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Theorem imasip 14558
Description: The inner product of an image structure. (Contributed 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 )
imasip.i  |-  .,  =  ( .i `  R )
imasip.w  |-  I  =  ( .i `  U
)
Assertion
Ref Expression
imasip  |-  ( ph  ->  I  =  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( p  .,  q )
>. } )
Distinct variable groups:    q, p, F    R, p, q    ph, p, q    V, p, q
Allowed substitution hints:    B( q, p)    U( q, p)    ., ( q, p)    I( q, p)    Z( q, p)

Proof of Theorem imasip
Dummy variables  u  v  w  x  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 2451 . . 3  |-  ( +g  `  R )  =  ( +g  `  R )
4 eqid 2451 . . 3  |-  ( .r
`  R )  =  ( .r `  R
)
5 eqid 2451 . . 3  |-  (Scalar `  R )  =  (Scalar `  R )
6 eqid 2451 . . 3  |-  ( Base `  (Scalar `  R )
)  =  ( Base `  (Scalar `  R )
)
7 eqid 2451 . . 3  |-  ( .s
`  R )  =  ( .s `  R
)
8 imasip.i . . 3  |-  .,  =  ( .i `  R )
9 eqid 2451 . . 3  |-  ( TopOpen `  R )  =  (
TopOpen `  R )
10 eqid 2451 . . 3  |-  ( dist `  R )  =  (
dist `  R )
11 eqid 2451 . . 3  |-  ( le
`  R )  =  ( le `  R
)
12 imasbas.f . . . 4  |-  ( ph  ->  F : V -onto-> B
)
13 imasbas.r . . . 4  |-  ( ph  ->  R  e.  Z )
14 eqid 2451 . . . 4  |-  ( +g  `  U )  =  ( +g  `  U )
151, 2, 12, 13, 3, 14imasplusg 14554 . . 3  |-  ( ph  ->  ( +g  `  U
)  =  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( F `  ( p ( +g  `  R
) q ) )
>. } )
16 eqid 2451 . . . 4  |-  ( .r
`  U )  =  ( .r `  U
)
171, 2, 12, 13, 4, 16imasmulr 14555 . . 3  |-  ( ph  ->  ( .r `  U
)  =  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( F `  ( p ( .r `  R
) q ) )
>. } )
18 eqid 2451 . . . 4  |-  ( .s
`  U )  =  ( .s `  U
)
191, 2, 12, 13, 5, 6, 7, 18imasvsca 14557 . . 3  |-  ( ph  ->  ( .s `  U
)  =  U_ q  e.  V  ( p  e.  ( Base `  (Scalar `  R ) ) ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p ( .s
`  R ) q ) ) ) )
20 eqidd 2452 . . 3  |-  ( ph  ->  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q )
>. ,  ( p  .,  q ) >. }  =  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q )
>. ,  ( p  .,  q ) >. } )
21 eqidd 2452 . . 3  |-  ( ph  ->  ( ( TopOpen `  R
) qTop  F )  =  ( ( TopOpen `  R ) qTop  F ) )
22 eqid 2451 . . . 4  |-  ( dist `  U )  =  (
dist `  U )
231, 2, 12, 13, 10, 22imasds 14550 . . 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* ,  `'  <  ) ) )
24 eqidd 2452 . . 3  |-  ( ph  ->  ( ( F  o.  ( le `  R ) )  o.  `' F
)  =  ( ( F  o.  ( le
`  R ) )  o.  `' F ) )
251, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 15, 17, 19, 20, 21, 23, 24, 12, 13imasval 14548 . 2  |-  ( ph  ->  U  =  ( ( { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  ( +g  `  U ) >. ,  <. ( .r `  ndx ) ,  ( .r `  U ) >. }  u.  {
<. (Scalar `  ndx ) ,  (Scalar `  R ) >. ,  <. ( .s `  ndx ) ,  ( .s
`  U ) >. ,  <. ( .i `  ndx ) ,  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( p  .,  q )
>. } >. } )  u. 
{ <. (TopSet `  ndx ) ,  ( ( TopOpen
`  R ) qTop  F
) >. ,  <. ( le `  ndx ) ,  ( ( F  o.  ( le `  R ) )  o.  `' F
) >. ,  <. ( dist `  ndx ) ,  ( dist `  U
) >. } ) )
26 eqid 2451 . . 3  |-  ( ( { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  ( +g  `  U ) >. ,  <. ( .r `  ndx ) ,  ( .r `  U ) >. }  u.  {
<. (Scalar `  ndx ) ,  (Scalar `  R ) >. ,  <. ( .s `  ndx ) ,  ( .s
`  U ) >. ,  <. ( .i `  ndx ) ,  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( p  .,  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 ) ,  (Scalar `  R ) >. ,  <. ( .s `  ndx ) ,  ( .s
`  U ) >. ,  <. ( .i `  ndx ) ,  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( p  .,  q )
>. } >. } )  u. 
{ <. (TopSet `  ndx ) ,  ( ( TopOpen
`  R ) qTop  F
) >. ,  <. ( le `  ndx ) ,  ( ( F  o.  ( le `  R ) )  o.  `' F
) >. ,  <. ( dist `  ndx ) ,  ( dist `  U
) >. } )
2726imasvalstr 14489 . 2  |-  ( ( { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  ( +g  `  U ) >. ,  <. ( .r `  ndx ) ,  ( .r `  U ) >. }  u.  {
<. (Scalar `  ndx ) ,  (Scalar `  R ) >. ,  <. ( .s `  ndx ) ,  ( .s
`  U ) >. ,  <. ( .i `  ndx ) ,  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( p  .,  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 ipid 14407 . 2  |-  .i  = Slot  ( .i `  ndx )
29 snsstp3 4121 . . . 4  |-  { <. ( .i `  ndx ) ,  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q )
>. ,  ( p  .,  q ) >. } >. } 
C_  { <. (Scalar ` 
ndx ) ,  (Scalar `  R ) >. ,  <. ( .s `  ndx ) ,  ( .s `  U ) >. ,  <. ( .i `  ndx ) ,  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q )
>. ,  ( p  .,  q ) >. } >. }
30 ssun2 3615 . . . 4  |-  { <. (Scalar `  ndx ) ,  (Scalar `  R ) >. ,  <. ( .s `  ndx ) ,  ( .s `  U ) >. ,  <. ( .i `  ndx ) ,  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q )
>. ,  ( p  .,  q ) >. } >. } 
C_  ( { <. (
Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) ,  ( +g  `  U
) >. ,  <. ( .r `  ndx ) ,  ( .r `  U
) >. }  u.  { <. (Scalar `  ndx ) ,  (Scalar `  R ) >. ,  <. ( .s `  ndx ) ,  ( .s
`  U ) >. ,  <. ( .i `  ndx ) ,  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( p  .,  q )
>. } >. } )
3129, 30sstri 3460 . . 3  |-  { <. ( .i `  ndx ) ,  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q )
>. ,  ( p  .,  q ) >. } >. } 
C_  ( { <. (
Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) ,  ( +g  `  U
) >. ,  <. ( .r `  ndx ) ,  ( .r `  U
) >. }  u.  { <. (Scalar `  ndx ) ,  (Scalar `  R ) >. ,  <. ( .s `  ndx ) ,  ( .s
`  U ) >. ,  <. ( .i `  ndx ) ,  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( p  .,  q )
>. } >. } )
32 ssun1 3614 . . 3  |-  ( {
<. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  ( +g  `  U ) >. ,  <. ( .r `  ndx ) ,  ( .r `  U ) >. }  u.  {
<. (Scalar `  ndx ) ,  (Scalar `  R ) >. ,  <. ( .s `  ndx ) ,  ( .s
`  U ) >. ,  <. ( .i `  ndx ) ,  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( p  .,  q )
>. } >. } )  C_  ( ( { <. (
Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) ,  ( +g  `  U
) >. ,  <. ( .r `  ndx ) ,  ( .r `  U
) >. }  u.  { <. (Scalar `  ndx ) ,  (Scalar `  R ) >. ,  <. ( .s `  ndx ) ,  ( .s
`  U ) >. ,  <. ( .i `  ndx ) ,  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( p  .,  q )
>. } >. } )  u. 
{ <. (TopSet `  ndx ) ,  ( ( TopOpen
`  R ) qTop  F
) >. ,  <. ( le `  ndx ) ,  ( ( F  o.  ( le `  R ) )  o.  `' F
) >. ,  <. ( dist `  ndx ) ,  ( dist `  U
) >. } )
3331, 32sstri 3460 . 2  |-  { <. ( .i `  ndx ) ,  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q )
>. ,  ( p  .,  q ) >. } >. } 
C_  ( ( {
<. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  ( +g  `  U ) >. ,  <. ( .r `  ndx ) ,  ( .r `  U ) >. }  u.  {
<. (Scalar `  ndx ) ,  (Scalar `  R ) >. ,  <. ( .s `  ndx ) ,  ( .s
`  U ) >. ,  <. ( .i `  ndx ) ,  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( p  .,  q )
>. } >. } )  u. 
{ <. (TopSet `  ndx ) ,  ( ( TopOpen
`  R ) qTop  F
) >. ,  <. ( le `  ndx ) ,  ( ( F  o.  ( le `  R ) )  o.  `' F
) >. ,  <. ( dist `  ndx ) ,  ( dist `  U
) >. } )
34 fvex 5796 . . . 4  |-  ( Base `  R )  e.  _V
352, 34syl6eqel 2545 . . 3  |-  ( ph  ->  V  e.  _V )
36 snex 4628 . . . . . 6  |-  { <. <.
( F `  p
) ,  ( F `
 q ) >. ,  ( p  .,  q ) >. }  e.  _V
3736rgenw 2888 . . . . 5  |-  A. q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( p  .,  q )
>. }  e.  _V
38 iunexg 6650 . . . . 5  |-  ( ( V  e.  _V  /\  A. q  e.  V  { <. <. ( F `  p ) ,  ( F `  q )
>. ,  ( p  .,  q ) >. }  e.  _V )  ->  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( p  .,  q )
>. }  e.  _V )
3935, 37, 38sylancl 662 . . . 4  |-  ( ph  ->  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q )
>. ,  ( p  .,  q ) >. }  e.  _V )
4039ralrimivw 2821 . . 3  |-  ( ph  ->  A. p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q )
>. ,  ( p  .,  q ) >. }  e.  _V )
41 iunexg 6650 . . 3  |-  ( ( V  e.  _V  /\  A. p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q )
>. ,  ( p  .,  q ) >. }  e.  _V )  ->  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( p  .,  q )
>. }  e.  _V )
4235, 40, 41syl2anc 661 . 2  |-  ( ph  ->  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q )
>. ,  ( p  .,  q ) >. }  e.  _V )
43 imasip.w . 2  |-  I  =  ( .i `  U
)
4425, 27, 28, 33, 42, 43strfv3 14308 1  |-  ( ph  ->  I  =  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( p  .,  q )
>. } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758   A.wral 2793   _Vcvv 3065    u. cun 3421   {csn 3972   {ctp 3976   <.cop 3978   U_ciun 4266   `'ccnv 4934    o. ccom 4939   -onto->wfo 5511   ` cfv 5513  (class class class)co 6187   1c1 9381   2c2 10469  ;cdc 10853   ndxcnx 14270   Basecbs 14273   +g cplusg 14337   .rcmulr 14338  Scalarcsca 14340   .scvsca 14341   .icip 14342  TopSetcts 14343   lecple 14344   distcds 14346   TopOpenctopn 14459   qTop cqtop 14540    "s cimas 14541
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 1952  ax-ext 2430  ax-rep 4498  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626  ax-un 6469  ax-cnex 9436  ax-resscn 9437  ax-1cn 9438  ax-icn 9439  ax-addcl 9440  ax-addrcl 9441  ax-mulcl 9442  ax-mulrcl 9443  ax-mulcom 9444  ax-addass 9445  ax-mulass 9446  ax-distr 9447  ax-i2m1 9448  ax-1ne0 9449  ax-1rid 9450  ax-rnegex 9451  ax-rrecex 9452  ax-cnre 9453  ax-pre-lttri 9454  ax-pre-lttrn 9455  ax-pre-ltadd 9456  ax-pre-mulgt0 9457
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 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-nel 2645  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-pss 3439  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4187  df-int 4224  df-iun 4268  df-br 4388  df-opab 4446  df-mpt 4447  df-tr 4481  df-eprel 4727  df-id 4731  df-po 4736  df-so 4737  df-fr 4774  df-we 4776  df-ord 4817  df-on 4818  df-lim 4819  df-suc 4820  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-riota 6148  df-ov 6190  df-oprab 6191  df-mpt2 6192  df-om 6574  df-1st 6674  df-2nd 6675  df-recs 6929  df-rdg 6963  df-1o 7017  df-oadd 7021  df-er 7198  df-en 7408  df-dom 7409  df-sdom 7410  df-fin 7411  df-sup 7789  df-pnf 9518  df-mnf 9519  df-xr 9520  df-ltxr 9521  df-le 9522  df-sub 9695  df-neg 9696  df-nn 10421  df-2 10478  df-3 10479  df-4 10480  df-5 10481  df-6 10482  df-7 10483  df-8 10484  df-9 10485  df-10 10486  df-n0 10678  df-z 10745  df-dec 10854  df-uz 10960  df-fz 11536  df-struct 14275  df-ndx 14276  df-slot 14277  df-base 14278  df-plusg 14350  df-mulr 14351  df-sca 14353  df-vsca 14354  df-ip 14355  df-tset 14356  df-ple 14357  df-ds 14359  df-imas 14545
This theorem is referenced by: (None)
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