ,  t >.  =  B  ->  ( <.
h ,  t >.  .~  <. s ,  r
>. 
<->  B  .~  <. s ,  r >. )
)
1715, 16anbi12d 457 . . . 4  |-  ( <.
h ,  t >.  =  B  ->  ( ( A  .~  <. h ,  t >.  /\  <. h ,  t >.  .~  <. s ,  r >. )  <->  ( A  .~  B  /\  B  .~  <. s ,  r
>. ) ) )
1817imbi1d 229 . . 3  |-  ( <.
h ,  t >.  =  B  ->  ( ( ( A  .~  <. h ,  t >.  /\  <. h ,  t >.  .~  <. s ,  r >. )  ->  A  .~  <. s ,  r >. )  <->  ( ( A  .~  B  /\  B  .~  <. s ,  r >. )  ->  A  .~  <. s ,  r >. )
) )
19 breq2 3849 . . . . 5  |-  ( <.
s ,  r >.  =  C  ->  ( B  .~  <. s ,  r
>. 
<->  B  .~  C ) )
2019anbi2d 452 . . . 4  |-  ( <.
s ,  r >.  =  C  ->  ( ( A  .~  B  /\  B  .~  <. s ,  r
>. )  <->  ( A  .~  B  /\  B  .~  C
) ) )
21 breq2 3849 . . . 4  |-  ( <.
s ,  r >.  =  C  ->  ( A  .~  <. s ,  r
>. 
<->  A  .~  C ) )
2220, 21imbi12d 232 . . 3  |-  ( <.
s ,  r >.  =  C  ->  ( ( ( A  .~  B  /\  B  .~  <. s ,  r >. )  ->  A  .~  <. s ,  r >. )  <->  ( ( A  .~  B  /\  B  .~  C )  ->  A  .~  C
) ) )
231ecopoveq 6387 . . . . . . . 8  |-  ( ( ( f  e.  S  /\  g  e.  S
)  /\  ( h  e.  S  /\  t  e.  S ) )  -> 
( <. f ,  g
>.  .~  <. h ,  t
>. 
<->  ( f  .+  t
)  =  ( g 
.+  h ) ) )
24233adant3 963 . . . . . . 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 6387 . . . . . . . 8  |-  ( ( ( h  e.  S  /\  t  e.  S
)  /\  ( s  e.  S  /\  r  e.  S ) )  -> 
( <. h ,  t
>.  .~  <. s ,  r
>. 
<->  ( h  .+  r
)  =  ( t 
.+  s ) ) )
26253adant1 961 . . . . . . 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 457 . . . . . 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 5661 . . . . . . 7  |-  ( ( ( f  .+  t
)  =  ( g 
.+  h )  /\  ( h  .+  r )  =  ( t  .+  s ) )  -> 
( ( f  .+  t )  .+  (
h  .+  r )
)  =  ( ( g  .+  h ) 
.+  ( t  .+  s ) ) )
29 simp2l 969 . . . . . . . . 9  |-  ( ( ( f  e.  S  /\  g  e.  S
)  /\  ( h  e.  S  /\  t  e.  S )  /\  (
s  e.  S  /\  r  e.  S )
)  ->  h  e.  S )
30 simp2r 970 . . . . . . . . 9  |-  ( ( ( f  e.  S  /\  g  e.  S
)  /\  ( h  e.  S  /\  t  e.  S )  /\  (
s  e.  S  /\  r  e.  S )
)  ->  t  e.  S )
31 simp1l 967 . . . . . . . . 9  |-  ( ( ( f  e.  S  /\  g  e.  S
)  /\  ( h  e.  S  /\  t  e.  S )  /\  (
s  e.  S  /\  r  e.  S )
)  ->  f  e.  S )
32 ecopopr.com . . . . . . . . . 10  |-  ( x 
.+  y )  =  ( y  .+  x
)
3332a1i 9 . . . . . . . . 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 ecopopr.ass . . . . . . . . . 10  |-  ( ( x  .+  y ) 
.+  z )  =  ( x  .+  (
y  .+  z )
)
3534a1i 9 . . . . . . . . 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 972 . . . . . . . . 9  |-  ( ( ( f  e.  S  /\  g  e.  S
)  /\  ( h  e.  S  /\  t  e.  S )  /\  (
s  e.  S  /\  r  e.  S )
)  ->  r  e.  S )
37 ecopopr.cl . . . . . . . . . 10  |-  ( ( x  e.  S  /\  y  e.  S )  ->  ( x  .+  y
)  e.  S )
3837adantl 271 . . . . . . . . 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 5830 . . . . . . . 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 968 . . . . . . . . . 10  |-  ( ( ( f  e.  S  /\  g  e.  S
)  /\  ( h  e.  S  /\  t  e.  S )  /\  (
s  e.  S  /\  r  e.  S )
)  ->  g  e.  S )
41 simp3l 971 . . . . . . . . . 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 5830 . . . . . . . . 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 5829 . . . . . . . . 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 2122 . . . . . . . 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 2102 . . . . . . 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 154 . . . . . 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 148 . . . . 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 ecopopr.can . . . . . . . . 9  |-  ( ( x  e.  S  /\  y  e.  S )  ->  ( ( x  .+  y )  =  ( x  .+  z )  ->  y  =  z ) )
49483adant3 963 . . . . . . . 8  |-  ( ( x  e.  S  /\  y  e.  S  /\  z  e.  S )  ->  ( ( x  .+  y )  =  ( x  .+  z )  ->  y  =  z ) )
50 oveq2 5660 . . . . . . . 8  |-  ( y  =  z  ->  (
x  .+  y )  =  ( x  .+  z ) )
5149, 50impbid1 140 . . . . . . 7  |-  ( ( x  e.  S  /\  y  e.  S  /\  z  e.  S )  ->  ( ( x  .+  y )  =  ( x  .+  z )  <-> 
y  =  z ) )
5251adantl 271 . . . . . 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 ) )
5337caovcl 5799 . . . . . . 7  |-  ( ( h  e.  S  /\  t  e.  S )  ->  ( h  .+  t
)  e.  S )
5429, 30, 53syl2anc 403 . . . . . 6  |-  ( ( ( f  e.  S  /\  g  e.  S
)  /\  ( h  e.  S  /\  t  e.  S )  /\  (
s  e.  S  /\  r  e.  S )
)  ->  ( h  .+  t )  e.  S
)
5537caovcl 5799 . . . . . . 7  |-  ( ( f  e.  S  /\  r  e.  S )  ->  ( f  .+  r
)  e.  S )
5631, 36, 55syl2anc 403 . . . . . 6  |-  ( ( ( f  e.  S  /\  g  e.  S
)  /\  ( h  e.  S  /\  t  e.  S )  /\  (
s  e.  S  /\  r  e.  S )
)  ->  ( f  .+  r )  e.  S
)
5738, 40, 41caovcld 5798 . . . . . 6  |-  ( ( ( f  e.  S  /\  g  e.  S
)  /\  ( h  e.  S  /\  t  e.  S )  /\  (
s  e.  S  /\  r  e.  S )
)  ->  ( g  .+  s )  e.  S
)
5852, 54, 56, 57caovcand 5807 . . . . 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 ) ) )
5947, 58sylibd 147 . . . 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
) ) )
601ecopoveq 6387 . . . . 5  |-  ( ( ( f  e.  S  /\  g  e.  S
)  /\  ( s  e.  S  /\  r  e.  S ) )  -> 
( <. f ,  g
>.  .~  <. s ,  r
>. 
<->  ( f  .+  r
)  =  ( g 
.+  s ) ) )
61603adant2 962 . . . 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 ) ) )
6259, 61sylibrd 167 . . 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 >. )
)
6310, 14, 18, 22, 623optocl 4516 . 2  |-  ( ( A  e.  ( S  X.  S )  /\  B  e.  ( S  X.  S )  /\  C  e.  ( S  X.  S
) )  ->  (
( A  .~  B  /\  B  .~  C )  ->  A  .~  C
) )
649, 63mpcom 36 1  |-  ( ( A  .~  B  /\  B  .~  C )  ->  A  .~  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    /\ w3a 924    = wceq 1289   E.wex 1426    e. wcel 1438   <.cop 3449   class class class wbr 3845   {copab 3898    X. cxp 4436  (class class class)co 5652
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-xp 4444  df-iota 4980  df-fv 5023  df-ov 5655
This theorem is referenced by:  ecopover  6390
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