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Theorem rrxnm 21908
Description: The norm 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
rrxnm  |-  ( I  e.  V  ->  (
f  e.  B  |->  ( sqr `  (RRfld  gsumg  ( x  e.  I  |->  ( ( f `  x ) ^ 2 ) ) ) ) )  =  ( norm `  H
) )
Distinct variable groups:    x, f, B    f, I, x    f, V, x
Allowed substitution hints:    H( x, f)

Proof of Theorem rrxnm
Dummy variables  g  h are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 recrng 18748 . . . . 5  |- RRfld  e.  *Ring
2 srngring 17614 . . . . 5  |-  (RRfld  e.  *Ring  -> RRfld 
e.  Ring )
31, 2ax-mp 5 . . . 4  |- RRfld  e.  Ring
4 eqid 2382 . . . . 5  |-  (RRfld freeLMod  I )  =  (RRfld freeLMod  I )
54frlmlmod 18871 . . . 4  |-  ( (RRfld 
e.  Ring  /\  I  e.  V )  ->  (RRfld freeLMod  I )  e.  LMod )
63, 5mpan 668 . . 3  |-  ( I  e.  V  ->  (RRfld freeLMod  I )  e.  LMod )
7 lmodgrp 17632 . . 3  |-  ( (RRfld freeLMod  I )  e.  LMod  ->  (RRfld freeLMod  I )  e.  Grp )
8 eqid 2382 . . . 4  |-  (toCHil `  (RRfld freeLMod  I ) )  =  (toCHil `  (RRfld freeLMod  I ) )
9 eqid 2382 . . . 4  |-  ( norm `  (toCHil `  (RRfld freeLMod  I ) ) )  =  (
norm `  (toCHil `  (RRfld freeLMod  I ) ) )
10 eqid 2382 . . . 4  |-  ( Base `  (RRfld freeLMod  I ) )  =  ( Base `  (RRfld freeLMod  I ) )
11 eqid 2382 . . . 4  |-  ( .i
`  (RRfld freeLMod  I ) )  =  ( .i `  (RRfld freeLMod  I ) )
128, 9, 10, 11tchnmfval 21756 . . 3  |-  ( (RRfld freeLMod  I )  e.  Grp  ->  (
norm `  (toCHil `  (RRfld freeLMod  I ) ) )  =  ( f  e.  (
Base `  (RRfld freeLMod  I ) )  |->  ( sqr `  (
f ( .i `  (RRfld freeLMod  I ) ) f ) ) ) )
136, 7, 123syl 20 . 2  |-  ( I  e.  V  ->  ( norm `  (toCHil `  (RRfld freeLMod  I ) ) )  =  ( f  e.  (
Base `  (RRfld freeLMod  I ) )  |->  ( sqr `  (
f ( .i `  (RRfld freeLMod  I ) ) f ) ) ) )
14 rrxval.r . . . 4  |-  H  =  (ℝ^ `  I )
1514rrxval 21904 . . 3  |-  ( I  e.  V  ->  H  =  (toCHil `  (RRfld freeLMod  I ) ) )
1615fveq2d 5778 . 2  |-  ( I  e.  V  ->  ( norm `  H )  =  ( norm `  (toCHil `  (RRfld freeLMod  I ) ) ) )
1715fveq2d 5778 . . . 4  |-  ( I  e.  V  ->  ( Base `  H )  =  ( Base `  (toCHil `  (RRfld freeLMod  I ) ) ) )
18 rrxbase.b . . . 4  |-  B  =  ( Base `  H
)
198, 10tchbas 21747 . . . 4  |-  ( Base `  (RRfld freeLMod  I ) )  =  ( Base `  (toCHil `  (RRfld freeLMod  I ) ) )
2017, 18, 193eqtr4g 2448 . . 3  |-  ( I  e.  V  ->  B  =  ( Base `  (RRfld freeLMod  I ) ) )
2114, 18rrxbase 21905 . . . . . . . 8  |-  ( I  e.  V  ->  B  =  { f  e.  ( RR  ^m  I )  |  f finSupp  0 } )
22 ssrab2 3499 . . . . . . . 8  |-  { f  e.  ( RR  ^m  I )  |  f finSupp 
0 }  C_  ( RR  ^m  I )
2321, 22syl6eqss 3467 . . . . . . 7  |-  ( I  e.  V  ->  B  C_  ( RR  ^m  I
) )
2423sselda 3417 . . . . . 6  |-  ( ( I  e.  V  /\  f  e.  B )  ->  f  e.  ( RR 
^m  I ) )
2515fveq2d 5778 . . . . . . . . 9  |-  ( I  e.  V  ->  ( .i `  H )  =  ( .i `  (toCHil `  (RRfld freeLMod  I ) ) ) )
2614, 18rrxip 21907 . . . . . . . . 9  |-  ( I  e.  V  ->  (
h  e.  ( RR 
^m  I ) ,  g  e.  ( RR 
^m  I )  |->  (RRfld  gsumg  ( x  e.  I  |->  ( ( h `  x
)  x.  ( g `
 x ) ) ) ) )  =  ( .i `  H
) )
278, 11tchip 21753 . . . . . . . . . 10  |-  ( .i
`  (RRfld freeLMod  I ) )  =  ( .i `  (toCHil `  (RRfld freeLMod  I ) ) )
2827a1i 11 . . . . . . . . 9  |-  ( I  e.  V  ->  ( .i `  (RRfld freeLMod  I ) )  =  ( .i `  (toCHil `  (RRfld freeLMod  I ) ) ) )
2925, 26, 283eqtr4rd 2434 . . . . . . . 8  |-  ( I  e.  V  ->  ( .i `  (RRfld freeLMod  I ) )  =  ( h  e.  ( RR  ^m  I
) ,  g  e.  ( RR  ^m  I
)  |->  (RRfld  gsumg  ( x  e.  I  |->  ( ( h `  x )  x.  (
g `  x )
) ) ) ) )
3029adantr 463 . . . . . . 7  |-  ( ( I  e.  V  /\  f  e.  ( RR  ^m  I ) )  -> 
( .i `  (RRfld freeLMod  I ) )  =  ( h  e.  ( RR 
^m  I ) ,  g  e.  ( RR 
^m  I )  |->  (RRfld  gsumg  ( x  e.  I  |->  ( ( h `  x
)  x.  ( g `
 x ) ) ) ) ) )
31 simprl 754 . . . . . . . . . . . . 13  |-  ( ( ( I  e.  V  /\  f  e.  ( RR  ^m  I ) )  /\  ( h  =  f  /\  g  =  f ) )  ->  h  =  f )
3231fveq1d 5776 . . . . . . . . . . . 12  |-  ( ( ( I  e.  V  /\  f  e.  ( RR  ^m  I ) )  /\  ( h  =  f  /\  g  =  f ) )  -> 
( h `  x
)  =  ( f `
 x ) )
33 simprr 755 . . . . . . . . . . . . 13  |-  ( ( ( I  e.  V  /\  f  e.  ( RR  ^m  I ) )  /\  ( h  =  f  /\  g  =  f ) )  -> 
g  =  f )
3433fveq1d 5776 . . . . . . . . . . . 12  |-  ( ( ( I  e.  V  /\  f  e.  ( RR  ^m  I ) )  /\  ( h  =  f  /\  g  =  f ) )  -> 
( g `  x
)  =  ( f `
 x ) )
3532, 34oveq12d 6214 . . . . . . . . . . 11  |-  ( ( ( I  e.  V  /\  f  e.  ( RR  ^m  I ) )  /\  ( h  =  f  /\  g  =  f ) )  -> 
( ( h `  x )  x.  (
g `  x )
)  =  ( ( f `  x )  x.  ( f `  x ) ) )
3635adantr 463 . . . . . . . . . 10  |-  ( ( ( ( I  e.  V  /\  f  e.  ( RR  ^m  I
) )  /\  (
h  =  f  /\  g  =  f )
)  /\  x  e.  I )  ->  (
( h `  x
)  x.  ( g `
 x ) )  =  ( ( f `
 x )  x.  ( f `  x
) ) )
37 elmapi 7359 . . . . . . . . . . . . . . 15  |-  ( f  e.  ( RR  ^m  I )  ->  f : I --> RR )
3837adantl 464 . . . . . . . . . . . . . 14  |-  ( ( I  e.  V  /\  f  e.  ( RR  ^m  I ) )  -> 
f : I --> RR )
3938ffvelrnda 5933 . . . . . . . . . . . . 13  |-  ( ( ( I  e.  V  /\  f  e.  ( RR  ^m  I ) )  /\  x  e.  I
)  ->  ( f `  x )  e.  RR )
4039recnd 9533 . . . . . . . . . . . 12  |-  ( ( ( I  e.  V  /\  f  e.  ( RR  ^m  I ) )  /\  x  e.  I
)  ->  ( f `  x )  e.  CC )
4140adantlr 712 . . . . . . . . . . 11  |-  ( ( ( ( I  e.  V  /\  f  e.  ( RR  ^m  I
) )  /\  (
h  =  f  /\  g  =  f )
)  /\  x  e.  I )  ->  (
f `  x )  e.  CC )
4241sqvald 12209 . . . . . . . . . 10  |-  ( ( ( ( I  e.  V  /\  f  e.  ( RR  ^m  I
) )  /\  (
h  =  f  /\  g  =  f )
)  /\  x  e.  I )  ->  (
( f `  x
) ^ 2 )  =  ( ( f `
 x )  x.  ( f `  x
) ) )
4336, 42eqtr4d 2426 . . . . . . . . 9  |-  ( ( ( ( I  e.  V  /\  f  e.  ( RR  ^m  I
) )  /\  (
h  =  f  /\  g  =  f )
)  /\  x  e.  I )  ->  (
( h `  x
)  x.  ( g `
 x ) )  =  ( ( f `
 x ) ^
2 ) )
4443mpteq2dva 4453 . . . . . . . 8  |-  ( ( ( I  e.  V  /\  f  e.  ( RR  ^m  I ) )  /\  ( h  =  f  /\  g  =  f ) )  -> 
( x  e.  I  |->  ( ( h `  x )  x.  (
g `  x )
) )  =  ( x  e.  I  |->  ( ( f `  x
) ^ 2 ) ) )
4544oveq2d 6212 . . . . . . 7  |-  ( ( ( I  e.  V  /\  f  e.  ( RR  ^m  I ) )  /\  ( h  =  f  /\  g  =  f ) )  -> 
(RRfld  gsumg  ( x  e.  I  |->  ( ( h `  x )  x.  (
g `  x )
) ) )  =  (RRfld  gsumg  ( x  e.  I  |->  ( ( f `  x ) ^ 2 ) ) ) )
46 simpr 459 . . . . . . 7  |-  ( ( I  e.  V  /\  f  e.  ( RR  ^m  I ) )  -> 
f  e.  ( RR 
^m  I ) )
47 ovex 6224 . . . . . . . 8  |-  (RRfld  gsumg  ( x  e.  I  |->  ( ( f `  x ) ^ 2 ) ) )  e.  _V
4847a1i 11 . . . . . . 7  |-  ( ( I  e.  V  /\  f  e.  ( RR  ^m  I ) )  -> 
(RRfld  gsumg  ( x  e.  I  |->  ( ( f `  x ) ^ 2 ) ) )  e. 
_V )
4930, 45, 46, 46, 48ovmpt2d 6329 . . . . . 6  |-  ( ( I  e.  V  /\  f  e.  ( RR  ^m  I ) )  -> 
( f ( .i
`  (RRfld freeLMod  I ) ) f )  =  (RRfld  gsumg  ( x  e.  I  |->  ( ( f `  x
) ^ 2 ) ) ) )
5024, 49syldan 468 . . . . 5  |-  ( ( I  e.  V  /\  f  e.  B )  ->  ( f ( .i
`  (RRfld freeLMod  I ) ) f )  =  (RRfld  gsumg  ( x  e.  I  |->  ( ( f `  x
) ^ 2 ) ) ) )
5150eqcomd 2390 . . . 4  |-  ( ( I  e.  V  /\  f  e.  B )  ->  (RRfld  gsumg  ( x  e.  I  |->  ( ( f `  x ) ^ 2 ) ) )  =  ( f ( .i
`  (RRfld freeLMod  I ) ) f ) )
5251fveq2d 5778 . . 3  |-  ( ( I  e.  V  /\  f  e.  B )  ->  ( sqr `  (RRfld  gsumg  (
x  e.  I  |->  ( ( f `  x
) ^ 2 ) ) ) )  =  ( sqr `  (
f ( .i `  (RRfld freeLMod  I ) ) f ) ) )
5320, 52mpteq12dva 4444 . 2  |-  ( I  e.  V  ->  (
f  e.  B  |->  ( sqr `  (RRfld  gsumg  ( x  e.  I  |->  ( ( f `  x ) ^ 2 ) ) ) ) )  =  ( f  e.  (
Base `  (RRfld freeLMod  I ) )  |->  ( sqr `  (
f ( .i `  (RRfld freeLMod  I ) ) f ) ) ) )
5413, 16, 533eqtr4rd 2434 1  |-  ( I  e.  V  ->  (
f  e.  B  |->  ( sqr `  (RRfld  gsumg  ( x  e.  I  |->  ( ( f `  x ) ^ 2 ) ) ) ) )  =  ( norm `  H
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1399    e. wcel 1826   {crab 2736   _Vcvv 3034   class class class wbr 4367    |-> cmpt 4425   -->wf 5492   ` cfv 5496  (class class class)co 6196    |-> cmpt2 6198    ^m cmap 7338   finSupp cfsupp 7744   CCcc 9401   RRcr 9402   0cc0 9403    x. cmul 9408   2c2 10502   ^cexp 12069   sqrcsqrt 13068   Basecbs 14634   .icip 14707    gsumg cgsu 14848   Grpcgrp 16170   Ringcrg 17311   *Ringcsr 17606   LModclmod 17625  RRfldcrefld 18731   freeLMod cfrlm 18868   normcnm 21182  toCHilctch 21699  ℝ^crrx 21900
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480  ax-pre-sup 9481  ax-addf 9482  ax-mulf 9483
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-1st 6699  df-2nd 6700  df-supp 6818  df-tpos 6873  df-recs 6960  df-rdg 6994  df-1o 7048  df-oadd 7052  df-er 7229  df-map 7340  df-ixp 7389  df-en 7436  df-dom 7437  df-sdom 7438  df-fin 7439  df-fsupp 7745  df-sup 7816  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-div 10124  df-nn 10453  df-2 10511  df-3 10512  df-4 10513  df-5 10514  df-6 10515  df-7 10516  df-8 10517  df-9 10518  df-10 10519  df-n0 10713  df-z 10782  df-dec 10896  df-uz 11002  df-rp 11140  df-fz 11594  df-seq 12011  df-exp 12070  df-cj 12934  df-re 12935  df-im 12936  df-sqrt 13070  df-abs 13071  df-struct 14636  df-ndx 14637  df-slot 14638  df-base 14639  df-sets 14640  df-ress 14641  df-plusg 14715  df-mulr 14716  df-starv 14717  df-sca 14718  df-vsca 14719  df-ip 14720  df-tset 14721  df-ple 14722  df-ds 14724  df-unif 14725  df-hom 14726  df-cco 14727  df-0g 14849  df-prds 14855  df-pws 14857  df-mgm 15989  df-sgrp 16028  df-mnd 16038  df-mhm 16083  df-grp 16174  df-minusg 16175  df-sbg 16176  df-subg 16315  df-ghm 16382  df-cmn 16917  df-mgp 17255  df-ur 17267  df-ring 17313  df-cring 17314  df-oppr 17385  df-dvdsr 17403  df-unit 17404  df-invr 17434  df-dvr 17445  df-rnghom 17477  df-drng 17511  df-field 17512  df-subrg 17540  df-staf 17607  df-srng 17608  df-lmod 17627  df-lss 17692  df-sra 17931  df-rgmod 17932  df-cnfld 18534  df-refld 18732  df-dsmm 18854  df-frlm 18869  df-nm 21188  df-tng 21190  df-tch 21701  df-rrx 21902
This theorem is referenced by:  rrxds  21910
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