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Theorem imassca 14571
Description: The scalar field of an image structure. (Contributed by Mario Carneiro, 23-Feb-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
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 2452 . . . 4  |-  ( +g  `  R )  =  ( +g  `  R )
4 eqid 2452 . . . 4  |-  ( .r
`  R )  =  ( .r `  R
)
5 imassca.g . . . 4  |-  G  =  (Scalar `  R )
6 eqid 2452 . . . 4  |-  ( Base `  G )  =  (
Base `  G )
7 eqid 2452 . . . 4  |-  ( .s
`  R )  =  ( .s `  R
)
8 eqid 2452 . . . 4  |-  ( .i
`  R )  =  ( .i `  R
)
9 eqid 2452 . . . 4  |-  ( TopOpen `  R )  =  (
TopOpen `  R )
10 eqid 2452 . . . 4  |-  ( dist `  R )  =  (
dist `  R )
11 eqid 2452 . . . 4  |-  ( le
`  R )  =  ( le `  R
)
12 imasbas.f . . . . 5  |-  ( ph  ->  F : V -onto-> B
)
13 imasbas.r . . . . 5  |-  ( ph  ->  R  e.  Z )
14 eqid 2452 . . . . 5  |-  ( +g  `  U )  =  ( +g  `  U )
151, 2, 12, 13, 3, 14imasplusg 14569 . . . 4  |-  ( ph  ->  ( +g  `  U
)  =  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( F `  ( p ( +g  `  R
) q ) )
>. } )
16 eqid 2452 . . . . 5  |-  ( .r
`  U )  =  ( .r `  U
)
171, 2, 12, 13, 4, 16imasmulr 14570 . . . 4  |-  ( ph  ->  ( .r `  U
)  =  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( F `  ( p ( .r `  R
) q ) )
>. } )
18 eqidd 2453 . . . 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 ) ) ) )
19 eqidd 2453 . . . 4  |-  ( ph  ->  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q )
>. ,  ( p
( .i `  R
) q ) >. }  =  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( p ( .i `  R ) q )
>. } )
20 eqidd 2453 . . . 4  |-  ( ph  ->  ( ( TopOpen `  R
) qTop  F )  =  ( ( TopOpen `  R ) qTop  F ) )
21 eqid 2452 . . . . 5  |-  ( dist `  U )  =  (
dist `  U )
221, 2, 12, 13, 10, 21imasds 14565 . . . 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* ,  `'  <  ) ) )
23 eqidd 2453 . . . 4  |-  ( ph  ->  ( ( F  o.  ( le `  R ) )  o.  `' F
)  =  ( ( F  o.  ( le
`  R ) )  o.  `' F ) )
241, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 15, 17, 18, 19, 20, 22, 23, 12, 13imasval 14563 . . 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 ) ) ) >. ,  <. ( .i `  ndx ) ,  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( p ( .i `  R ) q )
>. } >. } )  u. 
{ <. (TopSet `  ndx ) ,  ( ( TopOpen
`  R ) qTop  F
) >. ,  <. ( le `  ndx ) ,  ( ( F  o.  ( le `  R ) )  o.  `' F
) >. ,  <. ( dist `  ndx ) ,  ( dist `  U
) >. } ) )
2524fveq2d 5798 . 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 ) ) ) >. ,  <. ( .i `  ndx ) ,  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( p ( .i `  R ) q )
>. } >. } )  u. 
{ <. (TopSet `  ndx ) ,  ( ( TopOpen
`  R ) qTop  F
) >. ,  <. ( le `  ndx ) ,  ( ( F  o.  ( le `  R ) )  o.  `' F
) >. ,  <. ( dist `  ndx ) ,  ( dist `  U
) >. } ) ) )
26 fvex 5804 . . . 4  |-  (Scalar `  R )  e.  _V
275, 26eqeltri 2536 . . 3  |-  G  e. 
_V
28 eqid 2452 . . . . 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 ) ) ) >. ,  <. ( .i `  ndx ) ,  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( p ( .i `  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 ) ) ) >. ,  <. ( .i `  ndx ) ,  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( p ( .i `  R ) q )
>. } >. } )  u. 
{ <. (TopSet `  ndx ) ,  ( ( TopOpen
`  R ) qTop  F
) >. ,  <. ( le `  ndx ) ,  ( ( F  o.  ( le `  R ) )  o.  `' F
) >. ,  <. ( dist `  ndx ) ,  ( dist `  U
) >. } )
2928imasvalstr 14504 . . . 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 ) ) ) >. ,  <. ( .i `  ndx ) ,  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( p ( .i `  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 >.
30 scaid 14413 . . . 4  |- Scalar  = Slot  (Scalar ` 
ndx )
31 snsstp1 4127 . . . . . 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 ) ) ) >. ,  <. ( .i `  ndx ) , 
U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q )
>. ,  ( p
( .i `  R
) q ) >. } >. }
32 ssun2 3623 . . . . . 6  |-  { <. (Scalar `  ndx ) ,  G >. ,  <. ( .s `  ndx ) ,  U_ q  e.  V  ( p  e.  ( Base `  G
) ,  x  e. 
{ ( F `  q ) }  |->  ( F `  ( p ( .s `  R
) q ) ) ) >. ,  <. ( .i `  ndx ) , 
U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q )
>. ,  ( p
( .i `  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 ) ) ) >. ,  <. ( .i `  ndx ) ,  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( p ( .i `  R ) q )
>. } >. } )
3331, 32sstri 3468 . . . . 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 ) ) ) >. ,  <. ( .i `  ndx ) ,  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( p ( .i `  R ) q )
>. } >. } )
34 ssun1 3622 . . . . 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 ) ) ) >. ,  <. ( .i `  ndx ) ,  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( p ( .i `  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 ) ) ) >. ,  <. ( .i `  ndx ) ,  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( p ( .i `  R ) q )
>. } >. } )  u. 
{ <. (TopSet `  ndx ) ,  ( ( TopOpen
`  R ) qTop  F
) >. ,  <. ( le `  ndx ) ,  ( ( F  o.  ( le `  R ) )  o.  `' F
) >. ,  <. ( dist `  ndx ) ,  ( dist `  U
) >. } )
3533, 34sstri 3468 . . . 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 ) ) ) >. ,  <. ( .i `  ndx ) ,  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( p ( .i `  R ) q )
>. } >. } )  u. 
{ <. (TopSet `  ndx ) ,  ( ( TopOpen
`  R ) qTop  F
) >. ,  <. ( le `  ndx ) ,  ( ( F  o.  ( le `  R ) )  o.  `' F
) >. ,  <. ( dist `  ndx ) ,  ( dist `  U
) >. } )
3629, 30, 35strfv 14321 . . 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 ) ) ) >. ,  <. ( .i `  ndx ) ,  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( p ( .i `  R ) q )
>. } >. } )  u. 
{ <. (TopSet `  ndx ) ,  ( ( TopOpen
`  R ) qTop  F
) >. ,  <. ( le `  ndx ) ,  ( ( F  o.  ( le `  R ) )  o.  `' F
) >. ,  <. ( dist `  ndx ) ,  ( dist `  U
) >. } ) ) )
3727, 36ax-mp 5 . 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 ) ) ) >. ,  <. ( .i `  ndx ) ,  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( p ( .i `  R ) q )
>. } >. } )  u. 
{ <. (TopSet `  ndx ) ,  ( ( TopOpen
`  R ) qTop  F
) >. ,  <. ( le `  ndx ) ,  ( ( F  o.  ( le `  R ) )  o.  `' F
) >. ,  <. ( dist `  ndx ) ,  ( dist `  U
) >. } ) )
3825, 37syl6reqr 2512 1  |-  ( ph  ->  G  =  (Scalar `  U ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758   _Vcvv 3072    u. cun 3429   {csn 3980   {ctp 3984   <.cop 3986   U_ciun 4274   `'ccnv 4942    o. ccom 4947   -onto->wfo 5519   ` cfv 5521  (class class class)co 6195    |-> cmpt2 6197   1c1 9389   2c2 10477  ;cdc 10861   ndxcnx 14284   Basecbs 14287   +g cplusg 14352   .rcmulr 14353  Scalarcsca 14355   .scvsca 14356   .icip 14357  TopSetcts 14358   lecple 14359   distcds 14361   TopOpenctopn 14474   qTop cqtop 14555    "s cimas 14556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-cnex 9444  ax-resscn 9445  ax-1cn 9446  ax-icn 9447  ax-addcl 9448  ax-addrcl 9449  ax-mulcl 9450  ax-mulrcl 9451  ax-mulcom 9452  ax-addass 9453  ax-mulass 9454  ax-distr 9455  ax-i2m1 9456  ax-1ne0 9457  ax-1rid 9458  ax-rnegex 9459  ax-rrecex 9460  ax-cnre 9461  ax-pre-lttri 9462  ax-pre-lttrn 9463  ax-pre-ltadd 9464  ax-pre-mulgt0 9465
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-int 4232  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-om 6582  df-1st 6682  df-2nd 6683  df-recs 6937  df-rdg 6971  df-1o 7025  df-oadd 7029  df-er 7206  df-en 7416  df-dom 7417  df-sdom 7418  df-fin 7419  df-sup 7797  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529  df-le 9530  df-sub 9703  df-neg 9704  df-nn 10429  df-2 10486  df-3 10487  df-4 10488  df-5 10489  df-6 10490  df-7 10491  df-8 10492  df-9 10493  df-10 10494  df-n0 10686  df-z 10753  df-dec 10862  df-uz 10968  df-fz 11550  df-struct 14289  df-ndx 14290  df-slot 14291  df-base 14292  df-plusg 14365  df-mulr 14366  df-sca 14368  df-vsca 14369  df-ip 14370  df-tset 14371  df-ple 14372  df-ds 14374  df-imas 14560
This theorem is referenced by:  divssca  14598  xpssca  14630
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