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Theorem imasvscaf 13441
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 13439 . 2  |-  ( ph  -> 
.xb  Fn  ( K  X.  B ) )
111, 2, 3, 4, 5, 6, 7, 8imasvsca 13423 . . 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 5451 . . . . . . . . . . . . . 14  |-  ( F : V -onto-> B  ->  F : V --> B )
143, 13syl 15 . . . . . . . . . . . . 13  |-  ( ph  ->  F : V --> B )
15 ffvelrn 5663 . . . . . . . . . . . . 13  |-  ( ( F : V --> B  /\  ( p  .x.  q )  e.  V )  -> 
( F `  (
p  .x.  q )
)  e.  B )
1614, 15sylan 457 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( p  .x.  q )  e.  V
)  ->  ( F `  ( p  .x.  q
) )  e.  B
)
1712, 16syldan 456 . . . . . . . . . . 11  |-  ( (
ph  /\  ( p  e.  K  /\  q  e.  V ) )  -> 
( F `  (
p  .x.  q )
)  e.  B )
1817ralrimivw 2627 . . . . . . . . . 10  |-  ( (
ph  /\  ( p  e.  K  /\  q  e.  V ) )  ->  A. x  e.  { ( F `  q ) }  ( F `  ( p  .x.  q ) )  e.  B )
1918anass1rs 782 . . . . . . . . 9  |-  ( ( ( ph  /\  q  e.  V )  /\  p  e.  K )  ->  A. x  e.  { ( F `  q ) }  ( F `  ( p  .x.  q ) )  e.  B )
2019ralrimiva 2626 . . . . . . . 8  |-  ( (
ph  /\  q  e.  V )  ->  A. p  e.  K  A. x  e.  { ( F `  q ) }  ( F `  ( p  .x.  q ) )  e.  B )
21 eqid 2283 . . . . . . . . 9  |-  ( p  e.  K ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p 
.x.  q ) ) )  =  ( p  e.  K ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p 
.x.  q ) ) )
2221fmpt2 6191 . . . . . . . 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 )
2320, 22sylib 188 . . . . . . 7  |-  ( (
ph  /\  q  e.  V )  ->  (
p  e.  K ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p  .x.  q ) ) ) : ( K  X.  { ( F `  q ) } ) --> B )
24 fssxp 5400 . . . . . . 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 ) )
2523, 24syl 15 . . . . . 6  |-  ( (
ph  /\  q  e.  V )  ->  (
p  e.  K ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p  .x.  q ) ) )  C_  (
( K  X.  {
( F `  q
) } )  X.  B ) )
26 ffvelrn 5663 . . . . . . . . 9  |-  ( ( F : V --> B  /\  q  e.  V )  ->  ( F `  q
)  e.  B )
2714, 26sylan 457 . . . . . . . 8  |-  ( (
ph  /\  q  e.  V )  ->  ( F `  q )  e.  B )
2827snssd 3760 . . . . . . 7  |-  ( (
ph  /\  q  e.  V )  ->  { ( F `  q ) }  C_  B )
29 xpss2 4796 . . . . . . 7  |-  ( { ( F `  q
) }  C_  B  ->  ( K  X.  {
( F `  q
) } )  C_  ( K  X.  B
) )
30 xpss1 4795 . . . . . . 7  |-  ( ( K  X.  { ( F `  q ) } )  C_  ( K  X.  B )  -> 
( ( K  X.  { ( F `  q ) } )  X.  B )  C_  ( ( K  X.  B )  X.  B
) )
3128, 29, 303syl 18 . . . . . 6  |-  ( (
ph  /\  q  e.  V )  ->  (
( K  X.  {
( F `  q
) } )  X.  B )  C_  (
( K  X.  B
)  X.  B ) )
3225, 31sstrd 3189 . . . . 5  |-  ( (
ph  /\  q  e.  V )  ->  (
p  e.  K ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p  .x.  q ) ) )  C_  (
( K  X.  B
)  X.  B ) )
3332ralrimiva 2626 . . . 4  |-  ( ph  ->  A. q  e.  V  ( p  e.  K ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p  .x.  q ) ) )  C_  (
( K  X.  B
)  X.  B ) )
34 iunss 3943 . . . 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 ) )
3533, 34sylibr 203 . . 3  |-  ( ph  ->  U_ q  e.  V  ( p  e.  K ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p  .x.  q ) ) )  C_  (
( K  X.  B
)  X.  B ) )
3611, 35eqsstrd 3212 . 2  |-  ( ph  -> 
.xb  C_  ( ( K  X.  B )  X.  B ) )
37 dff2 5672 . 2  |-  (  .xb  : ( K  X.  B
) --> B  <->  (  .xb  Fn  ( K  X.  B
)  /\  .xb  C_  (
( K  X.  B
)  X.  B ) ) )
3810, 36, 37sylanbrc 645 1  |-  ( ph  -> 
.xb  : ( K  X.  B ) --> B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543    C_ wss 3152   {csn 3640   U_ciun 3905    X. cxp 4687    Fn wfn 5250   -->wf 5251   -onto->wfo 5253   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   Basecbs 13148  Scalarcsca 13211   .scvsca 13212    "s cimas 13407
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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