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Theorem xpsval 15429
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 3096 . . . 4  |-  ( R  e.  V  ->  R  e.  _V )
42, 3syl 17 . . 3  |-  ( ph  ->  R  e.  _V )
5 xpsval.2 . . . 4  |-  ( ph  ->  S  e.  W )
6 elex 3096 . . . 4  |-  ( S  e.  W  ->  S  e.  _V )
75, 6syl 17 . . 3  |-  ( ph  ->  S  e.  _V )
8 fveq2 5881 . . . . . . . . 9  |-  ( r  =  R  ->  ( Base `  r )  =  ( Base `  R
) )
9 xpsval.x . . . . . . . . 9  |-  X  =  ( Base `  R
)
108, 9syl6eqr 2488 . . . . . . . 8  |-  ( r  =  R  ->  ( Base `  r )  =  X )
11 fveq2 5881 . . . . . . . . 9  |-  ( s  =  S  ->  ( Base `  s )  =  ( Base `  S
) )
12 xpsval.y . . . . . . . . 9  |-  Y  =  ( Base `  S
)
1311, 12syl6eqr 2488 . . . . . . . 8  |-  ( s  =  S  ->  ( Base `  s )  =  Y )
14 mpt2eq12 6365 . . . . . . . 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 479 . . . . . . 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 2488 . . . . . 6  |-  ( ( r  =  R  /\  s  =  S )  ->  ( x  e.  (
Base `  r ) ,  y  e.  ( Base `  s )  |->  `' ( { x }  +c  { y } ) )  =  F )
1817cnveqd 5030 . . . . 5  |-  ( ( r  =  R  /\  s  =  S )  ->  `' ( x  e.  ( Base `  r
) ,  y  e.  ( Base `  s
)  |->  `' ( { x }  +c  {
y } ) )  =  `' F )
19 fveq2 5881 . . . . . . . . 9  |-  ( r  =  R  ->  (Scalar `  r )  =  (Scalar `  R ) )
2019adantr 466 . . . . . . . 8  |-  ( ( r  =  R  /\  s  =  S )  ->  (Scalar `  r )  =  (Scalar `  R )
)
21 xpsval.k . . . . . . . 8  |-  G  =  (Scalar `  R )
2220, 21syl6eqr 2488 . . . . . . 7  |-  ( ( r  =  R  /\  s  =  S )  ->  (Scalar `  r )  =  G )
23 sneq 4012 . . . . . . . . 9  |-  ( r  =  R  ->  { r }  =  { R } )
24 sneq 4012 . . . . . . . . 9  |-  ( s  =  S  ->  { s }  =  { S } )
2523, 24oveqan12d 6324 . . . . . . . 8  |-  ( ( r  =  R  /\  s  =  S )  ->  ( { r }  +c  { s } )  =  ( { R }  +c  { S } ) )
2625cnveqd 5030 . . . . . . 7  |-  ( ( r  =  R  /\  s  =  S )  ->  `' ( { r }  +c  { s } )  =  `' ( { R }  +c  { S } ) )
2722, 26oveq12d 6323 . . . . . 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 2488 . . . . 5  |-  ( ( r  =  R  /\  s  =  S )  ->  ( (Scalar `  r
) X_s `' ( { r }  +c  { s } ) )  =  U )
3018, 29oveq12d 6323 . . . 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 15367 . . . 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 6333 . . . 4  |-  ( `' F  "s  U )  e.  _V
3330, 31, 32ovmpt2a 6441 . . 3  |-  ( ( R  e.  _V  /\  S  e.  _V )  ->  ( R  X.s  S )  =  ( `' F  "s  U ) )
344, 7, 33syl2anc 665 . 2  |-  ( ph  ->  ( R  X.s  S )  =  ( `' F  "s  U ) )
351, 34syl5eq 2482 1  |-  ( ph  ->  T  =  ( `' F  "s  U ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870   _Vcvv 3087   {csn 4002   `'ccnv 4853   ` cfv 5601  (class class class)co 6305    |-> cmpt2 6307    +c ccda 8595   Basecbs 15084  Scalarcsca 15155   X_scprds 15303    "s cimas 15361    X.s cxps 15363
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-iota 5565  df-fun 5603  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-xps 15367
This theorem is referenced by:  xpsbas  15431  xpsadd  15433  xpsmul  15434  xpssca  15435  xpsvsca  15436  xpsless  15437  xpsle  15438  xpsmnd  16527  xpsgrp  16756  xpstps  20756  xpstopnlem2  20757  xpsdsfn  21323  xpsxmet  21326  xpsdsval  21327  xpsmet  21328  xpsxms  21480  xpsms  21481
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