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Theorem xpsval 14510
Description: Value of the binary structure product function. (Contributed by Mario Carneiro, 14-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 )
xpsval.f  |-  F  =  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) )
xpsval.k  |-  G  =  (Scalar `  R )
xpsval.u  |-  U  =  ( G X_s `' ( { R }  +c  { S }
) )
Assertion
Ref Expression
xpsval  |-  ( ph  ->  T  =  ( `' F  "s  U ) )
Distinct variable groups:    x, y    x, W    x, X, y   
x, R    x, Y, y
Allowed substitution hints:    ph( x, y)    R( y)    S( x, y)    T( x, y)    U( x, y)    F( x, y)    G( x, y)    V( x, y)    W( y)

Proof of Theorem xpsval
Dummy variables  s 
r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xpsval.t . 2  |-  T  =  ( R  X.s  S )
2 xpsval.1 . . . 4  |-  ( ph  ->  R  e.  V )
3 elex 2981 . . . 4  |-  ( R  e.  V  ->  R  e.  _V )
42, 3syl 16 . . 3  |-  ( ph  ->  R  e.  _V )
5 xpsval.2 . . . 4  |-  ( ph  ->  S  e.  W )
6 elex 2981 . . . 4  |-  ( S  e.  W  ->  S  e.  _V )
75, 6syl 16 . . 3  |-  ( ph  ->  S  e.  _V )
8 fveq2 5691 . . . . . . . . 9  |-  ( r  =  R  ->  ( Base `  r )  =  ( Base `  R
) )
9 xpsval.x . . . . . . . . 9  |-  X  =  ( Base `  R
)
108, 9syl6eqr 2493 . . . . . . . 8  |-  ( r  =  R  ->  ( Base `  r )  =  X )
11 fveq2 5691 . . . . . . . . 9  |-  ( s  =  S  ->  ( Base `  s )  =  ( Base `  S
) )
12 xpsval.y . . . . . . . . 9  |-  Y  =  ( Base `  S
)
1311, 12syl6eqr 2493 . . . . . . . 8  |-  ( s  =  S  ->  ( Base `  s )  =  Y )
14 mpt2eq12 6146 . . . . . . . 8  |-  ( ( ( Base `  r
)  =  X  /\  ( Base `  s )  =  Y )  ->  (
x  e.  ( Base `  r ) ,  y  e.  ( Base `  s
)  |->  `' ( { x }  +c  {
y } ) )  =  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  {
y } ) ) )
1510, 13, 14syl2an 477 . . . . . . 7  |-  ( ( r  =  R  /\  s  =  S )  ->  ( x  e.  (
Base `  r ) ,  y  e.  ( Base `  s )  |->  `' ( { x }  +c  { y } ) )  =  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) ) )
16 xpsval.f . . . . . . 7  |-  F  =  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) )
1715, 16syl6eqr 2493 . . . . . 6  |-  ( ( r  =  R  /\  s  =  S )  ->  ( x  e.  (
Base `  r ) ,  y  e.  ( Base `  s )  |->  `' ( { x }  +c  { y } ) )  =  F )
1817cnveqd 5015 . . . . 5  |-  ( ( r  =  R  /\  s  =  S )  ->  `' ( x  e.  ( Base `  r
) ,  y  e.  ( Base `  s
)  |->  `' ( { x }  +c  {
y } ) )  =  `' F )
19 fveq2 5691 . . . . . . . . 9  |-  ( r  =  R  ->  (Scalar `  r )  =  (Scalar `  R ) )
2019adantr 465 . . . . . . . 8  |-  ( ( r  =  R  /\  s  =  S )  ->  (Scalar `  r )  =  (Scalar `  R )
)
21 xpsval.k . . . . . . . 8  |-  G  =  (Scalar `  R )
2220, 21syl6eqr 2493 . . . . . . 7  |-  ( ( r  =  R  /\  s  =  S )  ->  (Scalar `  r )  =  G )
23 sneq 3887 . . . . . . . . 9  |-  ( r  =  R  ->  { r }  =  { R } )
24 sneq 3887 . . . . . . . . 9  |-  ( s  =  S  ->  { s }  =  { S } )
2523, 24oveqan12d 6110 . . . . . . . 8  |-  ( ( r  =  R  /\  s  =  S )  ->  ( { r }  +c  { s } )  =  ( { R }  +c  { S } ) )
2625cnveqd 5015 . . . . . . 7  |-  ( ( r  =  R  /\  s  =  S )  ->  `' ( { r }  +c  { s } )  =  `' ( { R }  +c  { S } ) )
2722, 26oveq12d 6109 . . . . . 6  |-  ( ( r  =  R  /\  s  =  S )  ->  ( (Scalar `  r
) X_s `' ( { r }  +c  { s } ) )  =  ( G X_s `' ( { R }  +c  { S }
) ) )
28 xpsval.u . . . . . 6  |-  U  =  ( G X_s `' ( { R }  +c  { S }
) )
2927, 28syl6eqr 2493 . . . . 5  |-  ( ( r  =  R  /\  s  =  S )  ->  ( (Scalar `  r
) X_s `' ( { r }  +c  { s } ) )  =  U )
3018, 29oveq12d 6109 . . . 4  |-  ( ( r  =  R  /\  s  =  S )  ->  ( `' ( x  e.  ( Base `  r
) ,  y  e.  ( Base `  s
)  |->  `' ( { x }  +c  {
y } ) ) 
"s  ( (Scalar `  r
) X_s `' ( { r }  +c  { s } ) ) )  =  ( `' F  "s  U ) )
31 df-xps 14448 . . . 4  |-  X.s  =  ( r  e.  _V , 
s  e.  _V  |->  ( `' ( x  e.  ( Base `  r
) ,  y  e.  ( Base `  s
)  |->  `' ( { x }  +c  {
y } ) ) 
"s  ( (Scalar `  r
) X_s `' ( { r }  +c  { s } ) ) ) )
32 ovex 6116 . . . 4  |-  ( `' F  "s  U )  e.  _V
3330, 31, 32ovmpt2a 6221 . . 3  |-  ( ( R  e.  _V  /\  S  e.  _V )  ->  ( R  X.s  S )  =  ( `' F  "s  U ) )
344, 7, 33syl2anc 661 . 2  |-  ( ph  ->  ( R  X.s  S )  =  ( `' F  "s  U ) )
351, 34syl5eq 2487 1  |-  ( ph  ->  T  =  ( `' F  "s  U ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   _Vcvv 2972   {csn 3877   `'ccnv 4839   ` cfv 5418  (class class class)co 6091    e. cmpt2 6093    +c ccda 8336   Basecbs 14174  Scalarcsca 14241   X_scprds 14384    "s cimas 14442    X.s cxps 14444
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-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-iota 5381  df-fun 5420  df-fv 5426  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-xps 14448
This theorem is referenced by:  xpsbas  14512  xpsadd  14514  xpsmul  14515  xpssca  14516  xpsvsca  14517  xpsless  14518  xpsle  14519  xpsmnd  15461  xpsgrp  15674  xpstps  19383  xpstopnlem2  19384  xpsdsfn  19952  xpsxmet  19955  xpsdsval  19956  xpsmet  19957  xpsxms  20109  xpsms  20110
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