MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  imasvsca Structured version   Unicode version

Theorem imasvsca 13777
Description: The scalar multiplication operation of an image structure. (Contributed by Mario Carneiro, 23-Feb-2015.)
Hypotheses
Ref Expression
imasbas.u  |-  ( ph  ->  U  =  ( F 
"s  R ) )
imasbas.v  |-  ( ph  ->  V  =  ( Base `  R ) )
imasbas.f  |-  ( ph  ->  F : V -onto-> B
)
imasbas.r  |-  ( ph  ->  R  e.  Z )
imassca.g  |-  G  =  (Scalar `  R )
imasvsca.k  |-  K  =  ( Base `  G
)
imasvsca.q  |-  .x.  =  ( .s `  R )
imasvsca.s  |-  .xb  =  ( .s `  U )
Assertion
Ref Expression
imasvsca  |-  ( ph  -> 
.xb  =  U_ q  e.  V  ( p  e.  K ,  x  e. 
{ ( F `  q ) }  |->  ( F `  ( p 
.x.  q ) ) ) )
Distinct variable groups:    q, p, x, F    R, p, q, x    x, U    x, B    ph, p, q, x    K, p, x    V, p, q
Allowed substitution hints:    B( q, p)    .xb (
x, q, p)    .x. ( x, q, p)    U( q, p)    G( x, q, p)    K( q)    V( x)    Z( x, q, p)

Proof of Theorem imasvsca
Dummy variables  u  v  w  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 imasbas.u . . 3  |-  ( ph  ->  U  =  ( F 
"s  R ) )
2 imasbas.v . . 3  |-  ( ph  ->  V  =  ( Base `  R ) )
3 eqid 2442 . . 3  |-  ( +g  `  R )  =  ( +g  `  R )
4 eqid 2442 . . 3  |-  ( .r
`  R )  =  ( .r `  R
)
5 imassca.g . . 3  |-  G  =  (Scalar `  R )
6 imasvsca.k . . 3  |-  K  =  ( Base `  G
)
7 imasvsca.q . . 3  |-  .x.  =  ( .s `  R )
8 eqid 2442 . . 3  |-  ( TopOpen `  R )  =  (
TopOpen `  R )
9 eqid 2442 . . 3  |-  ( dist `  R )  =  (
dist `  R )
10 eqid 2442 . . 3  |-  ( le
`  R )  =  ( le `  R
)
11 imasbas.f . . . 4  |-  ( ph  ->  F : V -onto-> B
)
12 imasbas.r . . . 4  |-  ( ph  ->  R  e.  Z )
13 eqid 2442 . . . 4  |-  ( +g  `  U )  =  ( +g  `  U )
141, 2, 11, 12, 3, 13imasplusg 13774 . . 3  |-  ( ph  ->  ( +g  `  U
)  =  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( F `  ( p ( +g  `  R
) q ) )
>. } )
15 eqid 2442 . . . 4  |-  ( .r
`  U )  =  ( .r `  U
)
161, 2, 11, 12, 4, 15imasmulr 13775 . . 3  |-  ( ph  ->  ( .r `  U
)  =  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( F `  ( p ( .r `  R
) q ) )
>. } )
17 eqidd 2443 . . 3  |-  ( ph  ->  U_ q  e.  V  ( p  e.  K ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p  .x.  q ) ) )  =  U_ q  e.  V  (
p  e.  K ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p  .x.  q ) ) ) )
18 eqidd 2443 . . 3  |-  ( ph  ->  ( ( TopOpen `  R
) qTop  F )  =  ( ( TopOpen `  R ) qTop  F ) )
19 eqid 2442 . . . 4  |-  ( dist `  U )  =  (
dist `  U )
201, 2, 11, 12, 9, 19imasds 13770 . . 3  |-  ( ph  ->  ( dist `  U
)  =  ( x  e.  B ,  y  e.  B  |->  sup ( U_ u  e.  NN  ran  ( z  e.  {
w  e.  ( ( V  X.  V )  ^m  ( 1 ... u ) )  |  ( ( F `  ( 1st `  ( w `
 1 ) ) )  =  x  /\  ( F `  ( 2nd `  ( w `  u
) ) )  =  y  /\  A. v  e.  ( 1 ... (
u  -  1 ) ) ( F `  ( 2nd `  ( w `
 v ) ) )  =  ( F `
 ( 1st `  (
w `  ( v  +  1 ) ) ) ) ) } 
|->  ( RR* s  gsumg  ( (
dist `  R )  o.  z ) ) ) ,  RR* ,  `'  <  ) ) )
21 eqidd 2443 . . 3  |-  ( ph  ->  ( ( F  o.  ( le `  R ) )  o.  `' F
)  =  ( ( F  o.  ( le
`  R ) )  o.  `' F ) )
221, 2, 3, 4, 5, 6, 7, 8, 9, 10, 14, 16, 17, 18, 20, 21, 11, 12imasval 13768 . 2  |-  ( ph  ->  U  =  ( ( { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  ( +g  `  U ) >. ,  <. ( .r `  ndx ) ,  ( .r `  U ) >. }  u.  {
<. (Scalar `  ndx ) ,  G >. ,  <. ( .s `  ndx ) , 
U_ q  e.  V  ( p  e.  K ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p  .x.  q ) ) ) >. } )  u.  { <. (TopSet ` 
ndx ) ,  ( ( TopOpen `  R ) qTop  F ) >. ,  <. ( le `  ndx ) ,  ( ( F  o.  ( le `  R ) )  o.  `' F
) >. ,  <. ( dist `  ndx ) ,  ( dist `  U
) >. } ) )
23 eqid 2442 . . 3  |-  ( ( { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  ( +g  `  U ) >. ,  <. ( .r `  ndx ) ,  ( .r `  U ) >. }  u.  {
<. (Scalar `  ndx ) ,  G >. ,  <. ( .s `  ndx ) , 
U_ q  e.  V  ( p  e.  K ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p  .x.  q ) ) ) >. } )  u.  { <. (TopSet ` 
ndx ) ,  ( ( TopOpen `  R ) qTop  F ) >. ,  <. ( le `  ndx ) ,  ( ( F  o.  ( le `  R ) )  o.  `' F
) >. ,  <. ( dist `  ndx ) ,  ( dist `  U
) >. } )  =  ( ( { <. (
Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) ,  ( +g  `  U
) >. ,  <. ( .r `  ndx ) ,  ( .r `  U
) >. }  u.  { <. (Scalar `  ndx ) ,  G >. ,  <. ( .s `  ndx ) , 
U_ q  e.  V  ( p  e.  K ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p  .x.  q ) ) ) >. } )  u.  { <. (TopSet ` 
ndx ) ,  ( ( TopOpen `  R ) qTop  F ) >. ,  <. ( le `  ndx ) ,  ( ( F  o.  ( le `  R ) )  o.  `' F
) >. ,  <. ( dist `  ndx ) ,  ( dist `  U
) >. } )
2423imasvalstr 13706 . 2  |-  ( ( { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  ( +g  `  U ) >. ,  <. ( .r `  ndx ) ,  ( .r `  U ) >. }  u.  {
<. (Scalar `  ndx ) ,  G >. ,  <. ( .s `  ndx ) , 
U_ q  e.  V  ( p  e.  K ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p  .x.  q ) ) ) >. } )  u.  { <. (TopSet ` 
ndx ) ,  ( ( TopOpen `  R ) qTop  F ) >. ,  <. ( le `  ndx ) ,  ( ( F  o.  ( le `  R ) )  o.  `' F
) >. ,  <. ( dist `  ndx ) ,  ( dist `  U
) >. } ) Struct  <. 1 , ; 1 2 >.
25 vscaid 13623 . 2  |-  .s  = Slot  ( .s `  ndx )
26 snsspr2 3972 . . . 4  |-  { <. ( .s `  ndx ) ,  U_ q  e.  V  ( p  e.  K ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p  .x.  q ) ) ) >. }  C_  {
<. (Scalar `  ndx ) ,  G >. ,  <. ( .s `  ndx ) , 
U_ q  e.  V  ( p  e.  K ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p  .x.  q ) ) ) >. }
27 ssun2 3497 . . . 4  |-  { <. (Scalar `  ndx ) ,  G >. ,  <. ( .s `  ndx ) ,  U_ q  e.  V  ( p  e.  K ,  x  e. 
{ ( F `  q ) }  |->  ( F `  ( p 
.x.  q ) ) ) >. }  C_  ( { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  ( +g  `  U ) >. ,  <. ( .r `  ndx ) ,  ( .r `  U ) >. }  u.  {
<. (Scalar `  ndx ) ,  G >. ,  <. ( .s `  ndx ) , 
U_ q  e.  V  ( p  e.  K ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p  .x.  q ) ) ) >. } )
2826, 27sstri 3343 . . 3  |-  { <. ( .s `  ndx ) ,  U_ q  e.  V  ( p  e.  K ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p  .x.  q ) ) ) >. }  C_  ( { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  ( +g  `  U ) >. ,  <. ( .r `  ndx ) ,  ( .r `  U ) >. }  u.  {
<. (Scalar `  ndx ) ,  G >. ,  <. ( .s `  ndx ) , 
U_ q  e.  V  ( p  e.  K ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p  .x.  q ) ) ) >. } )
29 ssun1 3496 . . 3  |-  ( {
<. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  ( +g  `  U ) >. ,  <. ( .r `  ndx ) ,  ( .r `  U ) >. }  u.  {
<. (Scalar `  ndx ) ,  G >. ,  <. ( .s `  ndx ) , 
U_ q  e.  V  ( p  e.  K ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p  .x.  q ) ) ) >. } ) 
C_  ( ( {
<. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  ( +g  `  U ) >. ,  <. ( .r `  ndx ) ,  ( .r `  U ) >. }  u.  {
<. (Scalar `  ndx ) ,  G >. ,  <. ( .s `  ndx ) , 
U_ q  e.  V  ( p  e.  K ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p  .x.  q ) ) ) >. } )  u.  { <. (TopSet ` 
ndx ) ,  ( ( TopOpen `  R ) qTop  F ) >. ,  <. ( le `  ndx ) ,  ( ( F  o.  ( le `  R ) )  o.  `' F
) >. ,  <. ( dist `  ndx ) ,  ( dist `  U
) >. } )
3028, 29sstri 3343 . 2  |-  { <. ( .s `  ndx ) ,  U_ q  e.  V  ( p  e.  K ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p  .x.  q ) ) ) >. }  C_  ( ( { <. (
Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) ,  ( +g  `  U
) >. ,  <. ( .r `  ndx ) ,  ( .r `  U
) >. }  u.  { <. (Scalar `  ndx ) ,  G >. ,  <. ( .s `  ndx ) , 
U_ q  e.  V  ( p  e.  K ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p  .x.  q ) ) ) >. } )  u.  { <. (TopSet ` 
ndx ) ,  ( ( TopOpen `  R ) qTop  F ) >. ,  <. ( le `  ndx ) ,  ( ( F  o.  ( le `  R ) )  o.  `' F
) >. ,  <. ( dist `  ndx ) ,  ( dist `  U
) >. } )
31 fvex 5771 . . . 4  |-  ( Base `  R )  e.  _V
322, 31syl6eqel 2530 . . 3  |-  ( ph  ->  V  e.  _V )
33 fvex 5771 . . . . . 6  |-  ( Base `  G )  e.  _V
346, 33eqeltri 2512 . . . . 5  |-  K  e. 
_V
35 snex 4434 . . . . 5  |-  { ( F `  q ) }  e.  _V
3634, 35mpt2ex 6454 . . . 4  |-  ( p  e.  K ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p 
.x.  q ) ) )  e.  _V
3736rgenw 2779 . . 3  |-  A. q  e.  V  ( p  e.  K ,  x  e. 
{ ( F `  q ) }  |->  ( F `  ( p 
.x.  q ) ) )  e.  _V
38 iunexg 6016 . . 3  |-  ( ( V  e.  _V  /\  A. q  e.  V  ( p  e.  K ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p  .x.  q ) ) )  e.  _V )  ->  U_ q  e.  V  ( p  e.  K ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p  .x.  q ) ) )  e.  _V )
3932, 37, 38sylancl 645 . 2  |-  ( ph  ->  U_ q  e.  V  ( p  e.  K ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p  .x.  q ) ) )  e.  _V )
40 imasvsca.s . 2  |-  .xb  =  ( .s `  U )
4122, 24, 25, 30, 39, 40strfv3 13533 1  |-  ( ph  -> 
.xb  =  U_ q  e.  V  ( p  e.  K ,  x  e. 
{ ( F `  q ) }  |->  ( F `  ( p 
.x.  q ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1727   A.wral 2711   _Vcvv 2962    u. cun 3304   {csn 3838   {cpr 3839   {ctp 3840   <.cop 3841   U_ciun 4117   `'ccnv 4906    o. ccom 4911   -onto->wfo 5481   ` cfv 5483  (class class class)co 6110    e. cmpt2 6112   1c1 9022   2c2 10080  ;cdc 10413   ndxcnx 13497   Basecbs 13500   +g cplusg 13560   .rcmulr 13561  Scalarcsca 13563   .scvsca 13564  TopSetcts 13566   lecple 13567   distcds 13569   TopOpenctopn 13680   qTop cqtop 13760    "s cimas 13761
This theorem is referenced by:  imastset  13778  imasle  13779  imasvscafn  13793  imasvscaval  13794  imasvscaf  13795
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730  ax-cnex 9077  ax-resscn 9078  ax-1cn 9079  ax-icn 9080  ax-addcl 9081  ax-addrcl 9082  ax-mulcl 9083  ax-mulrcl 9084  ax-mulcom 9085  ax-addass 9086  ax-mulass 9087  ax-distr 9088  ax-i2m1 9089  ax-1ne0 9090  ax-1rid 9091  ax-rnegex 9092  ax-rrecex 9093  ax-cnre 9094  ax-pre-lttri 9095  ax-pre-lttrn 9096  ax-pre-ltadd 9097  ax-pre-mulgt0 9098
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2716  df-rex 2717  df-reu 2718  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-pss 3322  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-tp 3846  df-op 3847  df-uni 4040  df-int 4075  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-tr 4328  df-eprel 4523  df-id 4527  df-po 4532  df-so 4533  df-fr 4570  df-we 4572  df-ord 4613  df-on 4614  df-lim 4615  df-suc 4616  df-om 4875  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-1st 6378  df-2nd 6379  df-riota 6578  df-recs 6662  df-rdg 6697  df-1o 6753  df-oadd 6757  df-er 6934  df-en 7139  df-dom 7140  df-sdom 7141  df-fin 7142  df-sup 7475  df-pnf 9153  df-mnf 9154  df-xr 9155  df-ltxr 9156  df-le 9157  df-sub 9324  df-neg 9325  df-nn 10032  df-2 10089  df-3 10090  df-4 10091  df-5 10092  df-6 10093  df-7 10094  df-8 10095  df-9 10096  df-10 10097  df-n0 10253  df-z 10314  df-dec 10414  df-uz 10520  df-fz 11075  df-struct 13502  df-ndx 13503  df-slot 13504  df-base 13505  df-plusg 13573  df-mulr 13574  df-sca 13576  df-vsca 13577  df-tset 13579  df-ple 13580  df-ds 13582  df-imas 13765
  Copyright terms: Public domain W3C validator