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Theorem rrxip 21688
Description: The inner product of the generalized real Euclidean spaces. (Contributed by Thierry Arnoux, 16-Jun-2019.)
Hypotheses
Ref Expression
rrxval.r  |-  H  =  (ℝ^ `  I )
rrxbase.b  |-  B  =  ( Base `  H
)
Assertion
Ref Expression
rrxip  |-  ( I  e.  V  ->  (
f  e.  ( RR 
^m  I ) ,  g  e.  ( RR 
^m  I )  |->  (RRfld  gsumg  ( x  e.  I  |->  ( ( f `  x
)  x.  ( g `
 x ) ) ) ) )  =  ( .i `  H
) )
Distinct variable groups:    f, g, x, B    f, I, g, x    f, V, g, x
Allowed substitution hints:    H( x, f, g)

Proof of Theorem rrxip
StepHypRef Expression
1 rrxval.r . . . 4  |-  H  =  (ℝ^ `  I )
2 rrxbase.b . . . 4  |-  B  =  ( Base `  H
)
31, 2rrxprds 21687 . . 3  |-  ( I  e.  V  ->  H  =  (toCHil `  ( (RRfld X_s ( I  X.  { ( (subringAlg  ` RRfld ) `  RR ) } ) )s  B ) ) )
43fveq2d 5856 . 2  |-  ( I  e.  V  ->  ( .i `  H )  =  ( .i `  (toCHil `  ( (RRfld X_s ( I  X.  {
( (subringAlg  ` RRfld ) `  RR ) } ) )s  B ) ) ) )
5 eqid 2441 . . . 4  |-  (toCHil `  ( (RRfld X_s ( I  X.  {
( (subringAlg  ` RRfld ) `  RR ) } ) )s  B ) )  =  (toCHil `  ( (RRfld X_s ( I  X.  {
( (subringAlg  ` RRfld ) `  RR ) } ) )s  B ) )
6 eqid 2441 . . . 4  |-  ( .i
`  ( (RRfld X_s (
I  X.  { ( (subringAlg  ` RRfld ) `  RR ) } ) )s  B ) )  =  ( .i
`  ( (RRfld X_s (
I  X.  { ( (subringAlg  ` RRfld ) `  RR ) } ) )s  B ) )
75, 6tchip 21534 . . 3  |-  ( .i
`  ( (RRfld X_s (
I  X.  { ( (subringAlg  ` RRfld ) `  RR ) } ) )s  B ) )  =  ( .i
`  (toCHil `  ( (RRfld X_s ( I  X.  { ( (subringAlg  ` RRfld ) `  RR ) } ) )s  B ) ) )
8 fvex 5862 . . . . . 6  |-  ( Base `  H )  e.  _V
92, 8eqeltri 2525 . . . . 5  |-  B  e. 
_V
10 eqid 2441 . . . . . 6  |-  ( (RRfld X_s ( I  X.  { ( (subringAlg  ` RRfld ) `  RR ) } ) )s  B )  =  ( (RRfld X_s (
I  X.  { ( (subringAlg  ` RRfld ) `  RR ) } ) )s  B )
11 eqid 2441 . . . . . 6  |-  ( .i
`  (RRfld X_s ( I  X.  {
( (subringAlg  ` RRfld ) `  RR ) } ) ) )  =  ( .i
`  (RRfld X_s ( I  X.  {
( (subringAlg  ` RRfld ) `  RR ) } ) ) )
1210, 11ressip 14649 . . . . 5  |-  ( B  e.  _V  ->  ( .i `  (RRfld X_s ( I  X.  {
( (subringAlg  ` RRfld ) `  RR ) } ) ) )  =  ( .i
`  ( (RRfld X_s (
I  X.  { ( (subringAlg  ` RRfld ) `  RR ) } ) )s  B ) ) )
139, 12ax-mp 5 . . . 4  |-  ( .i
`  (RRfld X_s ( I  X.  {
( (subringAlg  ` RRfld ) `  RR ) } ) ) )  =  ( .i
`  ( (RRfld X_s (
I  X.  { ( (subringAlg  ` RRfld ) `  RR ) } ) )s  B ) )
14 eqid 2441 . . . . . 6  |-  (RRfld X_s (
I  X.  { ( (subringAlg  ` RRfld ) `  RR ) } ) )  =  (RRfld X_s ( I  X.  {
( (subringAlg  ` RRfld ) `  RR ) } ) )
15 refld 18522 . . . . . . 7  |- RRfld  e. Field
1615a1i 11 . . . . . 6  |-  ( I  e.  V  -> RRfld  e. Field )
17 snex 4674 . . . . . . 7  |-  { ( (subringAlg  ` RRfld ) `  RR ) }  e.  _V
18 xpexg 6583 . . . . . . 7  |-  ( ( I  e.  V  /\  { ( (subringAlg  ` RRfld ) `  RR ) }  e.  _V )  ->  ( I  X.  { ( (subringAlg  ` RRfld ) `  RR ) } )  e. 
_V )
1917, 18mpan2 671 . . . . . 6  |-  ( I  e.  V  ->  (
I  X.  { ( (subringAlg  ` RRfld ) `  RR ) } )  e.  _V )
20 eqid 2441 . . . . . 6  |-  ( Base `  (RRfld X_s ( I  X.  {
( (subringAlg  ` RRfld ) `  RR ) } ) ) )  =  ( Base `  (RRfld X_s ( I  X.  {
( (subringAlg  ` RRfld ) `  RR ) } ) ) )
21 fvex 5862 . . . . . . . . 9  |-  ( (subringAlg  ` RRfld
) `  RR )  e.  _V
2221snnz 4129 . . . . . . . 8  |-  { ( (subringAlg  ` RRfld ) `  RR ) }  =/=  (/)
23 dmxp 5207 . . . . . . . 8  |-  ( { ( (subringAlg  ` RRfld ) `  RR ) }  =/=  (/)  ->  dom  ( I  X.  { ( (subringAlg  ` RRfld ) `  RR ) } )  =  I )
2422, 23ax-mp 5 . . . . . . 7  |-  dom  (
I  X.  { ( (subringAlg  ` RRfld ) `  RR ) } )  =  I
2524a1i 11 . . . . . 6  |-  ( I  e.  V  ->  dom  ( I  X.  { ( (subringAlg  ` RRfld ) `  RR ) } )  =  I )
2614, 16, 19, 20, 25, 11prdsip 14730 . . . . 5  |-  ( I  e.  V  ->  ( .i `  (RRfld X_s ( I  X.  {
( (subringAlg  ` RRfld ) `  RR ) } ) ) )  =  ( f  e.  ( Base `  (RRfld X_s ( I  X.  { ( (subringAlg  ` RRfld ) `  RR ) } ) ) ) ,  g  e.  (
Base `  (RRfld X_s ( I  X.  {
( (subringAlg  ` RRfld ) `  RR ) } ) ) )  |->  (RRfld  gsumg  ( x  e.  I  |->  ( ( f `  x ) ( .i
`  ( ( I  X.  { ( (subringAlg  ` RRfld
) `  RR ) } ) `  x
) ) ( g `
 x ) ) ) ) ) )
2714, 16, 19, 20, 25prdsbas 14726 . . . . . . 7  |-  ( I  e.  V  ->  ( Base `  (RRfld X_s ( I  X.  {
( (subringAlg  ` RRfld ) `  RR ) } ) ) )  =  X_ x  e.  I  ( Base `  ( ( I  X.  { ( (subringAlg  ` RRfld ) `  RR ) } ) `  x ) ) )
28 eqidd 2442 . . . . . . . . . . 11  |-  ( x  e.  I  ->  (
(subringAlg  ` RRfld ) `  RR )  =  ( (subringAlg  ` RRfld
) `  RR )
)
29 rebase 18509 . . . . . . . . . . . . 13  |-  RR  =  ( Base ` RRfld )
3029eqimssi 3540 . . . . . . . . . . . 12  |-  RR  C_  ( Base ` RRfld )
3130a1i 11 . . . . . . . . . . 11  |-  ( x  e.  I  ->  RR  C_  ( Base ` RRfld ) )
3228, 31srabase 17692 . . . . . . . . . 10  |-  ( x  e.  I  ->  ( Base ` RRfld )  =  (
Base `  ( (subringAlg  ` RRfld
) `  RR )
) )
3329a1i 11 . . . . . . . . . 10  |-  ( x  e.  I  ->  RR  =  ( Base ` RRfld ) )
3421fvconst2 6107 . . . . . . . . . . 11  |-  ( x  e.  I  ->  (
( I  X.  {
( (subringAlg  ` RRfld ) `  RR ) } ) `  x )  =  ( (subringAlg  ` RRfld ) `  RR ) )
3534fveq2d 5856 . . . . . . . . . 10  |-  ( x  e.  I  ->  ( Base `  ( ( I  X.  { ( (subringAlg  ` RRfld
) `  RR ) } ) `  x
) )  =  (
Base `  ( (subringAlg  ` RRfld
) `  RR )
) )
3632, 33, 353eqtr4rd 2493 . . . . . . . . 9  |-  ( x  e.  I  ->  ( Base `  ( ( I  X.  { ( (subringAlg  ` RRfld
) `  RR ) } ) `  x
) )  =  RR )
3736adantl 466 . . . . . . . 8  |-  ( ( I  e.  V  /\  x  e.  I )  ->  ( Base `  (
( I  X.  {
( (subringAlg  ` RRfld ) `  RR ) } ) `  x ) )  =  RR )
3837ixpeq2dva 7482 . . . . . . 7  |-  ( I  e.  V  ->  X_ x  e.  I  ( Base `  ( ( I  X.  { ( (subringAlg  ` RRfld ) `  RR ) } ) `  x ) )  = 
X_ x  e.  I  RR )
39 reex 9581 . . . . . . . 8  |-  RR  e.  _V
40 ixpconstg 7476 . . . . . . . 8  |-  ( ( I  e.  V  /\  RR  e.  _V )  ->  X_ x  e.  I  RR  =  ( RR  ^m  I ) )
4139, 40mpan2 671 . . . . . . 7  |-  ( I  e.  V  ->  X_ x  e.  I  RR  =  ( RR  ^m  I ) )
4227, 38, 413eqtrd 2486 . . . . . 6  |-  ( I  e.  V  ->  ( Base `  (RRfld X_s ( I  X.  {
( (subringAlg  ` RRfld ) `  RR ) } ) ) )  =  ( RR 
^m  I ) )
43 remulr 18514 . . . . . . . . . . 11  |-  x.  =  ( .r ` RRfld )
4434, 31sraip 17697 . . . . . . . . . . 11  |-  ( x  e.  I  ->  ( .r ` RRfld )  =  ( .i `  ( ( I  X.  { ( (subringAlg  ` RRfld ) `  RR ) } ) `  x
) ) )
4543, 44syl5req 2495 . . . . . . . . . 10  |-  ( x  e.  I  ->  ( .i `  ( ( I  X.  { ( (subringAlg  ` RRfld
) `  RR ) } ) `  x
) )  =  x.  )
4645oveqd 6294 . . . . . . . . 9  |-  ( x  e.  I  ->  (
( f `  x
) ( .i `  ( ( I  X.  { ( (subringAlg  ` RRfld ) `  RR ) } ) `  x ) ) ( g `  x ) )  =  ( ( f `  x )  x.  ( g `  x ) ) )
4746mpteq2ia 4515 . . . . . . . 8  |-  ( x  e.  I  |->  ( ( f `  x ) ( .i `  (
( I  X.  {
( (subringAlg  ` RRfld ) `  RR ) } ) `  x ) ) ( g `  x ) ) )  =  ( x  e.  I  |->  ( ( f `  x
)  x.  ( g `
 x ) ) )
4847a1i 11 . . . . . . 7  |-  ( I  e.  V  ->  (
x  e.  I  |->  ( ( f `  x
) ( .i `  ( ( I  X.  { ( (subringAlg  ` RRfld ) `  RR ) } ) `  x ) ) ( g `  x ) ) )  =  ( x  e.  I  |->  ( ( f `  x
)  x.  ( g `
 x ) ) ) )
4948oveq2d 6293 . . . . . 6  |-  ( I  e.  V  ->  (RRfld  gsumg  (
x  e.  I  |->  ( ( f `  x
) ( .i `  ( ( I  X.  { ( (subringAlg  ` RRfld ) `  RR ) } ) `  x ) ) ( g `  x ) ) ) )  =  (RRfld  gsumg  ( x  e.  I  |->  ( ( f `  x )  x.  (
g `  x )
) ) ) )
5042, 42, 49mpt2eq123dv 6340 . . . . 5  |-  ( I  e.  V  ->  (
f  e.  ( Base `  (RRfld X_s ( I  X.  {
( (subringAlg  ` RRfld ) `  RR ) } ) ) ) ,  g  e.  ( Base `  (RRfld X_s ( I  X.  { ( (subringAlg  ` RRfld ) `  RR ) } ) ) ) 
|->  (RRfld  gsumg  ( x  e.  I  |->  ( ( f `  x ) ( .i
`  ( ( I  X.  { ( (subringAlg  ` RRfld
) `  RR ) } ) `  x
) ) ( g `
 x ) ) ) ) )  =  ( f  e.  ( RR  ^m  I ) ,  g  e.  ( RR  ^m  I ) 
|->  (RRfld  gsumg  ( x  e.  I  |->  ( ( f `  x )  x.  (
g `  x )
) ) ) ) )
5126, 50eqtrd 2482 . . . 4  |-  ( I  e.  V  ->  ( .i `  (RRfld X_s ( I  X.  {
( (subringAlg  ` RRfld ) `  RR ) } ) ) )  =  ( f  e.  ( RR  ^m  I ) ,  g  e.  ( RR  ^m  I )  |->  (RRfld  gsumg  ( x  e.  I  |->  ( ( f `  x )  x.  ( g `  x ) ) ) ) ) )
5213, 51syl5eqr 2496 . . 3  |-  ( I  e.  V  ->  ( .i `  ( (RRfld X_s (
I  X.  { ( (subringAlg  ` RRfld ) `  RR ) } ) )s  B ) )  =  ( f  e.  ( RR  ^m  I ) ,  g  e.  ( RR  ^m  I )  |->  (RRfld  gsumg  ( x  e.  I  |->  ( ( f `  x )  x.  ( g `  x ) ) ) ) ) )
537, 52syl5eqr 2496 . 2  |-  ( I  e.  V  ->  ( .i `  (toCHil `  (
(RRfld X_s ( I  X.  {
( (subringAlg  ` RRfld ) `  RR ) } ) )s  B ) ) )  =  ( f  e.  ( RR  ^m  I ) ,  g  e.  ( RR  ^m  I ) 
|->  (RRfld  gsumg  ( x  e.  I  |->  ( ( f `  x )  x.  (
g `  x )
) ) ) ) )
544, 53eqtr2d 2483 1  |-  ( I  e.  V  ->  (
f  e.  ( RR 
^m  I ) ,  g  e.  ( RR 
^m  I )  |->  (RRfld  gsumg  ( x  e.  I  |->  ( ( f `  x
)  x.  ( g `
 x ) ) ) ) )  =  ( .i `  H
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1381    e. wcel 1802    =/= wne 2636   _Vcvv 3093    C_ wss 3458   (/)c0 3767   {csn 4010    |-> cmpt 4491    X. cxp 4983   dom cdm 4985   ` cfv 5574  (class class class)co 6277    |-> cmpt2 6279    ^m cmap 7418   X_cixp 7467   RRcr 9489    x. cmul 9495   Basecbs 14504   ↾s cress 14505   .rcmulr 14570   .icip 14574    gsumg cgsu 14710   X_scprds 14715  Fieldcfield 17265  subringAlg csra 17682  RRfldcrefld 18507  toCHilctch 21480  ℝ^crrx 21681
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567  ax-pre-sup 9568  ax-addf 9569  ax-mulf 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-int 4268  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-1st 6781  df-2nd 6782  df-tpos 6953  df-recs 7040  df-rdg 7074  df-1o 7128  df-oadd 7132  df-er 7309  df-map 7420  df-ixp 7468  df-en 7515  df-dom 7516  df-sdom 7517  df-fin 7518  df-sup 7899  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-div 10208  df-nn 10538  df-2 10595  df-3 10596  df-4 10597  df-5 10598  df-6 10599  df-7 10600  df-8 10601  df-9 10602  df-10 10603  df-n0 10797  df-z 10866  df-dec 10980  df-uz 11086  df-rp 11225  df-fz 11677  df-seq 12082  df-exp 12141  df-cj 12906  df-re 12907  df-im 12908  df-sqrt 13042  df-abs 13043  df-struct 14506  df-ndx 14507  df-slot 14508  df-base 14509  df-sets 14510  df-ress 14511  df-plusg 14582  df-mulr 14583  df-starv 14584  df-sca 14585  df-vsca 14586  df-ip 14587  df-tset 14588  df-ple 14589  df-ds 14591  df-unif 14592  df-hom 14593  df-cco 14594  df-0g 14711  df-prds 14717  df-pws 14719  df-mgm 15741  df-sgrp 15780  df-mnd 15790  df-grp 15926  df-minusg 15927  df-subg 16067  df-cmn 16669  df-mgp 17010  df-ur 17022  df-ring 17068  df-cring 17069  df-oppr 17140  df-dvdsr 17158  df-unit 17159  df-invr 17189  df-dvr 17200  df-drng 17266  df-field 17267  df-subrg 17295  df-sra 17686  df-rgmod 17687  df-cnfld 18289  df-refld 18508  df-dsmm 18630  df-frlm 18645  df-tng 20971  df-tch 21482  df-rrx 21683
This theorem is referenced by:  rrxnm  21689
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