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Theorem rrxnm 21586
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 18452 . . . . 5  |- RRfld  e.  *Ring
2 srngrng 17301 . . . . 5  |-  (RRfld  e.  *Ring  -> RRfld 
e.  Ring )
31, 2ax-mp 5 . . . 4  |- RRfld  e.  Ring
4 eqid 2467 . . . . 5  |-  (RRfld freeLMod  I )  =  (RRfld freeLMod  I )
54frlmlmod 18575 . . . 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 17319 . . 3  |-  ( (RRfld freeLMod  I )  e.  LMod  ->  (RRfld freeLMod  I )  e.  Grp )
8 eqid 2467 . . . 4  |-  (toCHil `  (RRfld freeLMod  I ) )  =  (toCHil `  (RRfld freeLMod  I ) )
9 eqid 2467 . . . 4  |-  ( norm `  (toCHil `  (RRfld freeLMod  I ) ) )  =  (
norm `  (toCHil `  (RRfld freeLMod  I ) ) )
10 eqid 2467 . . . 4  |-  ( Base `  (RRfld freeLMod  I ) )  =  ( Base `  (RRfld freeLMod  I ) )
11 eqid 2467 . . . 4  |-  ( .i
`  (RRfld freeLMod  I ) )  =  ( .i `  (RRfld freeLMod  I ) )
128, 9, 10, 11tchnmfval 21434 . . 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 21582 . . 3  |-  ( I  e.  V  ->  H  =  (toCHil `  (RRfld freeLMod  I ) ) )
1615fveq2d 5870 . 2  |-  ( I  e.  V  ->  ( norm `  H )  =  ( norm `  (toCHil `  (RRfld freeLMod  I ) ) ) )
17 rrxbase.b . . . 4  |-  B  =  ( Base `  H
)
1815fveq2d 5870 . . . . 5  |-  ( I  e.  V  ->  ( Base `  H )  =  ( Base `  (toCHil `  (RRfld freeLMod  I ) ) ) )
198, 10tchbas 21425 . . . . 5  |-  ( Base `  (RRfld freeLMod  I ) )  =  ( Base `  (toCHil `  (RRfld freeLMod  I ) ) )
2018, 19syl6eqr 2526 . . . 4  |-  ( I  e.  V  ->  ( Base `  H )  =  ( Base `  (RRfld freeLMod  I ) ) )
2117, 20syl5eq 2520 . . 3  |-  ( I  e.  V  ->  B  =  ( Base `  (RRfld freeLMod  I ) ) )
2214, 17rrxbase 21583 . . . . . . . 8  |-  ( I  e.  V  ->  B  =  { f  e.  ( RR  ^m  I )  |  f finSupp  0 } )
23 ssrab2 3585 . . . . . . . 8  |-  { f  e.  ( RR  ^m  I )  |  f finSupp 
0 }  C_  ( RR  ^m  I )
2422, 23syl6eqss 3554 . . . . . . 7  |-  ( I  e.  V  ->  B  C_  ( RR  ^m  I
) )
2524sselda 3504 . . . . . 6  |-  ( ( I  e.  V  /\  f  e.  B )  ->  f  e.  ( RR 
^m  I ) )
2615fveq2d 5870 . . . . . . . . 9  |-  ( I  e.  V  ->  ( .i `  H )  =  ( .i `  (toCHil `  (RRfld freeLMod  I ) ) ) )
2714, 17rrxip 21585 . . . . . . . . 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 21431 . . . . . . . . . 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 2519 . . . . . . . 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 5868 . . . . . . . . . . . 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 5868 . . . . . . . . . . . 12  |-  ( ( ( I  e.  V  /\  f  e.  ( RR  ^m  I ) )  /\  ( h  =  f  /\  g  =  f ) )  -> 
( g `  x
)  =  ( f `
 x ) )
3633, 35oveq12d 6302 . . . . . . . . . . 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 7440 . . . . . . . . . . . . . . 15  |-  ( f  e.  ( RR  ^m  I )  ->  f : I --> RR )
3938adantl 466 . . . . . . . . . . . . . 14  |-  ( ( I  e.  V  /\  f  e.  ( RR  ^m  I ) )  -> 
f : I --> RR )
4039ffvelrnda 6021 . . . . . . . . . . . . 13  |-  ( ( ( I  e.  V  /\  f  e.  ( RR  ^m  I ) )  /\  x  e.  I
)  ->  ( f `  x )  e.  RR )
4140recnd 9622 . . . . . . . . . . . 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 12275 . . . . . . . . . 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 2511 . . . . . . . . 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 4533 . . . . . . . 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 6300 . . . . . . 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 6309 . . . . . . . 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 6414 . . . . . 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 2475 . . . 4  |-  ( ( I  e.  V  /\  f  e.  B )  ->  (RRfld  gsumg  ( x  e.  I  |->  ( ( f `  x ) ^ 2 ) ) )  =  ( f ( .i
`  (RRfld freeLMod  I ) ) f ) )
5352fveq2d 5870 . . 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 4524 . 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 2519 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 1379    e. wcel 1767   {crab 2818   _Vcvv 3113   class class class wbr 4447    |-> cmpt 4505   -->wf 5584   ` cfv 5588  (class class class)co 6284    |-> cmpt2 6286    ^m cmap 7420   finSupp cfsupp 7829   CCcc 9490   RRcr 9491   0cc0 9492    x. cmul 9497   2c2 10585   ^cexp 12134   sqrcsqrt 13029   Basecbs 14490   .icip 14560    gsumg cgsu 14696   Grpcgrp 15727   Ringcrg 17000   *Ringcsr 17293   LModclmod 17312  RRfldcrefld 18435   freeLMod cfrlm 18572   normcnm 20860  toCHilctch 21377  ℝ^crrx 21578
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 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 9570  ax-addf 9571  ax-mulf 9572
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 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-supp 6902  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-fsupp 7830  df-sup 7901  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-nn 10537  df-2 10594  df-3 10595  df-4 10596  df-5 10597  df-6 10598  df-7 10599  df-8 10600  df-9 10601  df-10 10602  df-n0 10796  df-z 10865  df-dec 10977  df-uz 11083  df-rp 11221  df-fz 11673  df-seq 12076  df-exp 12135  df-cj 12895  df-re 12896  df-im 12897  df-sqrt 13031  df-abs 13032  df-struct 14492  df-ndx 14493  df-slot 14494  df-base 14495  df-sets 14496  df-ress 14497  df-plusg 14568  df-mulr 14569  df-starv 14570  df-sca 14571  df-vsca 14572  df-ip 14573  df-tset 14574  df-ple 14575  df-ds 14577  df-unif 14578  df-hom 14579  df-cco 14580  df-0g 14697  df-prds 14703  df-pws 14705  df-mnd 15732  df-mhm 15786  df-grp 15867  df-minusg 15868  df-sbg 15869  df-subg 16003  df-ghm 16070  df-cmn 16606  df-mgp 16944  df-ur 16956  df-rng 17002  df-cring 17003  df-oppr 17073  df-dvdsr 17091  df-unit 17092  df-invr 17122  df-dvr 17133  df-rnghom 17165  df-drng 17198  df-field 17199  df-subrg 17227  df-staf 17294  df-srng 17295  df-lmod 17314  df-lss 17379  df-sra 17618  df-rgmod 17619  df-cnfld 18220  df-refld 18436  df-dsmm 18558  df-frlm 18573  df-nm 20866  df-tng 20868  df-tch 21379  df-rrx 21580
This theorem is referenced by:  rrxds  21588
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