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Theorem xpsaddlem 13479
Description: Lemma for xpsadd 13480 and xpsmul 13481. (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 5863 . . . . 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 13471 . . . . . 6  |-  ( ( A  e.  X  /\  B  e.  Y )  ->  ( A F B )  =  `' ( { A }  +c  { B } ) )
62, 3, 5syl2anc 642 . . . . 5  |-  ( ph  ->  ( A F B )  =  `' ( { A }  +c  { B } ) )
71, 6syl5eqr 2331 . . . 4  |-  ( ph  ->  ( F `  <. A ,  B >. )  =  `' ( { A }  +c  { B }
) )
8 opelxpi 4723 . . . . . 6  |-  ( ( A  e.  X  /\  B  e.  Y )  -> 
<. A ,  B >.  e.  ( X  X.  Y
) )
92, 3, 8syl2anc 642 . . . . 5  |-  ( ph  -> 
<. A ,  B >.  e.  ( X  X.  Y
) )
104xpsff1o2 13475 . . . . . . 7  |-  F :
( X  X.  Y
)
-1-1-onto-> ran  F
11 f1of 5474 . . . . . . 7  |-  ( F : ( X  X.  Y ) -1-1-onto-> ran  F  ->  F : ( X  X.  Y ) --> ran  F
)
1210, 11ax-mp 8 . . . . . 6  |-  F :
( X  X.  Y
) --> ran  F
1312ffvelrni 5666 . . . . 5  |-  ( <. A ,  B >.  e.  ( X  X.  Y
)  ->  ( F `  <. A ,  B >. )  e.  ran  F
)
149, 13syl 15 . . . 4  |-  ( ph  ->  ( F `  <. A ,  B >. )  e.  ran  F )
157, 14eqeltrrd 2360 . . 3  |-  ( ph  ->  `' ( { A }  +c  { B }
)  e.  ran  F
)
16 df-ov 5863 . . . . 5  |-  ( C F D )  =  ( F `  <. C ,  D >. )
17 xpsadd.5 . . . . . 6  |-  ( ph  ->  C  e.  X )
18 xpsadd.6 . . . . . 6  |-  ( ph  ->  D  e.  Y )
194xpsfval 13471 . . . . . 6  |-  ( ( C  e.  X  /\  D  e.  Y )  ->  ( C F D )  =  `' ( { C }  +c  { D } ) )
2017, 18, 19syl2anc 642 . . . . 5  |-  ( ph  ->  ( C F D )  =  `' ( { C }  +c  { D } ) )
2116, 20syl5eqr 2331 . . . 4  |-  ( ph  ->  ( F `  <. C ,  D >. )  =  `' ( { C }  +c  { D }
) )
22 opelxpi 4723 . . . . . 6  |-  ( ( C  e.  X  /\  D  e.  Y )  -> 
<. C ,  D >.  e.  ( X  X.  Y
) )
2317, 18, 22syl2anc 642 . . . . 5  |-  ( ph  -> 
<. C ,  D >.  e.  ( X  X.  Y
) )
2412ffvelrni 5666 . . . . 5  |-  ( <. C ,  D >.  e.  ( X  X.  Y
)  ->  ( F `  <. C ,  D >. )  e.  ran  F
)
2523, 24syl 15 . . . 4  |-  ( ph  ->  ( F `  <. C ,  D >. )  e.  ran  F )
2621, 25eqeltrrd 2360 . . 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 1279 . 2  |-  ( ph  ->  ( ( `' F `  `' ( { A }  +c  { B }
) )  .xb  ( `' F `  `' ( { C }  +c  { D } ) ) )  =  ( `' F `  ( `' ( { A }  +c  { B } ) ( E `  U
) `' ( { C }  +c  { D } ) ) ) )
29 f1ocnvfv 5796 . . . . 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 644 . . . 4  |-  ( ph  ->  ( ( F `  <. A ,  B >. )  =  `' ( { A }  +c  { B } )  ->  ( `' F `  `' ( { A }  +c  { B } ) )  =  <. A ,  B >. ) )
317, 30mpd 14 . . 3  |-  ( ph  ->  ( `' F `  `' ( { A }  +c  { B }
) )  =  <. A ,  B >. )
32 f1ocnvfv 5796 . . . . 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 644 . . . 4  |-  ( ph  ->  ( ( F `  <. C ,  D >. )  =  `' ( { C }  +c  { D } )  ->  ( `' F `  `' ( { C }  +c  { D } ) )  =  <. C ,  D >. ) )
3421, 33mpd 14 . . 3  |-  ( ph  ->  ( `' F `  `' ( { C }  +c  { D }
) )  =  <. C ,  D >. )
3531, 34oveq12d 5878 . 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 13463 . . . . . . 7  |-  ( ( R  e.  V  /\  S  e.  W )  ->  `' ( { R }  +c  { S }
)  Fn  2o )
3936, 37, 38syl2anc 642 . . . . . 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 2285 . . . . . . . 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 13477 . . . . . . 7  |-  ( ph  ->  ran  F  =  (
Base `  U )
)
4615, 45eleqtrd 2361 . . . . . 6  |-  ( ph  ->  `' ( { A }  +c  { B }
)  e.  ( Base `  U ) )
4726, 45eleqtrd 2361 . . . . . 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 1182 . . . . 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 13463 . . . . . . . 8  |-  ( ( ( A  .x.  C
)  e.  X  /\  ( B  .X.  D )  e.  Y )  ->  `' ( { ( A  .x.  C ) }  +c  { ( B  .X.  D ) } )  Fn  2o )
5350, 51, 52syl2anc 642 . . . . . . 7  |-  ( ph  ->  `' ( { ( A  .x.  C ) }  +c  { ( B  .X.  D ) } )  Fn  2o )
54 dffn5 5570 . . . . . . 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 188 . . . . . 6  |-  ( ph  ->  `' ( { ( A  .x.  C ) }  +c  { ( B  .X.  D ) } )  =  ( k  e.  2o  |->  ( `' ( { ( A  .x.  C ) }  +c  { ( B  .X.  D ) } ) `  k
) ) )
56 iftrue 3573 . . . . . . . . . . . . 13  |-  ( k  =  (/)  ->  if ( k  =  (/) ,  R ,  S )  =  R )
5756fveq2d 5531 . . . . . . . . . . . 12  |-  ( k  =  (/)  ->  ( E `
 if ( k  =  (/) ,  R ,  S ) )  =  ( E `  R
) )
58 xpsaddlem.m . . . . . . . . . . . 12  |-  .x.  =  ( E `  R )
5957, 58syl6eqr 2335 . . . . . . . . . . 11  |-  ( k  =  (/)  ->  ( E `
 if ( k  =  (/) ,  R ,  S ) )  = 
.x.  )
60 iftrue 3573 . . . . . . . . . . 11  |-  ( k  =  (/)  ->  if ( k  =  (/) ,  A ,  B )  =  A )
61 iftrue 3573 . . . . . . . . . . 11  |-  ( k  =  (/)  ->  if ( k  =  (/) ,  C ,  D )  =  C )
6259, 60, 61oveq123d 5881 . . . . . . . . . 10  |-  ( k  =  (/)  ->  ( if ( k  =  (/) ,  A ,  B ) ( E `  if ( k  =  (/) ,  R ,  S ) ) if ( k  =  (/) ,  C ,  D ) )  =  ( A  .x.  C
) )
63 iftrue 3573 . . . . . . . . . 10  |-  ( k  =  (/)  ->  if ( k  =  (/) ,  ( A  .x.  C ) ,  ( B  .X.  D ) )  =  ( A  .x.  C
) )
6462, 63eqtr4d 2320 . . . . . . . . 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 3574 . . . . . . . . . . . . 13  |-  ( -.  k  =  (/)  ->  if ( k  =  (/) ,  R ,  S )  =  S )
6665fveq2d 5531 . . . . . . . . . . . 12  |-  ( -.  k  =  (/)  ->  ( E `  if (
k  =  (/) ,  R ,  S ) )  =  ( E `  S
) )
67 xpsaddlem.n . . . . . . . . . . . 12  |-  .X.  =  ( E `  S )
6866, 67syl6eqr 2335 . . . . . . . . . . 11  |-  ( -.  k  =  (/)  ->  ( E `  if (
k  =  (/) ,  R ,  S ) )  = 
.X.  )
69 iffalse 3574 . . . . . . . . . . 11  |-  ( -.  k  =  (/)  ->  if ( k  =  (/) ,  A ,  B )  =  B )
70 iffalse 3574 . . . . . . . . . . 11  |-  ( -.  k  =  (/)  ->  if ( k  =  (/) ,  C ,  D )  =  D )
7168, 69, 70oveq123d 5881 . . . . . . . . . 10  |-  ( -.  k  =  (/)  ->  ( if ( k  =  (/) ,  A ,  B ) ( E `  if ( k  =  (/) ,  R ,  S ) ) if ( k  =  (/) ,  C ,  D ) )  =  ( B  .X.  D
) )
72 iffalse 3574 . . . . . . . . . 10  |-  ( -.  k  =  (/)  ->  if ( k  =  (/) ,  ( A  .x.  C
) ,  ( B 
.X.  D ) )  =  ( B  .X.  D ) )
7371, 72eqtr4d 2320 . . . . . . . . 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 156 . . . . . . . 8  |-  ( if ( k  =  (/) ,  A ,  B ) ( E `  if ( k  =  (/) ,  R ,  S ) ) if ( k  =  (/) ,  C ,  D ) )  =  if ( k  =  (/) ,  ( A  .x.  C ) ,  ( B  .X.  D )
)
7536adantr 451 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  2o )  ->  R  e.  V )
7637adantr 451 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  2o )  ->  S  e.  W )
77 simpr 447 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  2o )  ->  k  e.  2o )
78 xpscfv 13466 . . . . . . . . . . 11  |-  ( ( R  e.  V  /\  S  e.  W  /\  k  e.  2o )  ->  ( `' ( { R }  +c  { S } ) `  k
)  =  if ( k  =  (/) ,  R ,  S ) )
7975, 76, 77, 78syl3anc 1182 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  2o )  ->  ( `' ( { R }  +c  { S } ) `
 k )  =  if ( k  =  (/) ,  R ,  S
) )
8079fveq2d 5531 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  2o )  ->  ( E `
 ( `' ( { R }  +c  { S } ) `  k ) )  =  ( E `  if ( k  =  (/) ,  R ,  S ) ) )
812adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  2o )  ->  A  e.  X )
823adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  2o )  ->  B  e.  Y )
83 xpscfv 13466 . . . . . . . . . 10  |-  ( ( A  e.  X  /\  B  e.  Y  /\  k  e.  2o )  ->  ( `' ( { A }  +c  { B } ) `  k
)  =  if ( k  =  (/) ,  A ,  B ) )
8481, 82, 77, 83syl3anc 1182 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  2o )  ->  ( `' ( { A }  +c  { B } ) `
 k )  =  if ( k  =  (/) ,  A ,  B
) )
8517adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  2o )  ->  C  e.  X )
8618adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  2o )  ->  D  e.  Y )
87 xpscfv 13466 . . . . . . . . . 10  |-  ( ( C  e.  X  /\  D  e.  Y  /\  k  e.  2o )  ->  ( `' ( { C }  +c  { D } ) `  k
)  =  if ( k  =  (/) ,  C ,  D ) )
8885, 86, 77, 87syl3anc 1182 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  2o )  ->  ( `' ( { C }  +c  { D } ) `
 k )  =  if ( k  =  (/) ,  C ,  D
) )
8980, 84, 88oveq123d 5881 . . . . . . . 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 451 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  2o )  ->  ( A 
.x.  C )  e.  X )
9151adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  2o )  ->  ( B 
.X.  D )  e.  Y )
92 xpscfv 13466 . . . . . . . . 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 1182 . . . . . . . 8  |-  ( (
ph  /\  k  e.  2o )  ->  ( `' ( { ( A 
.x.  C ) }  +c  { ( B 
.X.  D ) } ) `  k )  =  if ( k  =  (/) ,  ( A 
.x.  C ) ,  ( B  .X.  D
) ) )
9474, 89, 933eqtr4a 2343 . . . . . . 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 4108 . . . . . 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 2320 . . . . 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 2320 . . . 4  |-  ( ph  ->  ( `' ( { A }  +c  { B } ) ( E `
 U ) `' ( { C }  +c  { D } ) )  =  `' ( { ( A  .x.  C ) }  +c  { ( B  .X.  D
) } ) )
9897fveq2d 5531 . . 3  |-  ( ph  ->  ( `' F `  ( `' ( { A }  +c  { B }
) ( E `  U ) `' ( { C }  +c  { D } ) ) )  =  ( `' F `  `' ( { ( A  .x.  C ) }  +c  { ( B  .X.  D
) } ) ) )
99 df-ov 5863 . . . . 5  |-  ( ( A  .x.  C ) F ( B  .X.  D ) )  =  ( F `  <. ( A  .x.  C ) ,  ( B  .X.  D ) >. )
1004xpsfval 13471 . . . . . 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 642 . . . . 5  |-  ( ph  ->  ( ( A  .x.  C ) F ( B  .X.  D )
)  =  `' ( { ( A  .x.  C ) }  +c  { ( B  .X.  D
) } ) )
10299, 101syl5eqr 2331 . . . 4  |-  ( ph  ->  ( F `  <. ( A  .x.  C ) ,  ( B  .X.  D ) >. )  =  `' ( { ( A  .x.  C ) }  +c  { ( B  .X.  D ) } ) )
103 opelxpi 4723 . . . . . 6  |-  ( ( ( A  .x.  C
)  e.  X  /\  ( B  .X.  D )  e.  Y )  ->  <. ( A  .x.  C
) ,  ( B 
.X.  D ) >.  e.  ( X  X.  Y
) )
10450, 51, 103syl2anc 642 . . . . 5  |-  ( ph  -> 
<. ( A  .x.  C
) ,  ( B 
.X.  D ) >.  e.  ( X  X.  Y
) )
105 f1ocnvfv 5796 . . . . 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 644 . . . 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 14 . . 3  |-  ( ph  ->  ( `' F `  `' ( { ( A  .x.  C ) }  +c  { ( B  .X.  D ) } ) )  = 
<. ( A  .x.  C
) ,  ( B 
.X.  D ) >.
)
10898, 107eqtrd 2317 . 2  |-  ( ph  ->  ( `' F `  ( `' ( { A }  +c  { B }
) ( E `  U ) `' ( { C }  +c  { D } ) ) )  =  <. ( A  .x.  C ) ,  ( B  .X.  D
) >. )
10928, 35, 1083eqtr3d 2325 1  |-  ( ph  ->  ( <. A ,  B >. 
.xb  <. C ,  D >. )  =  <. ( A  .x.  C ) ,  ( B  .X.  D
) >. )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1625    e. wcel 1686   (/)c0 3457   ifcif 3567   {csn 3642   <.cop 3645    e. cmpt 4079    X. cxp 4689   `'ccnv 4690   ran crn 4692    Fn wfn 5252   -->wf 5253   -1-1-onto->wf1o 5256   ` cfv 5257  (class class class)co 5860    e. cmpt2 5862   2oc2o 6475    +c ccda 7795   Basecbs 13150  Scalarcsca 13213   X_scprds 13348    X.s cxps 13411
This theorem is referenced by:  xpsadd  13480  xpsmul  13481
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-cnex 8795  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815  ax-pre-mulgt0 8816
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-1st 6124  df-2nd 6125  df-riota 6306  df-recs 6390  df-rdg 6425  df-1o 6481  df-2o 6482  df-oadd 6485  df-er 6662  df-map 6776  df-ixp 6820  df-en 6866  df-dom 6867  df-sdom 6868  df-fin 6869  df-sup 7196  df-cda 7796  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874  df-le 8875  df-sub 9041  df-neg 9042  df-nn 9749  df-2 9806  df-3 9807  df-4 9808  df-5 9809  df-6 9810  df-7 9811  df-8 9812  df-9 9813  df-10 9814  df-n0 9968  df-z 10027  df-dec 10127  df-uz 10233  df-fz 10785  df-struct 13152  df-ndx 13153  df-slot 13154  df-base 13155  df-plusg 13223  df-mulr 13224  df-sca 13226  df-vsca 13227  df-tset 13229  df-ple 13230  df-ds 13232  df-hom 13234  df-cco 13235  df-prds 13350
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