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Theorem imasvscaf 13691
Description: The image structure's scalar multiplication is closed in the base set. (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 ) ) ) )
imasvscaf.c  |-  ( (
ph  /\  ( p  e.  K  /\  q  e.  V ) )  -> 
( p  .x.  q
)  e.  V )
Assertion
Ref Expression
imasvscaf  |-  ( ph  -> 
.xb  : ( K  X.  B ) --> B )
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
Allowed substitution hints:    B( a)    R( a)    .x. ( a)    U( q, p, a)    G( q, p, a)    Z( q, p, a)

Proof of Theorem imasvscaf
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 imasvscaf.u . . 3  |-  ( ph  ->  U  =  ( F 
"s  R ) )
2 imasvscaf.v . . 3  |-  ( ph  ->  V  =  ( Base `  R ) )
3 imasvscaf.f . . 3  |-  ( ph  ->  F : V -onto-> B
)
4 imasvscaf.r . . 3  |-  ( ph  ->  R  e.  Z )
5 imasvscaf.g . . 3  |-  G  =  (Scalar `  R )
6 imasvscaf.k . . 3  |-  K  =  ( Base `  G
)
7 imasvscaf.q . . 3  |-  .x.  =  ( .s `  R )
8 imasvscaf.s . . 3  |-  .xb  =  ( .s `  U )
9 imasvscaf.e . . 3  |-  ( (
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 13689 . 2  |-  ( ph  -> 
.xb  Fn  ( K  X.  B ) )
111, 2, 3, 4, 5, 6, 7, 8imasvsca 13673 . . 3  |-  ( ph  -> 
.xb  =  U_ q  e.  V  ( p  e.  K ,  x  e. 
{ ( F `  q ) }  |->  ( F `  ( p 
.x.  q ) ) ) )
12 imasvscaf.c . . . . . . . . . . . 12  |-  ( (
ph  /\  ( p  e.  K  /\  q  e.  V ) )  -> 
( p  .x.  q
)  e.  V )
13 fof 5593 . . . . . . . . . . . . . 14  |-  ( F : V -onto-> B  ->  F : V --> B )
143, 13syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  F : V --> B )
1514ffvelrnda 5809 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( p  .x.  q )  e.  V
)  ->  ( F `  ( p  .x.  q
) )  e.  B
)
1612, 15syldan 457 . . . . . . . . . . 11  |-  ( (
ph  /\  ( p  e.  K  /\  q  e.  V ) )  -> 
( F `  (
p  .x.  q )
)  e.  B )
1716ralrimivw 2733 . . . . . . . . . 10  |-  ( (
ph  /\  ( p  e.  K  /\  q  e.  V ) )  ->  A. x  e.  { ( F `  q ) }  ( F `  ( p  .x.  q ) )  e.  B )
1817anass1rs 783 . . . . . . . . 9  |-  ( ( ( ph  /\  q  e.  V )  /\  p  e.  K )  ->  A. x  e.  { ( F `  q ) }  ( F `  ( p  .x.  q ) )  e.  B )
1918ralrimiva 2732 . . . . . . . 8  |-  ( (
ph  /\  q  e.  V )  ->  A. p  e.  K  A. x  e.  { ( F `  q ) }  ( F `  ( p  .x.  q ) )  e.  B )
20 eqid 2387 . . . . . . . . 9  |-  ( p  e.  K ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p 
.x.  q ) ) )  =  ( p  e.  K ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p 
.x.  q ) ) )
2120fmpt2 6357 . . . . . . . 8  |-  ( A. p  e.  K  A. x  e.  { ( F `  q ) }  ( F `  ( p  .x.  q ) )  e.  B  <->  ( p  e.  K ,  x  e. 
{ ( F `  q ) }  |->  ( F `  ( p 
.x.  q ) ) ) : ( K  X.  { ( F `
 q ) } ) --> B )
2219, 21sylib 189 . . . . . . 7  |-  ( (
ph  /\  q  e.  V )  ->  (
p  e.  K ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p  .x.  q ) ) ) : ( K  X.  { ( F `  q ) } ) --> B )
23 fssxp 5542 . . . . . . 7  |-  ( ( p  e.  K ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p  .x.  q ) ) ) : ( K  X.  { ( F `  q ) } ) --> B  -> 
( p  e.  K ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p  .x.  q ) ) )  C_  (
( K  X.  {
( F `  q
) } )  X.  B ) )
2422, 23syl 16 . . . . . 6  |-  ( (
ph  /\  q  e.  V )  ->  (
p  e.  K ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p  .x.  q ) ) )  C_  (
( K  X.  {
( F `  q
) } )  X.  B ) )
2514ffvelrnda 5809 . . . . . . . 8  |-  ( (
ph  /\  q  e.  V )  ->  ( F `  q )  e.  B )
2625snssd 3886 . . . . . . 7  |-  ( (
ph  /\  q  e.  V )  ->  { ( F `  q ) }  C_  B )
27 xpss2 4925 . . . . . . 7  |-  ( { ( F `  q
) }  C_  B  ->  ( K  X.  {
( F `  q
) } )  C_  ( K  X.  B
) )
28 xpss1 4924 . . . . . . 7  |-  ( ( K  X.  { ( F `  q ) } )  C_  ( K  X.  B )  -> 
( ( K  X.  { ( F `  q ) } )  X.  B )  C_  ( ( K  X.  B )  X.  B
) )
2926, 27, 283syl 19 . . . . . 6  |-  ( (
ph  /\  q  e.  V )  ->  (
( K  X.  {
( F `  q
) } )  X.  B )  C_  (
( K  X.  B
)  X.  B ) )
3024, 29sstrd 3301 . . . . 5  |-  ( (
ph  /\  q  e.  V )  ->  (
p  e.  K ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p  .x.  q ) ) )  C_  (
( K  X.  B
)  X.  B ) )
3130ralrimiva 2732 . . . 4  |-  ( ph  ->  A. q  e.  V  ( p  e.  K ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p  .x.  q ) ) )  C_  (
( K  X.  B
)  X.  B ) )
32 iunss 4073 . . . 4  |-  ( U_ q  e.  V  (
p  e.  K ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p  .x.  q ) ) )  C_  (
( K  X.  B
)  X.  B )  <->  A. q  e.  V  ( p  e.  K ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p  .x.  q ) ) )  C_  (
( K  X.  B
)  X.  B ) )
3331, 32sylibr 204 . . 3  |-  ( ph  ->  U_ q  e.  V  ( p  e.  K ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p  .x.  q ) ) )  C_  (
( K  X.  B
)  X.  B ) )
3411, 33eqsstrd 3325 . 2  |-  ( ph  -> 
.xb  C_  ( ( K  X.  B )  X.  B ) )
35 dff2 5820 . 2  |-  (  .xb  : ( K  X.  B
) --> B  <->  (  .xb  Fn  ( K  X.  B
)  /\  .xb  C_  (
( K  X.  B
)  X.  B ) ) )
3610, 34, 35sylanbrc 646 1  |-  ( ph  -> 
.xb  : ( K  X.  B ) --> B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   A.wral 2649    C_ wss 3263   {csn 3757   U_ciun 4035    X. cxp 4816    Fn wfn 5389   -->wf 5390   -onto->wfo 5392   ` cfv 5394  (class class class)co 6020    e. cmpt2 6022   Basecbs 13396  Scalarcsca 13459   .scvsca 13460    "s cimas 13657
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-oadd 6664  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-sup 7381  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-nn 9933  df-2 9990  df-3 9991  df-4 9992  df-5 9993  df-6 9994  df-7 9995  df-8 9996  df-9 9997  df-10 9998  df-n0 10154  df-z 10215  df-dec 10315  df-uz 10421  df-fz 10976  df-struct 13398  df-ndx 13399  df-slot 13400  df-base 13401  df-plusg 13469  df-mulr 13470  df-sca 13472  df-vsca 13473  df-tset 13475  df-ple 13476  df-ds 13478  df-imas 13661
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