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Theorem rrxip 23178
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 23177 . . 3 |- ( I e. V -> H = (toCHil ` ( (RRfld X_s ( I X. { ( (subringAlg ` RRfld ) ` RR ) } ) ) ↾s B ) ) )
43fveq2d 6195 . 2 |- ( I e. V -> ( .i ` H ) = ( .i ` (toCHil ` ( (RRfld X_s ( I X. { ( (subringAlg ` RRfld ) ` RR ) } ) ) ↾s B ) ) ) )
5 eqid 2622 . . . 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 2622 . . . 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 23024 . . 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 6201 . . . . . 6 |- ( Base ` H ) e. _V
92, 8eqeltri 2697 . . . . 5 |- B e. _V
10 eqid 2622 . . . . . 6 |- ( (RRfld X_s ( I X. { ( (subringAlg ` RRfld ) ` RR ) } ) ) ↾s B ) = ( (RRfld X_s ( I X. { ( (subringAlg ` RRfld ) ` RR ) } ) ) ↾s B )
11 eqid 2622 . . . . . 6 |- ( .i ` (RRfld X_s ( I X. { ( (subringAlg ` RRfld ) ` RR ) } ) ) ) = ( .i ` (RRfld X_s ( I X. { ( (subringAlg ` RRfld ) ` RR ) } ) ) )
1210, 11ressip 16033 . . . . 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 2622 . . . . . 6 |- (RRfld X_s ( I X. { ( (subringAlg ` RRfld ) ` RR ) } ) ) = (RRfld X_s ( I X. { ( (subringAlg ` RRfld ) ` RR ) } ) )
15 refld 19965 . . . . . . 7 |- RRfld e. Field
1615a1i 11 . . . . . 6 |- ( I e. V -> RRfld e. Field )
17 snex 4908 . . . . . . 7 |- { ( (subringAlg ` RRfld ) ` RR ) } e. _V
18 xpexg 6960 . . . . . . 7 |- ( ( I e. V /\ { ( (subringAlg ` RRfld ) ` RR ) } e. _V ) -> ( I X. { ( (subringAlg ` RRfld ) ` RR ) } ) e. _V )
1917, 18mpan2 707 . . . . . 6 |- ( I e. V -> ( I X. { ( (subringAlg ` RRfld ) ` RR ) } ) e. _V )
20 eqid 2622 . . . . . 6 |- ( Base ` (RRfld X_s ( I X. { ( (subringAlg ` RRfld ) ` RR ) } ) ) ) = ( Base ` (RRfld X_s ( I X. { ( (subringAlg ` RRfld ) ` RR ) } ) ) )
21 fvex 6201 . . . . . . . . 9 |- ( (subringAlg ` RRfld ) ` RR ) e. _V
2221snnz 4309 . . . . . . . 8 |- { ( (subringAlg ` RRfld ) ` RR ) } =/= (/)
23 dmxp 5344 . . . . . . . 8 |- ( { ( (subringAlg ` RRfld ) ` RR ) } =/= (/) -> dom ( I X. { ( (subringAlg ` RRfld ) ` RR ) } ) = I )
2422, 23ax-mp 5 . . . . . . 7 |- dom ( I X. { ( (subringAlg ` RRfld ) ` RR ) } ) = I
2524a1i 11 . . . . . 6 |- ( I e. V -> dom ( I X. { ( (subringAlg ` RRfld ) ` RR ) } ) = I )
2614, 16, 19, 20, 25, 11prdsip 16121 . . . . 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 ) ) ) ) ) )
2714, 16, 19, 20, 25prdsbas 16117 . . . . . . 7 |- ( I e. V -> ( Base ` (RRfld X_s ( I X. { ( (subringAlg ` RRfld ) ` RR ) } ) ) ) = X_ x e. I ( Base ` ( ( I X. { ( (subringAlg ` RRfld ) ` RR ) } ) ` x ) ) )
28 eqidd 2623 . . . . . . . . . . 11 |- ( x e. I -> ( (subringAlg ` RRfld ) ` RR ) = ( (subringAlg ` RRfld ) ` RR ) )
29 rebase 19952 . . . . . . . . . . . . 13 |- RR = ( Base ` RRfld )
3029eqimssi 3659 . . . . . . . . . . . 12 |- RR C_ ( Base ` RRfld )
3130a1i 11 . . . . . . . . . . 11 |- ( x e. I -> RR C_ ( Base ` RRfld ) )
3228, 31srabase 19178 . . . . . . . . . 10 |- ( x e. I -> ( Base ` RRfld ) = ( Base ` ( (subringAlg ` RRfld ) ` RR ) ) )
3329a1i 11 . . . . . . . . . 10 |- ( x e. I -> RR = ( Base ` RRfld ) )
3421fvconst2 6469 . . . . . . . . . . 11 |- ( x e. I -> ( ( I X. { ( (subringAlg ` RRfld ) ` RR ) } ) ` x ) = ( (subringAlg ` RRfld ) ` RR ) )
3534fveq2d 6195 . . . . . . . . . 10 |- ( x e. I -> ( Base ` ( ( I X. { ( (subringAlg ` RRfld ) ` RR ) } ) ` x ) ) = ( Base ` ( (subringAlg ` RRfld ) ` RR ) ) )
3632, 33, 353eqtr4rd 2667 . . . . . . . . 9 |- ( x e. I -> ( Base ` ( ( I X. { ( (subringAlg ` RRfld ) ` RR ) } ) ` x ) ) = RR )
3736adantl 482 . . . . . . . 8 |- ( ( I e. V /\ x e. I ) -> ( Base ` ( ( I X. { ( (subringAlg ` RRfld ) ` RR ) } ) ` x ) ) = RR )
3837ixpeq2dva 7923 . . . . . . 7 |- ( I e. V -> X_ x e. I ( Base ` ( ( I X. { ( (subringAlg ` RRfld ) ` RR ) } ) ` x ) ) = X_ x e. I RR )
39 reex 10027 . . . . . . . 8 |- RR e. _V
40 ixpconstg 7917 . . . . . . . 8 |- ( ( I e. V /\ RR e. _V ) -> X_ x e. I RR = ( RR ^m I ) )
4139, 40mpan2 707 . . . . . . 7 |- ( I e. V -> X_ x e. I RR = ( RR ^m I ) )
4227, 38, 413eqtrd 2660 . . . . . 6 |- ( I e. V -> ( Base ` (RRfld X_s ( I X. { ( (subringAlg ` RRfld ) ` RR ) } ) ) ) = ( RR ^m I ) )
43 remulr 19957 . . . . . . . . . . 11 |- x. = ( .r ` RRfld )
4434, 31sraip 19183 . . . . . . . . . . 11 |- ( x e. I -> ( .r ` RRfld ) = ( .i ` ( ( I X. { ( (subringAlg ` RRfld ) ` RR ) } ) ` x ) ) )
4543, 44syl5req 2669 . . . . . . . . . 10 |- ( x e. I -> ( .i ` ( ( I X. { ( (subringAlg ` RRfld ) ` RR ) } ) ` x ) ) = x. )
4645oveqd 6667 . . . . . . . . 9 |- ( x e. I -> ( ( f ` x ) ( .i ` ( ( I X. { ( (subringAlg ` RRfld ) ` RR ) } ) ` x ) ) ( g ` x ) ) = ( ( f ` x ) x. ( g ` x ) ) )
4746mpteq2ia 4740 . . . . . . . 8 |- ( x e. I |-> ( ( f ` x ) ( .i ` ( ( I X. { ( (subringAlg ` RRfld ) ` RR ) } ) ` x ) ) ( g ` x ) ) ) = ( x e. I |-> ( ( f ` x ) x. ( g ` x ) ) )
4847a1i 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 ) ) ) )
4948oveq2d 6666 . . . . . 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 ) ) ) ) )
5042, 42, 49mpt2eq123dv 6717 . . . . 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 ) ) ) ) ) )
5126, 50eqtrd 2656 . . . 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 ) ) ) ) ) )
5213, 51syl5eqr 2670 . . 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 ) ) ) ) ) )
537, 52syl5eqr 2670 . 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 ) ) ) ) ) )
544, 53eqtr2d 2657 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 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   C_ wss 3574   (/)c0 3915   {csn 4177   |-> cmpt 4729   X. cxp 5112   dom cdm 5114   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   ^m cmap 7857   X_cixp 7908   RRcr 9935   x. cmul 9941   Basecbs 15857   ↾s cress 15858   .rcmulr 15942   .icip 15946   gsumg cgsu 16101   X_scprds 16106  Fieldcfield 18748  subringAlg csra 19168  RRfldcrefld 19950  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-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-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-grp 17425  df-minusg 17426  df-subg 17591  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-drng 18749  df-field 18750  df-subrg 18778  df-sra 19172  df-rgmod 19173  df-cnfld 19747  df-refld 19951  df-dsmm 20076  df-frlm 20091  df-tng 22389  df-tch 22969  df-rrx 23173
This theorem is referenced by:  rrxnm  23179
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