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Theorem th3q 6767
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 4721 . . . 4  |-  ( ( A  e.  S  /\  B  e.  S )  -> 
<. A ,  B >.  e.  ( S  X.  S
) )
2 th3q.1 . . . . 5  |-  .~  e.  _V
32ecelqsi 6715 . . . 4  |-  ( <. A ,  B >.  e.  ( S  X.  S
)  ->  [ <. A ,  B >. ]  .~  e.  ( ( S  X.  S ) /.  .~  ) )
41, 3syl 15 . . 3  |-  ( ( A  e.  S  /\  B  e.  S )  ->  [ <. A ,  B >. ]  .~  e.  ( ( S  X.  S
) /.  .~  )
)
5 opelxpi 4721 . . . 4  |-  ( ( C  e.  S  /\  D  e.  S )  -> 
<. C ,  D >.  e.  ( S  X.  S
) )
62ecelqsi 6715 . . . 4  |-  ( <. C ,  D >.  e.  ( S  X.  S
)  ->  [ <. C ,  D >. ]  .~  e.  ( ( S  X.  S ) /.  .~  ) )
75, 6syl 15 . . 3  |-  ( ( C  e.  S  /\  D  e.  S )  ->  [ <. C ,  D >. ]  .~  e.  ( ( S  X.  S
) /.  .~  )
)
84, 7anim12i 549 . 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 2283 . . . 4  |-  [ <. A ,  B >. ]  .~  =  [ <. A ,  B >. ]  .~
10 eqid 2283 . . . 4  |-  [ <. C ,  D >. ]  .~  =  [ <. C ,  D >. ]  .~
119, 10pm3.2i 441 . . 3  |-  ( [
<. A ,  B >. ]  .~  =  [ <. A ,  B >. ]  .~  /\ 
[ <. C ,  D >. ]  .~  =  [ <. C ,  D >. ]  .~  )
12 eqid 2283 . . 3  |-  [ (
<. A ,  B >.  .+ 
<. C ,  D >. ) ]  .~  =  [
( <. A ,  B >.  .+  <. C ,  D >. ) ]  .~
13 opeq12 3798 . . . . . 6  |-  ( ( w  =  A  /\  v  =  B )  -> 
<. w ,  v >.  =  <. A ,  B >. )
14 eceq1 6696 . . . . . . . . 9  |-  ( <.
w ,  v >.  =  <. A ,  B >.  ->  [ <. w ,  v >. ]  .~  =  [ <. A ,  B >. ]  .~  )
1514eqeq2d 2294 . . . . . . . 8  |-  ( <.
w ,  v >.  =  <. A ,  B >.  ->  ( [ <. A ,  B >. ]  .~  =  [ <. w ,  v
>. ]  .~  <->  [ <. A ,  B >. ]  .~  =  [ <. A ,  B >. ]  .~  ) )
1615anbi1d 685 . . . . . . 7  |-  ( <.
w ,  v >.  =  <. A ,  B >.  ->  ( ( [
<. A ,  B >. ]  .~  =  [ <. w ,  v >. ]  .~  /\ 
[ <. C ,  D >. ]  .~  =  [ <. C ,  D >. ]  .~  )  <->  ( [ <. A ,  B >. ]  .~  =  [ <. A ,  B >. ]  .~  /\ 
[ <. C ,  D >. ]  .~  =  [ <. C ,  D >. ]  .~  ) ) )
17 oveq1 5865 . . . . . . . . 9  |-  ( <.
w ,  v >.  =  <. A ,  B >.  ->  ( <. w ,  v >.  .+  <. C ,  D >. )  =  ( <. A ,  B >.  .+  <. C ,  D >. ) )
18 eceq1 6696 . . . . . . . . 9  |-  ( (
<. w ,  v >.  .+  <. C ,  D >. )  =  ( <. A ,  B >.  .+ 
<. C ,  D >. )  ->  [ ( <.
w ,  v >.  .+  <. C ,  D >. ) ]  .~  =  [ ( <. A ,  B >.  .+  <. C ,  D >. ) ]  .~  )
1917, 18syl 15 . . . . . . . 8  |-  ( <.
w ,  v >.  =  <. A ,  B >.  ->  [ ( <.
w ,  v >.  .+  <. C ,  D >. ) ]  .~  =  [ ( <. A ,  B >.  .+  <. C ,  D >. ) ]  .~  )
2019eqeq2d 2294 . . . . . . 7  |-  ( <.
w ,  v >.  =  <. A ,  B >.  ->  ( [ (
<. A ,  B >.  .+ 
<. C ,  D >. ) ]  .~  =  [
( <. w ,  v
>.  .+  <. C ,  D >. ) ]  .~  <->  [ ( <. A ,  B >.  .+ 
<. C ,  D >. ) ]  .~  =  [
( <. A ,  B >.  .+  <. C ,  D >. ) ]  .~  )
)
2116, 20anbi12d 691 . . . . . 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 >. ) ]  .~  ) ) )
2213, 21syl 15 . . . . 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 >. ) ]  .~  ) ) )
2322spc2egv 2870 . . . 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 >. ) ]  .~  )
) )
24 opeq12 3798 . . . . . . 7  |-  ( ( u  =  C  /\  t  =  D )  -> 
<. u ,  t >.  =  <. C ,  D >. )
25 eceq1 6696 . . . . . . . . . 10  |-  ( <.
u ,  t >.  =  <. C ,  D >.  ->  [ <. u ,  t >. ]  .~  =  [ <. C ,  D >. ]  .~  )
2625eqeq2d 2294 . . . . . . . . 9  |-  ( <.
u ,  t >.  =  <. C ,  D >.  ->  ( [ <. C ,  D >. ]  .~  =  [ <. u ,  t
>. ]  .~  <->  [ <. C ,  D >. ]  .~  =  [ <. C ,  D >. ]  .~  ) )
2726anbi2d 684 . . . . . . . 8  |-  ( <.
u ,  t >.  =  <. C ,  D >.  ->  ( ( [
<. A ,  B >. ]  .~  =  [ <. w ,  v >. ]  .~  /\ 
[ <. C ,  D >. ]  .~  =  [ <. u ,  t >. ]  .~  )  <->  ( [ <. A ,  B >. ]  .~  =  [ <. w ,  v >. ]  .~  /\ 
[ <. C ,  D >. ]  .~  =  [ <. C ,  D >. ]  .~  ) ) )
28 oveq2 5866 . . . . . . . . . 10  |-  ( <.
u ,  t >.  =  <. C ,  D >.  ->  ( <. w ,  v >.  .+  <. u ,  t >. )  =  ( <. w ,  v >.  .+  <. C ,  D >. )
)
29 eceq1 6696 . . . . . . . . . 10  |-  ( (
<. w ,  v >.  .+  <. u ,  t
>. )  =  ( <. w ,  v >.  .+  <. C ,  D >. )  ->  [ ( <. w ,  v >.  .+  <. u ,  t
>. ) ]  .~  =  [ ( <. w ,  v >.  .+  <. C ,  D >. ) ]  .~  )
3028, 29syl 15 . . . . . . . . 9  |-  ( <.
u ,  t >.  =  <. C ,  D >.  ->  [ ( <.
w ,  v >.  .+  <. u ,  t
>. ) ]  .~  =  [ ( <. w ,  v >.  .+  <. C ,  D >. ) ]  .~  )
3130eqeq2d 2294 . . . . . . . 8  |-  ( <.
u ,  t >.  =  <. C ,  D >.  ->  ( [ (
<. A ,  B >.  .+ 
<. C ,  D >. ) ]  .~  =  [
( <. w ,  v
>.  .+  <. u ,  t
>. ) ]  .~  <->  [ ( <. A ,  B >.  .+ 
<. C ,  D >. ) ]  .~  =  [
( <. w ,  v
>.  .+  <. C ,  D >. ) ]  .~  )
)
3227, 31anbi12d 691 . . . . . . 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 >. ) ]  .~  ) ) )
3324, 32syl 15 . . . . . 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 >. ) ]  .~  ) ) )
3433spc2egv 2870 . . . . 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 >. ) ]  .~  ) ) )
35342eximdv 1610 . . . 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 >. ) ]  .~  ) ) )
3623, 35sylan9 638 . . 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
>. ) ]  .~  )
) )
3711, 12, 36mp2ani 659 . 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
>. ) ]  .~  )
)
38 ecexg 6664 . . . 4  |-  (  .~  e.  _V  ->  [ ( <. A ,  B >.  .+ 
<. C ,  D >. ) ]  .~  e.  _V )
392, 38ax-mp 8 . . 3  |-  [ (
<. A ,  B >.  .+ 
<. C ,  D >. ) ]  .~  e.  _V
40 eqeq1 2289 . . . . . . . 8  |-  ( x  =  [ <. A ,  B >. ]  .~  ->  ( x  =  [ <. w ,  v >. ]  .~  <->  [
<. A ,  B >. ]  .~  =  [ <. w ,  v >. ]  .~  ) )
41 eqeq1 2289 . . . . . . . 8  |-  ( y  =  [ <. C ,  D >. ]  .~  ->  ( y  =  [ <. u ,  t >. ]  .~  <->  [
<. C ,  D >. ]  .~  =  [ <. u ,  t >. ]  .~  ) )
4240, 41bi2anan9 843 . . . . . . 7  |-  ( ( x  =  [ <. A ,  B >. ]  .~  /\  y  =  [ <. C ,  D >. ]  .~  )  ->  ( ( x  =  [ <. w ,  v >. ]  .~  /\  y  =  [ <. u ,  t >. ]  .~  ) 
<->  ( [ <. A ,  B >. ]  .~  =  [ <. w ,  v
>. ]  .~  /\  [ <. C ,  D >. ]  .~  =  [ <. u ,  t >. ]  .~  ) ) )
43 eqeq1 2289 . . . . . . 7  |-  ( z  =  [ ( <. A ,  B >.  .+ 
<. C ,  D >. ) ]  .~  ->  (
z  =  [ (
<. w ,  v >.  .+  <. u ,  t
>. ) ]  .~  <->  [ ( <. A ,  B >.  .+ 
<. C ,  D >. ) ]  .~  =  [
( <. w ,  v
>.  .+  <. u ,  t
>. ) ]  .~  )
)
4442, 43bi2anan9 843 . . . . . 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
>. ) ]  .~  )
) )
45443impa 1146 . . . . 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
>. ) ]  .~  )
) )
46454exbidv 1616 . . . 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
>. ) ]  .~  )
) )
47 th3q.2 . . . . 5  |-  .~  Er  ( S  X.  S
)
48 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 >. ) ) )
492, 47, 48th3qlem2 6765 . . . 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
>. ) ]  .~  )
)
50 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
>. ) ]  .~  )
) }
5146, 49, 50ovig 5969 . . 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 >. ) ]  .~  ) )
5239, 51mp3an3 1266 . 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 >. ) ]  .~  )
)
538, 37, 52sylc 56 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    <-> wb 176    /\ wa 358    /\ w3a 934   E.wex 1528    = wceq 1623    e. wcel 1684   _Vcvv 2788   <.cop 3643   class class class wbr 4023    X. cxp 4687  (class class class)co 5858   {coprab 5859    Er wer 6657   [cec 6658   /.cqs 6659
This theorem is referenced by:  ovec  6768
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-er 6660  df-ec 6662  df-qs 6666
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