Theorem frlmip 20117

Description: The inner product of a free module. (Contributed by Thierry Arnoux, 20-Jun-2019.)

Hypotheses

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

Assertion

Ref	Expression
frlmip	|- ( ( I e. W /\ R e. V ) -> ( f e. ( B ^m I ) , g e. ( B ^m I ) |-> ( R gsumg ( x e. I |-> ( ( f ` x ) .x. ( g ` x ) ) ) ) ) = ( .i ` Y ) )

Distinct variable groups:   B, f, g, x   f, I, g, x   R, f, g, x   f, V, g, x   f, W, g, x

Allowed substitution hints:   .x. ( x, f, g)   Y( x, f, g)

Proof of Theorem frlmip

Step	Hyp	Ref	Expression
1		frlmphl.y	. . . 4 |- Y = ( R freeLMod I )
2		eqid 2622	. . . . . . 7 |- ( R freeLMod I ) = ( R freeLMod I )
3		eqid 2622	. . . . . . 7 |- ( Base ` ( R freeLMod I ) ) = ( Base ` ( R freeLMod I ) )
4	2, 3	frlmpws 20094	. . . . . 6 |- ( ( R e. V /\ I e. W ) -> ( R freeLMod I ) = ( ( (ringLMod ` R ) ^s I ) ↾s ( Base ` ( R freeLMod I ) ) ) )
5	4	ancoms 469	. . . . 5 |- ( ( I e. W /\ R e. V ) -> ( R freeLMod I ) = ( ( (ringLMod ` R ) ^s I ) ↾s ( Base ` ( R freeLMod I ) ) ) )
6		frlmphl.b	. . . . . . . . . . 11 |- B = ( Base ` R )
7	6	ressid 15935	. . . . . . . . . 10 |- ( R e. V -> ( R ↾s B ) = R )
8		eqidd 2623	. . . . . . . . . . 11 |- ( R e. V -> ( (subringAlg ` R ) ` B ) = ( (subringAlg ` R ) ` B ) )
9	6	eqimssi 3659	. . . . . . . . . . . 12 |- B C_ ( Base ` R )
10	9	a1i 11	. . . . . . . . . . 11 |- ( R e. V -> B C_ ( Base ` R ) )
11	8, 10	srasca 19181	. . . . . . . . . 10 |- ( R e. V -> ( R ↾s B ) = (Scalar ` ( (subringAlg ` R ) ` B ) ) )
12	7, 11	eqtr3d 2658	. . . . . . . . 9 |- ( R e. V -> R = (Scalar ` ( (subringAlg ` R ) ` B ) ) )
13	12	oveq1d 6665	. . . . . . . 8 |- ( R e. V -> ( R X_s ( I X. { ( (subringAlg ` R ) ` B ) } ) ) = ( (Scalar ` ( (subringAlg ` R ) ` B ) ) X_s ( I X. { ( (subringAlg ` R ) ` B ) } ) ) )
14	13	adantl 482	. . . . . . 7 |- ( ( I e. W /\ R e. V ) -> ( R X_s ( I X. { ( (subringAlg ` R ) ` B ) } ) ) = ( (Scalar ` ( (subringAlg ` R ) ` B ) ) X_s ( I X. { ( (subringAlg ` R ) ` B ) } ) ) )
15		fvex 6201	. . . . . . . . 9 |- ( (subringAlg ` R ) ` B ) e. _V
16		rlmval 19191	. . . . . . . . . . . 12 |- (ringLMod ` R ) = ( (subringAlg ` R ) ` ( Base ` R ) )
17	6	fveq2i 6194	. . . . . . . . . . . 12 |- ( (subringAlg ` R ) ` B ) = ( (subringAlg ` R ) ` ( Base ` R ) )
18	16, 17	eqtr4i 2647	. . . . . . . . . . 11 |- (ringLMod ` R ) = ( (subringAlg ` R ) ` B )
19	18	oveq1i 6660	. . . . . . . . . 10 |- ( (ringLMod ` R ) ^s I ) = ( ( (subringAlg ` R ) ` B ) ^s I )
20		eqid 2622	. . . . . . . . . 10 |- (Scalar ` ( (subringAlg ` R ) ` B ) ) = (Scalar ` ( (subringAlg ` R ) ` B ) )
21	19, 20	pwsval 16146	. . . . . . . . 9 |- ( ( ( (subringAlg ` R ) ` B ) e. _V /\ I e. W ) -> ( (ringLMod ` R ) ^s I ) = ( (Scalar ` ( (subringAlg ` R ) ` B ) ) X_s ( I X. { ( (subringAlg ` R ) ` B ) } ) ) )
22	15, 21	mpan 706	. . . . . . . 8 |- ( I e. W -> ( (ringLMod ` R ) ^s I ) = ( (Scalar ` ( (subringAlg ` R ) ` B ) ) X_s ( I X. { ( (subringAlg ` R ) ` B ) } ) ) )
23	22	adantr 481	. . . . . . 7 |- ( ( I e. W /\ R e. V ) -> ( (ringLMod ` R ) ^s I ) = ( (Scalar ` ( (subringAlg ` R ) ` B ) ) X_s ( I X. { ( (subringAlg ` R ) ` B ) } ) ) )
24	14, 23	eqtr4d 2659	. . . . . 6 |- ( ( I e. W /\ R e. V ) -> ( R X_s ( I X. { ( (subringAlg ` R ) ` B ) } ) ) = ( (ringLMod ` R ) ^s I ) )
25	1	fveq2i 6194	. . . . . . 7 |- ( Base ` Y ) = ( Base ` ( R freeLMod I ) )
26	25	a1i 11	. . . . . 6 |- ( ( I e. W /\ R e. V ) -> ( Base ` Y ) = ( Base ` ( R freeLMod I ) ) )
27	24, 26	oveq12d 6668	. . . . 5 |- ( ( I e. W /\ R e. V ) -> ( ( R X_s ( I X. { ( (subringAlg ` R ) ` B ) } ) ) ↾s ( Base ` Y ) ) = ( ( (ringLMod ` R ) ^s I ) ↾s ( Base ` ( R freeLMod I ) ) ) )
28	5, 27	eqtr4d 2659	. . . 4 |- ( ( I e. W /\ R e. V ) -> ( R freeLMod I ) = ( ( R X_s ( I X. { ( (subringAlg ` R ) ` B ) } ) ) ↾s ( Base ` Y ) ) )
29	1, 28	syl5eq 2668	. . 3 |- ( ( I e. W /\ R e. V ) -> Y = ( ( R X_s ( I X. { ( (subringAlg ` R ) ` B ) } ) ) ↾s ( Base ` Y ) ) )
30	29	fveq2d 6195	. 2 |- ( ( I e. W /\ R e. V ) -> ( .i ` Y ) = ( .i ` ( ( R X_s ( I X. { ( (subringAlg ` R ) ` B ) } ) ) ↾s ( Base ` Y ) ) ) )
31		fvex 6201	. . . 4 |- ( Base ` Y ) e. _V
32		eqid 2622	. . . . 5 |- ( ( R X_s ( I X. { ( (subringAlg ` R ) ` B ) } ) ) ↾s ( Base ` Y ) ) = ( ( R X_s ( I X. { ( (subringAlg ` R ) ` B ) } ) ) ↾s ( Base ` Y ) )
33		eqid 2622	. . . . 5 |- ( .i ` ( R X_s ( I X. { ( (subringAlg ` R ) ` B ) } ) ) ) = ( .i ` ( R X_s ( I X. { ( (subringAlg ` R ) ` B ) } ) ) )
34	32, 33	ressip 16033	. . . 4 |- ( ( Base ` Y ) e. _V -> ( .i ` ( R X_s ( I X. { ( (subringAlg ` R ) ` B ) } ) ) ) = ( .i ` ( ( R X_s ( I X. { ( (subringAlg ` R ) ` B ) } ) ) ↾s ( Base ` Y ) ) ) )
35	31, 34	ax-mp 5	. . 3 |- ( .i ` ( R X_s ( I X. { ( (subringAlg ` R ) ` B ) } ) ) ) = ( .i ` ( ( R X_s ( I X. { ( (subringAlg ` R ) ` B ) } ) ) ↾s ( Base ` Y ) ) )
36		eqid 2622	. . . . 5 |- ( R X_s ( I X. { ( (subringAlg ` R ) ` B ) } ) ) = ( R X_s ( I X. { ( (subringAlg ` R ) ` B ) } ) )
37		simpr 477	. . . . 5 |- ( ( I e. W /\ R e. V ) -> R e. V )
38		snex 4908	. . . . . . 7 |- { ( (subringAlg ` R ) ` B ) } e. _V
39		xpexg 6960	. . . . . . 7 |- ( ( I e. W /\ { ( (subringAlg ` R ) ` B ) } e. _V ) -> ( I X. { ( (subringAlg ` R ) ` B ) } ) e. _V )
40	38, 39	mpan2 707	. . . . . 6 |- ( I e. W -> ( I X. { ( (subringAlg ` R ) ` B ) } ) e. _V )
41	40	adantr 481	. . . . 5 |- ( ( I e. W /\ R e. V ) -> ( I X. { ( (subringAlg ` R ) ` B ) } ) e. _V )
42		eqid 2622	. . . . 5 |- ( Base ` ( R X_s ( I X. { ( (subringAlg ` R ) ` B ) } ) ) ) = ( Base ` ( R X_s ( I X. { ( (subringAlg ` R ) ` B ) } ) ) )
43	15	snnz 4309	. . . . . . 7 |- { ( (subringAlg ` R ) ` B ) } =/= (/)
44		dmxp 5344	. . . . . . 7 |- ( { ( (subringAlg ` R ) ` B ) } =/= (/) -> dom ( I X. { ( (subringAlg ` R ) ` B ) } ) = I )
45	43, 44	ax-mp 5	. . . . . 6 |- dom ( I X. { ( (subringAlg ` R ) ` B ) } ) = I
46	45	a1i 11	. . . . 5 |- ( ( I e. W /\ R e. V ) -> dom ( I X. { ( (subringAlg ` R ) ` B ) } ) = I )
47	36, 37, 41, 42, 46, 33	prdsip 16121	. . . 4 |- ( ( I e. W /\ R e. V ) -> ( .i ` ( R X_s ( I X. { ( (subringAlg ` R ) ` B ) } ) ) ) = ( f e. ( Base ` ( R X_s ( I X. { ( (subringAlg ` R ) ` B ) } ) ) ) , g e. ( Base ` ( R X_s ( I X. { ( (subringAlg ` R ) ` B ) } ) ) ) |-> ( R gsumg ( x e. I |-> ( ( f ` x ) ( .i ` ( ( I X. { ( (subringAlg ` R ) ` B ) } ) ` x ) ) ( g ` x ) ) ) ) ) )
48	36, 37, 41, 42, 46	prdsbas 16117	. . . . . 6 |- ( ( I e. W /\ R e. V ) -> ( Base ` ( R X_s ( I X. { ( (subringAlg ` R ) ` B ) } ) ) ) = X_ x e. I ( Base ` ( ( I X. { ( (subringAlg ` R ) ` B ) } ) ` x ) ) )
49		eqidd 2623	. . . . . . . . . 10 |- ( x e. I -> ( (subringAlg ` R ) ` B ) = ( (subringAlg ` R ) ` B ) )
50	9	a1i 11	. . . . . . . . . 10 |- ( x e. I -> B C_ ( Base ` R ) )
51	49, 50	srabase 19178	. . . . . . . . 9 |- ( x e. I -> ( Base ` R ) = ( Base ` ( (subringAlg ` R ) ` B ) ) )
52	6	a1i 11	. . . . . . . . 9 |- ( x e. I -> B = ( Base ` R ) )
53	15	fvconst2 6469	. . . . . . . . . 10 |- ( x e. I -> ( ( I X. { ( (subringAlg ` R ) ` B ) } ) ` x ) = ( (subringAlg ` R ) ` B ) )
54	53	fveq2d 6195	. . . . . . . . 9 |- ( x e. I -> ( Base ` ( ( I X. { ( (subringAlg ` R ) ` B ) } ) ` x ) ) = ( Base ` ( (subringAlg ` R ) ` B ) ) )
55	51, 52, 54	3eqtr4rd 2667	. . . . . . . 8 |- ( x e. I -> ( Base ` ( ( I X. { ( (subringAlg ` R ) ` B ) } ) ` x ) ) = B )
56	55	adantl 482	. . . . . . 7 |- ( ( ( I e. W /\ R e. V ) /\ x e. I ) -> ( Base ` ( ( I X. { ( (subringAlg ` R ) ` B ) } ) ` x ) ) = B )
57	56	ixpeq2dva 7923	. . . . . 6 |- ( ( I e. W /\ R e. V ) -> X_ x e. I ( Base ` ( ( I X. { ( (subringAlg ` R ) ` B ) } ) ` x ) ) = X_ x e. I B )
58		fvex 6201	. . . . . . . . 9 |- ( Base ` R ) e. _V
59	6, 58	eqeltri 2697	. . . . . . . 8 |- B e. _V
60		ixpconstg 7917	. . . . . . . 8 |- ( ( I e. W /\ B e. _V ) -> X_ x e. I B = ( B ^m I ) )
61	59, 60	mpan2 707	. . . . . . 7 |- ( I e. W -> X_ x e. I B = ( B ^m I ) )
62	61	adantr 481	. . . . . 6 |- ( ( I e. W /\ R e. V ) -> X_ x e. I B = ( B ^m I ) )
63	48, 57, 62	3eqtrd 2660	. . . . 5 |- ( ( I e. W /\ R e. V ) -> ( Base ` ( R X_s ( I X. { ( (subringAlg ` R ) ` B ) } ) ) ) = ( B ^m I ) )
64		frlmphl.t	. . . . . . . . . 10 |- .x. = ( .r ` R )
65	53, 50	sraip 19183	. . . . . . . . . 10 |- ( x e. I -> ( .r ` R ) = ( .i ` ( ( I X. { ( (subringAlg ` R ) ` B ) } ) ` x ) ) )
66	64, 65	syl5req 2669	. . . . . . . . 9 |- ( x e. I -> ( .i ` ( ( I X. { ( (subringAlg ` R ) ` B ) } ) ` x ) ) = .x. )
67	66	oveqd 6667	. . . . . . . 8 |- ( x e. I -> ( ( f ` x ) ( .i ` ( ( I X. { ( (subringAlg ` R ) ` B ) } ) ` x ) ) ( g ` x ) ) = ( ( f ` x ) .x. ( g ` x ) ) )
68	67	mpteq2ia 4740	. . . . . . 7 |- ( x e. I |-> ( ( f ` x ) ( .i ` ( ( I X. { ( (subringAlg ` R ) ` B ) } ) ` x ) ) ( g ` x ) ) ) = ( x e. I |-> ( ( f ` x ) .x. ( g ` x ) ) )
69	68	oveq2i 6661	. . . . . 6 |- ( R gsumg ( x e. I |-> ( ( f ` x ) ( .i ` ( ( I X. { ( (subringAlg ` R ) ` B ) } ) ` x ) ) ( g ` x ) ) ) ) = ( R gsumg ( x e. I |-> ( ( f ` x ) .x. ( g ` x ) ) ) )
70	69	a1i 11	. . . . 5 |- ( ( I e. W /\ R e. V ) -> ( R gsumg ( x e. I |-> ( ( f ` x ) ( .i ` ( ( I X. { ( (subringAlg ` R ) ` B ) } ) ` x ) ) ( g ` x ) ) ) ) = ( R gsumg ( x e. I |-> ( ( f ` x ) .x. ( g ` x ) ) ) ) )
71	63, 63, 70	mpt2eq123dv 6717	. . . 4 |- ( ( I e. W /\ R e. V ) -> ( f e. ( Base ` ( R X_s ( I X. { ( (subringAlg ` R ) ` B ) } ) ) ) , g e. ( Base ` ( R X_s ( I X. { ( (subringAlg ` R ) ` B ) } ) ) ) |-> ( R gsumg ( x e. I |-> ( ( f ` x ) ( .i ` ( ( I X. { ( (subringAlg ` R ) ` B ) } ) ` x ) ) ( g ` x ) ) ) ) ) = ( f e. ( B ^m I ) , g e. ( B ^m I ) |-> ( R gsumg ( x e. I |-> ( ( f ` x ) .x. ( g ` x ) ) ) ) ) )
72	47, 71	eqtrd 2656	. . 3 |- ( ( I e. W /\ R e. V ) -> ( .i ` ( R X_s ( I X. { ( (subringAlg ` R ) ` B ) } ) ) ) = ( f e. ( B ^m I ) , g e. ( B ^m I ) |-> ( R gsumg ( x e. I |-> ( ( f ` x ) .x. ( g ` x ) ) ) ) ) )
73	35, 72	syl5eqr 2670	. 2 |- ( ( I e. W /\ R e. V ) -> ( .i ` ( ( R X_s ( I X. { ( (subringAlg ` R ) ` B ) } ) ) ↾s ( Base ` Y ) ) ) = ( f e. ( B ^m I ) , g e. ( B ^m I ) |-> ( R gsumg ( x e. I |-> ( ( f ` x ) .x. ( g ` x ) ) ) ) ) )
74	30, 73	eqtr2d 2657	1 |- ( ( I e. W /\ R e. V ) -> ( f e. ( B ^m I ) , g e. ( B ^m I ) |-> ( R gsumg ( x e. I |-> ( ( f ` x ) .x. ( g ` x ) ) ) ) ) = ( .i ` Y ) )

Syntax hints:   -> wi 4   /\ wa 384   = 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   Basecbs 15857   ↾_s cress 15858   .rcmulr 15942  Scalarcsca 15944   .icip 15946   gsum_g cgsu 16101   X__scprds 16106   ^_s cpws 16107  subringAlg csra 19168  ringLModcrglmod 19169   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-om 7066  df-1st 7168  df-2nd 7169  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-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-prds 16108  df-pws 16110  df-sra 19172  df-rgmod 19173  df-dsmm 20076  df-frlm 20091

This theorem is referenced by:  frlmipval  20118  frlmphl  20120