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Theorem rrxip 20794
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 20793 . . 3  |-  ( I  e.  V  ->  H  =  (toCHil `  ( (RRfld X_s ( I  X.  { ( (subringAlg  ` RRfld ) `  RR ) } ) )s  B ) ) )
43fveq2d 5692 . 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 20640 . . 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 5698 . . . . . 6  |-  ( Base `  H )  e.  _V
92, 8eqeltri 2511 . . . . 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 14314 . . . . 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 17949 . . . . . . 7  |- RRfld  e. Field
1615a1i 11 . . . . . 6  |-  ( I  e.  V  -> RRfld  e. Field )
17 snex 4530 . . . . . . 7  |-  { ( (subringAlg  ` RRfld ) `  RR ) }  e.  _V
18 xpexg 6506 . . . . . . 7  |-  ( ( I  e.  V  /\  { ( (subringAlg  ` RRfld ) `  RR ) }  e.  _V )  ->  ( I  X.  { ( (subringAlg  ` RRfld ) `  RR ) } )  e. 
_V )
1917, 18mpan2 666 . . . . . 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 5698 . . . . . . . . 9  |-  ( (subringAlg  ` RRfld
) `  RR )  e.  _V
22 snnzb 3937 . . . . . . . . 9  |-  ( ( (subringAlg  ` RRfld ) `  RR )  e.  _V  <->  { (
(subringAlg  ` RRfld ) `  RR ) }  =/=  (/) )
2321, 22mpbi 208 . . . . . . . 8  |-  { ( (subringAlg  ` RRfld ) `  RR ) }  =/=  (/)
24 dmxp 5054 . . . . . . . 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 14395 . . . . 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 14391 . . . . . . 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 2442 . . . . . . . . . . 11  |-  ( x  e.  I  ->  (
(subringAlg  ` RRfld ) `  RR )  =  ( (subringAlg  ` RRfld
) `  RR )
)
30 ssid 3372 . . . . . . . . . . . . . 14  |-  RR  C_  RR
31 rebase 17936 . . . . . . . . . . . . . 14  |-  RR  =  ( Base ` RRfld )
3230, 31sseqtri 3385 . . . . . . . . . . . . 13  |-  RR  C_  ( Base ` RRfld )
3332rgenw 2781 . . . . . . . . . . . 12  |-  A. x  e.  I  RR  C_  ( Base ` RRfld )
3433rspec 2778 . . . . . . . . . . 11  |-  ( x  e.  I  ->  RR  C_  ( Base ` RRfld ) )
3529, 34srabase 17237 . . . . . . . . . 10  |-  ( x  e.  I  ->  ( Base ` RRfld )  =  (
Base `  ( (subringAlg  ` RRfld
) `  RR )
) )
3631a1i 11 . . . . . . . . . 10  |-  ( x  e.  I  ->  RR  =  ( Base ` RRfld ) )
37 fvconst2g 5928 . . . . . . . . . . . 12  |-  ( ( ( (subringAlg  ` RRfld ) `  RR )  e.  _V  /\  x  e.  I )  ->  ( ( I  X.  { ( (subringAlg  ` RRfld
) `  RR ) } ) `  x
)  =  ( (subringAlg  ` RRfld
) `  RR )
)
3821, 37mpan 665 . . . . . . . . . . 11  |-  ( x  e.  I  ->  (
( I  X.  {
( (subringAlg  ` RRfld ) `  RR ) } ) `  x )  =  ( (subringAlg  ` RRfld ) `  RR ) )
3938fveq2d 5692 . . . . . . . . . 10  |-  ( x  e.  I  ->  ( Base `  ( ( I  X.  { ( (subringAlg  ` RRfld
) `  RR ) } ) `  x
) )  =  (
Base `  ( (subringAlg  ` RRfld
) `  RR )
) )
4035, 36, 393eqtr4rd 2484 . . . . . . . . 9  |-  ( x  e.  I  ->  ( Base `  ( ( I  X.  { ( (subringAlg  ` RRfld
) `  RR ) } ) `  x
) )  =  RR )
4140adantl 463 . . . . . . . 8  |-  ( ( I  e.  V  /\  x  e.  I )  ->  ( Base `  (
( I  X.  {
( (subringAlg  ` RRfld ) `  RR ) } ) `  x ) )  =  RR )
4241ixpeq2dva 7274 . . . . . . 7  |-  ( I  e.  V  ->  X_ x  e.  I  ( Base `  ( ( I  X.  { ( (subringAlg  ` RRfld ) `  RR ) } ) `  x ) )  = 
X_ x  e.  I  RR )
43 reex 9369 . . . . . . . 8  |-  RR  e.  _V
44 ixpconstg 7268 . . . . . . . 8  |-  ( ( I  e.  V  /\  RR  e.  _V )  ->  X_ x  e.  I  RR  =  ( RR  ^m  I ) )
4543, 44mpan2 666 . . . . . . 7  |-  ( I  e.  V  ->  X_ x  e.  I  RR  =  ( RR  ^m  I ) )
4628, 42, 453eqtrd 2477 . . . . . 6  |-  ( I  e.  V  ->  ( Base `  (RRfld X_s ( I  X.  {
( (subringAlg  ` RRfld ) `  RR ) } ) ) )  =  ( RR 
^m  I ) )
47 remulr 17941 . . . . . . . . . . 11  |-  x.  =  ( .r ` RRfld )
4838, 34sraip 17242 . . . . . . . . . . 11  |-  ( x  e.  I  ->  ( .r ` RRfld )  =  ( .i `  ( ( I  X.  { ( (subringAlg  ` RRfld ) `  RR ) } ) `  x
) ) )
4947, 48syl5req 2486 . . . . . . . . . 10  |-  ( x  e.  I  ->  ( .i `  ( ( I  X.  { ( (subringAlg  ` RRfld
) `  RR ) } ) `  x
) )  =  x.  )
5049oveqd 6107 . . . . . . . . 9  |-  ( x  e.  I  ->  (
( f `  x
) ( .i `  ( ( I  X.  { ( (subringAlg  ` RRfld ) `  RR ) } ) `  x ) ) ( g `  x ) )  =  ( ( f `  x )  x.  ( g `  x ) ) )
5150mpteq2ia 4371 . . . . . . . 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 6106 . . . . . 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 6147 . . . . 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 2473 . . . 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 2487 . . 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 2487 . 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 2474 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 1364    e. wcel 1761    =/= wne 2604   _Vcvv 2970    C_ wss 3325   (/)c0 3634   {csn 3874    e. cmpt 4347    X. cxp 4834   dom cdm 4836   ` cfv 5415  (class class class)co 6090    e. cmpt2 6092    ^m cmap 7210   X_cixp 7259   RRcr 9277    x. cmul 9283   Basecbs 14170   ↾s cress 14171   .rcmulr 14235   .icip 14239    gsumg cgsu 14375   X_scprds 14380  Fieldcfield 16813  subringAlg csra 17227  RRfldcrefld 17934  toCHilctch 20586  ℝ^crrx 20787
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356  ax-addf 9357  ax-mulf 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-tpos 6744  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-map 7212  df-ixp 7260  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-sup 7687  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-rp 10988  df-fz 11434  df-seq 11803  df-exp 11862  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-mulr 14248  df-starv 14249  df-sca 14250  df-vsca 14251  df-ip 14252  df-tset 14253  df-ple 14254  df-ds 14256  df-unif 14257  df-hom 14258  df-cco 14259  df-0g 14376  df-prds 14382  df-pws 14384  df-mnd 15411  df-grp 15538  df-minusg 15539  df-subg 15671  df-cmn 16272  df-mgp 16582  df-ur 16594  df-rng 16637  df-cring 16638  df-oppr 16705  df-dvdsr 16723  df-unit 16724  df-invr 16754  df-dvr 16765  df-drng 16814  df-field 16815  df-subrg 16843  df-sra 17231  df-rgmod 17232  df-cnfld 17719  df-refld 17935  df-dsmm 18057  df-frlm 18072  df-tng 20077  df-tch 20588  df-rrx 20789
This theorem is referenced by:  rrxnm  20795
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