Theorem lflvscl 34364

Description: Closure of a scalar product with a functional. Note that this is the scalar product for a right vector space with the scalar after the vector; reversing these fails closure. (Contributed by NM, 9-Oct-2014.) (Revised by Mario Carneiro, 22-Apr-2015.)

Hypotheses

Ref	Expression
lflsccl.v	|- V = ( Base ` W )
lflsccl.d	|- D = (Scalar ` W )
lflsccl.k	|- K = ( Base ` D )
lflsccl.t	|- .x. = ( .r ` D )
lflsccl.f	|- F = (LFnl ` W )
lflsccl.w	|- ( ph -> W e. LMod )
lflsccl.g	|- ( ph -> G e. F )
lflsccl.r	|- ( ph -> R e. K )

Assertion

Ref	Expression
lflvscl	|- ( ph -> ( G oF .x. ( V X. { R } ) ) e. F )

Proof of Theorem lflvscl

Dummy variables x r y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lflsccl.v	. . 3 |- V = ( Base ` W )
2	1	a1i 11	. 2 |- ( ph -> V = ( Base ` W ) )
3		eqidd 2623	. 2 |- ( ph -> ( +g ` W ) = ( +g ` W ) )
4		lflsccl.d	. . 3 |- D = (Scalar ` W )
5	4	a1i 11	. 2 |- ( ph -> D = (Scalar ` W ) )
6		eqidd 2623	. 2 |- ( ph -> ( .s ` W ) = ( .s ` W ) )
7		lflsccl.k	. . 3 |- K = ( Base ` D )
8	7	a1i 11	. 2 |- ( ph -> K = ( Base ` D ) )
9		eqidd 2623	. 2 |- ( ph -> ( +g ` D ) = ( +g ` D ) )
10		lflsccl.t	. . 3 |- .x. = ( .r ` D )
11	10	a1i 11	. 2 |- ( ph -> .x. = ( .r ` D ) )
12		lflsccl.f	. . 3 |- F = (LFnl ` W )
13	12	a1i 11	. 2 |- ( ph -> F = (LFnl ` W ) )
14		lflsccl.w	. . . . 5 |- ( ph -> W e. LMod )
15	4	lmodring 18871	. . . . 5 |- ( W e. LMod -> D e. Ring )
16	14, 15	syl 17	. . . 4 |- ( ph -> D e. Ring )
17	7, 10	ringcl 18561	. . . . 5 |- ( ( D e. Ring /\ x e. K /\ y e. K ) -> ( x .x. y ) e. K )
18	17	3expb 1266	. . . 4 |- ( ( D e. Ring /\ ( x e. K /\ y e. K ) ) -> ( x .x. y ) e. K )
19	16, 18	sylan 488	. . 3 |- ( ( ph /\ ( x e. K /\ y e. K ) ) -> ( x .x. y ) e. K )
20		lflsccl.g	. . . 4 |- ( ph -> G e. F )
21	4, 7, 1, 12	lflf 34350	. . . 4 |- ( ( W e. LMod /\ G e. F ) -> G : V --> K )
22	14, 20, 21	syl2anc 693	. . 3 |- ( ph -> G : V --> K )
23		lflsccl.r	. . . 4 |- ( ph -> R e. K )
24		fconst6g 6094	. . . 4 |- ( R e. K -> ( V X. { R } ) : V --> K )
25	23, 24	syl 17	. . 3 |- ( ph -> ( V X. { R } ) : V --> K )
26		fvex 6201	. . . . 5 |- ( Base ` W ) e. _V
27	1, 26	eqeltri 2697	. . . 4 |- V e. _V
28	27	a1i 11	. . 3 |- ( ph -> V e. _V )
29		inidm 3822	. . 3 |- ( V i^i V ) = V
30	19, 22, 25, 28, 28, 29	off 6912	. 2 |- ( ph -> ( G oF .x. ( V X. { R } ) ) : V --> K )
31	14	adantr 481	. . . . . 6 |- ( ( ph /\ ( r e. K /\ x e. V /\ y e. V ) ) -> W e. LMod )
32	20	adantr 481	. . . . . 6 |- ( ( ph /\ ( r e. K /\ x e. V /\ y e. V ) ) -> G e. F )
33		simpr1 1067	. . . . . 6 |- ( ( ph /\ ( r e. K /\ x e. V /\ y e. V ) ) -> r e. K )
34		simpr2 1068	. . . . . 6 |- ( ( ph /\ ( r e. K /\ x e. V /\ y e. V ) ) -> x e. V )
35		simpr3 1069	. . . . . 6 |- ( ( ph /\ ( r e. K /\ x e. V /\ y e. V ) ) -> y e. V )
36		eqid 2622	. . . . . . 7 |- ( +g ` W ) = ( +g ` W )
37		eqid 2622	. . . . . . 7 |- ( .s ` W ) = ( .s ` W )
38		eqid 2622	. . . . . . 7 |- ( +g ` D ) = ( +g ` D )
39	1, 36, 4, 37, 7, 38, 10, 12	lfli 34348	. . . . . 6 |- ( ( W e. LMod /\ G e. F /\ ( r e. K /\ x e. V /\ y e. V ) ) -> ( G ` ( ( r ( .s ` W ) x ) ( +g ` W ) y ) ) = ( ( r .x. ( G ` x ) ) ( +g ` D ) ( G ` y ) ) )
40	31, 32, 33, 34, 35, 39	syl113anc 1338	. . . . 5 |- ( ( ph /\ ( r e. K /\ x e. V /\ y e. V ) ) -> ( G ` ( ( r ( .s ` W ) x ) ( +g ` W ) y ) ) = ( ( r .x. ( G ` x ) ) ( +g ` D ) ( G ` y ) ) )
41	40	oveq1d 6665	. . . 4 |- ( ( ph /\ ( r e. K /\ x e. V /\ y e. V ) ) -> ( ( G ` ( ( r ( .s ` W ) x ) ( +g ` W ) y ) ) .x. R ) = ( ( ( r .x. ( G ` x ) ) ( +g ` D ) ( G ` y ) ) .x. R ) )
42	16	adantr 481	. . . . 5 |- ( ( ph /\ ( r e. K /\ x e. V /\ y e. V ) ) -> D e. Ring )
43	4, 7, 1, 12	lflcl 34351	. . . . . . 7 |- ( ( W e. LMod /\ G e. F /\ x e. V ) -> ( G ` x ) e. K )
44	31, 32, 34, 43	syl3anc 1326	. . . . . 6 |- ( ( ph /\ ( r e. K /\ x e. V /\ y e. V ) ) -> ( G ` x ) e. K )
45	7, 10	ringcl 18561	. . . . . 6 |- ( ( D e. Ring /\ r e. K /\ ( G ` x ) e. K ) -> ( r .x. ( G ` x ) ) e. K )
46	42, 33, 44, 45	syl3anc 1326	. . . . 5 |- ( ( ph /\ ( r e. K /\ x e. V /\ y e. V ) ) -> ( r .x. ( G ` x ) ) e. K )
47	4, 7, 1, 12	lflcl 34351	. . . . . 6 |- ( ( W e. LMod /\ G e. F /\ y e. V ) -> ( G ` y ) e. K )
48	31, 32, 35, 47	syl3anc 1326	. . . . 5 |- ( ( ph /\ ( r e. K /\ x e. V /\ y e. V ) ) -> ( G ` y ) e. K )
49	23	adantr 481	. . . . 5 |- ( ( ph /\ ( r e. K /\ x e. V /\ y e. V ) ) -> R e. K )
50	7, 38, 10	ringdir 18567	. . . . 5 |- ( ( D e. Ring /\ ( ( r .x. ( G ` x ) ) e. K /\ ( G ` y ) e. K /\ R e. K ) ) -> ( ( ( r .x. ( G ` x ) ) ( +g ` D ) ( G ` y ) ) .x. R ) = ( ( ( r .x. ( G ` x ) ) .x. R ) ( +g ` D ) ( ( G ` y ) .x. R ) ) )
51	42, 46, 48, 49, 50	syl13anc 1328	. . . 4 |- ( ( ph /\ ( r e. K /\ x e. V /\ y e. V ) ) -> ( ( ( r .x. ( G ` x ) ) ( +g ` D ) ( G ` y ) ) .x. R ) = ( ( ( r .x. ( G ` x ) ) .x. R ) ( +g ` D ) ( ( G ` y ) .x. R ) ) )
52	7, 10	ringass 18564	. . . . . 6 |- ( ( D e. Ring /\ ( r e. K /\ ( G ` x ) e. K /\ R e. K ) ) -> ( ( r .x. ( G ` x ) ) .x. R ) = ( r .x. ( ( G ` x ) .x. R ) ) )
53	42, 33, 44, 49, 52	syl13anc 1328	. . . . 5 |- ( ( ph /\ ( r e. K /\ x e. V /\ y e. V ) ) -> ( ( r .x. ( G ` x ) ) .x. R ) = ( r .x. ( ( G ` x ) .x. R ) ) )
54	53	oveq1d 6665	. . . 4 |- ( ( ph /\ ( r e. K /\ x e. V /\ y e. V ) ) -> ( ( ( r .x. ( G ` x ) ) .x. R ) ( +g ` D ) ( ( G ` y ) .x. R ) ) = ( ( r .x. ( ( G ` x ) .x. R ) ) ( +g ` D ) ( ( G ` y ) .x. R ) ) )
55	41, 51, 54	3eqtrd 2660	. . 3 |- ( ( ph /\ ( r e. K /\ x e. V /\ y e. V ) ) -> ( ( G ` ( ( r ( .s ` W ) x ) ( +g ` W ) y ) ) .x. R ) = ( ( r .x. ( ( G ` x ) .x. R ) ) ( +g ` D ) ( ( G ` y ) .x. R ) ) )
56	1, 4, 37, 7	lmodvscl 18880	. . . . . 6 |- ( ( W e. LMod /\ r e. K /\ x e. V ) -> ( r ( .s ` W ) x ) e. V )
57	31, 33, 34, 56	syl3anc 1326	. . . . 5 |- ( ( ph /\ ( r e. K /\ x e. V /\ y e. V ) ) -> ( r ( .s ` W ) x ) e. V )
58	1, 36	lmodvacl 18877	. . . . 5 |- ( ( W e. LMod /\ ( r ( .s ` W ) x ) e. V /\ y e. V ) -> ( ( r ( .s ` W ) x ) ( +g ` W ) y ) e. V )
59	31, 57, 35, 58	syl3anc 1326	. . . 4 |- ( ( ph /\ ( r e. K /\ x e. V /\ y e. V ) ) -> ( ( r ( .s ` W ) x ) ( +g ` W ) y ) e. V )
60		ffn 6045	. . . . . 6 |- ( G : V --> K -> G Fn V )
61	22, 60	syl 17	. . . . 5 |- ( ph -> G Fn V )
62		eqidd 2623	. . . . 5 |- ( ( ph /\ ( ( r ( .s ` W ) x ) ( +g ` W ) y ) e. V ) -> ( G ` ( ( r ( .s ` W ) x ) ( +g ` W ) y ) ) = ( G ` ( ( r ( .s ` W ) x ) ( +g ` W ) y ) ) )
63	28, 23, 61, 62	ofc2 6921	. . . 4 |- ( ( ph /\ ( ( r ( .s ` W ) x ) ( +g ` W ) y ) e. V ) -> ( ( G oF .x. ( V X. { R } ) ) ` ( ( r ( .s ` W ) x ) ( +g ` W ) y ) ) = ( ( G ` ( ( r ( .s ` W ) x ) ( +g ` W ) y ) ) .x. R ) )
64	59, 63	syldan 487	. . 3 |- ( ( ph /\ ( r e. K /\ x e. V /\ y e. V ) ) -> ( ( G oF .x. ( V X. { R } ) ) ` ( ( r ( .s ` W ) x ) ( +g ` W ) y ) ) = ( ( G ` ( ( r ( .s ` W ) x ) ( +g ` W ) y ) ) .x. R ) )
65		eqidd 2623	. . . . . . 7 |- ( ( ph /\ x e. V ) -> ( G ` x ) = ( G ` x ) )
66	28, 23, 61, 65	ofc2 6921	. . . . . 6 |- ( ( ph /\ x e. V ) -> ( ( G oF .x. ( V X. { R } ) ) ` x ) = ( ( G ` x ) .x. R ) )
67	34, 66	syldan 487	. . . . 5 |- ( ( ph /\ ( r e. K /\ x e. V /\ y e. V ) ) -> ( ( G oF .x. ( V X. { R } ) ) ` x ) = ( ( G ` x ) .x. R ) )
68	67	oveq2d 6666	. . . 4 |- ( ( ph /\ ( r e. K /\ x e. V /\ y e. V ) ) -> ( r .x. ( ( G oF .x. ( V X. { R } ) ) ` x ) ) = ( r .x. ( ( G ` x ) .x. R ) ) )
69		eqidd 2623	. . . . . 6 |- ( ( ph /\ y e. V ) -> ( G ` y ) = ( G ` y ) )
70	28, 23, 61, 69	ofc2 6921	. . . . 5 |- ( ( ph /\ y e. V ) -> ( ( G oF .x. ( V X. { R } ) ) ` y ) = ( ( G ` y ) .x. R ) )
71	35, 70	syldan 487	. . . 4 |- ( ( ph /\ ( r e. K /\ x e. V /\ y e. V ) ) -> ( ( G oF .x. ( V X. { R } ) ) ` y ) = ( ( G ` y ) .x. R ) )
72	68, 71	oveq12d 6668	. . 3 |- ( ( ph /\ ( r e. K /\ x e. V /\ y e. V ) ) -> ( ( r .x. ( ( G oF .x. ( V X. { R } ) ) ` x ) ) ( +g ` D ) ( ( G oF .x. ( V X. { R } ) ) ` y ) ) = ( ( r .x. ( ( G ` x ) .x. R ) ) ( +g ` D ) ( ( G ` y ) .x. R ) ) )
73	55, 64, 72	3eqtr4d 2666	. 2 |- ( ( ph /\ ( r e. K /\ x e. V /\ y e. V ) ) -> ( ( G oF .x. ( V X. { R } ) ) ` ( ( r ( .s ` W ) x ) ( +g ` W ) y ) ) = ( ( r .x. ( ( G oF .x. ( V X. { R } ) ) ` x ) ) ( +g ` D ) ( ( G oF .x. ( V X. { R } ) ) ` y ) ) )
74	2, 3, 5, 6, 8, 9, 11, 13, 30, 73, 14	islfld 34349	1 |- ( ph -> ( G oF .x. ( V X. { R } ) ) e. F )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   {csn 4177   X. cxp 5112   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   oFcof 6895   Basecbs 15857   +g cplusg 15941   .rcmulr 15942  Scalarcsca 15944   .scvsca 15945   Ringcrg 18547   LModclmod 18863  LFnlclfn 34344

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-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-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  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-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-plusg 15954  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-mgp 18490  df-ring 18549  df-lmod 18865  df-lfl 34345

This theorem is referenced by:  lkrsc  34384  lfl1dim  34408  ldualvscl  34426  ldualvsass  34428