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Theorem rrxip 20906
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 20905 . . 3  |-  ( I  e.  V  ->  H  =  (toCHil `  ( (RRfld X_s ( I  X.  { ( (subringAlg  ` RRfld ) `  RR ) } ) )s  B ) ) )
43fveq2d 5707 . 2  |-  ( I  e.  V  ->  ( .i `  H )  =  ( .i `  (toCHil `  ( (RRfld X_s ( I  X.  {
( (subringAlg  ` RRfld ) `  RR ) } ) )s  B ) ) ) )
5 eqid 2443 . . . 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 2443 . . . 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 20752 . . 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 5713 . . . . . 6  |-  ( Base `  H )  e.  _V
92, 8eqeltri 2513 . . . . 5  |-  B  e. 
_V
10 eqid 2443 . . . . . 6  |-  ( (RRfld X_s ( I  X.  { ( (subringAlg  ` RRfld ) `  RR ) } ) )s  B )  =  ( (RRfld X_s (
I  X.  { ( (subringAlg  ` RRfld ) `  RR ) } ) )s  B )
11 eqid 2443 . . . . . 6  |-  ( .i
`  (RRfld X_s ( I  X.  {
( (subringAlg  ` RRfld ) `  RR ) } ) ) )  =  ( .i
`  (RRfld X_s ( I  X.  {
( (subringAlg  ` RRfld ) `  RR ) } ) ) )
1210, 11ressip 14330 . . . . 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 2443 . . . . . 6  |-  (RRfld X_s (
I  X.  { ( (subringAlg  ` RRfld ) `  RR ) } ) )  =  (RRfld X_s ( I  X.  {
( (subringAlg  ` RRfld ) `  RR ) } ) )
15 refld 18061 . . . . . . 7  |- RRfld  e. Field
1615a1i 11 . . . . . 6  |-  ( I  e.  V  -> RRfld  e. Field )
17 snex 4545 . . . . . . 7  |-  { ( (subringAlg  ` RRfld ) `  RR ) }  e.  _V
18 xpexg 6519 . . . . . . 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 2443 . . . . . 6  |-  ( Base `  (RRfld X_s ( I  X.  {
( (subringAlg  ` RRfld ) `  RR ) } ) ) )  =  ( Base `  (RRfld X_s ( I  X.  {
( (subringAlg  ` RRfld ) `  RR ) } ) ) )
21 fvex 5713 . . . . . . . . 9  |-  ( (subringAlg  ` RRfld
) `  RR )  e.  _V
22 snnzb 3952 . . . . . . . . 9  |-  ( ( (subringAlg  ` RRfld ) `  RR )  e.  _V  <->  { (
(subringAlg  ` RRfld ) `  RR ) }  =/=  (/) )
2321, 22mpbi 208 . . . . . . . 8  |-  { ( (subringAlg  ` RRfld ) `  RR ) }  =/=  (/)
24 dmxp 5070 . . . . . . . 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 14411 . . . . 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 14407 . . . . . . 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 2444 . . . . . . . . . . 11  |-  ( x  e.  I  ->  (
(subringAlg  ` RRfld ) `  RR )  =  ( (subringAlg  ` RRfld
) `  RR )
)
30 ssid 3387 . . . . . . . . . . . . . 14  |-  RR  C_  RR
31 rebase 18048 . . . . . . . . . . . . . 14  |-  RR  =  ( Base ` RRfld )
3230, 31sseqtri 3400 . . . . . . . . . . . . 13  |-  RR  C_  ( Base ` RRfld )
3332rgenw 2795 . . . . . . . . . . . 12  |-  A. x  e.  I  RR  C_  ( Base ` RRfld )
3433rspec 2792 . . . . . . . . . . 11  |-  ( x  e.  I  ->  RR  C_  ( Base ` RRfld ) )
3529, 34srabase 17271 . . . . . . . . . 10  |-  ( x  e.  I  ->  ( Base ` RRfld )  =  (
Base `  ( (subringAlg  ` RRfld
) `  RR )
) )
3631a1i 11 . . . . . . . . . 10  |-  ( x  e.  I  ->  RR  =  ( Base ` RRfld ) )
37 fvconst2g 5943 . . . . . . . . . . . 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 5707 . . . . . . . . . 10  |-  ( x  e.  I  ->  ( Base `  ( ( I  X.  { ( (subringAlg  ` RRfld
) `  RR ) } ) `  x
) )  =  (
Base `  ( (subringAlg  ` RRfld
) `  RR )
) )
4035, 36, 393eqtr4rd 2486 . . . . . . . . 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 7290 . . . . . . 7  |-  ( I  e.  V  ->  X_ x  e.  I  ( Base `  ( ( I  X.  { ( (subringAlg  ` RRfld ) `  RR ) } ) `  x ) )  = 
X_ x  e.  I  RR )
43 reex 9385 . . . . . . . 8  |-  RR  e.  _V
44 ixpconstg 7284 . . . . . . . 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 2479 . . . . . 6  |-  ( I  e.  V  ->  ( Base `  (RRfld X_s ( I  X.  {
( (subringAlg  ` RRfld ) `  RR ) } ) ) )  =  ( RR 
^m  I ) )
47 remulr 18053 . . . . . . . . . . 11  |-  x.  =  ( .r ` RRfld )
4838, 34sraip 17276 . . . . . . . . . . 11  |-  ( x  e.  I  ->  ( .r ` RRfld )  =  ( .i `  ( ( I  X.  { ( (subringAlg  ` RRfld ) `  RR ) } ) `  x
) ) )
4947, 48syl5req 2488 . . . . . . . . . 10  |-  ( x  e.  I  ->  ( .i `  ( ( I  X.  { ( (subringAlg  ` RRfld
) `  RR ) } ) `  x
) )  =  x.  )
5049oveqd 6120 . . . . . . . . 9  |-  ( x  e.  I  ->  (
( f `  x
) ( .i `  ( ( I  X.  { ( (subringAlg  ` RRfld ) `  RR ) } ) `  x ) ) ( g `  x ) )  =  ( ( f `  x )  x.  ( g `  x ) ) )
5150mpteq2ia 4386 . . . . . . . 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 6119 . . . . . 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 6160 . . . . 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 2475 . . . 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 2489 . . 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 2489 . 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 2476 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 1369    e. wcel 1756    =/= wne 2618   _Vcvv 2984    C_ wss 3340   (/)c0 3649   {csn 3889    e. cmpt 4362    X. cxp 4850   dom cdm 4852   ` cfv 5430  (class class class)co 6103    e. cmpt2 6105    ^m cmap 7226   X_cixp 7275   RRcr 9293    x. cmul 9299   Basecbs 14186   ↾s cress 14187   .rcmulr 14251   .icip 14255    gsumg cgsu 14391   X_scprds 14396  Fieldcfield 16845  subringAlg csra 17261  RRfldcrefld 18046  toCHilctch 20698  ℝ^crrx 20899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371  ax-pre-sup 9372  ax-addf 9373  ax-mulf 9374
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-int 4141  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-om 6489  df-1st 6589  df-2nd 6590  df-tpos 6757  df-recs 6844  df-rdg 6878  df-1o 6932  df-oadd 6936  df-er 7113  df-map 7228  df-ixp 7276  df-en 7323  df-dom 7324  df-sdom 7325  df-fin 7326  df-sup 7703  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-div 10006  df-nn 10335  df-2 10392  df-3 10393  df-4 10394  df-5 10395  df-6 10396  df-7 10397  df-8 10398  df-9 10399  df-10 10400  df-n0 10592  df-z 10659  df-dec 10768  df-uz 10874  df-rp 11004  df-fz 11450  df-seq 11819  df-exp 11878  df-cj 12600  df-re 12601  df-im 12602  df-sqr 12736  df-abs 12737  df-struct 14188  df-ndx 14189  df-slot 14190  df-base 14191  df-sets 14192  df-ress 14193  df-plusg 14263  df-mulr 14264  df-starv 14265  df-sca 14266  df-vsca 14267  df-ip 14268  df-tset 14269  df-ple 14270  df-ds 14272  df-unif 14273  df-hom 14274  df-cco 14275  df-0g 14392  df-prds 14398  df-pws 14400  df-mnd 15427  df-grp 15557  df-minusg 15558  df-subg 15690  df-cmn 16291  df-mgp 16604  df-ur 16616  df-rng 16659  df-cring 16660  df-oppr 16727  df-dvdsr 16745  df-unit 16746  df-invr 16776  df-dvr 16787  df-drng 16846  df-field 16847  df-subrg 16875  df-sra 17265  df-rgmod 17266  df-cnfld 17831  df-refld 18047  df-dsmm 18169  df-frlm 18184  df-tng 20189  df-tch 20700  df-rrx 20901
This theorem is referenced by:  rrxnm  20907
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