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Theorem xpsaddlem 14511
Description: Lemma for xpsadd 14512 and xpsmul 14513. (Contributed by Mario Carneiro, 15-Aug-2015.)
Hypotheses
Ref Expression
xpsval.t  |-  T  =  ( R  X.s  S )
xpsval.x  |-  X  =  ( Base `  R
)
xpsval.y  |-  Y  =  ( Base `  S
)
xpsval.1  |-  ( ph  ->  R  e.  V )
xpsval.2  |-  ( ph  ->  S  e.  W )
xpsadd.3  |-  ( ph  ->  A  e.  X )
xpsadd.4  |-  ( ph  ->  B  e.  Y )
xpsadd.5  |-  ( ph  ->  C  e.  X )
xpsadd.6  |-  ( ph  ->  D  e.  Y )
xpsadd.7  |-  ( ph  ->  ( A  .x.  C
)  e.  X )
xpsadd.8  |-  ( ph  ->  ( B  .X.  D
)  e.  Y )
xpsaddlem.m  |-  .x.  =  ( E `  R )
xpsaddlem.n  |-  .X.  =  ( E `  S )
xpsaddlem.p  |-  .xb  =  ( E `  T )
xpsaddlem.f  |-  F  =  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) )
xpsaddlem.u  |-  U  =  ( (Scalar `  R
) X_s `' ( { R }  +c  { S }
) )
xpsaddlem.1  |-  ( (
ph  /\  `' ( { A }  +c  { B } )  e.  ran  F  /\  `' ( { C }  +c  { D } )  e.  ran  F )  ->  ( ( `' F `  `' ( { A }  +c  { B } ) ) 
.xb  ( `' F `  `' ( { C }  +c  { D }
) ) )  =  ( `' F `  ( `' ( { A }  +c  { B }
) ( E `  U ) `' ( { C }  +c  { D } ) ) ) )
xpsaddlem.2  |-  ( ( `' ( { R }  +c  { S }
)  Fn  2o  /\  `' ( { A }  +c  { B }
)  e.  ( Base `  U )  /\  `' ( { C }  +c  { D } )  e.  ( Base `  U
) )  ->  ( `' ( { A }  +c  { B }
) ( E `  U ) `' ( { C }  +c  { D } ) )  =  ( k  e.  2o  |->  ( ( `' ( { A }  +c  { B } ) `
 k ) ( E `  ( `' ( { R }  +c  { S } ) `
 k ) ) ( `' ( { C }  +c  { D } ) `  k
) ) ) )
Assertion
Ref Expression
xpsaddlem  |-  ( ph  ->  ( <. A ,  B >. 
.xb  <. C ,  D >. )  =  <. ( A  .x.  C ) ,  ( B  .X.  D
) >. )
Distinct variable groups:    x, k,
y, A    B, k, x, y    C, k, x, y    D, k, x, y    S, k    U, k    x, W    ph, k    .x. , k, x, y    .X. , k, x, y   
k, X, x, y    R, k, x    k, Y, x, y
Allowed substitution hints:    ph( x, y)    R( y)    S( x, y)    .xb (
x, y, k)    T( x, y, k)    U( x, y)    E( x, y, k)    F( x, y, k)    V( x, y, k)    W( y, k)

Proof of Theorem xpsaddlem
StepHypRef Expression
1 df-ov 6092 . . . . 5  |-  ( A F B )  =  ( F `  <. A ,  B >. )
2 xpsadd.3 . . . . . 6  |-  ( ph  ->  A  e.  X )
3 xpsadd.4 . . . . . 6  |-  ( ph  ->  B  e.  Y )
4 xpsaddlem.f . . . . . . 7  |-  F  =  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) )
54xpsfval 14503 . . . . . 6  |-  ( ( A  e.  X  /\  B  e.  Y )  ->  ( A F B )  =  `' ( { A }  +c  { B } ) )
62, 3, 5syl2anc 661 . . . . 5  |-  ( ph  ->  ( A F B )  =  `' ( { A }  +c  { B } ) )
71, 6syl5eqr 2487 . . . 4  |-  ( ph  ->  ( F `  <. A ,  B >. )  =  `' ( { A }  +c  { B }
) )
8 opelxpi 4869 . . . . . 6  |-  ( ( A  e.  X  /\  B  e.  Y )  -> 
<. A ,  B >.  e.  ( X  X.  Y
) )
92, 3, 8syl2anc 661 . . . . 5  |-  ( ph  -> 
<. A ,  B >.  e.  ( X  X.  Y
) )
104xpsff1o2 14507 . . . . . . 7  |-  F :
( X  X.  Y
)
-1-1-onto-> ran  F
11 f1of 5639 . . . . . . 7  |-  ( F : ( X  X.  Y ) -1-1-onto-> ran  F  ->  F : ( X  X.  Y ) --> ran  F
)
1210, 11ax-mp 5 . . . . . 6  |-  F :
( X  X.  Y
) --> ran  F
1312ffvelrni 5840 . . . . 5  |-  ( <. A ,  B >.  e.  ( X  X.  Y
)  ->  ( F `  <. A ,  B >. )  e.  ran  F
)
149, 13syl 16 . . . 4  |-  ( ph  ->  ( F `  <. A ,  B >. )  e.  ran  F )
157, 14eqeltrrd 2516 . . 3  |-  ( ph  ->  `' ( { A }  +c  { B }
)  e.  ran  F
)
16 df-ov 6092 . . . . 5  |-  ( C F D )  =  ( F `  <. C ,  D >. )
17 xpsadd.5 . . . . . 6  |-  ( ph  ->  C  e.  X )
18 xpsadd.6 . . . . . 6  |-  ( ph  ->  D  e.  Y )
194xpsfval 14503 . . . . . 6  |-  ( ( C  e.  X  /\  D  e.  Y )  ->  ( C F D )  =  `' ( { C }  +c  { D } ) )
2017, 18, 19syl2anc 661 . . . . 5  |-  ( ph  ->  ( C F D )  =  `' ( { C }  +c  { D } ) )
2116, 20syl5eqr 2487 . . . 4  |-  ( ph  ->  ( F `  <. C ,  D >. )  =  `' ( { C }  +c  { D }
) )
22 opelxpi 4869 . . . . . 6  |-  ( ( C  e.  X  /\  D  e.  Y )  -> 
<. C ,  D >.  e.  ( X  X.  Y
) )
2317, 18, 22syl2anc 661 . . . . 5  |-  ( ph  -> 
<. C ,  D >.  e.  ( X  X.  Y
) )
2412ffvelrni 5840 . . . . 5  |-  ( <. C ,  D >.  e.  ( X  X.  Y
)  ->  ( F `  <. C ,  D >. )  e.  ran  F
)
2523, 24syl 16 . . . 4  |-  ( ph  ->  ( F `  <. C ,  D >. )  e.  ran  F )
2621, 25eqeltrrd 2516 . . 3  |-  ( ph  ->  `' ( { C }  +c  { D }
)  e.  ran  F
)
27 xpsaddlem.1 . . 3  |-  ( (
ph  /\  `' ( { A }  +c  { B } )  e.  ran  F  /\  `' ( { C }  +c  { D } )  e.  ran  F )  ->  ( ( `' F `  `' ( { A }  +c  { B } ) ) 
.xb  ( `' F `  `' ( { C }  +c  { D }
) ) )  =  ( `' F `  ( `' ( { A }  +c  { B }
) ( E `  U ) `' ( { C }  +c  { D } ) ) ) )
2815, 26, 27mpd3an23 1316 . 2  |-  ( ph  ->  ( ( `' F `  `' ( { A }  +c  { B }
) )  .xb  ( `' F `  `' ( { C }  +c  { D } ) ) )  =  ( `' F `  ( `' ( { A }  +c  { B } ) ( E `  U
) `' ( { C }  +c  { D } ) ) ) )
29 f1ocnvfv 5983 . . . . 5  |-  ( ( F : ( X  X.  Y ) -1-1-onto-> ran  F  /\  <. A ,  B >.  e.  ( X  X.  Y ) )  -> 
( ( F `  <. A ,  B >. )  =  `' ( { A }  +c  { B } )  ->  ( `' F `  `' ( { A }  +c  { B } ) )  =  <. A ,  B >. ) )
3010, 9, 29sylancr 663 . . . 4  |-  ( ph  ->  ( ( F `  <. A ,  B >. )  =  `' ( { A }  +c  { B } )  ->  ( `' F `  `' ( { A }  +c  { B } ) )  =  <. A ,  B >. ) )
317, 30mpd 15 . . 3  |-  ( ph  ->  ( `' F `  `' ( { A }  +c  { B }
) )  =  <. A ,  B >. )
32 f1ocnvfv 5983 . . . . 5  |-  ( ( F : ( X  X.  Y ) -1-1-onto-> ran  F  /\  <. C ,  D >.  e.  ( X  X.  Y ) )  -> 
( ( F `  <. C ,  D >. )  =  `' ( { C }  +c  { D } )  ->  ( `' F `  `' ( { C }  +c  { D } ) )  =  <. C ,  D >. ) )
3310, 23, 32sylancr 663 . . . 4  |-  ( ph  ->  ( ( F `  <. C ,  D >. )  =  `' ( { C }  +c  { D } )  ->  ( `' F `  `' ( { C }  +c  { D } ) )  =  <. C ,  D >. ) )
3421, 33mpd 15 . . 3  |-  ( ph  ->  ( `' F `  `' ( { C }  +c  { D }
) )  =  <. C ,  D >. )
3531, 34oveq12d 6107 . 2  |-  ( ph  ->  ( ( `' F `  `' ( { A }  +c  { B }
) )  .xb  ( `' F `  `' ( { C }  +c  { D } ) ) )  =  ( <. A ,  B >.  .xb 
<. C ,  D >. ) )
36 xpsval.1 . . . . . . 7  |-  ( ph  ->  R  e.  V )
37 xpsval.2 . . . . . . 7  |-  ( ph  ->  S  e.  W )
38 xpscfn 14495 . . . . . . 7  |-  ( ( R  e.  V  /\  S  e.  W )  ->  `' ( { R }  +c  { S }
)  Fn  2o )
3936, 37, 38syl2anc 661 . . . . . 6  |-  ( ph  ->  `' ( { R }  +c  { S }
)  Fn  2o )
40 xpsval.t . . . . . . . 8  |-  T  =  ( R  X.s  S )
41 xpsval.x . . . . . . . 8  |-  X  =  ( Base `  R
)
42 xpsval.y . . . . . . . 8  |-  Y  =  ( Base `  S
)
43 eqid 2441 . . . . . . . 8  |-  (Scalar `  R )  =  (Scalar `  R )
44 xpsaddlem.u . . . . . . . 8  |-  U  =  ( (Scalar `  R
) X_s `' ( { R }  +c  { S }
) )
4540, 41, 42, 36, 37, 4, 43, 44xpslem 14509 . . . . . . 7  |-  ( ph  ->  ran  F  =  (
Base `  U )
)
4615, 45eleqtrd 2517 . . . . . 6  |-  ( ph  ->  `' ( { A }  +c  { B }
)  e.  ( Base `  U ) )
4726, 45eleqtrd 2517 . . . . . 6  |-  ( ph  ->  `' ( { C }  +c  { D }
)  e.  ( Base `  U ) )
48 xpsaddlem.2 . . . . . 6  |-  ( ( `' ( { R }  +c  { S }
)  Fn  2o  /\  `' ( { A }  +c  { B }
)  e.  ( Base `  U )  /\  `' ( { C }  +c  { D } )  e.  ( Base `  U
) )  ->  ( `' ( { A }  +c  { B }
) ( E `  U ) `' ( { C }  +c  { D } ) )  =  ( k  e.  2o  |->  ( ( `' ( { A }  +c  { B } ) `
 k ) ( E `  ( `' ( { R }  +c  { S } ) `
 k ) ) ( `' ( { C }  +c  { D } ) `  k
) ) ) )
4939, 46, 47, 48syl3anc 1218 . . . . 5  |-  ( ph  ->  ( `' ( { A }  +c  { B } ) ( E `
 U ) `' ( { C }  +c  { D } ) )  =  ( k  e.  2o  |->  ( ( `' ( { A }  +c  { B }
) `  k )
( E `  ( `' ( { R }  +c  { S }
) `  k )
) ( `' ( { C }  +c  { D } ) `  k ) ) ) )
50 xpsadd.7 . . . . . . . 8  |-  ( ph  ->  ( A  .x.  C
)  e.  X )
51 xpsadd.8 . . . . . . . 8  |-  ( ph  ->  ( B  .X.  D
)  e.  Y )
52 xpscfn 14495 . . . . . . . 8  |-  ( ( ( A  .x.  C
)  e.  X  /\  ( B  .X.  D )  e.  Y )  ->  `' ( { ( A  .x.  C ) }  +c  { ( B  .X.  D ) } )  Fn  2o )
5350, 51, 52syl2anc 661 . . . . . . 7  |-  ( ph  ->  `' ( { ( A  .x.  C ) }  +c  { ( B  .X.  D ) } )  Fn  2o )
54 dffn5 5735 . . . . . . 7  |-  ( `' ( { ( A 
.x.  C ) }  +c  { ( B 
.X.  D ) } )  Fn  2o  <->  `' ( { ( A  .x.  C ) }  +c  { ( B  .X.  D
) } )  =  ( k  e.  2o  |->  ( `' ( { ( A  .x.  C ) }  +c  { ( B  .X.  D ) } ) `  k
) ) )
5553, 54sylib 196 . . . . . 6  |-  ( ph  ->  `' ( { ( A  .x.  C ) }  +c  { ( B  .X.  D ) } )  =  ( k  e.  2o  |->  ( `' ( { ( A  .x.  C ) }  +c  { ( B  .X.  D ) } ) `  k
) ) )
56 iftrue 3795 . . . . . . . . . . . . 13  |-  ( k  =  (/)  ->  if ( k  =  (/) ,  R ,  S )  =  R )
5756fveq2d 5693 . . . . . . . . . . . 12  |-  ( k  =  (/)  ->  ( E `
 if ( k  =  (/) ,  R ,  S ) )  =  ( E `  R
) )
58 xpsaddlem.m . . . . . . . . . . . 12  |-  .x.  =  ( E `  R )
5957, 58syl6eqr 2491 . . . . . . . . . . 11  |-  ( k  =  (/)  ->  ( E `
 if ( k  =  (/) ,  R ,  S ) )  = 
.x.  )
60 iftrue 3795 . . . . . . . . . . 11  |-  ( k  =  (/)  ->  if ( k  =  (/) ,  A ,  B )  =  A )
61 iftrue 3795 . . . . . . . . . . 11  |-  ( k  =  (/)  ->  if ( k  =  (/) ,  C ,  D )  =  C )
6259, 60, 61oveq123d 6110 . . . . . . . . . 10  |-  ( k  =  (/)  ->  ( if ( k  =  (/) ,  A ,  B ) ( E `  if ( k  =  (/) ,  R ,  S ) ) if ( k  =  (/) ,  C ,  D ) )  =  ( A  .x.  C
) )
63 iftrue 3795 . . . . . . . . . 10  |-  ( k  =  (/)  ->  if ( k  =  (/) ,  ( A  .x.  C ) ,  ( B  .X.  D ) )  =  ( A  .x.  C
) )
6462, 63eqtr4d 2476 . . . . . . . . 9  |-  ( k  =  (/)  ->  ( if ( k  =  (/) ,  A ,  B ) ( E `  if ( k  =  (/) ,  R ,  S ) ) if ( k  =  (/) ,  C ,  D ) )  =  if ( k  =  (/) ,  ( A  .x.  C ) ,  ( B  .X.  D )
) )
65 iffalse 3797 . . . . . . . . . . . . 13  |-  ( -.  k  =  (/)  ->  if ( k  =  (/) ,  R ,  S )  =  S )
6665fveq2d 5693 . . . . . . . . . . . 12  |-  ( -.  k  =  (/)  ->  ( E `  if (
k  =  (/) ,  R ,  S ) )  =  ( E `  S
) )
67 xpsaddlem.n . . . . . . . . . . . 12  |-  .X.  =  ( E `  S )
6866, 67syl6eqr 2491 . . . . . . . . . . 11  |-  ( -.  k  =  (/)  ->  ( E `  if (
k  =  (/) ,  R ,  S ) )  = 
.X.  )
69 iffalse 3797 . . . . . . . . . . 11  |-  ( -.  k  =  (/)  ->  if ( k  =  (/) ,  A ,  B )  =  B )
70 iffalse 3797 . . . . . . . . . . 11  |-  ( -.  k  =  (/)  ->  if ( k  =  (/) ,  C ,  D )  =  D )
7168, 69, 70oveq123d 6110 . . . . . . . . . 10  |-  ( -.  k  =  (/)  ->  ( if ( k  =  (/) ,  A ,  B ) ( E `  if ( k  =  (/) ,  R ,  S ) ) if ( k  =  (/) ,  C ,  D ) )  =  ( B  .X.  D
) )
72 iffalse 3797 . . . . . . . . . 10  |-  ( -.  k  =  (/)  ->  if ( k  =  (/) ,  ( A  .x.  C
) ,  ( B 
.X.  D ) )  =  ( B  .X.  D ) )
7371, 72eqtr4d 2476 . . . . . . . . 9  |-  ( -.  k  =  (/)  ->  ( if ( k  =  (/) ,  A ,  B ) ( E `  if ( k  =  (/) ,  R ,  S ) ) if ( k  =  (/) ,  C ,  D ) )  =  if ( k  =  (/) ,  ( A  .x.  C ) ,  ( B  .X.  D )
) )
7464, 73pm2.61i 164 . . . . . . . 8  |-  ( if ( k  =  (/) ,  A ,  B ) ( E `  if ( k  =  (/) ,  R ,  S ) ) if ( k  =  (/) ,  C ,  D ) )  =  if ( k  =  (/) ,  ( A  .x.  C ) ,  ( B  .X.  D )
)
7536adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  2o )  ->  R  e.  V )
7637adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  2o )  ->  S  e.  W )
77 simpr 461 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  2o )  ->  k  e.  2o )
78 xpscfv 14498 . . . . . . . . . . 11  |-  ( ( R  e.  V  /\  S  e.  W  /\  k  e.  2o )  ->  ( `' ( { R }  +c  { S } ) `  k
)  =  if ( k  =  (/) ,  R ,  S ) )
7975, 76, 77, 78syl3anc 1218 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  2o )  ->  ( `' ( { R }  +c  { S } ) `
 k )  =  if ( k  =  (/) ,  R ,  S
) )
8079fveq2d 5693 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  2o )  ->  ( E `
 ( `' ( { R }  +c  { S } ) `  k ) )  =  ( E `  if ( k  =  (/) ,  R ,  S ) ) )
812adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  2o )  ->  A  e.  X )
823adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  2o )  ->  B  e.  Y )
83 xpscfv 14498 . . . . . . . . . 10  |-  ( ( A  e.  X  /\  B  e.  Y  /\  k  e.  2o )  ->  ( `' ( { A }  +c  { B } ) `  k
)  =  if ( k  =  (/) ,  A ,  B ) )
8481, 82, 77, 83syl3anc 1218 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  2o )  ->  ( `' ( { A }  +c  { B } ) `
 k )  =  if ( k  =  (/) ,  A ,  B
) )
8517adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  2o )  ->  C  e.  X )
8618adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  2o )  ->  D  e.  Y )
87 xpscfv 14498 . . . . . . . . . 10  |-  ( ( C  e.  X  /\  D  e.  Y  /\  k  e.  2o )  ->  ( `' ( { C }  +c  { D } ) `  k
)  =  if ( k  =  (/) ,  C ,  D ) )
8885, 86, 77, 87syl3anc 1218 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  2o )  ->  ( `' ( { C }  +c  { D } ) `
 k )  =  if ( k  =  (/) ,  C ,  D
) )
8980, 84, 88oveq123d 6110 . . . . . . . 8  |-  ( (
ph  /\  k  e.  2o )  ->  ( ( `' ( { A }  +c  { B }
) `  k )
( E `  ( `' ( { R }  +c  { S }
) `  k )
) ( `' ( { C }  +c  { D } ) `  k ) )  =  ( if ( k  =  (/) ,  A ,  B ) ( E `
 if ( k  =  (/) ,  R ,  S ) ) if ( k  =  (/) ,  C ,  D ) ) )
9050adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  2o )  ->  ( A 
.x.  C )  e.  X )
9151adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  2o )  ->  ( B 
.X.  D )  e.  Y )
92 xpscfv 14498 . . . . . . . . 9  |-  ( ( ( A  .x.  C
)  e.  X  /\  ( B  .X.  D )  e.  Y  /\  k  e.  2o )  ->  ( `' ( { ( A  .x.  C ) }  +c  { ( B  .X.  D ) } ) `  k
)  =  if ( k  =  (/) ,  ( A  .x.  C ) ,  ( B  .X.  D ) ) )
9390, 91, 77, 92syl3anc 1218 . . . . . . . 8  |-  ( (
ph  /\  k  e.  2o )  ->  ( `' ( { ( A 
.x.  C ) }  +c  { ( B 
.X.  D ) } ) `  k )  =  if ( k  =  (/) ,  ( A 
.x.  C ) ,  ( B  .X.  D
) ) )
9474, 89, 933eqtr4a 2499 . . . . . . 7  |-  ( (
ph  /\  k  e.  2o )  ->  ( ( `' ( { A }  +c  { B }
) `  k )
( E `  ( `' ( { R }  +c  { S }
) `  k )
) ( `' ( { C }  +c  { D } ) `  k ) )  =  ( `' ( { ( A  .x.  C
) }  +c  {
( B  .X.  D
) } ) `  k ) )
9594mpteq2dva 4376 . . . . . 6  |-  ( ph  ->  ( k  e.  2o  |->  ( ( `' ( { A }  +c  { B } ) `  k ) ( E `
 ( `' ( { R }  +c  { S } ) `  k ) ) ( `' ( { C }  +c  { D }
) `  k )
) )  =  ( k  e.  2o  |->  ( `' ( { ( A  .x.  C ) }  +c  { ( B  .X.  D ) } ) `  k
) ) )
9655, 95eqtr4d 2476 . . . . 5  |-  ( ph  ->  `' ( { ( A  .x.  C ) }  +c  { ( B  .X.  D ) } )  =  ( k  e.  2o  |->  ( ( `' ( { A }  +c  { B } ) `  k
) ( E `  ( `' ( { R }  +c  { S }
) `  k )
) ( `' ( { C }  +c  { D } ) `  k ) ) ) )
9749, 96eqtr4d 2476 . . . 4  |-  ( ph  ->  ( `' ( { A }  +c  { B } ) ( E `
 U ) `' ( { C }  +c  { D } ) )  =  `' ( { ( A  .x.  C ) }  +c  { ( B  .X.  D
) } ) )
9897fveq2d 5693 . . 3  |-  ( ph  ->  ( `' F `  ( `' ( { A }  +c  { B }
) ( E `  U ) `' ( { C }  +c  { D } ) ) )  =  ( `' F `  `' ( { ( A  .x.  C ) }  +c  { ( B  .X.  D
) } ) ) )
99 df-ov 6092 . . . . 5  |-  ( ( A  .x.  C ) F ( B  .X.  D ) )  =  ( F `  <. ( A  .x.  C ) ,  ( B  .X.  D ) >. )
1004xpsfval 14503 . . . . . 6  |-  ( ( ( A  .x.  C
)  e.  X  /\  ( B  .X.  D )  e.  Y )  -> 
( ( A  .x.  C ) F ( B  .X.  D )
)  =  `' ( { ( A  .x.  C ) }  +c  { ( B  .X.  D
) } ) )
10150, 51, 100syl2anc 661 . . . . 5  |-  ( ph  ->  ( ( A  .x.  C ) F ( B  .X.  D )
)  =  `' ( { ( A  .x.  C ) }  +c  { ( B  .X.  D
) } ) )
10299, 101syl5eqr 2487 . . . 4  |-  ( ph  ->  ( F `  <. ( A  .x.  C ) ,  ( B  .X.  D ) >. )  =  `' ( { ( A  .x.  C ) }  +c  { ( B  .X.  D ) } ) )
103 opelxpi 4869 . . . . . 6  |-  ( ( ( A  .x.  C
)  e.  X  /\  ( B  .X.  D )  e.  Y )  ->  <. ( A  .x.  C
) ,  ( B 
.X.  D ) >.  e.  ( X  X.  Y
) )
10450, 51, 103syl2anc 661 . . . . 5  |-  ( ph  -> 
<. ( A  .x.  C
) ,  ( B 
.X.  D ) >.  e.  ( X  X.  Y
) )
105 f1ocnvfv 5983 . . . . 5  |-  ( ( F : ( X  X.  Y ) -1-1-onto-> ran  F  /\  <. ( A  .x.  C ) ,  ( B  .X.  D ) >.  e.  ( X  X.  Y ) )  -> 
( ( F `  <. ( A  .x.  C
) ,  ( B 
.X.  D ) >.
)  =  `' ( { ( A  .x.  C ) }  +c  { ( B  .X.  D
) } )  -> 
( `' F `  `' ( { ( A  .x.  C ) }  +c  { ( B  .X.  D ) } ) )  = 
<. ( A  .x.  C
) ,  ( B 
.X.  D ) >.
) )
10610, 104, 105sylancr 663 . . . 4  |-  ( ph  ->  ( ( F `  <. ( A  .x.  C
) ,  ( B 
.X.  D ) >.
)  =  `' ( { ( A  .x.  C ) }  +c  { ( B  .X.  D
) } )  -> 
( `' F `  `' ( { ( A  .x.  C ) }  +c  { ( B  .X.  D ) } ) )  = 
<. ( A  .x.  C
) ,  ( B 
.X.  D ) >.
) )
107102, 106mpd 15 . . 3  |-  ( ph  ->  ( `' F `  `' ( { ( A  .x.  C ) }  +c  { ( B  .X.  D ) } ) )  = 
<. ( A  .x.  C
) ,  ( B 
.X.  D ) >.
)
10898, 107eqtrd 2473 . 2  |-  ( ph  ->  ( `' F `  ( `' ( { A }  +c  { B }
) ( E `  U ) `' ( { C }  +c  { D } ) ) )  =  <. ( A  .x.  C ) ,  ( B  .X.  D
) >. )
10928, 35, 1083eqtr3d 2481 1  |-  ( ph  ->  ( <. A ,  B >. 
.xb  <. C ,  D >. )  =  <. ( A  .x.  C ) ,  ( B  .X.  D
) >. )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   (/)c0 3635   ifcif 3789   {csn 3875   <.cop 3881    e. cmpt 4348    X. cxp 4836   `'ccnv 4837   ran crn 4839    Fn wfn 5411   -->wf 5412   -1-1-onto->wf1o 5415   ` cfv 5416  (class class class)co 6089    e. cmpt2 6091   2oc2o 6912    +c ccda 8334   Basecbs 14172  Scalarcsca 14239   X_scprds 14382    X.s cxps 14442
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-int 4127  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-om 6475  df-1st 6575  df-2nd 6576  df-recs 6830  df-rdg 6864  df-1o 6918  df-2o 6919  df-oadd 6922  df-er 7099  df-map 7214  df-ixp 7262  df-en 7309  df-dom 7310  df-sdom 7311  df-fin 7312  df-sup 7689  df-cda 8335  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-nn 10321  df-2 10378  df-3 10379  df-4 10380  df-5 10381  df-6 10382  df-7 10383  df-8 10384  df-9 10385  df-10 10386  df-n0 10578  df-z 10645  df-dec 10754  df-uz 10860  df-fz 11436  df-struct 14174  df-ndx 14175  df-slot 14176  df-base 14177  df-plusg 14249  df-mulr 14250  df-sca 14252  df-vsca 14253  df-ip 14254  df-tset 14255  df-ple 14256  df-ds 14258  df-hom 14260  df-cco 14261  df-prds 14384
This theorem is referenced by:  xpsadd  14512  xpsmul  14513
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