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Theorem imasvscaval 13440
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 13439 . . . . . 6  |-  ( ph  -> 
.xb  Fn  ( K  X.  B ) )
11 fnfun 5341 . . . . . 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 2284 . . . . . . . 8  |-  ( q  =  Y  ->  K  =  K )
15 fveq2 5525 . . . . . . . . 9  |-  ( q  =  Y  ->  ( F `  q )  =  ( F `  Y ) )
1615sneqd 3653 . . . . . . . 8  |-  ( q  =  Y  ->  { ( F `  q ) }  =  { ( F `  Y ) } )
17 oveq2 5866 . . . . . . . . 9  |-  ( q  =  Y  ->  (
p  .x.  q )  =  ( p  .x.  Y ) )
1817fveq2d 5529 . . . . . . . 8  |-  ( q  =  Y  ->  ( F `  ( p  .x.  q ) )  =  ( F `  (
p  .x.  Y )
) )
1914, 16, 18mpt2eq123dv 5910 . . . . . . 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 3946 . . . . . 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 13423 . . . . . 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 3215 . . . 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 5539 . . . . . . 7  |-  ( F `
 Y )  e. 
_V
2726snid 3667 . . . . . 6  |-  ( F `
 Y )  e. 
{ ( F `  Y ) }
28 opelxpi 4721 . . . . . 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 2283 . . . . . 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 5539 . . . . . 6  |-  ( F `
 ( p  .x.  Y ) )  e. 
_V
3230, 31dmmpt2 6194 . . . . 5  |-  dom  (
p  e.  K ,  x  e.  { ( F `  Y ) }  |->  ( F `  ( p  .x.  Y ) ) )  =  ( K  X.  { ( F `  Y ) } )
3329, 32syl6eleqr 2374 . . . 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 5543 . . . 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 5861 . . 3  |-  ( X 
.xb  ( F `  Y ) )  =  (  .xb  `  <. X , 
( F `  Y
) >. )
37 df-ov 5861 . . 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 2340 . 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 5865 . . . . 5  |-  ( p  =  X  ->  (
p  .x.  Y )  =  ( X  .x.  Y ) )
4039fveq2d 5529 . . . 4  |-  ( p  =  X  ->  ( F `  ( p  .x.  Y ) )  =  ( F `  ( X  .x.  Y ) ) )
41 eqidd 2284 . . . 4  |-  ( x  =  ( F `  Y )  ->  ( F `  ( X  .x.  Y ) )  =  ( F `  ( X  .x.  Y ) ) )
42 fvex 5539 . . . 4  |-  ( F `
 ( X  .x.  Y ) )  e. 
_V
4340, 41, 30, 42ovmpt2 5983 . . 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 2315 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 1623    e. wcel 1684    C_ wss 3152   {csn 3640   <.cop 3643   U_ciun 3905    X. cxp 4687   dom cdm 4689   Fun wfun 5249    Fn wfn 5250   -onto->wfo 5253   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   Basecbs 13148  Scalarcsca 13211   .scvsca 13212    "s cimas 13407
This theorem is referenced by:  xpsvsca  13481
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-fz 10783  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-imas 13411
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