Theorem rrxnm 23179

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.

Step	Hyp	Ref	Expression
1		recrng 19967	. . . . 5 |- RRfld e. *Ring
2		srngring 18852	. . . . 5 |- (RRfld e. *Ring -> RRfld e. Ring )
3	1, 2	ax-mp 5	. . . 4 |- RRfld e. Ring
4		eqid 2622	. . . . 5 |- (RRfld freeLMod I ) = (RRfld freeLMod I )
5	4	frlmlmod 20093	. . . 4 |- ( (RRfld e. Ring /\ I e. V ) -> (RRfld freeLMod I ) e. LMod )
6	3, 5	mpan 706	. . 3 |- ( I e. V -> (RRfld freeLMod I ) e. LMod )
7		lmodgrp 18870	. . 3 |- ( (RRfld freeLMod I ) e. LMod -> (RRfld freeLMod I ) e. Grp )
8		eqid 2622	. . . 4 |- (toCHil ` (RRfld freeLMod I ) ) = (toCHil ` (RRfld freeLMod I ) )
9		eqid 2622	. . . 4 |- ( norm ` (toCHil ` (RRfld freeLMod I ) ) ) = ( norm ` (toCHil ` (RRfld freeLMod I ) ) )
10		eqid 2622	. . . 4 |- ( Base ` (RRfld freeLMod I ) ) = ( Base ` (RRfld freeLMod I ) )
11		eqid 2622	. . . 4 |- ( .i ` (RRfld freeLMod I ) ) = ( .i ` (RRfld freeLMod I ) )
12	8, 9, 10, 11	tchnmfval 23027	. . 3 |- ( (RRfld freeLMod I ) e. Grp -> ( norm ` (toCHil ` (RRfld freeLMod I ) ) ) = ( f e. ( Base ` (RRfld freeLMod I ) ) |-> ( sqr ` ( f ( .i ` (RRfld freeLMod I ) ) f ) ) ) )
13	6, 7, 12	3syl 18	. 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 )
15	14	rrxval 23175	. . 3 |- ( I e. V -> H = (toCHil ` (RRfld freeLMod I ) ) )
16	15	fveq2d 6195	. 2 |- ( I e. V -> ( norm ` H ) = ( norm ` (toCHil ` (RRfld freeLMod I ) ) ) )
17	15	fveq2d 6195	. . . 4 |- ( I e. V -> ( Base ` H ) = ( Base ` (toCHil ` (RRfld freeLMod I ) ) ) )
18		rrxbase.b	. . . 4 |- B = ( Base ` H )
19	8, 10	tchbas 23018	. . . 4 |- ( Base ` (RRfld freeLMod I ) ) = ( Base ` (toCHil ` (RRfld freeLMod I ) ) )
20	17, 18, 19	3eqtr4g 2681	. . 3 |- ( I e. V -> B = ( Base ` (RRfld freeLMod I ) ) )
21	14, 18	rrxbase 23176	. . . . . . . 8 |- ( I e. V -> B = { f e. ( RR ^m I ) | f finSupp 0 } )
22		ssrab2 3687	. . . . . . . 8 |- { f e. ( RR ^m I ) | f finSupp 0 } C_ ( RR ^m I )
23	21, 22	syl6eqss 3655	. . . . . . 7 |- ( I e. V -> B C_ ( RR ^m I ) )
24	23	sselda 3603	. . . . . 6 |- ( ( I e. V /\ f e. B ) -> f e. ( RR ^m I ) )
25	15	fveq2d 6195	. . . . . . . . 9 |- ( I e. V -> ( .i ` H ) = ( .i ` (toCHil ` (RRfld freeLMod I ) ) ) )
26	14, 18	rrxip 23178	. . . . . . . . 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 ) )
27	8, 11	tchip 23024	. . . . . . . . . 10 |- ( .i ` (RRfld freeLMod I ) ) = ( .i ` (toCHil ` (RRfld freeLMod I ) ) )
28	27	a1i 11	. . . . . . . . 9 |- ( I e. V -> ( .i ` (RRfld freeLMod I ) ) = ( .i ` (toCHil ` (RRfld freeLMod I ) ) ) )
29	25, 26, 28	3eqtr4rd 2667	. . . . . . . 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 ) ) ) ) ) )
30	29	adantr 481	. . . . . . 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 794	. . . . . . . . . . . . 13 |- ( ( ( I e. V /\ f e. ( RR ^m I ) ) /\ ( h = f /\ g = f ) ) -> h = f )
32	31	fveq1d 6193	. . . . . . . . . . . 12 |- ( ( ( I e. V /\ f e. ( RR ^m I ) ) /\ ( h = f /\ g = f ) ) -> ( h ` x ) = ( f ` x ) )
33		simprr 796	. . . . . . . . . . . . 13 |- ( ( ( I e. V /\ f e. ( RR ^m I ) ) /\ ( h = f /\ g = f ) ) -> g = f )
34	33	fveq1d 6193	. . . . . . . . . . . 12 |- ( ( ( I e. V /\ f e. ( RR ^m I ) ) /\ ( h = f /\ g = f ) ) -> ( g ` x ) = ( f ` x ) )
35	32, 34	oveq12d 6668	. . . . . . . . . . 11 |- ( ( ( I e. V /\ f e. ( RR ^m I ) ) /\ ( h = f /\ g = f ) ) -> ( ( h ` x ) x. ( g ` x ) ) = ( ( f ` x ) x. ( f ` x ) ) )
36	35	adantr 481	. . . . . . . . . 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 7879	. . . . . . . . . . . . . . 15 |- ( f e. ( RR ^m I ) -> f : I --> RR )
38	37	adantl 482	. . . . . . . . . . . . . 14 |- ( ( I e. V /\ f e. ( RR ^m I ) ) -> f : I --> RR )
39	38	ffvelrnda 6359	. . . . . . . . . . . . 13 |- ( ( ( I e. V /\ f e. ( RR ^m I ) ) /\ x e. I ) -> ( f ` x ) e. RR )
40	39	recnd 10068	. . . . . . . . . . . 12 |- ( ( ( I e. V /\ f e. ( RR ^m I ) ) /\ x e. I ) -> ( f ` x ) e. CC )
41	40	adantlr 751	. . . . . . . . . . 11 |- ( ( ( ( I e. V /\ f e. ( RR ^m I ) ) /\ ( h = f /\ g = f ) ) /\ x e. I ) -> ( f ` x ) e. CC )
42	41	sqvald 13005	. . . . . . . . . 10 |- ( ( ( ( I e. V /\ f e. ( RR ^m I ) ) /\ ( h = f /\ g = f ) ) /\ x e. I ) -> ( ( f ` x ) ^ 2 ) = ( ( f ` x ) x. ( f ` x ) ) )
43	36, 42	eqtr4d 2659	. . . . . . . . 9 |- ( ( ( ( I e. V /\ f e. ( RR ^m I ) ) /\ ( h = f /\ g = f ) ) /\ x e. I ) -> ( ( h ` x ) x. ( g ` x ) ) = ( ( f ` x ) ^ 2 ) )
44	43	mpteq2dva 4744	. . . . . . . 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 ) ) )
45	44	oveq2d 6666	. . . . . . 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 477	. . . . . . 7 |- ( ( I e. V /\ f e. ( RR ^m I ) ) -> f e. ( RR ^m I ) )
47		ovexd 6680	. . . . . . 7 |- ( ( I e. V /\ f e. ( RR ^m I ) ) -> (RRfld gsumg ( x e. I |-> ( ( f ` x ) ^ 2 ) ) ) e. _V )
48	30, 45, 46, 46, 47	ovmpt2d 6788	. . . . . 6 |- ( ( I e. V /\ f e. ( RR ^m I ) ) -> ( f ( .i ` (RRfld freeLMod I ) ) f ) = (RRfld gsumg ( x e. I |-> ( ( f ` x ) ^ 2 ) ) ) )
49	24, 48	syldan 487	. . . . 5 |- ( ( I e. V /\ f e. B ) -> ( f ( .i ` (RRfld freeLMod I ) ) f ) = (RRfld gsumg ( x e. I |-> ( ( f ` x ) ^ 2 ) ) ) )
50	49	eqcomd 2628	. . . 4 |- ( ( I e. V /\ f e. B ) -> (RRfld gsumg ( x e. I |-> ( ( f ` x ) ^ 2 ) ) ) = ( f ( .i ` (RRfld freeLMod I ) ) f ) )
51	50	fveq2d 6195	. . 3 |- ( ( I e. V /\ f e. B ) -> ( sqr ` (RRfld gsumg ( x e. I |-> ( ( f ` x ) ^ 2 ) ) ) ) = ( sqr ` ( f ( .i ` (RRfld freeLMod I ) ) f ) ) )
52	20, 51	mpteq12dva 4732	. 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 ) ) ) )
53	13, 16, 52	3eqtr4rd 2667	1 |- ( I e. V -> ( f e. B |-> ( sqr ` (RRfld gsumg ( x e. I |-> ( ( f ` x ) ^ 2 ) ) ) ) ) = ( norm ` H ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   class class class wbr 4653   |-> cmpt 4729   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   ^m cmap 7857   finSupp cfsupp 8275   CCcc 9934   RRcr 9935   0cc0 9936   x. cmul 9941   2c2 11070   ^cexp 12860   sqrcsqrt 13973   Basecbs 15857   .icip 15946   gsum_g cgsu 16101   Grpcgrp 17422   Ringcrg 18547   *Ringcsr 18844   LModclmod 18863  RRfldcrefld 19950   freeLMod cfrlm 20090   normcnm 22381  toCHilctch 22967  ℝ^crrx 23171

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-sup 8348  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-rp 11833  df-fz 12327  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-0g 16102  df-prds 16108  df-pws 16110  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-grp 17425  df-minusg 17426  df-sbg 17427  df-subg 17591  df-ghm 17658  df-cmn 18195  df-mgp 18490  df-ur 18502  df-ring 18549  df-cring 18550  df-oppr 18623  df-dvdsr 18641  df-unit 18642  df-invr 18672  df-dvr 18683  df-rnghom 18715  df-drng 18749  df-field 18750  df-subrg 18778  df-staf 18845  df-srng 18846  df-lmod 18865  df-lss 18933  df-sra 19172  df-rgmod 19173  df-cnfld 19747  df-refld 19951  df-dsmm 20076  df-frlm 20091  df-nm 22387  df-tng 22389  df-tch 22969  df-rrx 23173

This theorem is referenced by:  rrxds  23181