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Theorem imassca 13745
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 2436 . . . 4  |-  ( +g  `  R )  =  ( +g  `  R )
4 eqid 2436 . . . 4  |-  ( .r
`  R )  =  ( .r `  R
)
5 imassca.g . . . 4  |-  G  =  (Scalar `  R )
6 eqid 2436 . . . 4  |-  ( Base `  G )  =  (
Base `  G )
7 eqid 2436 . . . 4  |-  ( .s
`  R )  =  ( .s `  R
)
8 eqid 2436 . . . 4  |-  ( TopOpen `  R )  =  (
TopOpen `  R )
9 eqid 2436 . . . 4  |-  ( dist `  R )  =  (
dist `  R )
10 eqid 2436 . . . 4  |-  ( le
`  R )  =  ( le `  R
)
11 imasbas.f . . . . 5  |-  ( ph  ->  F : V -onto-> B
)
12 imasbas.r . . . . 5  |-  ( ph  ->  R  e.  Z )
13 eqid 2436 . . . . 5  |-  ( +g  `  U )  =  ( +g  `  U )
141, 2, 11, 12, 3, 13imasplusg 13743 . . . 4  |-  ( ph  ->  ( +g  `  U
)  =  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( F `  ( p ( +g  `  R
) q ) )
>. } )
15 eqid 2436 . . . . 5  |-  ( .r
`  U )  =  ( .r `  U
)
161, 2, 11, 12, 4, 15imasmulr 13744 . . . 4  |-  ( ph  ->  ( .r `  U
)  =  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( F `  ( p ( .r `  R
) q ) )
>. } )
17 eqidd 2437 . . . 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 2437 . . . 4  |-  ( ph  ->  ( ( TopOpen `  R
) qTop  F )  =  ( ( TopOpen `  R ) qTop  F ) )
19 eqid 2436 . . . . 5  |-  ( dist `  U )  =  (
dist `  U )
201, 2, 11, 12, 9, 19imasds 13739 . . . 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 2437 . . . 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 13737 . . 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 5732 . 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 5742 . . . 4  |-  (Scalar `  R )  e.  _V
255, 24eqeltri 2506 . . 3  |-  G  e. 
_V
26 eqid 2436 . . . . 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 13675 . . . 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 13590 . . . 4  |- Scalar  = Slot  (Scalar ` 
ndx )
29 snsspr1 3947 . . . . . 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 3511 . . . . . 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 3357 . . . . 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 3510 . . . . 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 3357 . . . 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 13501 . . 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 2487 1  |-  ( ph  ->  G  =  (Scalar `  U ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   _Vcvv 2956    u. cun 3318   {csn 3814   {cpr 3815   {ctp 3816   <.cop 3817   U_ciun 4093   `'ccnv 4877    o. ccom 4882   -onto->wfo 5452   ` cfv 5454  (class class class)co 6081    e. cmpt2 6083   1c1 8991   2c2 10049  ;cdc 10382   ndxcnx 13466   Basecbs 13469   +g cplusg 13529   .rcmulr 13530  Scalarcsca 13532   .scvsca 13533  TopSetcts 13535   lecple 13536   distcds 13538   TopOpenctopn 13649   qTop cqtop 13729    "s cimas 13730
This theorem is referenced by:  divssca  13771  xpssca  13803
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-7 10063  df-8 10064  df-9 10065  df-10 10066  df-n0 10222  df-z 10283  df-dec 10383  df-uz 10489  df-fz 11044  df-struct 13471  df-ndx 13472  df-slot 13473  df-base 13474  df-plusg 13542  df-mulr 13543  df-sca 13545  df-vsca 13546  df-tset 13548  df-ple 13549  df-ds 13551  df-imas 13734
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