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Theorem imasvscaf 14794
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 14792 . 2  |-  ( ph  -> 
.xb  Fn  ( K  X.  B ) )
111, 2, 3, 4, 5, 6, 7, 8imasvsca 14775 . . 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 5795 . . . . . . . . . . . . . 14  |-  ( F : V -onto-> B  ->  F : V --> B )
143, 13syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  F : V --> B )
1514ffvelrnda 6021 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( p  .x.  q )  e.  V
)  ->  ( F `  ( p  .x.  q
) )  e.  B
)
1612, 15syldan 470 . . . . . . . . . . 11  |-  ( (
ph  /\  ( p  e.  K  /\  q  e.  V ) )  -> 
( F `  (
p  .x.  q )
)  e.  B )
1716ralrimivw 2879 . . . . . . . . . 10  |-  ( (
ph  /\  ( p  e.  K  /\  q  e.  V ) )  ->  A. x  e.  { ( F `  q ) }  ( F `  ( p  .x.  q ) )  e.  B )
1817anass1rs 805 . . . . . . . . 9  |-  ( ( ( ph  /\  q  e.  V )  /\  p  e.  K )  ->  A. x  e.  { ( F `  q ) }  ( F `  ( p  .x.  q ) )  e.  B )
1918ralrimiva 2878 . . . . . . . 8  |-  ( (
ph  /\  q  e.  V )  ->  A. p  e.  K  A. x  e.  { ( F `  q ) }  ( F `  ( p  .x.  q ) )  e.  B )
20 eqid 2467 . . . . . . . . 9  |-  ( p  e.  K ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p 
.x.  q ) ) )  =  ( p  e.  K ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p 
.x.  q ) ) )
2120fmpt2 6851 . . . . . . . 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 196 . . . . . . 7  |-  ( (
ph  /\  q  e.  V )  ->  (
p  e.  K ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p  .x.  q ) ) ) : ( K  X.  { ( F `  q ) } ) --> B )
23 fssxp 5743 . . . . . . 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 6021 . . . . . . . 8  |-  ( (
ph  /\  q  e.  V )  ->  ( F `  q )  e.  B )
2625snssd 4172 . . . . . . 7  |-  ( (
ph  /\  q  e.  V )  ->  { ( F `  q ) }  C_  B )
27 xpss2 5112 . . . . . . 7  |-  ( { ( F `  q
) }  C_  B  ->  ( K  X.  {
( F `  q
) } )  C_  ( K  X.  B
) )
28 xpss1 5111 . . . . . . 7  |-  ( ( K  X.  { ( F `  q ) } )  C_  ( K  X.  B )  -> 
( ( K  X.  { ( F `  q ) } )  X.  B )  C_  ( ( K  X.  B )  X.  B
) )
2926, 27, 283syl 20 . . . . . 6  |-  ( (
ph  /\  q  e.  V )  ->  (
( K  X.  {
( F `  q
) } )  X.  B )  C_  (
( K  X.  B
)  X.  B ) )
3024, 29sstrd 3514 . . . . 5  |-  ( (
ph  /\  q  e.  V )  ->  (
p  e.  K ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p  .x.  q ) ) )  C_  (
( K  X.  B
)  X.  B ) )
3130ralrimiva 2878 . . . 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 4366 . . . 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 212 . . 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 3538 . 2  |-  ( ph  -> 
.xb  C_  ( ( K  X.  B )  X.  B ) )
35 dff2 6033 . 2  |-  (  .xb  : ( K  X.  B
) --> B  <->  (  .xb  Fn  ( K  X.  B
)  /\  .xb  C_  (
( K  X.  B
)  X.  B ) ) )
3610, 34, 35sylanbrc 664 1  |-  ( ph  -> 
.xb  : ( K  X.  B ) --> B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2814    C_ wss 3476   {csn 4027   U_ciun 4325    X. cxp 4997    Fn wfn 5583   -->wf 5584   -onto->wfo 5586   ` cfv 5588  (class class class)co 6284    |-> cmpt2 6286   Basecbs 14490  Scalarcsca 14558   .scvsca 14559    "s cimas 14759
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-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  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-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-sup 7901  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-nn 10537  df-2 10594  df-3 10595  df-4 10596  df-5 10597  df-6 10598  df-7 10599  df-8 10600  df-9 10601  df-10 10602  df-n0 10796  df-z 10865  df-dec 10977  df-uz 11083  df-fz 11673  df-struct 14492  df-ndx 14493  df-slot 14494  df-base 14495  df-plusg 14568  df-mulr 14569  df-sca 14571  df-vsca 14572  df-ip 14573  df-tset 14574  df-ple 14575  df-ds 14577  df-imas 14763
This theorem is referenced by: (None)
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