Theorem frlmipval 20118

Description: The inner product of a free module. (Contributed by Thierry Arnoux, 21-Jun-2019.)

Hypotheses

Ref	Expression
frlmphl.y	|- Y = ( R freeLMod I )
frlmphl.b	|- B = ( Base ` R )
frlmphl.t	|- .x. = ( .r ` R )
frlmphl.v	|- V = ( Base ` Y )
frlmphl.j	|- ., = ( .i ` Y )

Assertion

Ref	Expression
frlmipval	|- ( ( ( I e. W /\ R e. X ) /\ ( F e. V /\ G e. V ) ) -> ( F ., G ) = ( R gsumg ( F oF .x. G ) ) )

Proof of Theorem frlmipval

Dummy variables f g x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		frlmphl.y	. . . . . . 7 |- Y = ( R freeLMod I )
2		frlmphl.b	. . . . . . 7 |- B = ( Base ` R )
3		frlmphl.v	. . . . . . 7 |- V = ( Base ` Y )
4	1, 2, 3	frlmbasmap 20103	. . . . . 6 |- ( ( I e. W /\ F e. V ) -> F e. ( B ^m I ) )
5	4	ad2ant2r 783	. . . . 5 |- ( ( ( I e. W /\ R e. X ) /\ ( F e. V /\ G e. V ) ) -> F e. ( B ^m I ) )
6		elmapi 7879	. . . . 5 |- ( F e. ( B ^m I ) -> F : I --> B )
7		ffn 6045	. . . . 5 |- ( F : I --> B -> F Fn I )
8	5, 6, 7	3syl 18	. . . 4 |- ( ( ( I e. W /\ R e. X ) /\ ( F e. V /\ G e. V ) ) -> F Fn I )
9	1, 2, 3	frlmbasmap 20103	. . . . . 6 |- ( ( I e. W /\ G e. V ) -> G e. ( B ^m I ) )
10	9	ad2ant2rl 785	. . . . 5 |- ( ( ( I e. W /\ R e. X ) /\ ( F e. V /\ G e. V ) ) -> G e. ( B ^m I ) )
11		elmapi 7879	. . . . 5 |- ( G e. ( B ^m I ) -> G : I --> B )
12		ffn 6045	. . . . 5 |- ( G : I --> B -> G Fn I )
13	10, 11, 12	3syl 18	. . . 4 |- ( ( ( I e. W /\ R e. X ) /\ ( F e. V /\ G e. V ) ) -> G Fn I )
14		simpll 790	. . . 4 |- ( ( ( I e. W /\ R e. X ) /\ ( F e. V /\ G e. V ) ) -> I e. W )
15		inidm 3822	. . . 4 |- ( I i^i I ) = I
16		eqidd 2623	. . . 4 |- ( ( ( ( I e. W /\ R e. X ) /\ ( F e. V /\ G e. V ) ) /\ x e. I ) -> ( F ` x ) = ( F ` x ) )
17		eqidd 2623	. . . 4 |- ( ( ( ( I e. W /\ R e. X ) /\ ( F e. V /\ G e. V ) ) /\ x e. I ) -> ( G ` x ) = ( G ` x ) )
18	8, 13, 14, 14, 15, 16, 17	offval 6904	. . 3 |- ( ( ( I e. W /\ R e. X ) /\ ( F e. V /\ G e. V ) ) -> ( F oF .x. G ) = ( x e. I |-> ( ( F ` x ) .x. ( G ` x ) ) ) )
19	18	oveq2d 6666	. 2 |- ( ( ( I e. W /\ R e. X ) /\ ( F e. V /\ G e. V ) ) -> ( R gsumg ( F oF .x. G ) ) = ( R gsumg ( x e. I |-> ( ( F ` x ) .x. ( G ` x ) ) ) ) )
20		ovexd 6680	. . 3 |- ( ( ( I e. W /\ R e. X ) /\ ( F e. V /\ G e. V ) ) -> ( R gsumg ( x e. I |-> ( ( F ` x ) .x. ( G ` x ) ) ) ) e. _V )
21		fveq1 6190	. . . . . . 7 |- ( f = F -> ( f ` x ) = ( F ` x ) )
22	21	oveq1d 6665	. . . . . 6 |- ( f = F -> ( ( f ` x ) .x. ( g ` x ) ) = ( ( F ` x ) .x. ( g ` x ) ) )
23	22	mpteq2dv 4745	. . . . 5 |- ( f = F -> ( x e. I |-> ( ( f ` x ) .x. ( g ` x ) ) ) = ( x e. I |-> ( ( F ` x ) .x. ( g ` x ) ) ) )
24	23	oveq2d 6666	. . . 4 |- ( f = F -> ( R gsumg ( x e. I |-> ( ( f ` x ) .x. ( g ` x ) ) ) ) = ( R gsumg ( x e. I |-> ( ( F ` x ) .x. ( g ` x ) ) ) ) )
25		fveq1 6190	. . . . . . 7 |- ( g = G -> ( g ` x ) = ( G ` x ) )
26	25	oveq2d 6666	. . . . . 6 |- ( g = G -> ( ( F ` x ) .x. ( g ` x ) ) = ( ( F ` x ) .x. ( G ` x ) ) )
27	26	mpteq2dv 4745	. . . . 5 |- ( g = G -> ( x e. I |-> ( ( F ` x ) .x. ( g ` x ) ) ) = ( x e. I |-> ( ( F ` x ) .x. ( G ` x ) ) ) )
28	27	oveq2d 6666	. . . 4 |- ( g = G -> ( R gsumg ( x e. I |-> ( ( F ` x ) .x. ( g ` x ) ) ) ) = ( R gsumg ( x e. I |-> ( ( F ` x ) .x. ( G ` x ) ) ) ) )
29		eqid 2622	. . . 4 |- ( f e. ( B ^m I ) , g e. ( B ^m I ) |-> ( R gsumg ( x e. I |-> ( ( f ` x ) .x. ( g ` x ) ) ) ) ) = ( f e. ( B ^m I ) , g e. ( B ^m I ) |-> ( R gsumg ( x e. I |-> ( ( f ` x ) .x. ( g ` x ) ) ) ) )
30	24, 28, 29	ovmpt2g 6795	. . 3 |- ( ( F e. ( B ^m I ) /\ G e. ( B ^m I ) /\ ( R gsumg ( x e. I |-> ( ( F ` x ) .x. ( G ` x ) ) ) ) e. _V ) -> ( F ( f e. ( B ^m I ) , g e. ( B ^m I ) |-> ( R gsumg ( x e. I |-> ( ( f ` x ) .x. ( g ` x ) ) ) ) ) G ) = ( R gsumg ( x e. I |-> ( ( F ` x ) .x. ( G ` x ) ) ) ) )
31	5, 10, 20, 30	syl3anc 1326	. 2 |- ( ( ( I e. W /\ R e. X ) /\ ( F e. V /\ G e. V ) ) -> ( F ( f e. ( B ^m I ) , g e. ( B ^m I ) |-> ( R gsumg ( x e. I |-> ( ( f ` x ) .x. ( g ` x ) ) ) ) ) G ) = ( R gsumg ( x e. I |-> ( ( F ` x ) .x. ( G ` x ) ) ) ) )
32		frlmphl.t	. . . . . 6 |- .x. = ( .r ` R )
33	1, 2, 32	frlmip 20117	. . . . 5 |- ( ( I e. W /\ R e. X ) -> ( f e. ( B ^m I ) , g e. ( B ^m I ) |-> ( R gsumg ( x e. I |-> ( ( f ` x ) .x. ( g ` x ) ) ) ) ) = ( .i ` Y ) )
34	33	adantr 481	. . . 4 |- ( ( ( I e. W /\ R e. X ) /\ ( F e. V /\ G e. V ) ) -> ( f e. ( B ^m I ) , g e. ( B ^m I ) |-> ( R gsumg ( x e. I |-> ( ( f ` x ) .x. ( g ` x ) ) ) ) ) = ( .i ` Y ) )
35		frlmphl.j	. . . 4 |- ., = ( .i ` Y )
36	34, 35	syl6eqr 2674	. . 3 |- ( ( ( I e. W /\ R e. X ) /\ ( F e. V /\ G e. V ) ) -> ( f e. ( B ^m I ) , g e. ( B ^m I ) |-> ( R gsumg ( x e. I |-> ( ( f ` x ) .x. ( g ` x ) ) ) ) ) = ., )
37	36	oveqd 6667	. 2 |- ( ( ( I e. W /\ R e. X ) /\ ( F e. V /\ G e. V ) ) -> ( F ( f e. ( B ^m I ) , g e. ( B ^m I ) |-> ( R gsumg ( x e. I |-> ( ( f ` x ) .x. ( g ` x ) ) ) ) ) G ) = ( F ., G ) )
38	19, 31, 37	3eqtr2rd 2663	1 |- ( ( ( I e. W /\ R e. X ) /\ ( F e. V /\ G e. V ) ) -> ( F ., G ) = ( R gsumg ( F oF .x. G ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   |-> cmpt 4729   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   oFcof 6895   ^m cmap 7857   Basecbs 15857   .rcmulr 15942   .icip 15946   gsum_g cgsu 16101   freeLMod cfrlm 20090

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

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-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-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  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-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-fz 12327  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-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-hom 15966  df-cco 15967  df-0g 16102  df-prds 16108  df-pws 16110  df-sra 19172  df-rgmod 19173  df-dsmm 20076  df-frlm 20091

This theorem is referenced by:  frlmphl  20120  rrxcph  23180