MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rrxnm Structured version   Unicode version

Theorem rrxnm 20894
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 18050 . . . . 5  |- RRfld  e.  *Ring
2 srngrng 16936 . . . . 5  |-  (RRfld  e.  *Ring  -> RRfld 
e.  Ring )
31, 2ax-mp 5 . . . 4  |- RRfld  e.  Ring
4 eqid 2442 . . . . 5  |-  (RRfld freeLMod  I )  =  (RRfld freeLMod  I )
54frlmlmod 18173 . . . 4  |-  ( (RRfld 
e.  Ring  /\  I  e.  V )  ->  (RRfld freeLMod  I )  e.  LMod )
63, 5mpan 670 . . 3  |-  ( I  e.  V  ->  (RRfld freeLMod  I )  e.  LMod )
7 lmodgrp 16954 . . 3  |-  ( (RRfld freeLMod  I )  e.  LMod  ->  (RRfld freeLMod  I )  e.  Grp )
8 eqid 2442 . . . 4  |-  (toCHil `  (RRfld freeLMod  I ) )  =  (toCHil `  (RRfld freeLMod  I ) )
9 eqid 2442 . . . 4  |-  ( norm `  (toCHil `  (RRfld freeLMod  I ) ) )  =  (
norm `  (toCHil `  (RRfld freeLMod  I ) ) )
10 eqid 2442 . . . 4  |-  ( Base `  (RRfld freeLMod  I ) )  =  ( Base `  (RRfld freeLMod  I ) )
11 eqid 2442 . . . 4  |-  ( .i
`  (RRfld freeLMod  I ) )  =  ( .i `  (RRfld freeLMod  I ) )
128, 9, 10, 11tchnmfval 20742 . . 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 20890 . . 3  |-  ( I  e.  V  ->  H  =  (toCHil `  (RRfld freeLMod  I ) ) )
1615fveq2d 5694 . 2  |-  ( I  e.  V  ->  ( norm `  H )  =  ( norm `  (toCHil `  (RRfld freeLMod  I ) ) ) )
17 rrxbase.b . . . 4  |-  B  =  ( Base `  H
)
1815fveq2d 5694 . . . . 5  |-  ( I  e.  V  ->  ( Base `  H )  =  ( Base `  (toCHil `  (RRfld freeLMod  I ) ) ) )
198, 10tchbas 20733 . . . . 5  |-  ( Base `  (RRfld freeLMod  I ) )  =  ( Base `  (toCHil `  (RRfld freeLMod  I ) ) )
2018, 19syl6eqr 2492 . . . 4  |-  ( I  e.  V  ->  ( Base `  H )  =  ( Base `  (RRfld freeLMod  I ) ) )
2117, 20syl5eq 2486 . . 3  |-  ( I  e.  V  ->  B  =  ( Base `  (RRfld freeLMod  I ) ) )
2214, 17rrxbase 20891 . . . . . . . 8  |-  ( I  e.  V  ->  B  =  { f  e.  ( RR  ^m  I )  |  f finSupp  0 } )
23 ssrab2 3436 . . . . . . . 8  |-  { f  e.  ( RR  ^m  I )  |  f finSupp 
0 }  C_  ( RR  ^m  I )
2422, 23syl6eqss 3405 . . . . . . 7  |-  ( I  e.  V  ->  B  C_  ( RR  ^m  I
) )
2524sselda 3355 . . . . . 6  |-  ( ( I  e.  V  /\  f  e.  B )  ->  f  e.  ( RR 
^m  I ) )
2615fveq2d 5694 . . . . . . . . 9  |-  ( I  e.  V  ->  ( .i `  H )  =  ( .i `  (toCHil `  (RRfld freeLMod  I ) ) ) )
2714, 17rrxip 20893 . . . . . . . . 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
) )
288, 11tchip 20739 . . . . . . . . . 10  |-  ( .i
`  (RRfld freeLMod  I ) )  =  ( .i `  (toCHil `  (RRfld freeLMod  I ) ) )
2928a1i 11 . . . . . . . . 9  |-  ( I  e.  V  ->  ( .i `  (RRfld freeLMod  I ) )  =  ( .i `  (toCHil `  (RRfld freeLMod  I ) ) ) )
3026, 27, 293eqtr4rd 2485 . . . . . . . 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 )
) ) ) ) )
3130adantr 465 . . . . . . 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 ) ) ) ) ) )
32 simprl 755 . . . . . . . . . . . . 13  |-  ( ( ( I  e.  V  /\  f  e.  ( RR  ^m  I ) )  /\  ( h  =  f  /\  g  =  f ) )  ->  h  =  f )
3332fveq1d 5692 . . . . . . . . . . . 12  |-  ( ( ( I  e.  V  /\  f  e.  ( RR  ^m  I ) )  /\  ( h  =  f  /\  g  =  f ) )  -> 
( h `  x
)  =  ( f `
 x ) )
34 simprr 756 . . . . . . . . . . . . 13  |-  ( ( ( I  e.  V  /\  f  e.  ( RR  ^m  I ) )  /\  ( h  =  f  /\  g  =  f ) )  -> 
g  =  f )
3534fveq1d 5692 . . . . . . . . . . . 12  |-  ( ( ( I  e.  V  /\  f  e.  ( RR  ^m  I ) )  /\  ( h  =  f  /\  g  =  f ) )  -> 
( g `  x
)  =  ( f `
 x ) )
3633, 35oveq12d 6108 . . . . . . . . . . 11  |-  ( ( ( I  e.  V  /\  f  e.  ( RR  ^m  I ) )  /\  ( h  =  f  /\  g  =  f ) )  -> 
( ( h `  x )  x.  (
g `  x )
)  =  ( ( f `  x )  x.  ( f `  x ) ) )
3736adantr 465 . . . . . . . . . 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
) ) )
38 elmapi 7233 . . . . . . . . . . . . . . 15  |-  ( f  e.  ( RR  ^m  I )  ->  f : I --> RR )
3938adantl 466 . . . . . . . . . . . . . 14  |-  ( ( I  e.  V  /\  f  e.  ( RR  ^m  I ) )  -> 
f : I --> RR )
4039ffvelrnda 5842 . . . . . . . . . . . . 13  |-  ( ( ( I  e.  V  /\  f  e.  ( RR  ^m  I ) )  /\  x  e.  I
)  ->  ( f `  x )  e.  RR )
4140recnd 9411 . . . . . . . . . . . 12  |-  ( ( ( I  e.  V  /\  f  e.  ( RR  ^m  I ) )  /\  x  e.  I
)  ->  ( f `  x )  e.  CC )
4241adantlr 714 . . . . . . . . . . 11  |-  ( ( ( ( I  e.  V  /\  f  e.  ( RR  ^m  I
) )  /\  (
h  =  f  /\  g  =  f )
)  /\  x  e.  I )  ->  (
f `  x )  e.  CC )
4342sqvald 12004 . . . . . . . . . 10  |-  ( ( ( ( I  e.  V  /\  f  e.  ( RR  ^m  I
) )  /\  (
h  =  f  /\  g  =  f )
)  /\  x  e.  I )  ->  (
( f `  x
) ^ 2 )  =  ( ( f `
 x )  x.  ( f `  x
) ) )
4437, 43eqtr4d 2477 . . . . . . . . 9  |-  ( ( ( ( I  e.  V  /\  f  e.  ( RR  ^m  I
) )  /\  (
h  =  f  /\  g  =  f )
)  /\  x  e.  I )  ->  (
( h `  x
)  x.  ( g `
 x ) )  =  ( ( f `
 x ) ^
2 ) )
4544mpteq2dva 4377 . . . . . . . 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 ) ) )
4645oveq2d 6106 . . . . . . 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 ) ) ) )
47 simpr 461 . . . . . . 7  |-  ( ( I  e.  V  /\  f  e.  ( RR  ^m  I ) )  -> 
f  e.  ( RR 
^m  I ) )
48 ovex 6115 . . . . . . . 8  |-  (RRfld  gsumg  ( x  e.  I  |->  ( ( f `  x ) ^ 2 ) ) )  e.  _V
4948a1i 11 . . . . . . 7  |-  ( ( I  e.  V  /\  f  e.  ( RR  ^m  I ) )  -> 
(RRfld  gsumg  ( x  e.  I  |->  ( ( f `  x ) ^ 2 ) ) )  e. 
_V )
5031, 46, 47, 47, 49ovmpt2d 6217 . . . . . 6  |-  ( ( I  e.  V  /\  f  e.  ( RR  ^m  I ) )  -> 
( f ( .i
`  (RRfld freeLMod  I ) ) f )  =  (RRfld  gsumg  ( x  e.  I  |->  ( ( f `  x
) ^ 2 ) ) ) )
5125, 50syldan 470 . . . . 5  |-  ( ( I  e.  V  /\  f  e.  B )  ->  ( f ( .i
`  (RRfld freeLMod  I ) ) f )  =  (RRfld  gsumg  ( x  e.  I  |->  ( ( f `  x
) ^ 2 ) ) ) )
5251eqcomd 2447 . . . 4  |-  ( ( I  e.  V  /\  f  e.  B )  ->  (RRfld  gsumg  ( x  e.  I  |->  ( ( f `  x ) ^ 2 ) ) )  =  ( f ( .i
`  (RRfld freeLMod  I ) ) f ) )
5352fveq2d 5694 . . 3  |-  ( ( I  e.  V  /\  f  e.  B )  ->  ( sqr `  (RRfld  gsumg  (
x  e.  I  |->  ( ( f `  x
) ^ 2 ) ) ) )  =  ( sqr `  (
f ( .i `  (RRfld freeLMod  I ) ) f ) ) )
5421, 53mpteq12dva 4368 . 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 ) ) ) )
5513, 16, 543eqtr4rd 2485 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 369    = wceq 1369    e. wcel 1756   {crab 2718   _Vcvv 2971   class class class wbr 4291    e. cmpt 4349   -->wf 5413   ` cfv 5417  (class class class)co 6090    e. cmpt2 6092    ^m cmap 7213   finSupp cfsupp 7619   CCcc 9279   RRcr 9280   0cc0 9281    x. cmul 9286   2c2 10370   ^cexp 11864   sqrcsqr 12721   Basecbs 14173   .icip 14242    gsumg cgsu 14378   Grpcgrp 15409   Ringcrg 16644   *Ringcsr 16928   LModclmod 16947  RRfldcrefld 18033   freeLMod cfrlm 18170   normcnm 20168  toCHilctch 20685  ℝ^crrx 20886
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 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358  ax-pre-sup 9359  ax-addf 9360  ax-mulf 9361
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 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-int 4128  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-supp 6690  df-tpos 6744  df-recs 6831  df-rdg 6865  df-1o 6919  df-oadd 6923  df-er 7100  df-map 7215  df-ixp 7263  df-en 7310  df-dom 7311  df-sdom 7312  df-fin 7313  df-fsupp 7620  df-sup 7690  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-div 9993  df-nn 10322  df-2 10379  df-3 10380  df-4 10381  df-5 10382  df-6 10383  df-7 10384  df-8 10385  df-9 10386  df-10 10387  df-n0 10579  df-z 10646  df-dec 10755  df-uz 10861  df-rp 10991  df-fz 11437  df-seq 11806  df-exp 11865  df-cj 12587  df-re 12588  df-im 12589  df-sqr 12723  df-abs 12724  df-struct 14175  df-ndx 14176  df-slot 14177  df-base 14178  df-sets 14179  df-ress 14180  df-plusg 14250  df-mulr 14251  df-starv 14252  df-sca 14253  df-vsca 14254  df-ip 14255  df-tset 14256  df-ple 14257  df-ds 14259  df-unif 14260  df-hom 14261  df-cco 14262  df-0g 14379  df-prds 14385  df-pws 14387  df-mnd 15414  df-mhm 15463  df-grp 15544  df-minusg 15545  df-sbg 15546  df-subg 15677  df-ghm 15744  df-cmn 16278  df-mgp 16591  df-ur 16603  df-rng 16646  df-cring 16647  df-oppr 16714  df-dvdsr 16732  df-unit 16733  df-invr 16763  df-dvr 16774  df-rnghom 16805  df-drng 16833  df-field 16834  df-subrg 16862  df-staf 16929  df-srng 16930  df-lmod 16949  df-lss 17013  df-sra 17252  df-rgmod 17253  df-cnfld 17818  df-refld 18034  df-dsmm 18156  df-frlm 18171  df-nm 20174  df-tng 20176  df-tch 20687  df-rrx 20888
This theorem is referenced by:  rrxds  20896
  Copyright terms: Public domain W3C validator