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Theorem xpsval 14830
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 3122 . . . 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 3122 . . . 4  |-  ( S  e.  W  ->  S  e.  _V )
75, 6syl 16 . . 3  |-  ( ph  ->  S  e.  _V )
8 fveq2 5866 . . . . . . . . 9  |-  ( r  =  R  ->  ( Base `  r )  =  ( Base `  R
) )
9 xpsval.x . . . . . . . . 9  |-  X  =  ( Base `  R
)
108, 9syl6eqr 2526 . . . . . . . 8  |-  ( r  =  R  ->  ( Base `  r )  =  X )
11 fveq2 5866 . . . . . . . . 9  |-  ( s  =  S  ->  ( Base `  s )  =  ( Base `  S
) )
12 xpsval.y . . . . . . . . 9  |-  Y  =  ( Base `  S
)
1311, 12syl6eqr 2526 . . . . . . . 8  |-  ( s  =  S  ->  ( Base `  s )  =  Y )
14 mpt2eq12 6342 . . . . . . . 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 2526 . . . . . 6  |-  ( ( r  =  R  /\  s  =  S )  ->  ( x  e.  (
Base `  r ) ,  y  e.  ( Base `  s )  |->  `' ( { x }  +c  { y } ) )  =  F )
1817cnveqd 5178 . . . . 5  |-  ( ( r  =  R  /\  s  =  S )  ->  `' ( x  e.  ( Base `  r
) ,  y  e.  ( Base `  s
)  |->  `' ( { x }  +c  {
y } ) )  =  `' F )
19 fveq2 5866 . . . . . . . . 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 2526 . . . . . . 7  |-  ( ( r  =  R  /\  s  =  S )  ->  (Scalar `  r )  =  G )
23 sneq 4037 . . . . . . . . 9  |-  ( r  =  R  ->  { r }  =  { R } )
24 sneq 4037 . . . . . . . . 9  |-  ( s  =  S  ->  { s }  =  { S } )
2523, 24oveqan12d 6304 . . . . . . . 8  |-  ( ( r  =  R  /\  s  =  S )  ->  ( { r }  +c  { s } )  =  ( { R }  +c  { S } ) )
2625cnveqd 5178 . . . . . . 7  |-  ( ( r  =  R  /\  s  =  S )  ->  `' ( { r }  +c  { s } )  =  `' ( { R }  +c  { S } ) )
2722, 26oveq12d 6303 . . . . . 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 2526 . . . . 5  |-  ( ( r  =  R  /\  s  =  S )  ->  ( (Scalar `  r
) X_s `' ( { r }  +c  { s } ) )  =  U )
3018, 29oveq12d 6303 . . . 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 14768 . . . 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 6310 . . . 4  |-  ( `' F  "s  U )  e.  _V
3330, 31, 32ovmpt2a 6418 . . 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 2520 1  |-  ( ph  ->  T  =  ( `' F  "s  U ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3113   {csn 4027   `'ccnv 4998   ` cfv 5588  (class class class)co 6285    |-> cmpt2 6287    +c ccda 8548   Basecbs 14493  Scalarcsca 14561   X_scprds 14704    "s cimas 14762    X.s cxps 14764
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5551  df-fun 5590  df-fv 5596  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-xps 14768
This theorem is referenced by:  xpsbas  14832  xpsadd  14834  xpsmul  14835  xpssca  14836  xpsvsca  14837  xpsless  14838  xpsle  14839  xpsmnd  15782  xpsgrp  16003  xpstps  20138  xpstopnlem2  20139  xpsdsfn  20707  xpsxmet  20710  xpsdsval  20711  xpsmet  20712  xpsxms  20864  xpsms  20865
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