Theorem lkr0f 34381

Description: The kernel of the zero functional is the set of all vectors. (Contributed by NM, 17-Apr-2014.)

Hypotheses

Ref	Expression
lkr0f.d	|- D = (Scalar ` W )
lkr0f.o	|- .0. = ( 0g ` D )
lkr0f.v	|- V = ( Base ` W )
lkr0f.f	|- F = (LFnl ` W )
lkr0f.k	|- K = (LKer ` W )

Assertion

Ref	Expression
lkr0f	|- ( ( W e. LMod /\ G e. F ) -> ( ( K ` G ) = V <-> G = ( V X. { .0. } ) ) )

Proof of Theorem lkr0f

Step	Hyp	Ref	Expression
1		lkr0f.d	. . . . . . 7 |- D = (Scalar ` W )
2		eqid 2622	. . . . . . 7 |- ( Base ` D ) = ( Base ` D )
3		lkr0f.v	. . . . . . 7 |- V = ( Base ` W )
4		lkr0f.f	. . . . . . 7 |- F = (LFnl ` W )
5	1, 2, 3, 4	lflf 34350	. . . . . 6 |- ( ( W e. LMod /\ G e. F ) -> G : V --> ( Base ` D ) )
6		ffn 6045	. . . . . 6 |- ( G : V --> ( Base ` D ) -> G Fn V )
7	5, 6	syl 17	. . . . 5 |- ( ( W e. LMod /\ G e. F ) -> G Fn V )
8	7	adantr 481	. . . 4 |- ( ( ( W e. LMod /\ G e. F ) /\ ( K ` G ) = V ) -> G Fn V )
9		lkr0f.o	. . . . . . 7 |- .0. = ( 0g ` D )
10		lkr0f.k	. . . . . . 7 |- K = (LKer ` W )
11	1, 9, 4, 10	lkrval 34375	. . . . . 6 |- ( ( W e. LMod /\ G e. F ) -> ( K ` G ) = ( `' G " { .0. } ) )
12	11	eqeq1d 2624	. . . . 5 |- ( ( W e. LMod /\ G e. F ) -> ( ( K ` G ) = V <-> ( `' G " { .0. } ) = V ) )
13	12	biimpa 501	. . . 4 |- ( ( ( W e. LMod /\ G e. F ) /\ ( K ` G ) = V ) -> ( `' G " { .0. } ) = V )
14		fvex 6201	. . . . . . 7 |- ( 0g ` D ) e. _V
15	9, 14	eqeltri 2697	. . . . . 6 |- .0. e. _V
16	15	fconst2 6470	. . . . 5 |- ( G : V --> { .0. } <-> G = ( V X. { .0. } ) )
17		fconst4 6478	. . . . 5 |- ( G : V --> { .0. } <-> ( G Fn V /\ ( `' G " { .0. } ) = V ) )
18	16, 17	bitr3i 266	. . . 4 |- ( G = ( V X. { .0. } ) <-> ( G Fn V /\ ( `' G " { .0. } ) = V ) )
19	8, 13, 18	sylanbrc 698	. . 3 |- ( ( ( W e. LMod /\ G e. F ) /\ ( K ` G ) = V ) -> G = ( V X. { .0. } ) )
20	19	ex 450	. 2 |- ( ( W e. LMod /\ G e. F ) -> ( ( K ` G ) = V -> G = ( V X. { .0. } ) ) )
21	18	biimpi 206	. . . . . 6 |- ( G = ( V X. { .0. } ) -> ( G Fn V /\ ( `' G " { .0. } ) = V ) )
22	21	adantl 482	. . . . 5 |- ( ( W e. LMod /\ G = ( V X. { .0. } ) ) -> ( G Fn V /\ ( `' G " { .0. } ) = V ) )
23		simpr 477	. . . . . . . . 9 |- ( ( W e. LMod /\ G = ( V X. { .0. } ) ) -> G = ( V X. { .0. } ) )
24		eqid 2622	. . . . . . . . . . 11 |- (LFnl ` W ) = (LFnl ` W )
25	1, 9, 3, 24	lfl0f 34356	. . . . . . . . . 10 |- ( W e. LMod -> ( V X. { .0. } ) e. (LFnl ` W ) )
26	25	adantr 481	. . . . . . . . 9 |- ( ( W e. LMod /\ G = ( V X. { .0. } ) ) -> ( V X. { .0. } ) e. (LFnl ` W ) )
27	23, 26	eqeltrd 2701	. . . . . . . 8 |- ( ( W e. LMod /\ G = ( V X. { .0. } ) ) -> G e. (LFnl ` W ) )
28	1, 9, 24, 10	lkrval 34375	. . . . . . . 8 |- ( ( W e. LMod /\ G e. (LFnl ` W ) ) -> ( K ` G ) = ( `' G " { .0. } ) )
29	27, 28	syldan 487	. . . . . . 7 |- ( ( W e. LMod /\ G = ( V X. { .0. } ) ) -> ( K ` G ) = ( `' G " { .0. } ) )
30	29	eqeq1d 2624	. . . . . 6 |- ( ( W e. LMod /\ G = ( V X. { .0. } ) ) -> ( ( K ` G ) = V <-> ( `' G " { .0. } ) = V ) )
31		ffn 6045	. . . . . . . . 9 |- ( G : V --> { .0. } -> G Fn V )
32	16, 31	sylbir 225	. . . . . . . 8 |- ( G = ( V X. { .0. } ) -> G Fn V )
33	32	adantl 482	. . . . . . 7 |- ( ( W e. LMod /\ G = ( V X. { .0. } ) ) -> G Fn V )
34	33	biantrurd 529	. . . . . 6 |- ( ( W e. LMod /\ G = ( V X. { .0. } ) ) -> ( ( `' G " { .0. } ) = V <-> ( G Fn V /\ ( `' G " { .0. } ) = V ) ) )
35	30, 34	bitrd 268	. . . . 5 |- ( ( W e. LMod /\ G = ( V X. { .0. } ) ) -> ( ( K ` G ) = V <-> ( G Fn V /\ ( `' G " { .0. } ) = V ) ) )
36	22, 35	mpbird 247	. . . 4 |- ( ( W e. LMod /\ G = ( V X. { .0. } ) ) -> ( K ` G ) = V )
37	36	ex 450	. . 3 |- ( W e. LMod -> ( G = ( V X. { .0. } ) -> ( K ` G ) = V ) )
38	37	adantr 481	. 2 |- ( ( W e. LMod /\ G e. F ) -> ( G = ( V X. { .0. } ) -> ( K ` G ) = V ) )
39	20, 38	impbid 202	1 |- ( ( W e. LMod /\ G e. F ) -> ( ( K ` G ) = V <-> G = ( V X. { .0. } ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   {csn 4177   X. cxp 5112   `'ccnv 5113   "cima 5117   Fn wfn 5883   -->wf 5884   ` cfv 5888   Basecbs 15857  Scalarcsca 15944   0gc0g 16100   LModclmod 18863  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-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-mgp 18490  df-ring 18549  df-lmod 18865  df-lfl 34345  df-lkr 34373

This theorem is referenced by:  lkrscss  34385  eqlkr  34386  lkrshp  34392  lkrshp3  34393  lkrshpor  34394  lfl1dim  34408  lfl1dim2N  34409  lkr0f2  34448  lclkrlem1  36795  lclkrlem2j  36805  lclkr  36822  lclkrs  36828  mapd0  36954