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Theorem th3q 6242
Description: Theorem 3Q of [Enderton] p. 60, extended to operations on ordered pairs. (Contributed by NM, 4-Aug-1995.) (Revised by Mario Carneiro, 19-Dec-2013.)
Hypotheses
Ref Expression
th3q.1  |-  .~  e.  _V
th3q.2  |-  .~  Er  ( S  X.  S
)
th3q.4  |-  ( ( ( ( w  e.  S  /\  v  e.  S )  /\  (
u  e.  S  /\  t  e.  S )
)  /\  ( (
s  e.  S  /\  f  e.  S )  /\  ( g  e.  S  /\  h  e.  S
) ) )  -> 
( ( <. w ,  v >.  .~  <. u ,  t >.  /\  <. s ,  f >.  .~  <. g ,  h >. )  ->  ( <. w ,  v
>.  .+  <. s ,  f
>. )  .~  ( <. u ,  t >.  .+  <. g ,  h >. ) ) )
th3q.5  |-  G  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  ( ( S  X.  S ) /.  .~  )  /\  y  e.  ( ( S  X.  S
) /.  .~  )
)  /\  E. w E. v E. u E. t ( ( x  =  [ <. w ,  v >. ]  .~  /\  y  =  [ <. u ,  t >. ]  .~  )  /\  z  =  [
( <. w ,  v
>.  .+  <. u ,  t
>. ) ]  .~  )
) }
Assertion
Ref Expression
th3q  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( [ <. A ,  B >. ]  .~  G [ <. C ,  D >. ]  .~  )  =  [ ( <. A ,  B >.  .+  <. C ,  D >. ) ]  .~  )
Distinct variable groups:    x, y, z, w, v, u, t, s, f, g, h, 
.~    x, S, y, z, w, v, u, t, s, f, g, h   
x, A, y, z, w, v, u, t, s, f    x, B, y, z, w, v, u, t, s, f   
x, C, y, z, w, v, u, t   
x, D, y, z, w, v, u, t   
x,  .+ , y, z, w, v, u, t, s, f, g, h
Allowed substitution hints:    A( g, h)    B( g, h)    C( f,
g, h, s)    D( f, g, h, s)    G( x, y, z, w, v, u, t, f, g, h, s)

Proof of Theorem th3q
StepHypRef Expression
1 opelxpi 4404 . . . 4  |-  ( ( A  e.  S  /\  B  e.  S )  -> 
<. A ,  B >.  e.  ( S  X.  S
) )
2 th3q.1 . . . . 5  |-  .~  e.  _V
32ecelqsi 6191 . . . 4  |-  ( <. A ,  B >.  e.  ( S  X.  S
)  ->  [ <. A ,  B >. ]  .~  e.  ( ( S  X.  S ) /.  .~  ) )
41, 3syl 14 . . 3  |-  ( ( A  e.  S  /\  B  e.  S )  ->  [ <. A ,  B >. ]  .~  e.  ( ( S  X.  S
) /.  .~  )
)
5 opelxpi 4404 . . . 4  |-  ( ( C  e.  S  /\  D  e.  S )  -> 
<. C ,  D >.  e.  ( S  X.  S
) )
62ecelqsi 6191 . . . 4  |-  ( <. C ,  D >.  e.  ( S  X.  S
)  ->  [ <. C ,  D >. ]  .~  e.  ( ( S  X.  S ) /.  .~  ) )
75, 6syl 14 . . 3  |-  ( ( C  e.  S  /\  D  e.  S )  ->  [ <. C ,  D >. ]  .~  e.  ( ( S  X.  S
) /.  .~  )
)
84, 7anim12i 325 . 2  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( [ <. A ,  B >. ]  .~  e.  ( ( S  X.  S ) /.  .~  )  /\  [ <. C ,  D >. ]  .~  e.  ( ( S  X.  S ) /.  .~  ) ) )
9 eqid 2056 . . . 4  |-  [ <. A ,  B >. ]  .~  =  [ <. A ,  B >. ]  .~
10 eqid 2056 . . . 4  |-  [ <. C ,  D >. ]  .~  =  [ <. C ,  D >. ]  .~
119, 10pm3.2i 261 . . 3  |-  ( [
<. A ,  B >. ]  .~  =  [ <. A ,  B >. ]  .~  /\ 
[ <. C ,  D >. ]  .~  =  [ <. C ,  D >. ]  .~  )
12 eqid 2056 . . 3  |-  [ (
<. A ,  B >.  .+ 
<. C ,  D >. ) ]  .~  =  [
( <. A ,  B >.  .+  <. C ,  D >. ) ]  .~
13 opeq12 3579 . . . . . 6  |-  ( ( w  =  A  /\  v  =  B )  -> 
<. w ,  v >.  =  <. A ,  B >. )
14 eceq1 6172 . . . . . . . . 9  |-  ( <.
w ,  v >.  =  <. A ,  B >.  ->  [ <. w ,  v >. ]  .~  =  [ <. A ,  B >. ]  .~  )
1514eqeq2d 2067 . . . . . . . 8  |-  ( <.
w ,  v >.  =  <. A ,  B >.  ->  ( [ <. A ,  B >. ]  .~  =  [ <. w ,  v
>. ]  .~  <->  [ <. A ,  B >. ]  .~  =  [ <. A ,  B >. ]  .~  ) )
1615anbi1d 446 . . . . . . 7  |-  ( <.
w ,  v >.  =  <. A ,  B >.  ->  ( ( [
<. A ,  B >. ]  .~  =  [ <. w ,  v >. ]  .~  /\ 
[ <. C ,  D >. ]  .~  =  [ <. C ,  D >. ]  .~  )  <->  ( [ <. A ,  B >. ]  .~  =  [ <. A ,  B >. ]  .~  /\ 
[ <. C ,  D >. ]  .~  =  [ <. C ,  D >. ]  .~  ) ) )
17 oveq1 5547 . . . . . . . . 9  |-  ( <.
w ,  v >.  =  <. A ,  B >.  ->  ( <. w ,  v >.  .+  <. C ,  D >. )  =  ( <. A ,  B >.  .+  <. C ,  D >. ) )
1817eceq1d 6173 . . . . . . . 8  |-  ( <.
w ,  v >.  =  <. A ,  B >.  ->  [ ( <.
w ,  v >.  .+  <. C ,  D >. ) ]  .~  =  [ ( <. A ,  B >.  .+  <. C ,  D >. ) ]  .~  )
1918eqeq2d 2067 . . . . . . 7  |-  ( <.
w ,  v >.  =  <. A ,  B >.  ->  ( [ (
<. A ,  B >.  .+ 
<. C ,  D >. ) ]  .~  =  [
( <. w ,  v
>.  .+  <. C ,  D >. ) ]  .~  <->  [ ( <. A ,  B >.  .+ 
<. C ,  D >. ) ]  .~  =  [
( <. A ,  B >.  .+  <. C ,  D >. ) ]  .~  )
)
2016, 19anbi12d 450 . . . . . 6  |-  ( <.
w ,  v >.  =  <. A ,  B >.  ->  ( ( ( [ <. A ,  B >. ]  .~  =  [ <. w ,  v >. ]  .~  /\  [ <. C ,  D >. ]  .~  =  [ <. C ,  D >. ]  .~  )  /\  [ ( <. A ,  B >.  .+  <. C ,  D >. ) ]  .~  =  [ ( <. w ,  v >.  .+  <. C ,  D >. ) ]  .~  )  <->  ( ( [ <. A ,  B >. ]  .~  =  [ <. A ,  B >. ]  .~  /\  [ <. C ,  D >. ]  .~  =  [ <. C ,  D >. ]  .~  )  /\  [ ( <. A ,  B >.  .+  <. C ,  D >. ) ]  .~  =  [ ( <. A ,  B >.  .+  <. C ,  D >. ) ]  .~  ) ) )
2113, 20syl 14 . . . . 5  |-  ( ( w  =  A  /\  v  =  B )  ->  ( ( ( [
<. A ,  B >. ]  .~  =  [ <. w ,  v >. ]  .~  /\ 
[ <. C ,  D >. ]  .~  =  [ <. C ,  D >. ]  .~  )  /\  [
( <. A ,  B >.  .+  <. C ,  D >. ) ]  .~  =  [ ( <. w ,  v >.  .+  <. C ,  D >. ) ]  .~  )  <->  ( ( [ <. A ,  B >. ]  .~  =  [ <. A ,  B >. ]  .~  /\  [ <. C ,  D >. ]  .~  =  [ <. C ,  D >. ]  .~  )  /\  [ ( <. A ,  B >.  .+  <. C ,  D >. ) ]  .~  =  [ ( <. A ,  B >.  .+  <. C ,  D >. ) ]  .~  ) ) )
2221spc2egv 2659 . . . 4  |-  ( ( A  e.  S  /\  B  e.  S )  ->  ( ( ( [
<. A ,  B >. ]  .~  =  [ <. A ,  B >. ]  .~  /\ 
[ <. C ,  D >. ]  .~  =  [ <. C ,  D >. ]  .~  )  /\  [
( <. A ,  B >.  .+  <. C ,  D >. ) ]  .~  =  [ ( <. A ,  B >.  .+  <. C ,  D >. ) ]  .~  )  ->  E. w E. v
( ( [ <. A ,  B >. ]  .~  =  [ <. w ,  v
>. ]  .~  /\  [ <. C ,  D >. ]  .~  =  [ <. C ,  D >. ]  .~  )  /\  [ ( <. A ,  B >.  .+ 
<. C ,  D >. ) ]  .~  =  [
( <. w ,  v
>.  .+  <. C ,  D >. ) ]  .~  )
) )
23 opeq12 3579 . . . . . . 7  |-  ( ( u  =  C  /\  t  =  D )  -> 
<. u ,  t >.  =  <. C ,  D >. )
24 eceq1 6172 . . . . . . . . . 10  |-  ( <.
u ,  t >.  =  <. C ,  D >.  ->  [ <. u ,  t >. ]  .~  =  [ <. C ,  D >. ]  .~  )
2524eqeq2d 2067 . . . . . . . . 9  |-  ( <.
u ,  t >.  =  <. C ,  D >.  ->  ( [ <. C ,  D >. ]  .~  =  [ <. u ,  t
>. ]  .~  <->  [ <. C ,  D >. ]  .~  =  [ <. C ,  D >. ]  .~  ) )
2625anbi2d 445 . . . . . . . 8  |-  ( <.
u ,  t >.  =  <. C ,  D >.  ->  ( ( [
<. A ,  B >. ]  .~  =  [ <. w ,  v >. ]  .~  /\ 
[ <. C ,  D >. ]  .~  =  [ <. u ,  t >. ]  .~  )  <->  ( [ <. A ,  B >. ]  .~  =  [ <. w ,  v >. ]  .~  /\ 
[ <. C ,  D >. ]  .~  =  [ <. C ,  D >. ]  .~  ) ) )
27 oveq2 5548 . . . . . . . . . 10  |-  ( <.
u ,  t >.  =  <. C ,  D >.  ->  ( <. w ,  v >.  .+  <. u ,  t >. )  =  ( <. w ,  v >.  .+  <. C ,  D >. )
)
2827eceq1d 6173 . . . . . . . . 9  |-  ( <.
u ,  t >.  =  <. C ,  D >.  ->  [ ( <.
w ,  v >.  .+  <. u ,  t
>. ) ]  .~  =  [ ( <. w ,  v >.  .+  <. C ,  D >. ) ]  .~  )
2928eqeq2d 2067 . . . . . . . 8  |-  ( <.
u ,  t >.  =  <. C ,  D >.  ->  ( [ (
<. A ,  B >.  .+ 
<. C ,  D >. ) ]  .~  =  [
( <. w ,  v
>.  .+  <. u ,  t
>. ) ]  .~  <->  [ ( <. A ,  B >.  .+ 
<. C ,  D >. ) ]  .~  =  [
( <. w ,  v
>.  .+  <. C ,  D >. ) ]  .~  )
)
3026, 29anbi12d 450 . . . . . . 7  |-  ( <.
u ,  t >.  =  <. C ,  D >.  ->  ( ( ( [ <. A ,  B >. ]  .~  =  [ <. w ,  v >. ]  .~  /\  [ <. C ,  D >. ]  .~  =  [ <. u ,  t
>. ]  .~  )  /\  [ ( <. A ,  B >.  .+  <. C ,  D >. ) ]  .~  =  [ ( <. w ,  v >.  .+  <. u ,  t >. ) ]  .~  )  <->  ( ( [ <. A ,  B >. ]  .~  =  [ <. w ,  v >. ]  .~  /\  [ <. C ,  D >. ]  .~  =  [ <. C ,  D >. ]  .~  )  /\  [ ( <. A ,  B >.  .+  <. C ,  D >. ) ]  .~  =  [ ( <. w ,  v >.  .+  <. C ,  D >. ) ]  .~  ) ) )
3123, 30syl 14 . . . . . 6  |-  ( ( u  =  C  /\  t  =  D )  ->  ( ( ( [
<. A ,  B >. ]  .~  =  [ <. w ,  v >. ]  .~  /\ 
[ <. C ,  D >. ]  .~  =  [ <. u ,  t >. ]  .~  )  /\  [
( <. A ,  B >.  .+  <. C ,  D >. ) ]  .~  =  [ ( <. w ,  v >.  .+  <. u ,  t >. ) ]  .~  )  <->  ( ( [ <. A ,  B >. ]  .~  =  [ <. w ,  v >. ]  .~  /\  [ <. C ,  D >. ]  .~  =  [ <. C ,  D >. ]  .~  )  /\  [ ( <. A ,  B >.  .+  <. C ,  D >. ) ]  .~  =  [ ( <. w ,  v >.  .+  <. C ,  D >. ) ]  .~  ) ) )
3231spc2egv 2659 . . . . 5  |-  ( ( C  e.  S  /\  D  e.  S )  ->  ( ( ( [
<. A ,  B >. ]  .~  =  [ <. w ,  v >. ]  .~  /\ 
[ <. C ,  D >. ]  .~  =  [ <. C ,  D >. ]  .~  )  /\  [
( <. A ,  B >.  .+  <. C ,  D >. ) ]  .~  =  [ ( <. w ,  v >.  .+  <. C ,  D >. ) ]  .~  )  ->  E. u E. t ( ( [
<. A ,  B >. ]  .~  =  [ <. w ,  v >. ]  .~  /\ 
[ <. C ,  D >. ]  .~  =  [ <. u ,  t >. ]  .~  )  /\  [
( <. A ,  B >.  .+  <. C ,  D >. ) ]  .~  =  [ ( <. w ,  v >.  .+  <. u ,  t >. ) ]  .~  ) ) )
33322eximdv 1778 . . . 4  |-  ( ( C  e.  S  /\  D  e.  S )  ->  ( E. w E. v ( ( [
<. A ,  B >. ]  .~  =  [ <. w ,  v >. ]  .~  /\ 
[ <. C ,  D >. ]  .~  =  [ <. C ,  D >. ]  .~  )  /\  [
( <. A ,  B >.  .+  <. C ,  D >. ) ]  .~  =  [ ( <. w ,  v >.  .+  <. C ,  D >. ) ]  .~  )  ->  E. w E. v E. u E. t ( ( [
<. A ,  B >. ]  .~  =  [ <. w ,  v >. ]  .~  /\ 
[ <. C ,  D >. ]  .~  =  [ <. u ,  t >. ]  .~  )  /\  [
( <. A ,  B >.  .+  <. C ,  D >. ) ]  .~  =  [ ( <. w ,  v >.  .+  <. u ,  t >. ) ]  .~  ) ) )
3422, 33sylan9 395 . . 3  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( ( ( [
<. A ,  B >. ]  .~  =  [ <. A ,  B >. ]  .~  /\ 
[ <. C ,  D >. ]  .~  =  [ <. C ,  D >. ]  .~  )  /\  [
( <. A ,  B >.  .+  <. C ,  D >. ) ]  .~  =  [ ( <. A ,  B >.  .+  <. C ,  D >. ) ]  .~  )  ->  E. w E. v E. u E. t ( ( [ <. A ,  B >. ]  .~  =  [ <. w ,  v
>. ]  .~  /\  [ <. C ,  D >. ]  .~  =  [ <. u ,  t >. ]  .~  )  /\  [ ( <. A ,  B >.  .+ 
<. C ,  D >. ) ]  .~  =  [
( <. w ,  v
>.  .+  <. u ,  t
>. ) ]  .~  )
) )
3511, 12, 34mp2ani 416 . 2  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  S  /\  D  e.  S ) )  ->  E. w E. v E. u E. t ( ( [ <. A ,  B >. ]  .~  =  [ <. w ,  v
>. ]  .~  /\  [ <. C ,  D >. ]  .~  =  [ <. u ,  t >. ]  .~  )  /\  [ ( <. A ,  B >.  .+ 
<. C ,  D >. ) ]  .~  =  [
( <. w ,  v
>.  .+  <. u ,  t
>. ) ]  .~  )
)
36 ecexg 6141 . . . 4  |-  (  .~  e.  _V  ->  [ ( <. A ,  B >.  .+ 
<. C ,  D >. ) ]  .~  e.  _V )
372, 36ax-mp 7 . . 3  |-  [ (
<. A ,  B >.  .+ 
<. C ,  D >. ) ]  .~  e.  _V
38 eqeq1 2062 . . . . . . . 8  |-  ( x  =  [ <. A ,  B >. ]  .~  ->  ( x  =  [ <. w ,  v >. ]  .~  <->  [
<. A ,  B >. ]  .~  =  [ <. w ,  v >. ]  .~  ) )
39 eqeq1 2062 . . . . . . . 8  |-  ( y  =  [ <. C ,  D >. ]  .~  ->  ( y  =  [ <. u ,  t >. ]  .~  <->  [
<. C ,  D >. ]  .~  =  [ <. u ,  t >. ]  .~  ) )
4038, 39bi2anan9 548 . . . . . . 7  |-  ( ( x  =  [ <. A ,  B >. ]  .~  /\  y  =  [ <. C ,  D >. ]  .~  )  ->  ( ( x  =  [ <. w ,  v >. ]  .~  /\  y  =  [ <. u ,  t >. ]  .~  ) 
<->  ( [ <. A ,  B >. ]  .~  =  [ <. w ,  v
>. ]  .~  /\  [ <. C ,  D >. ]  .~  =  [ <. u ,  t >. ]  .~  ) ) )
41 eqeq1 2062 . . . . . . 7  |-  ( z  =  [ ( <. A ,  B >.  .+ 
<. C ,  D >. ) ]  .~  ->  (
z  =  [ (
<. w ,  v >.  .+  <. u ,  t
>. ) ]  .~  <->  [ ( <. A ,  B >.  .+ 
<. C ,  D >. ) ]  .~  =  [
( <. w ,  v
>.  .+  <. u ,  t
>. ) ]  .~  )
)
4240, 41bi2anan9 548 . . . . . 6  |-  ( ( ( x  =  [ <. A ,  B >. ]  .~  /\  y  =  [ <. C ,  D >. ]  .~  )  /\  z  =  [ ( <. A ,  B >.  .+ 
<. C ,  D >. ) ]  .~  )  -> 
( ( ( x  =  [ <. w ,  v >. ]  .~  /\  y  =  [ <. u ,  t >. ]  .~  )  /\  z  =  [
( <. w ,  v
>.  .+  <. u ,  t
>. ) ]  .~  )  <->  ( ( [ <. A ,  B >. ]  .~  =  [ <. w ,  v
>. ]  .~  /\  [ <. C ,  D >. ]  .~  =  [ <. u ,  t >. ]  .~  )  /\  [ ( <. A ,  B >.  .+ 
<. C ,  D >. ) ]  .~  =  [
( <. w ,  v
>.  .+  <. u ,  t
>. ) ]  .~  )
) )
43423impa 1110 . . . . 5  |-  ( ( x  =  [ <. A ,  B >. ]  .~  /\  y  =  [ <. C ,  D >. ]  .~  /\  z  =  [ (
<. A ,  B >.  .+ 
<. C ,  D >. ) ]  .~  )  -> 
( ( ( x  =  [ <. w ,  v >. ]  .~  /\  y  =  [ <. u ,  t >. ]  .~  )  /\  z  =  [
( <. w ,  v
>.  .+  <. u ,  t
>. ) ]  .~  )  <->  ( ( [ <. A ,  B >. ]  .~  =  [ <. w ,  v
>. ]  .~  /\  [ <. C ,  D >. ]  .~  =  [ <. u ,  t >. ]  .~  )  /\  [ ( <. A ,  B >.  .+ 
<. C ,  D >. ) ]  .~  =  [
( <. w ,  v
>.  .+  <. u ,  t
>. ) ]  .~  )
) )
44434exbidv 1766 . . . 4  |-  ( ( x  =  [ <. A ,  B >. ]  .~  /\  y  =  [ <. C ,  D >. ]  .~  /\  z  =  [ (
<. A ,  B >.  .+ 
<. C ,  D >. ) ]  .~  )  -> 
( E. w E. v E. u E. t
( ( x  =  [ <. w ,  v
>. ]  .~  /\  y  =  [ <. u ,  t
>. ]  .~  )  /\  z  =  [ ( <. w ,  v >.  .+  <. u ,  t
>. ) ]  .~  )  <->  E. w E. v E. u E. t ( ( [ <. A ,  B >. ]  .~  =  [ <. w ,  v
>. ]  .~  /\  [ <. C ,  D >. ]  .~  =  [ <. u ,  t >. ]  .~  )  /\  [ ( <. A ,  B >.  .+ 
<. C ,  D >. ) ]  .~  =  [
( <. w ,  v
>.  .+  <. u ,  t
>. ) ]  .~  )
) )
45 th3q.2 . . . . 5  |-  .~  Er  ( S  X.  S
)
46 th3q.4 . . . . 5  |-  ( ( ( ( w  e.  S  /\  v  e.  S )  /\  (
u  e.  S  /\  t  e.  S )
)  /\  ( (
s  e.  S  /\  f  e.  S )  /\  ( g  e.  S  /\  h  e.  S
) ) )  -> 
( ( <. w ,  v >.  .~  <. u ,  t >.  /\  <. s ,  f >.  .~  <. g ,  h >. )  ->  ( <. w ,  v
>.  .+  <. s ,  f
>. )  .~  ( <. u ,  t >.  .+  <. g ,  h >. ) ) )
472, 45, 46th3qlem2 6240 . . . 4  |-  ( ( x  e.  ( ( S  X.  S ) /.  .~  )  /\  y  e.  ( ( S  X.  S ) /.  .~  ) )  ->  E* z E. w E. v E. u E. t ( ( x  =  [ <. w ,  v >. ]  .~  /\  y  =  [ <. u ,  t
>. ]  .~  )  /\  z  =  [ ( <. w ,  v >.  .+  <. u ,  t
>. ) ]  .~  )
)
48 th3q.5 . . . 4  |-  G  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  ( ( S  X.  S ) /.  .~  )  /\  y  e.  ( ( S  X.  S
) /.  .~  )
)  /\  E. w E. v E. u E. t ( ( x  =  [ <. w ,  v >. ]  .~  /\  y  =  [ <. u ,  t >. ]  .~  )  /\  z  =  [
( <. w ,  v
>.  .+  <. u ,  t
>. ) ]  .~  )
) }
4944, 47, 48ovig 5650 . . 3  |-  ( ( [ <. A ,  B >. ]  .~  e.  ( ( S  X.  S
) /.  .~  )  /\  [ <. C ,  D >. ]  .~  e.  ( ( S  X.  S
) /.  .~  )  /\  [ ( <. A ,  B >.  .+  <. C ,  D >. ) ]  .~  e.  _V )  ->  ( E. w E. v E. u E. t ( ( [ <. A ,  B >. ]  .~  =  [ <. w ,  v
>. ]  .~  /\  [ <. C ,  D >. ]  .~  =  [ <. u ,  t >. ]  .~  )  /\  [ ( <. A ,  B >.  .+ 
<. C ,  D >. ) ]  .~  =  [
( <. w ,  v
>.  .+  <. u ,  t
>. ) ]  .~  )  ->  ( [ <. A ,  B >. ]  .~  G [ <. C ,  D >. ]  .~  )  =  [ ( <. A ,  B >.  .+  <. C ,  D >. ) ]  .~  ) )
5037, 49mp3an3 1232 . 2  |-  ( ( [ <. A ,  B >. ]  .~  e.  ( ( S  X.  S
) /.  .~  )  /\  [ <. C ,  D >. ]  .~  e.  ( ( S  X.  S
) /.  .~  )
)  ->  ( E. w E. v E. u E. t ( ( [
<. A ,  B >. ]  .~  =  [ <. w ,  v >. ]  .~  /\ 
[ <. C ,  D >. ]  .~  =  [ <. u ,  t >. ]  .~  )  /\  [
( <. A ,  B >.  .+  <. C ,  D >. ) ]  .~  =  [ ( <. w ,  v >.  .+  <. u ,  t >. ) ]  .~  )  ->  ( [ <. A ,  B >. ]  .~  G [ <. C ,  D >. ]  .~  )  =  [
( <. A ,  B >.  .+  <. C ,  D >. ) ]  .~  )
)
518, 35, 50sylc 60 1  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( [ <. A ,  B >. ]  .~  G [ <. C ,  D >. ]  .~  )  =  [ ( <. A ,  B >.  .+  <. C ,  D >. ) ]  .~  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102    /\ w3a 896    = wceq 1259   E.wex 1397    e. wcel 1409   _Vcvv 2574   <.cop 3406   class class class wbr 3792    X. cxp 4371  (class class class)co 5540   {coprab 5541    Er wer 6134   [cec 6135   /.cqs 6136
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2788  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fv 4938  df-ov 5543  df-oprab 5544  df-er 6137  df-ec 6139  df-qs 6143
This theorem is referenced by:  oviec  6243
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