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Theorem imassca 13422
Description: The scalar field 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 )
Assertion
Ref Expression
imassca  |-  ( ph  ->  G  =  (Scalar `  U ) )

Proof of Theorem imassca
Dummy variables  g  h  i  n  p  q  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 imasbas.u . . . 4  |-  ( ph  ->  U  =  ( F 
"s  R ) )
2 imasbas.v . . . 4  |-  ( ph  ->  V  =  ( Base `  R ) )
3 eqid 2283 . . . 4  |-  ( +g  `  R )  =  ( +g  `  R )
4 eqid 2283 . . . 4  |-  ( .r
`  R )  =  ( .r `  R
)
5 imassca.g . . . 4  |-  G  =  (Scalar `  R )
6 eqid 2283 . . . 4  |-  ( Base `  G )  =  (
Base `  G )
7 eqid 2283 . . . 4  |-  ( .s
`  R )  =  ( .s `  R
)
8 eqid 2283 . . . 4  |-  ( TopOpen `  R )  =  (
TopOpen `  R )
9 eqid 2283 . . . 4  |-  ( dist `  R )  =  (
dist `  R )
10 eqid 2283 . . . 4  |-  ( le
`  R )  =  ( le `  R
)
11 imasbas.f . . . . 5  |-  ( ph  ->  F : V -onto-> B
)
12 imasbas.r . . . . 5  |-  ( ph  ->  R  e.  Z )
13 eqid 2283 . . . . 5  |-  ( +g  `  U )  =  ( +g  `  U )
141, 2, 11, 12, 3, 13imasplusg 13420 . . . 4  |-  ( ph  ->  ( +g  `  U
)  =  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( F `  ( p ( +g  `  R
) q ) )
>. } )
15 eqid 2283 . . . . 5  |-  ( .r
`  U )  =  ( .r `  U
)
161, 2, 11, 12, 4, 15imasmulr 13421 . . . 4  |-  ( ph  ->  ( .r `  U
)  =  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( F `  ( p ( .r `  R
) q ) )
>. } )
17 eqidd 2284 . . . 4  |-  ( ph  ->  U_ q  e.  V  ( p  e.  ( Base `  G ) ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p ( .s
`  R ) q ) ) )  = 
U_ q  e.  V  ( p  e.  ( Base `  G ) ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p ( .s
`  R ) q ) ) ) )
18 eqidd 2284 . . . 4  |-  ( ph  ->  ( ( TopOpen `  R
) qTop  F )  =  ( ( TopOpen `  R ) qTop  F ) )
19 eqid 2283 . . . . 5  |-  ( dist `  U )  =  (
dist `  U )
201, 2, 11, 12, 9, 19imasds 13416 . . . 4  |-  ( ph  ->  ( dist `  U
)  =  ( x  e.  B ,  y  e.  B  |->  sup ( U_ n  e.  NN  ran  ( g  e.  {
h  e.  ( ( V  X.  V )  ^m  ( 1 ... n ) )  |  ( ( F `  ( 1st `  ( h `
 1 ) ) )  =  x  /\  ( F `  ( 2nd `  ( h `  n
) ) )  =  y  /\  A. i  e.  ( 1 ... (
n  -  1 ) ) ( F `  ( 2nd `  ( h `
 i ) ) )  =  ( F `
 ( 1st `  (
h `  ( i  +  1 ) ) ) ) ) } 
|->  ( RR* s  gsumg  ( (
dist `  R )  o.  g ) ) ) ,  RR* ,  `'  <  ) ) )
21 eqidd 2284 . . . 4  |-  ( 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 13414 . . 3  |-  ( 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.  ( Base `  G ) ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p ( .s
`  R ) q ) ) ) >. } )  u.  { <. (TopSet `  ndx ) ,  ( ( TopOpen `  R
) qTop  F ) >. ,  <. ( le `  ndx ) ,  ( ( F  o.  ( le `  R ) )  o.  `' F ) >. ,  <. (
dist `  ndx ) ,  ( dist `  U
) >. } ) )
2322fveq2d 5529 . 2  |-  ( ph  ->  (Scalar `  U )  =  (Scalar `  ( ( { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  ( +g  `  U ) >. ,  <. ( .r `  ndx ) ,  ( .r `  U ) >. }  u.  {
<. (Scalar `  ndx ) ,  G >. ,  <. ( .s `  ndx ) , 
U_ q  e.  V  ( p  e.  ( Base `  G ) ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p ( .s
`  R ) q ) ) ) >. } )  u.  { <. (TopSet `  ndx ) ,  ( ( TopOpen `  R
) qTop  F ) >. ,  <. ( le `  ndx ) ,  ( ( F  o.  ( le `  R ) )  o.  `' F ) >. ,  <. (
dist `  ndx ) ,  ( dist `  U
) >. } ) ) )
24 fvex 5539 . . . 4  |-  (Scalar `  R )  e.  _V
255, 24eqeltri 2353 . . 3  |-  G  e. 
_V
26 eqid 2283 . . . . 5  |-  ( ( { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  ( +g  `  U ) >. ,  <. ( .r `  ndx ) ,  ( .r `  U ) >. }  u.  {
<. (Scalar `  ndx ) ,  G >. ,  <. ( .s `  ndx ) , 
U_ q  e.  V  ( p  e.  ( Base `  G ) ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p ( .s
`  R ) 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.  ( Base `  G ) ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p ( .s
`  R ) q ) ) ) >. } )  u.  { <. (TopSet `  ndx ) ,  ( ( TopOpen `  R
) qTop  F ) >. ,  <. ( le `  ndx ) ,  ( ( F  o.  ( le `  R ) )  o.  `' F ) >. ,  <. (
dist `  ndx ) ,  ( dist `  U
) >. } )
2726imasvalstr 13352 . . . 4  |-  ( ( { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  ( +g  `  U ) >. ,  <. ( .r `  ndx ) ,  ( .r `  U ) >. }  u.  {
<. (Scalar `  ndx ) ,  G >. ,  <. ( .s `  ndx ) , 
U_ q  e.  V  ( p  e.  ( Base `  G ) ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p ( .s
`  R ) q ) ) ) >. } )  u.  { <. (TopSet `  ndx ) ,  ( ( TopOpen `  R
) qTop  F ) >. ,  <. ( le `  ndx ) ,  ( ( F  o.  ( le `  R ) )  o.  `' F ) >. ,  <. (
dist `  ndx ) ,  ( dist `  U
) >. } ) Struct  <. 1 , ; 1 2 >.
28 scaid 13269 . . . 4  |- Scalar  = Slot  (Scalar ` 
ndx )
29 snsspr1 3764 . . . . . 6  |-  { <. (Scalar `  ndx ) ,  G >. }  C_  { <. (Scalar ` 
ndx ) ,  G >. ,  <. ( .s `  ndx ) ,  U_ q  e.  V  ( p  e.  ( Base `  G
) ,  x  e. 
{ ( F `  q ) }  |->  ( F `  ( p ( .s `  R
) q ) ) ) >. }
30 ssun2 3339 . . . . . 6  |-  { <. (Scalar `  ndx ) ,  G >. ,  <. ( .s `  ndx ) ,  U_ q  e.  V  ( p  e.  ( Base `  G
) ,  x  e. 
{ ( F `  q ) }  |->  ( F `  ( p ( .s `  R
) 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.  ( Base `  G ) ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p ( .s
`  R ) q ) ) ) >. } )
3129, 30sstri 3188 . . . . 5  |-  { <. (Scalar `  ndx ) ,  G >. }  C_  ( { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  ( +g  `  U ) >. ,  <. ( .r `  ndx ) ,  ( .r `  U ) >. }  u.  {
<. (Scalar `  ndx ) ,  G >. ,  <. ( .s `  ndx ) , 
U_ q  e.  V  ( p  e.  ( Base `  G ) ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p ( .s
`  R ) q ) ) ) >. } )
32 ssun1 3338 . . . . 5  |-  ( {
<. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  ( +g  `  U ) >. ,  <. ( .r `  ndx ) ,  ( .r `  U ) >. }  u.  {
<. (Scalar `  ndx ) ,  G >. ,  <. ( .s `  ndx ) , 
U_ q  e.  V  ( p  e.  ( Base `  G ) ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p ( .s
`  R ) 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.  ( Base `  G ) ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p ( .s
`  R ) q ) ) ) >. } )  u.  { <. (TopSet `  ndx ) ,  ( ( TopOpen `  R
) qTop  F ) >. ,  <. ( le `  ndx ) ,  ( ( F  o.  ( le `  R ) )  o.  `' F ) >. ,  <. (
dist `  ndx ) ,  ( dist `  U
) >. } )
3331, 32sstri 3188 . . . 4  |-  { <. (Scalar `  ndx ) ,  G >. }  C_  ( ( { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  ( +g  `  U ) >. ,  <. ( .r `  ndx ) ,  ( .r `  U ) >. }  u.  {
<. (Scalar `  ndx ) ,  G >. ,  <. ( .s `  ndx ) , 
U_ q  e.  V  ( p  e.  ( Base `  G ) ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p ( .s
`  R ) q ) ) ) >. } )  u.  { <. (TopSet `  ndx ) ,  ( ( TopOpen `  R
) qTop  F ) >. ,  <. ( le `  ndx ) ,  ( ( F  o.  ( le `  R ) )  o.  `' F ) >. ,  <. (
dist `  ndx ) ,  ( dist `  U
) >. } )
3427, 28, 33strfv 13180 . . 3  |-  ( G  e.  _V  ->  G  =  (Scalar `  ( ( { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  ( +g  `  U ) >. ,  <. ( .r `  ndx ) ,  ( .r `  U ) >. }  u.  {
<. (Scalar `  ndx ) ,  G >. ,  <. ( .s `  ndx ) , 
U_ q  e.  V  ( p  e.  ( Base `  G ) ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p ( .s
`  R ) q ) ) ) >. } )  u.  { <. (TopSet `  ndx ) ,  ( ( TopOpen `  R
) qTop  F ) >. ,  <. ( le `  ndx ) ,  ( ( F  o.  ( le `  R ) )  o.  `' F ) >. ,  <. (
dist `  ndx ) ,  ( dist `  U
) >. } ) ) )
3525, 34ax-mp 8 . 2  |-  G  =  (Scalar `  ( ( { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  ( +g  `  U ) >. ,  <. ( .r `  ndx ) ,  ( .r `  U ) >. }  u.  {
<. (Scalar `  ndx ) ,  G >. ,  <. ( .s `  ndx ) , 
U_ q  e.  V  ( p  e.  ( Base `  G ) ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p ( .s
`  R ) q ) ) ) >. } )  u.  { <. (TopSet `  ndx ) ,  ( ( TopOpen `  R
) qTop  F ) >. ,  <. ( le `  ndx ) ,  ( ( F  o.  ( le `  R ) )  o.  `' F ) >. ,  <. (
dist `  ndx ) ,  ( dist `  U
) >. } ) )
3623, 35syl6reqr 2334 1  |-  ( ph  ->  G  =  (Scalar `  U ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   _Vcvv 2788    u. cun 3150   {csn 3640   {cpr 3641   {ctp 3642   <.cop 3643   U_ciun 3905   `'ccnv 4688    o. ccom 4693   -onto->wfo 5253   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   1c1 8738   2c2 9795  ;cdc 10124   ndxcnx 13145   Basecbs 13148   +g cplusg 13208   .rcmulr 13209  Scalarcsca 13211   .scvsca 13212  TopSetcts 13214   lecple 13215   distcds 13217   TopOpenctopn 13326   qTop cqtop 13406    "s cimas 13407
This theorem is referenced by:  divssca  13448  xpssca  13480
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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