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Theorem rrxip 21573
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 21572 . . 3  |-  ( I  e.  V  ->  H  =  (toCHil `  ( (RRfld X_s ( I  X.  { ( (subringAlg  ` RRfld ) `  RR ) } ) )s  B ) ) )
43fveq2d 5869 . 2  |-  ( I  e.  V  ->  ( .i `  H )  =  ( .i `  (toCHil `  ( (RRfld X_s ( I  X.  {
( (subringAlg  ` RRfld ) `  RR ) } ) )s  B ) ) ) )
5 eqid 2467 . . . 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 2467 . . . 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 21419 . . 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 5875 . . . . . 6  |-  ( Base `  H )  e.  _V
92, 8eqeltri 2551 . . . . 5  |-  B  e. 
_V
10 eqid 2467 . . . . . 6  |-  ( (RRfld X_s ( I  X.  { ( (subringAlg  ` RRfld ) `  RR ) } ) )s  B )  =  ( (RRfld X_s (
I  X.  { ( (subringAlg  ` RRfld ) `  RR ) } ) )s  B )
11 eqid 2467 . . . . . 6  |-  ( .i
`  (RRfld X_s ( I  X.  {
( (subringAlg  ` RRfld ) `  RR ) } ) ) )  =  ( .i
`  (RRfld X_s ( I  X.  {
( (subringAlg  ` RRfld ) `  RR ) } ) ) )
1210, 11ressip 14634 . . . . 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 2467 . . . . . 6  |-  (RRfld X_s (
I  X.  { ( (subringAlg  ` RRfld ) `  RR ) } ) )  =  (RRfld X_s ( I  X.  {
( (subringAlg  ` RRfld ) `  RR ) } ) )
15 refld 18438 . . . . . . 7  |- RRfld  e. Field
1615a1i 11 . . . . . 6  |-  ( I  e.  V  -> RRfld  e. Field )
17 snex 4688 . . . . . . 7  |-  { ( (subringAlg  ` RRfld ) `  RR ) }  e.  _V
18 xpexg 6710 . . . . . . 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 2467 . . . . . 6  |-  ( Base `  (RRfld X_s ( I  X.  {
( (subringAlg  ` RRfld ) `  RR ) } ) ) )  =  ( Base `  (RRfld X_s ( I  X.  {
( (subringAlg  ` RRfld ) `  RR ) } ) ) )
21 fvex 5875 . . . . . . . . 9  |-  ( (subringAlg  ` RRfld
) `  RR )  e.  _V
22 snnzb 4092 . . . . . . . . 9  |-  ( ( (subringAlg  ` RRfld ) `  RR )  e.  _V  <->  { (
(subringAlg  ` RRfld ) `  RR ) }  =/=  (/) )
2321, 22mpbi 208 . . . . . . . 8  |-  { ( (subringAlg  ` RRfld ) `  RR ) }  =/=  (/)
24 dmxp 5220 . . . . . . . 8  |-  ( { ( (subringAlg  ` RRfld ) `  RR ) }  =/=  (/)  ->  dom  ( I  X.  { ( (subringAlg  ` RRfld ) `  RR ) } )  =  I )
2523, 24ax-mp 5 . . . . . . 7  |-  dom  (
I  X.  { ( (subringAlg  ` RRfld ) `  RR ) } )  =  I
2625a1i 11 . . . . . 6  |-  ( I  e.  V  ->  dom  ( I  X.  { ( (subringAlg  ` RRfld ) `  RR ) } )  =  I )
2714, 16, 19, 20, 26, 11prdsip 14715 . . . . 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 ) ) ) ) ) )
2814, 16, 19, 20, 26prdsbas 14711 . . . . . . 7  |-  ( I  e.  V  ->  ( Base `  (RRfld X_s ( I  X.  {
( (subringAlg  ` RRfld ) `  RR ) } ) ) )  =  X_ x  e.  I  ( Base `  ( ( I  X.  { ( (subringAlg  ` RRfld ) `  RR ) } ) `  x ) ) )
29 eqidd 2468 . . . . . . . . . . 11  |-  ( x  e.  I  ->  (
(subringAlg  ` RRfld ) `  RR )  =  ( (subringAlg  ` RRfld
) `  RR )
)
30 ssid 3523 . . . . . . . . . . . . . 14  |-  RR  C_  RR
31 rebase 18425 . . . . . . . . . . . . . 14  |-  RR  =  ( Base ` RRfld )
3230, 31sseqtri 3536 . . . . . . . . . . . . 13  |-  RR  C_  ( Base ` RRfld )
3332rgenw 2825 . . . . . . . . . . . 12  |-  A. x  e.  I  RR  C_  ( Base ` RRfld )
3433rspec 2832 . . . . . . . . . . 11  |-  ( x  e.  I  ->  RR  C_  ( Base ` RRfld ) )
3529, 34srabase 17619 . . . . . . . . . 10  |-  ( x  e.  I  ->  ( Base ` RRfld )  =  (
Base `  ( (subringAlg  ` RRfld
) `  RR )
) )
3631a1i 11 . . . . . . . . . 10  |-  ( x  e.  I  ->  RR  =  ( Base ` RRfld ) )
37 fvconst2g 6113 . . . . . . . . . . . 12  |-  ( ( ( (subringAlg  ` RRfld ) `  RR )  e.  _V  /\  x  e.  I )  ->  ( ( I  X.  { ( (subringAlg  ` RRfld
) `  RR ) } ) `  x
)  =  ( (subringAlg  ` RRfld
) `  RR )
)
3821, 37mpan 670 . . . . . . . . . . 11  |-  ( x  e.  I  ->  (
( I  X.  {
( (subringAlg  ` RRfld ) `  RR ) } ) `  x )  =  ( (subringAlg  ` RRfld ) `  RR ) )
3938fveq2d 5869 . . . . . . . . . 10  |-  ( x  e.  I  ->  ( Base `  ( ( I  X.  { ( (subringAlg  ` RRfld
) `  RR ) } ) `  x
) )  =  (
Base `  ( (subringAlg  ` RRfld
) `  RR )
) )
4035, 36, 393eqtr4rd 2519 . . . . . . . . 9  |-  ( x  e.  I  ->  ( Base `  ( ( I  X.  { ( (subringAlg  ` RRfld
) `  RR ) } ) `  x
) )  =  RR )
4140adantl 466 . . . . . . . 8  |-  ( ( I  e.  V  /\  x  e.  I )  ->  ( Base `  (
( I  X.  {
( (subringAlg  ` RRfld ) `  RR ) } ) `  x ) )  =  RR )
4241ixpeq2dva 7484 . . . . . . 7  |-  ( I  e.  V  ->  X_ x  e.  I  ( Base `  ( ( I  X.  { ( (subringAlg  ` RRfld ) `  RR ) } ) `  x ) )  = 
X_ x  e.  I  RR )
43 reex 9582 . . . . . . . 8  |-  RR  e.  _V
44 ixpconstg 7478 . . . . . . . 8  |-  ( ( I  e.  V  /\  RR  e.  _V )  ->  X_ x  e.  I  RR  =  ( RR  ^m  I ) )
4543, 44mpan2 671 . . . . . . 7  |-  ( I  e.  V  ->  X_ x  e.  I  RR  =  ( RR  ^m  I ) )
4628, 42, 453eqtrd 2512 . . . . . 6  |-  ( I  e.  V  ->  ( Base `  (RRfld X_s ( I  X.  {
( (subringAlg  ` RRfld ) `  RR ) } ) ) )  =  ( RR 
^m  I ) )
47 remulr 18430 . . . . . . . . . . 11  |-  x.  =  ( .r ` RRfld )
4838, 34sraip 17624 . . . . . . . . . . 11  |-  ( x  e.  I  ->  ( .r ` RRfld )  =  ( .i `  ( ( I  X.  { ( (subringAlg  ` RRfld ) `  RR ) } ) `  x
) ) )
4947, 48syl5req 2521 . . . . . . . . . 10  |-  ( x  e.  I  ->  ( .i `  ( ( I  X.  { ( (subringAlg  ` RRfld
) `  RR ) } ) `  x
) )  =  x.  )
5049oveqd 6300 . . . . . . . . 9  |-  ( x  e.  I  ->  (
( f `  x
) ( .i `  ( ( I  X.  { ( (subringAlg  ` RRfld ) `  RR ) } ) `  x ) ) ( g `  x ) )  =  ( ( f `  x )  x.  ( g `  x ) ) )
5150mpteq2ia 4529 . . . . . . . 8  |-  ( x  e.  I  |->  ( ( f `  x ) ( .i `  (
( I  X.  {
( (subringAlg  ` RRfld ) `  RR ) } ) `  x ) ) ( g `  x ) ) )  =  ( x  e.  I  |->  ( ( f `  x
)  x.  ( g `
 x ) ) )
5251a1i 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 ) ) ) )
5352oveq2d 6299 . . . . . 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 )
) ) ) )
5446, 46, 53mpt2eq123dv 6342 . . . . 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 )
) ) ) ) )
5527, 54eqtrd 2508 . . . 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 ) ) ) ) ) )
5613, 55syl5eqr 2522 . . 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 ) ) ) ) ) )
577, 56syl5eqr 2522 . 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 )
) ) ) ) )
584, 57eqtr2d 2509 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 1379    e. wcel 1767    =/= wne 2662   _Vcvv 3113    C_ wss 3476   (/)c0 3785   {csn 4027    |-> cmpt 4505    X. cxp 4997   dom cdm 4999   ` cfv 5587  (class class class)co 6283    |-> cmpt2 6285    ^m cmap 7420   X_cixp 7469   RRcr 9490    x. cmul 9496   Basecbs 14489   ↾s cress 14490   .rcmulr 14555   .icip 14559    gsumg cgsu 14695   X_scprds 14700  Fieldcfield 17192  subringAlg csra 17609  RRfldcrefld 18423  toCHilctch 21365  ℝ^crrx 21566
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-cnex 9547  ax-resscn 9548  ax-1cn 9549  ax-icn 9550  ax-addcl 9551  ax-addrcl 9552  ax-mulcl 9553  ax-mulrcl 9554  ax-mulcom 9555  ax-addass 9556  ax-mulass 9557  ax-distr 9558  ax-i2m1 9559  ax-1ne0 9560  ax-1rid 9561  ax-rnegex 9562  ax-rrecex 9563  ax-cnre 9564  ax-pre-lttri 9565  ax-pre-lttrn 9566  ax-pre-ltadd 9567  ax-pre-mulgt0 9568  ax-pre-sup 9569  ax-addf 9570  ax-mulf 9571
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-riota 6244  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-om 6680  df-1st 6784  df-2nd 6785  df-tpos 6955  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-map 7422  df-ixp 7470  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-sup 7900  df-pnf 9629  df-mnf 9630  df-xr 9631  df-ltxr 9632  df-le 9633  df-sub 9806  df-neg 9807  df-div 10206  df-nn 10536  df-2 10593  df-3 10594  df-4 10595  df-5 10596  df-6 10597  df-7 10598  df-8 10599  df-9 10600  df-10 10601  df-n0 10795  df-z 10864  df-dec 10976  df-uz 11082  df-rp 11220  df-fz 11672  df-seq 12075  df-exp 12134  df-cj 12894  df-re 12895  df-im 12896  df-sqrt 13030  df-abs 13031  df-struct 14491  df-ndx 14492  df-slot 14493  df-base 14494  df-sets 14495  df-ress 14496  df-plusg 14567  df-mulr 14568  df-starv 14569  df-sca 14570  df-vsca 14571  df-ip 14572  df-tset 14573  df-ple 14574  df-ds 14576  df-unif 14577  df-hom 14578  df-cco 14579  df-0g 14696  df-prds 14702  df-pws 14704  df-mnd 15731  df-grp 15864  df-minusg 15865  df-subg 16000  df-cmn 16603  df-mgp 16941  df-ur 16953  df-rng 16997  df-cring 16998  df-oppr 17068  df-dvdsr 17086  df-unit 17087  df-invr 17117  df-dvr 17128  df-drng 17193  df-field 17194  df-subrg 17222  df-sra 17613  df-rgmod 17614  df-cnfld 18208  df-refld 18424  df-dsmm 18546  df-frlm 18561  df-tng 20856  df-tch 21367  df-rrx 21568
This theorem is referenced by:  rrxnm  21574
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