Theorem lkrlss 34382

Description: The kernel of a linear functional is a subspace. (nlelshi 28919 analog.) (Contributed by NM, 16-Apr-2014.)

Hypotheses

Ref	Expression
lkrlss.f	|- F = (LFnl ` W )
lkrlss.k	|- K = (LKer ` W )
lkrlss.s	|- S = ( LSubSp ` W )

Assertion

Ref	Expression
lkrlss	|- ( ( W e. LMod /\ G e. F ) -> ( K ` G ) e. S )

Proof of Theorem lkrlss

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

Step	Hyp	Ref	Expression
1		eqid 2622	. . . 4 |- ( Base ` W ) = ( Base ` W )
2		eqid 2622	. . . 4 |- (Scalar ` W ) = (Scalar ` W )
3		eqid 2622	. . . 4 |- ( 0g ` (Scalar ` W ) ) = ( 0g ` (Scalar ` W ) )
4		lkrlss.f	. . . 4 |- F = (LFnl ` W )
5		lkrlss.k	. . . 4 |- K = (LKer ` W )
6	1, 2, 3, 4, 5	lkrval2 34377	. . 3 |- ( ( W e. LMod /\ G e. F ) -> ( K ` G ) = { x e. ( Base ` W ) | ( G ` x ) = ( 0g ` (Scalar ` W ) ) } )
7		ssrab2 3687	. . 3 |- { x e. ( Base ` W ) | ( G ` x ) = ( 0g ` (Scalar ` W ) ) } C_ ( Base ` W )
8	6, 7	syl6eqss 3655	. 2 |- ( ( W e. LMod /\ G e. F ) -> ( K ` G ) C_ ( Base ` W ) )
9		eqid 2622	. . . . . 6 |- ( 0g ` W ) = ( 0g ` W )
10	1, 9	lmod0vcl 18892	. . . . 5 |- ( W e. LMod -> ( 0g ` W ) e. ( Base ` W ) )
11	10	adantr 481	. . . 4 |- ( ( W e. LMod /\ G e. F ) -> ( 0g ` W ) e. ( Base ` W ) )
12	2, 3, 9, 4	lfl0 34352	. . . 4 |- ( ( W e. LMod /\ G e. F ) -> ( G ` ( 0g ` W ) ) = ( 0g ` (Scalar ` W ) ) )
13	1, 2, 3, 4, 5	ellkr 34376	. . . 4 |- ( ( W e. LMod /\ G e. F ) -> ( ( 0g ` W ) e. ( K ` G ) <-> ( ( 0g ` W ) e. ( Base ` W ) /\ ( G ` ( 0g ` W ) ) = ( 0g ` (Scalar ` W ) ) ) ) )
14	11, 12, 13	mpbir2and 957	. . 3 |- ( ( W e. LMod /\ G e. F ) -> ( 0g ` W ) e. ( K ` G ) )
15		ne0i 3921	. . 3 |- ( ( 0g ` W ) e. ( K ` G ) -> ( K ` G ) =/= (/) )
16	14, 15	syl 17	. 2 |- ( ( W e. LMod /\ G e. F ) -> ( K ` G ) =/= (/) )
17		simplll 798	. . . . . 6 |- ( ( ( ( W e. LMod /\ G e. F ) /\ r e. ( Base ` (Scalar ` W ) ) ) /\ ( x e. ( K ` G ) /\ y e. ( K ` G ) ) ) -> W e. LMod )
18		simplr 792	. . . . . . 7 |- ( ( ( ( W e. LMod /\ G e. F ) /\ r e. ( Base ` (Scalar ` W ) ) ) /\ ( x e. ( K ` G ) /\ y e. ( K ` G ) ) ) -> r e. ( Base ` (Scalar ` W ) ) )
19		simpllr 799	. . . . . . . 8 |- ( ( ( ( W e. LMod /\ G e. F ) /\ r e. ( Base ` (Scalar ` W ) ) ) /\ ( x e. ( K ` G ) /\ y e. ( K ` G ) ) ) -> G e. F )
20		simprl 794	. . . . . . . 8 |- ( ( ( ( W e. LMod /\ G e. F ) /\ r e. ( Base ` (Scalar ` W ) ) ) /\ ( x e. ( K ` G ) /\ y e. ( K ` G ) ) ) -> x e. ( K ` G ) )
21	1, 4, 5	lkrcl 34379	. . . . . . . 8 |- ( ( W e. LMod /\ G e. F /\ x e. ( K ` G ) ) -> x e. ( Base ` W ) )
22	17, 19, 20, 21	syl3anc 1326	. . . . . . 7 |- ( ( ( ( W e. LMod /\ G e. F ) /\ r e. ( Base ` (Scalar ` W ) ) ) /\ ( x e. ( K ` G ) /\ y e. ( K ` G ) ) ) -> x e. ( Base ` W ) )
23		eqid 2622	. . . . . . . 8 |- ( .s ` W ) = ( .s ` W )
24		eqid 2622	. . . . . . . 8 |- ( Base ` (Scalar ` W ) ) = ( Base ` (Scalar ` W ) )
25	1, 2, 23, 24	lmodvscl 18880	. . . . . . 7 |- ( ( W e. LMod /\ r e. ( Base ` (Scalar ` W ) ) /\ x e. ( Base ` W ) ) -> ( r ( .s ` W ) x ) e. ( Base ` W ) )
26	17, 18, 22, 25	syl3anc 1326	. . . . . 6 |- ( ( ( ( W e. LMod /\ G e. F ) /\ r e. ( Base ` (Scalar ` W ) ) ) /\ ( x e. ( K ` G ) /\ y e. ( K ` G ) ) ) -> ( r ( .s ` W ) x ) e. ( Base ` W ) )
27		simprr 796	. . . . . . 7 |- ( ( ( ( W e. LMod /\ G e. F ) /\ r e. ( Base ` (Scalar ` W ) ) ) /\ ( x e. ( K ` G ) /\ y e. ( K ` G ) ) ) -> y e. ( K ` G ) )
28	1, 4, 5	lkrcl 34379	. . . . . . 7 |- ( ( W e. LMod /\ G e. F /\ y e. ( K ` G ) ) -> y e. ( Base ` W ) )
29	17, 19, 27, 28	syl3anc 1326	. . . . . 6 |- ( ( ( ( W e. LMod /\ G e. F ) /\ r e. ( Base ` (Scalar ` W ) ) ) /\ ( x e. ( K ` G ) /\ y e. ( K ` G ) ) ) -> y e. ( Base ` W ) )
30		eqid 2622	. . . . . . 7 |- ( +g ` W ) = ( +g ` W )
31	1, 30	lmodvacl 18877	. . . . . 6 |- ( ( W e. LMod /\ ( r ( .s ` W ) x ) e. ( Base ` W ) /\ y e. ( Base ` W ) ) -> ( ( r ( .s ` W ) x ) ( +g ` W ) y ) e. ( Base ` W ) )
32	17, 26, 29, 31	syl3anc 1326	. . . . 5 |- ( ( ( ( W e. LMod /\ G e. F ) /\ r e. ( Base ` (Scalar ` W ) ) ) /\ ( x e. ( K ` G ) /\ y e. ( K ` G ) ) ) -> ( ( r ( .s ` W ) x ) ( +g ` W ) y ) e. ( Base ` W ) )
33		eqid 2622	. . . . . . . 8 |- ( +g ` (Scalar ` W ) ) = ( +g ` (Scalar ` W ) )
34		eqid 2622	. . . . . . . 8 |- ( .r ` (Scalar ` W ) ) = ( .r ` (Scalar ` W ) )
35	1, 30, 2, 23, 24, 33, 34, 4	lfli 34348	. . . . . . 7 |- ( ( W e. LMod /\ G e. F /\ ( r e. ( Base ` (Scalar ` W ) ) /\ x e. ( Base ` W ) /\ y e. ( Base ` W ) ) ) -> ( G ` ( ( r ( .s ` W ) x ) ( +g ` W ) y ) ) = ( ( r ( .r ` (Scalar ` W ) ) ( G ` x ) ) ( +g ` (Scalar ` W ) ) ( G ` y ) ) )
36	17, 19, 18, 22, 29, 35	syl113anc 1338	. . . . . 6 |- ( ( ( ( W e. LMod /\ G e. F ) /\ r e. ( Base ` (Scalar ` W ) ) ) /\ ( x e. ( K ` G ) /\ y e. ( K ` G ) ) ) -> ( G ` ( ( r ( .s ` W ) x ) ( +g ` W ) y ) ) = ( ( r ( .r ` (Scalar ` W ) ) ( G ` x ) ) ( +g ` (Scalar ` W ) ) ( G ` y ) ) )
37	2, 3, 4, 5	lkrf0 34380	. . . . . . . . . 10 |- ( ( W e. LMod /\ G e. F /\ x e. ( K ` G ) ) -> ( G ` x ) = ( 0g ` (Scalar ` W ) ) )
38	17, 19, 20, 37	syl3anc 1326	. . . . . . . . 9 |- ( ( ( ( W e. LMod /\ G e. F ) /\ r e. ( Base ` (Scalar ` W ) ) ) /\ ( x e. ( K ` G ) /\ y e. ( K ` G ) ) ) -> ( G ` x ) = ( 0g ` (Scalar ` W ) ) )
39	38	oveq2d 6666	. . . . . . . 8 |- ( ( ( ( W e. LMod /\ G e. F ) /\ r e. ( Base ` (Scalar ` W ) ) ) /\ ( x e. ( K ` G ) /\ y e. ( K ` G ) ) ) -> ( r ( .r ` (Scalar ` W ) ) ( G ` x ) ) = ( r ( .r ` (Scalar ` W ) ) ( 0g ` (Scalar ` W ) ) ) )
40	2	lmodring 18871	. . . . . . . . . 10 |- ( W e. LMod -> (Scalar ` W ) e. Ring )
41	17, 40	syl 17	. . . . . . . . 9 |- ( ( ( ( W e. LMod /\ G e. F ) /\ r e. ( Base ` (Scalar ` W ) ) ) /\ ( x e. ( K ` G ) /\ y e. ( K ` G ) ) ) -> (Scalar ` W ) e. Ring )
42	24, 34, 3	ringrz 18588	. . . . . . . . 9 |- ( ( (Scalar ` W ) e. Ring /\ r e. ( Base ` (Scalar ` W ) ) ) -> ( r ( .r ` (Scalar ` W ) ) ( 0g ` (Scalar ` W ) ) ) = ( 0g ` (Scalar ` W ) ) )
43	41, 18, 42	syl2anc 693	. . . . . . . 8 |- ( ( ( ( W e. LMod /\ G e. F ) /\ r e. ( Base ` (Scalar ` W ) ) ) /\ ( x e. ( K ` G ) /\ y e. ( K ` G ) ) ) -> ( r ( .r ` (Scalar ` W ) ) ( 0g ` (Scalar ` W ) ) ) = ( 0g ` (Scalar ` W ) ) )
44	39, 43	eqtrd 2656	. . . . . . 7 |- ( ( ( ( W e. LMod /\ G e. F ) /\ r e. ( Base ` (Scalar ` W ) ) ) /\ ( x e. ( K ` G ) /\ y e. ( K ` G ) ) ) -> ( r ( .r ` (Scalar ` W ) ) ( G ` x ) ) = ( 0g ` (Scalar ` W ) ) )
45	2, 3, 4, 5	lkrf0 34380	. . . . . . . 8 |- ( ( W e. LMod /\ G e. F /\ y e. ( K ` G ) ) -> ( G ` y ) = ( 0g ` (Scalar ` W ) ) )
46	17, 19, 27, 45	syl3anc 1326	. . . . . . 7 |- ( ( ( ( W e. LMod /\ G e. F ) /\ r e. ( Base ` (Scalar ` W ) ) ) /\ ( x e. ( K ` G ) /\ y e. ( K ` G ) ) ) -> ( G ` y ) = ( 0g ` (Scalar ` W ) ) )
47	44, 46	oveq12d 6668	. . . . . 6 |- ( ( ( ( W e. LMod /\ G e. F ) /\ r e. ( Base ` (Scalar ` W ) ) ) /\ ( x e. ( K ` G ) /\ y e. ( K ` G ) ) ) -> ( ( r ( .r ` (Scalar ` W ) ) ( G ` x ) ) ( +g ` (Scalar ` W ) ) ( G ` y ) ) = ( ( 0g ` (Scalar ` W ) ) ( +g ` (Scalar ` W ) ) ( 0g ` (Scalar ` W ) ) ) )
48	2	lmodfgrp 18872	. . . . . . . 8 |- ( W e. LMod -> (Scalar ` W ) e. Grp )
49	17, 48	syl 17	. . . . . . 7 |- ( ( ( ( W e. LMod /\ G e. F ) /\ r e. ( Base ` (Scalar ` W ) ) ) /\ ( x e. ( K ` G ) /\ y e. ( K ` G ) ) ) -> (Scalar ` W ) e. Grp )
50	24, 3	grpidcl 17450	. . . . . . . 8 |- ( (Scalar ` W ) e. Grp -> ( 0g ` (Scalar ` W ) ) e. ( Base ` (Scalar ` W ) ) )
51	49, 50	syl 17	. . . . . . 7 |- ( ( ( ( W e. LMod /\ G e. F ) /\ r e. ( Base ` (Scalar ` W ) ) ) /\ ( x e. ( K ` G ) /\ y e. ( K ` G ) ) ) -> ( 0g ` (Scalar ` W ) ) e. ( Base ` (Scalar ` W ) ) )
52	24, 33, 3	grplid 17452	. . . . . . 7 |- ( ( (Scalar ` W ) e. Grp /\ ( 0g ` (Scalar ` W ) ) e. ( Base ` (Scalar ` W ) ) ) -> ( ( 0g ` (Scalar ` W ) ) ( +g ` (Scalar ` W ) ) ( 0g ` (Scalar ` W ) ) ) = ( 0g ` (Scalar ` W ) ) )
53	49, 51, 52	syl2anc 693	. . . . . 6 |- ( ( ( ( W e. LMod /\ G e. F ) /\ r e. ( Base ` (Scalar ` W ) ) ) /\ ( x e. ( K ` G ) /\ y e. ( K ` G ) ) ) -> ( ( 0g ` (Scalar ` W ) ) ( +g ` (Scalar ` W ) ) ( 0g ` (Scalar ` W ) ) ) = ( 0g ` (Scalar ` W ) ) )
54	36, 47, 53	3eqtrd 2660	. . . . 5 |- ( ( ( ( W e. LMod /\ G e. F ) /\ r e. ( Base ` (Scalar ` W ) ) ) /\ ( x e. ( K ` G ) /\ y e. ( K ` G ) ) ) -> ( G ` ( ( r ( .s ` W ) x ) ( +g ` W ) y ) ) = ( 0g ` (Scalar ` W ) ) )
55	1, 2, 3, 4, 5	ellkr 34376	. . . . . 6 |- ( ( W e. LMod /\ G e. F ) -> ( ( ( r ( .s ` W ) x ) ( +g ` W ) y ) e. ( K ` G ) <-> ( ( ( r ( .s ` W ) x ) ( +g ` W ) y ) e. ( Base ` W ) /\ ( G ` ( ( r ( .s ` W ) x ) ( +g ` W ) y ) ) = ( 0g ` (Scalar ` W ) ) ) ) )
56	55	ad2antrr 762	. . . . 5 |- ( ( ( ( W e. LMod /\ G e. F ) /\ r e. ( Base ` (Scalar ` W ) ) ) /\ ( x e. ( K ` G ) /\ y e. ( K ` G ) ) ) -> ( ( ( r ( .s ` W ) x ) ( +g ` W ) y ) e. ( K ` G ) <-> ( ( ( r ( .s ` W ) x ) ( +g ` W ) y ) e. ( Base ` W ) /\ ( G ` ( ( r ( .s ` W ) x ) ( +g ` W ) y ) ) = ( 0g ` (Scalar ` W ) ) ) ) )
57	32, 54, 56	mpbir2and 957	. . . 4 |- ( ( ( ( W e. LMod /\ G e. F ) /\ r e. ( Base ` (Scalar ` W ) ) ) /\ ( x e. ( K ` G ) /\ y e. ( K ` G ) ) ) -> ( ( r ( .s ` W ) x ) ( +g ` W ) y ) e. ( K ` G ) )
58	57	ralrimivva 2971	. . 3 |- ( ( ( W e. LMod /\ G e. F ) /\ r e. ( Base ` (Scalar ` W ) ) ) -> A. x e. ( K ` G ) A. y e. ( K ` G ) ( ( r ( .s ` W ) x ) ( +g ` W ) y ) e. ( K ` G ) )
59	58	ralrimiva 2966	. 2 |- ( ( W e. LMod /\ G e. F ) -> A. r e. ( Base ` (Scalar ` W ) ) A. x e. ( K ` G ) A. y e. ( K ` G ) ( ( r ( .s ` W ) x ) ( +g ` W ) y ) e. ( K ` G ) )
60		lkrlss.s	. . 3 |- S = ( LSubSp ` W )
61	2, 24, 1, 30, 23, 60	islss 18935	. 2 |- ( ( K ` G ) e. S <-> ( ( K ` G ) C_ ( Base ` W ) /\ ( K ` G ) =/= (/) /\ A. r e. ( Base ` (Scalar ` W ) ) A. x e. ( K ` G ) A. y e. ( K ` G ) ( ( r ( .s ` W ) x ) ( +g ` W ) y ) e. ( K ` G ) ) )
62	8, 16, 59, 61	syl3anbrc 1246	1 |- ( ( W e. LMod /\ G e. F ) -> ( K ` G ) e. S )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   {crab 2916   C_ wss 3574   (/)c0 3915   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   .rcmulr 15942  Scalarcsca 15944   .scvsca 15945   0gc0g 16100   Grpcgrp 17422   Ringcrg 18547   LModclmod 18863   LSubSpclss 18932  LFnlclfn 34344  LKerclk 34372

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-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-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-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-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-sbg 17427  df-mgp 18490  df-ur 18502  df-ring 18549  df-lmod 18865  df-lss 18933  df-lfl 34345  df-lkr 34373

This theorem is referenced by:  lkrssv  34383  lkrlsp  34389  lkrlsp3  34391  lkrshp  34392  lclkrlem2f  36801  lclkrlem2n  36809  lclkrlem2v  36817  lcfrlem25  36856  lcfrlem35  36866