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Theorem imasvscaval 13456
Description: The value of an image structure's scalar multiplication. (Contributed by Mario Carneiro, 24-Feb-2015.)
Hypotheses
Ref Expression
imasvscaf.u  |-  ( ph  ->  U  =  ( F 
"s  R ) )
imasvscaf.v  |-  ( ph  ->  V  =  ( Base `  R ) )
imasvscaf.f  |-  ( ph  ->  F : V -onto-> B
)
imasvscaf.r  |-  ( ph  ->  R  e.  Z )
imasvscaf.g  |-  G  =  (Scalar `  R )
imasvscaf.k  |-  K  =  ( Base `  G
)
imasvscaf.q  |-  .x.  =  ( .s `  R )
imasvscaf.s  |-  .xb  =  ( .s `  U )
imasvscaf.e  |-  ( (
ph  /\  ( p  e.  K  /\  a  e.  V  /\  q  e.  V ) )  -> 
( ( F `  a )  =  ( F `  q )  ->  ( F `  ( p  .x.  a ) )  =  ( F `
 ( p  .x.  q ) ) ) )
Assertion
Ref Expression
imasvscaval  |-  ( (
ph  /\  X  e.  K  /\  Y  e.  V
)  ->  ( X  .xb  ( F `  Y
) )  =  ( F `  ( X 
.x.  Y ) ) )
Distinct variable groups:    p, a,
q, F    K, a, p, q    ph, a, p, q    B, p, q    R, p, q    .x. , p, q    .xb , a, p, q    V, a, p, q    X, p    Y, p, q
Allowed substitution hints:    B( a)    R( a)    .x. ( a)    U( q, p, a)    G( q, p, a)    X( q, a)    Y( a)    Z( q, p, a)

Proof of Theorem imasvscaval
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 imasvscaf.u . . . . . . 7  |-  ( ph  ->  U  =  ( F 
"s  R ) )
2 imasvscaf.v . . . . . . 7  |-  ( ph  ->  V  =  ( Base `  R ) )
3 imasvscaf.f . . . . . . 7  |-  ( ph  ->  F : V -onto-> B
)
4 imasvscaf.r . . . . . . 7  |-  ( ph  ->  R  e.  Z )
5 imasvscaf.g . . . . . . 7  |-  G  =  (Scalar `  R )
6 imasvscaf.k . . . . . . 7  |-  K  =  ( Base `  G
)
7 imasvscaf.q . . . . . . 7  |-  .x.  =  ( .s `  R )
8 imasvscaf.s . . . . . . 7  |-  .xb  =  ( .s `  U )
9 imasvscaf.e . . . . . . 7  |-  ( (
ph  /\  ( p  e.  K  /\  a  e.  V  /\  q  e.  V ) )  -> 
( ( F `  a )  =  ( F `  q )  ->  ( F `  ( p  .x.  a ) )  =  ( F `
 ( p  .x.  q ) ) ) )
101, 2, 3, 4, 5, 6, 7, 8, 9imasvscafn 13455 . . . . . 6  |-  ( ph  -> 
.xb  Fn  ( K  X.  B ) )
11 fnfun 5357 . . . . . 6  |-  (  .xb  Fn  ( K  X.  B
)  ->  Fun  .xb  )
1210, 11syl 15 . . . . 5  |-  ( ph  ->  Fun  .xb  )
13123ad2ant1 976 . . . 4  |-  ( (
ph  /\  X  e.  K  /\  Y  e.  V
)  ->  Fun  .xb  )
14 eqidd 2297 . . . . . . . 8  |-  ( q  =  Y  ->  K  =  K )
15 fveq2 5541 . . . . . . . . 9  |-  ( q  =  Y  ->  ( F `  q )  =  ( F `  Y ) )
1615sneqd 3666 . . . . . . . 8  |-  ( q  =  Y  ->  { ( F `  q ) }  =  { ( F `  Y ) } )
17 oveq2 5882 . . . . . . . . 9  |-  ( q  =  Y  ->  (
p  .x.  q )  =  ( p  .x.  Y ) )
1817fveq2d 5545 . . . . . . . 8  |-  ( q  =  Y  ->  ( F `  ( p  .x.  q ) )  =  ( F `  (
p  .x.  Y )
) )
1914, 16, 18mpt2eq123dv 5926 . . . . . . 7  |-  ( q  =  Y  ->  (
p  e.  K ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p  .x.  q ) ) )  =  ( p  e.  K ,  x  e.  { ( F `  Y ) }  |->  ( F `  ( p  .x.  Y ) ) ) )
2019ssiun2s 3962 . . . . . 6  |-  ( Y  e.  V  ->  (
p  e.  K ,  x  e.  { ( F `  Y ) }  |->  ( F `  ( p  .x.  Y ) ) )  C_  U_ q  e.  V  ( p  e.  K ,  x  e. 
{ ( F `  q ) }  |->  ( F `  ( p 
.x.  q ) ) ) )
21203ad2ant3 978 . . . . 5  |-  ( (
ph  /\  X  e.  K  /\  Y  e.  V
)  ->  ( p  e.  K ,  x  e. 
{ ( F `  Y ) }  |->  ( F `  ( p 
.x.  Y ) ) )  C_  U_ q  e.  V  ( p  e.  K ,  x  e. 
{ ( F `  q ) }  |->  ( F `  ( p 
.x.  q ) ) ) )
221, 2, 3, 4, 5, 6, 7, 8imasvsca 13439 . . . . . 6  |-  ( ph  -> 
.xb  =  U_ q  e.  V  ( p  e.  K ,  x  e. 
{ ( F `  q ) }  |->  ( F `  ( p 
.x.  q ) ) ) )
23223ad2ant1 976 . . . . 5  |-  ( (
ph  /\  X  e.  K  /\  Y  e.  V
)  ->  .xb  =  U_ q  e.  V  (
p  e.  K ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p  .x.  q ) ) ) )
2421, 23sseqtr4d 3228 . . . 4  |-  ( (
ph  /\  X  e.  K  /\  Y  e.  V
)  ->  ( p  e.  K ,  x  e. 
{ ( F `  Y ) }  |->  ( F `  ( p 
.x.  Y ) ) )  C_  .xb  )
25 simp2 956 . . . . . 6  |-  ( (
ph  /\  X  e.  K  /\  Y  e.  V
)  ->  X  e.  K )
26 fvex 5555 . . . . . . 7  |-  ( F `
 Y )  e. 
_V
2726snid 3680 . . . . . 6  |-  ( F `
 Y )  e. 
{ ( F `  Y ) }
28 opelxpi 4737 . . . . . 6  |-  ( ( X  e.  K  /\  ( F `  Y )  e.  { ( F `
 Y ) } )  ->  <. X , 
( F `  Y
) >.  e.  ( K  X.  { ( F `
 Y ) } ) )
2925, 27, 28sylancl 643 . . . . 5  |-  ( (
ph  /\  X  e.  K  /\  Y  e.  V
)  ->  <. X , 
( F `  Y
) >.  e.  ( K  X.  { ( F `
 Y ) } ) )
30 eqid 2296 . . . . . 6  |-  ( p  e.  K ,  x  e.  { ( F `  Y ) }  |->  ( F `  ( p 
.x.  Y ) ) )  =  ( p  e.  K ,  x  e.  { ( F `  Y ) }  |->  ( F `  ( p 
.x.  Y ) ) )
31 fvex 5555 . . . . . 6  |-  ( F `
 ( p  .x.  Y ) )  e. 
_V
3230, 31dmmpt2 6210 . . . . 5  |-  dom  (
p  e.  K ,  x  e.  { ( F `  Y ) }  |->  ( F `  ( p  .x.  Y ) ) )  =  ( K  X.  { ( F `  Y ) } )
3329, 32syl6eleqr 2387 . . . 4  |-  ( (
ph  /\  X  e.  K  /\  Y  e.  V
)  ->  <. X , 
( F `  Y
) >.  e.  dom  (
p  e.  K ,  x  e.  { ( F `  Y ) }  |->  ( F `  ( p  .x.  Y ) ) ) )
34 funssfv 5559 . . . 4  |-  ( ( Fun  .xb  /\  (
p  e.  K ,  x  e.  { ( F `  Y ) }  |->  ( F `  ( p  .x.  Y ) ) )  C_  .xb  /\  <. X ,  ( F `  Y ) >.  e.  dom  ( p  e.  K ,  x  e.  { ( F `  Y ) }  |->  ( F `  ( p  .x.  Y ) ) ) )  -> 
(  .xb  `  <. X , 
( F `  Y
) >. )  =  ( ( p  e.  K ,  x  e.  { ( F `  Y ) }  |->  ( F `  ( p  .x.  Y ) ) ) `  <. X ,  ( F `  Y ) >. )
)
3513, 24, 33, 34syl3anc 1182 . . 3  |-  ( (
ph  /\  X  e.  K  /\  Y  e.  V
)  ->  (  .xb  ` 
<. X ,  ( F `
 Y ) >.
)  =  ( ( p  e.  K ,  x  e.  { ( F `  Y ) }  |->  ( F `  ( p  .x.  Y ) ) ) `  <. X ,  ( F `  Y ) >. )
)
36 df-ov 5877 . . 3  |-  ( X 
.xb  ( F `  Y ) )  =  (  .xb  `  <. X , 
( F `  Y
) >. )
37 df-ov 5877 . . 3  |-  ( X ( p  e.  K ,  x  e.  { ( F `  Y ) }  |->  ( F `  ( p  .x.  Y ) ) ) ( F `
 Y ) )  =  ( ( p  e.  K ,  x  e.  { ( F `  Y ) }  |->  ( F `  ( p 
.x.  Y ) ) ) `  <. X , 
( F `  Y
) >. )
3835, 36, 373eqtr4g 2353 . 2  |-  ( (
ph  /\  X  e.  K  /\  Y  e.  V
)  ->  ( X  .xb  ( F `  Y
) )  =  ( X ( p  e.  K ,  x  e. 
{ ( F `  Y ) }  |->  ( F `  ( p 
.x.  Y ) ) ) ( F `  Y ) ) )
39 oveq1 5881 . . . . 5  |-  ( p  =  X  ->  (
p  .x.  Y )  =  ( X  .x.  Y ) )
4039fveq2d 5545 . . . 4  |-  ( p  =  X  ->  ( F `  ( p  .x.  Y ) )  =  ( F `  ( X  .x.  Y ) ) )
41 eqidd 2297 . . . 4  |-  ( x  =  ( F `  Y )  ->  ( F `  ( X  .x.  Y ) )  =  ( F `  ( X  .x.  Y ) ) )
42 fvex 5555 . . . 4  |-  ( F `
 ( X  .x.  Y ) )  e. 
_V
4340, 41, 30, 42ovmpt2 5999 . . 3  |-  ( ( X  e.  K  /\  ( F `  Y )  e.  { ( F `
 Y ) } )  ->  ( X
( p  e.  K ,  x  e.  { ( F `  Y ) }  |->  ( F `  ( p  .x.  Y ) ) ) ( F `
 Y ) )  =  ( F `  ( X  .x.  Y ) ) )
4425, 27, 43sylancl 643 . 2  |-  ( (
ph  /\  X  e.  K  /\  Y  e.  V
)  ->  ( X
( p  e.  K ,  x  e.  { ( F `  Y ) }  |->  ( F `  ( p  .x.  Y ) ) ) ( F `
 Y ) )  =  ( F `  ( X  .x.  Y ) ) )
4538, 44eqtrd 2328 1  |-  ( (
ph  /\  X  e.  K  /\  Y  e.  V
)  ->  ( X  .xb  ( F `  Y
) )  =  ( F `  ( X 
.x.  Y ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    C_ wss 3165   {csn 3653   <.cop 3656   U_ciun 3921    X. cxp 4703   dom cdm 4705   Fun wfun 5265    Fn wfn 5266   -onto->wfo 5269   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   Basecbs 13164  Scalarcsca 13227   .scvsca 13228    "s cimas 13423
This theorem is referenced by:  xpsvsca  13497
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-fz 10799  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-plusg 13237  df-mulr 13238  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-imas 13427
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