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Theorem xpsaddlem 13792
Description: Lemma for xpsadd 13793 and xpsmul 13794. (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 6076 . . . . 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 13784 . . . . . 6  |-  ( ( A  e.  X  /\  B  e.  Y )  ->  ( A F B )  =  `' ( { A }  +c  { B } ) )
62, 3, 5syl2anc 643 . . . . 5  |-  ( ph  ->  ( A F B )  =  `' ( { A }  +c  { B } ) )
71, 6syl5eqr 2481 . . . 4  |-  ( ph  ->  ( F `  <. A ,  B >. )  =  `' ( { A }  +c  { B }
) )
8 opelxpi 4902 . . . . . 6  |-  ( ( A  e.  X  /\  B  e.  Y )  -> 
<. A ,  B >.  e.  ( X  X.  Y
) )
92, 3, 8syl2anc 643 . . . . 5  |-  ( ph  -> 
<. A ,  B >.  e.  ( X  X.  Y
) )
104xpsff1o2 13788 . . . . . . 7  |-  F :
( X  X.  Y
)
-1-1-onto-> ran  F
11 f1of 5666 . . . . . . 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 5861 . . . . 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 2510 . . 3  |-  ( ph  ->  `' ( { A }  +c  { B }
)  e.  ran  F
)
16 df-ov 6076 . . . . 5  |-  ( C F D )  =  ( F `  <. C ,  D >. )
17 xpsadd.5 . . . . . 6  |-  ( ph  ->  C  e.  X )
18 xpsadd.6 . . . . . 6  |-  ( ph  ->  D  e.  Y )
194xpsfval 13784 . . . . . 6  |-  ( ( C  e.  X  /\  D  e.  Y )  ->  ( C F D )  =  `' ( { C }  +c  { D } ) )
2017, 18, 19syl2anc 643 . . . . 5  |-  ( ph  ->  ( C F D )  =  `' ( { C }  +c  { D } ) )
2116, 20syl5eqr 2481 . . . 4  |-  ( ph  ->  ( F `  <. C ,  D >. )  =  `' ( { C }  +c  { D }
) )
22 opelxpi 4902 . . . . . 6  |-  ( ( C  e.  X  /\  D  e.  Y )  -> 
<. C ,  D >.  e.  ( X  X.  Y
) )
2317, 18, 22syl2anc 643 . . . . 5  |-  ( ph  -> 
<. C ,  D >.  e.  ( X  X.  Y
) )
2412ffvelrni 5861 . . . . 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 2510 . . 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 1281 . 2  |-  ( ph  ->  ( ( `' F `  `' ( { A }  +c  { B }
) )  .xb  ( `' F `  `' ( { C }  +c  { D } ) ) )  =  ( `' F `  ( `' ( { A }  +c  { B } ) ( E `  U
) `' ( { C }  +c  { D } ) ) ) )
29 f1ocnvfv 6008 . . . . 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 645 . . . 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 6008 . . . . 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 645 . . . 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 6091 . 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 13776 . . . . . . 7  |-  ( ( R  e.  V  /\  S  e.  W )  ->  `' ( { R }  +c  { S }
)  Fn  2o )
3936, 37, 38syl2anc 643 . . . . . 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 2435 . . . . . . . 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 13790 . . . . . . 7  |-  ( ph  ->  ran  F  =  (
Base `  U )
)
4615, 45eleqtrd 2511 . . . . . 6  |-  ( ph  ->  `' ( { A }  +c  { B }
)  e.  ( Base `  U ) )
4726, 45eleqtrd 2511 . . . . . 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 1184 . . . . 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 13776 . . . . . . . 8  |-  ( ( ( A  .x.  C
)  e.  X  /\  ( B  .X.  D )  e.  Y )  ->  `' ( { ( A  .x.  C ) }  +c  { ( B  .X.  D ) } )  Fn  2o )
5350, 51, 52syl2anc 643 . . . . . . 7  |-  ( ph  ->  `' ( { ( A  .x.  C ) }  +c  { ( B  .X.  D ) } )  Fn  2o )
54 dffn5 5764 . . . . . . 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 189 . . . . . 6  |-  ( ph  ->  `' ( { ( A  .x.  C ) }  +c  { ( B  .X.  D ) } )  =  ( k  e.  2o  |->  ( `' ( { ( A  .x.  C ) }  +c  { ( B  .X.  D ) } ) `  k
) ) )
56 iftrue 3737 . . . . . . . . . . . . 13  |-  ( k  =  (/)  ->  if ( k  =  (/) ,  R ,  S )  =  R )
5756fveq2d 5724 . . . . . . . . . . . 12  |-  ( k  =  (/)  ->  ( E `
 if ( k  =  (/) ,  R ,  S ) )  =  ( E `  R
) )
58 xpsaddlem.m . . . . . . . . . . . 12  |-  .x.  =  ( E `  R )
5957, 58syl6eqr 2485 . . . . . . . . . . 11  |-  ( k  =  (/)  ->  ( E `
 if ( k  =  (/) ,  R ,  S ) )  = 
.x.  )
60 iftrue 3737 . . . . . . . . . . 11  |-  ( k  =  (/)  ->  if ( k  =  (/) ,  A ,  B )  =  A )
61 iftrue 3737 . . . . . . . . . . 11  |-  ( k  =  (/)  ->  if ( k  =  (/) ,  C ,  D )  =  C )
6259, 60, 61oveq123d 6094 . . . . . . . . . 10  |-  ( k  =  (/)  ->  ( if ( k  =  (/) ,  A ,  B ) ( E `  if ( k  =  (/) ,  R ,  S ) ) if ( k  =  (/) ,  C ,  D ) )  =  ( A  .x.  C
) )
63 iftrue 3737 . . . . . . . . . 10  |-  ( k  =  (/)  ->  if ( k  =  (/) ,  ( A  .x.  C ) ,  ( B  .X.  D ) )  =  ( A  .x.  C
) )
6462, 63eqtr4d 2470 . . . . . . . . 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 3738 . . . . . . . . . . . . 13  |-  ( -.  k  =  (/)  ->  if ( k  =  (/) ,  R ,  S )  =  S )
6665fveq2d 5724 . . . . . . . . . . . 12  |-  ( -.  k  =  (/)  ->  ( E `  if (
k  =  (/) ,  R ,  S ) )  =  ( E `  S
) )
67 xpsaddlem.n . . . . . . . . . . . 12  |-  .X.  =  ( E `  S )
6866, 67syl6eqr 2485 . . . . . . . . . . 11  |-  ( -.  k  =  (/)  ->  ( E `  if (
k  =  (/) ,  R ,  S ) )  = 
.X.  )
69 iffalse 3738 . . . . . . . . . . 11  |-  ( -.  k  =  (/)  ->  if ( k  =  (/) ,  A ,  B )  =  B )
70 iffalse 3738 . . . . . . . . . . 11  |-  ( -.  k  =  (/)  ->  if ( k  =  (/) ,  C ,  D )  =  D )
7168, 69, 70oveq123d 6094 . . . . . . . . . 10  |-  ( -.  k  =  (/)  ->  ( if ( k  =  (/) ,  A ,  B ) ( E `  if ( k  =  (/) ,  R ,  S ) ) if ( k  =  (/) ,  C ,  D ) )  =  ( B  .X.  D
) )
72 iffalse 3738 . . . . . . . . . 10  |-  ( -.  k  =  (/)  ->  if ( k  =  (/) ,  ( A  .x.  C
) ,  ( B 
.X.  D ) )  =  ( B  .X.  D ) )
7371, 72eqtr4d 2470 . . . . . . . . 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 158 . . . . . . . 8  |-  ( if ( k  =  (/) ,  A ,  B ) ( E `  if ( k  =  (/) ,  R ,  S ) ) if ( k  =  (/) ,  C ,  D ) )  =  if ( k  =  (/) ,  ( A  .x.  C ) ,  ( B  .X.  D )
)
7536adantr 452 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  2o )  ->  R  e.  V )
7637adantr 452 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  2o )  ->  S  e.  W )
77 simpr 448 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  2o )  ->  k  e.  2o )
78 xpscfv 13779 . . . . . . . . . . 11  |-  ( ( R  e.  V  /\  S  e.  W  /\  k  e.  2o )  ->  ( `' ( { R }  +c  { S } ) `  k
)  =  if ( k  =  (/) ,  R ,  S ) )
7975, 76, 77, 78syl3anc 1184 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  2o )  ->  ( `' ( { R }  +c  { S } ) `
 k )  =  if ( k  =  (/) ,  R ,  S
) )
8079fveq2d 5724 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  2o )  ->  ( E `
 ( `' ( { R }  +c  { S } ) `  k ) )  =  ( E `  if ( k  =  (/) ,  R ,  S ) ) )
812adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  2o )  ->  A  e.  X )
823adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  2o )  ->  B  e.  Y )
83 xpscfv 13779 . . . . . . . . . 10  |-  ( ( A  e.  X  /\  B  e.  Y  /\  k  e.  2o )  ->  ( `' ( { A }  +c  { B } ) `  k
)  =  if ( k  =  (/) ,  A ,  B ) )
8481, 82, 77, 83syl3anc 1184 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  2o )  ->  ( `' ( { A }  +c  { B } ) `
 k )  =  if ( k  =  (/) ,  A ,  B
) )
8517adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  2o )  ->  C  e.  X )
8618adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  2o )  ->  D  e.  Y )
87 xpscfv 13779 . . . . . . . . . 10  |-  ( ( C  e.  X  /\  D  e.  Y  /\  k  e.  2o )  ->  ( `' ( { C }  +c  { D } ) `  k
)  =  if ( k  =  (/) ,  C ,  D ) )
8885, 86, 77, 87syl3anc 1184 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  2o )  ->  ( `' ( { C }  +c  { D } ) `
 k )  =  if ( k  =  (/) ,  C ,  D
) )
8980, 84, 88oveq123d 6094 . . . . . . . 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 452 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  2o )  ->  ( A 
.x.  C )  e.  X )
9151adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  2o )  ->  ( B 
.X.  D )  e.  Y )
92 xpscfv 13779 . . . . . . . . 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 1184 . . . . . . . 8  |-  ( (
ph  /\  k  e.  2o )  ->  ( `' ( { ( A 
.x.  C ) }  +c  { ( B 
.X.  D ) } ) `  k )  =  if ( k  =  (/) ,  ( A 
.x.  C ) ,  ( B  .X.  D
) ) )
9474, 89, 933eqtr4a 2493 . . . . . . 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 4287 . . . . . 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 2470 . . . . 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 2470 . . . 4  |-  ( ph  ->  ( `' ( { A }  +c  { B } ) ( E `
 U ) `' ( { C }  +c  { D } ) )  =  `' ( { ( A  .x.  C ) }  +c  { ( B  .X.  D
) } ) )
9897fveq2d 5724 . . 3  |-  ( ph  ->  ( `' F `  ( `' ( { A }  +c  { B }
) ( E `  U ) `' ( { C }  +c  { D } ) ) )  =  ( `' F `  `' ( { ( A  .x.  C ) }  +c  { ( B  .X.  D
) } ) ) )
99 df-ov 6076 . . . . 5  |-  ( ( A  .x.  C ) F ( B  .X.  D ) )  =  ( F `  <. ( A  .x.  C ) ,  ( B  .X.  D ) >. )
1004xpsfval 13784 . . . . . 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 643 . . . . 5  |-  ( ph  ->  ( ( A  .x.  C ) F ( B  .X.  D )
)  =  `' ( { ( A  .x.  C ) }  +c  { ( B  .X.  D
) } ) )
10299, 101syl5eqr 2481 . . . 4  |-  ( ph  ->  ( F `  <. ( A  .x.  C ) ,  ( B  .X.  D ) >. )  =  `' ( { ( A  .x.  C ) }  +c  { ( B  .X.  D ) } ) )
103 opelxpi 4902 . . . . . 6  |-  ( ( ( A  .x.  C
)  e.  X  /\  ( B  .X.  D )  e.  Y )  ->  <. ( A  .x.  C
) ,  ( B 
.X.  D ) >.  e.  ( X  X.  Y
) )
10450, 51, 103syl2anc 643 . . . . 5  |-  ( ph  -> 
<. ( A  .x.  C
) ,  ( B 
.X.  D ) >.  e.  ( X  X.  Y
) )
105 f1ocnvfv 6008 . . . . 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 645 . . . 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 2467 . 2  |-  ( ph  ->  ( `' F `  ( `' ( { A }  +c  { B }
) ( E `  U ) `' ( { C }  +c  { D } ) ) )  =  <. ( A  .x.  C ) ,  ( B  .X.  D
) >. )
10928, 35, 1083eqtr3d 2475 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 359    /\ w3a 936    = wceq 1652    e. wcel 1725   (/)c0 3620   ifcif 3731   {csn 3806   <.cop 3809    e. cmpt 4258    X. cxp 4868   `'ccnv 4869   ran crn 4871    Fn wfn 5441   -->wf 5442   -1-1-onto->wf1o 5445   ` cfv 5446  (class class class)co 6073    e. cmpt2 6075   2oc2o 6710    +c ccda 8039   Basecbs 13461  Scalarcsca 13524   X_scprds 13661    X.s cxps 13724
This theorem is referenced by:  xpsadd  13793  xpsmul  13794
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-er 6897  df-map 7012  df-ixp 7056  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-cda 8040  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-7 10055  df-8 10056  df-9 10057  df-10 10058  df-n0 10214  df-z 10275  df-dec 10375  df-uz 10481  df-fz 11036  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-plusg 13534  df-mulr 13535  df-sca 13537  df-vsca 13538  df-tset 13540  df-ple 13541  df-ds 13543  df-hom 13545  df-cco 13546  df-prds 13663
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