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Theorem imasvsca 13709
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 2412 . . 3  |-  ( +g  `  R )  =  ( +g  `  R )
4 eqid 2412 . . 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 2412 . . 3  |-  ( TopOpen `  R )  =  (
TopOpen `  R )
9 eqid 2412 . . 3  |-  ( dist `  R )  =  (
dist `  R )
10 eqid 2412 . . 3  |-  ( le
`  R )  =  ( le `  R
)
11 imasbas.f . . . 4  |-  ( ph  ->  F : V -onto-> B
)
12 imasbas.r . . . 4  |-  ( ph  ->  R  e.  Z )
13 eqid 2412 . . . 4  |-  ( +g  `  U )  =  ( +g  `  U )
141, 2, 11, 12, 3, 13imasplusg 13706 . . 3  |-  ( ph  ->  ( +g  `  U
)  =  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( F `  ( p ( +g  `  R
) q ) )
>. } )
15 eqid 2412 . . . 4  |-  ( .r
`  U )  =  ( .r `  U
)
161, 2, 11, 12, 4, 15imasmulr 13707 . . 3  |-  ( ph  ->  ( .r `  U
)  =  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( F `  ( p ( .r `  R
) q ) )
>. } )
17 eqidd 2413 . . 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 2413 . . 3  |-  ( ph  ->  ( ( TopOpen `  R
) qTop  F )  =  ( ( TopOpen `  R ) qTop  F ) )
19 eqid 2412 . . . 4  |-  ( dist `  U )  =  (
dist `  U )
201, 2, 11, 12, 9, 19imasds 13702 . . 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 2413 . . 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 13700 . 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 2412 . . 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 13638 . 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 13555 . 2  |-  .s  = Slot  ( .s `  ndx )
26 snsspr2 3916 . . . 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 3479 . . . 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 3325 . . 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 3478 . . 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 3325 . 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 5709 . . . 4  |-  ( Base `  R )  e.  _V
322, 31syl6eqel 2500 . . 3  |-  ( ph  ->  V  e.  _V )
33 fvex 5709 . . . . . 6  |-  ( Base `  G )  e.  _V
346, 33eqeltri 2482 . . . . 5  |-  K  e. 
_V
35 snex 4373 . . . . 5  |-  { ( F `  q ) }  e.  _V
3634, 35mpt2ex 6392 . . . 4  |-  ( p  e.  K ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p 
.x.  q ) ) )  e.  _V
3736rgenw 2741 . . 3  |-  A. q  e.  V  ( p  e.  K ,  x  e. 
{ ( F `  q ) }  |->  ( F `  ( p 
.x.  q ) ) )  e.  _V
38 iunexg 5954 . . 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 644 . 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 13465 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 1649    e. wcel 1721   A.wral 2674   _Vcvv 2924    u. cun 3286   {csn 3782   {cpr 3783   {ctp 3784   <.cop 3785   U_ciun 4061   `'ccnv 4844    o. ccom 4849   -onto->wfo 5419   ` cfv 5421  (class class class)co 6048    e. cmpt2 6050   1c1 8955   2c2 10013  ;cdc 10346   ndxcnx 13429   Basecbs 13432   +g cplusg 13492   .rcmulr 13493  Scalarcsca 13495   .scvsca 13496  TopSetcts 13498   lecple 13499   distcds 13501   TopOpenctopn 13612   qTop cqtop 13692    "s cimas 13693
This theorem is referenced by:  imastset  13710  imasle  13711  imasvscafn  13725  imasvscaval  13726  imasvscaf  13727
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-1o 6691  df-oadd 6695  df-er 6872  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-sup 7412  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-nn 9965  df-2 10022  df-3 10023  df-4 10024  df-5 10025  df-6 10026  df-7 10027  df-8 10028  df-9 10029  df-10 10030  df-n0 10186  df-z 10247  df-dec 10347  df-uz 10453  df-fz 11008  df-struct 13434  df-ndx 13435  df-slot 13436  df-base 13437  df-plusg 13505  df-mulr 13506  df-sca 13508  df-vsca 13509  df-tset 13511  df-ple 13512  df-ds 13514  df-imas 13697
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