Theorem lfl0f 34356

Description: The zero function is a functional. (Contributed by NM, 16-Apr-2014.)

Hypotheses

Ref	Expression
lfl0f.d	|- D = (Scalar ` W )
lfl0f.o	|- .0. = ( 0g ` D )
lfl0f.v	|- V = ( Base ` W )
lfl0f.f	|- F = (LFnl ` W )

Assertion

Ref	Expression
lfl0f	|- ( W e. LMod -> ( V X. { .0. } ) e. F )

Proof of Theorem lfl0f

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

Step	Hyp	Ref	Expression
1		lfl0f.o	. . . . 5 |- .0. = ( 0g ` D )
2		fvex 6201	. . . . 5 |- ( 0g ` D ) e. _V
3	1, 2	eqeltri 2697	. . . 4 |- .0. e. _V
4	3	fconst 6091	. . 3 |- ( V X. { .0. } ) : V --> { .0. }
5		lfl0f.d	. . . . 5 |- D = (Scalar ` W )
6		eqid 2622	. . . . 5 |- ( Base ` D ) = ( Base ` D )
7	5, 6, 1	lmod0cl 18889	. . . 4 |- ( W e. LMod -> .0. e. ( Base ` D ) )
8	7	snssd 4340	. . 3 |- ( W e. LMod -> { .0. } C_ ( Base ` D ) )
9		fss 6056	. . 3 |- ( ( ( V X. { .0. } ) : V --> { .0. } /\ { .0. } C_ ( Base ` D ) ) -> ( V X. { .0. } ) : V --> ( Base ` D ) )
10	4, 8, 9	sylancr 695	. 2 |- ( W e. LMod -> ( V X. { .0. } ) : V --> ( Base ` D ) )
11	5	lmodring 18871	. . . . . . . . 9 |- ( W e. LMod -> D e. Ring )
12	11	ad2antrr 762	. . . . . . . 8 |- ( ( ( W e. LMod /\ ( r e. ( Base ` D ) /\ x e. V ) ) /\ y e. V ) -> D e. Ring )
13		simplrl 800	. . . . . . . 8 |- ( ( ( W e. LMod /\ ( r e. ( Base ` D ) /\ x e. V ) ) /\ y e. V ) -> r e. ( Base ` D ) )
14		eqid 2622	. . . . . . . . 9 |- ( .r ` D ) = ( .r ` D )
15	6, 14, 1	ringrz 18588	. . . . . . . 8 |- ( ( D e. Ring /\ r e. ( Base ` D ) ) -> ( r ( .r ` D ) .0. ) = .0. )
16	12, 13, 15	syl2anc 693	. . . . . . 7 |- ( ( ( W e. LMod /\ ( r e. ( Base ` D ) /\ x e. V ) ) /\ y e. V ) -> ( r ( .r ` D ) .0. ) = .0. )
17	16	oveq1d 6665	. . . . . 6 |- ( ( ( W e. LMod /\ ( r e. ( Base ` D ) /\ x e. V ) ) /\ y e. V ) -> ( ( r ( .r ` D ) .0. ) ( +g ` D ) .0. ) = ( .0. ( +g ` D ) .0. ) )
18		ringgrp 18552	. . . . . . . 8 |- ( D e. Ring -> D e. Grp )
19	12, 18	syl 17	. . . . . . 7 |- ( ( ( W e. LMod /\ ( r e. ( Base ` D ) /\ x e. V ) ) /\ y e. V ) -> D e. Grp )
20	6, 1	grpidcl 17450	. . . . . . . 8 |- ( D e. Grp -> .0. e. ( Base ` D ) )
21	19, 20	syl 17	. . . . . . 7 |- ( ( ( W e. LMod /\ ( r e. ( Base ` D ) /\ x e. V ) ) /\ y e. V ) -> .0. e. ( Base ` D ) )
22		eqid 2622	. . . . . . . 8 |- ( +g ` D ) = ( +g ` D )
23	6, 22, 1	grplid 17452	. . . . . . 7 |- ( ( D e. Grp /\ .0. e. ( Base ` D ) ) -> ( .0. ( +g ` D ) .0. ) = .0. )
24	19, 21, 23	syl2anc 693	. . . . . 6 |- ( ( ( W e. LMod /\ ( r e. ( Base ` D ) /\ x e. V ) ) /\ y e. V ) -> ( .0. ( +g ` D ) .0. ) = .0. )
25	17, 24	eqtrd 2656	. . . . 5 |- ( ( ( W e. LMod /\ ( r e. ( Base ` D ) /\ x e. V ) ) /\ y e. V ) -> ( ( r ( .r ` D ) .0. ) ( +g ` D ) .0. ) = .0. )
26		simplrr 801	. . . . . . . 8 |- ( ( ( W e. LMod /\ ( r e. ( Base ` D ) /\ x e. V ) ) /\ y e. V ) -> x e. V )
27	3	fvconst2 6469	. . . . . . . 8 |- ( x e. V -> ( ( V X. { .0. } ) ` x ) = .0. )
28	26, 27	syl 17	. . . . . . 7 |- ( ( ( W e. LMod /\ ( r e. ( Base ` D ) /\ x e. V ) ) /\ y e. V ) -> ( ( V X. { .0. } ) ` x ) = .0. )
29	28	oveq2d 6666	. . . . . 6 |- ( ( ( W e. LMod /\ ( r e. ( Base ` D ) /\ x e. V ) ) /\ y e. V ) -> ( r ( .r ` D ) ( ( V X. { .0. } ) ` x ) ) = ( r ( .r ` D ) .0. ) )
30	3	fvconst2 6469	. . . . . . 7 |- ( y e. V -> ( ( V X. { .0. } ) ` y ) = .0. )
31	30	adantl 482	. . . . . 6 |- ( ( ( W e. LMod /\ ( r e. ( Base ` D ) /\ x e. V ) ) /\ y e. V ) -> ( ( V X. { .0. } ) ` y ) = .0. )
32	29, 31	oveq12d 6668	. . . . 5 |- ( ( ( W e. LMod /\ ( r e. ( Base ` D ) /\ x e. V ) ) /\ y e. V ) -> ( ( r ( .r ` D ) ( ( V X. { .0. } ) ` x ) ) ( +g ` D ) ( ( V X. { .0. } ) ` y ) ) = ( ( r ( .r ` D ) .0. ) ( +g ` D ) .0. ) )
33		simpll 790	. . . . . . 7 |- ( ( ( W e. LMod /\ ( r e. ( Base ` D ) /\ x e. V ) ) /\ y e. V ) -> W e. LMod )
34		lfl0f.v	. . . . . . . . 9 |- V = ( Base ` W )
35		eqid 2622	. . . . . . . . 9 |- ( .s ` W ) = ( .s ` W )
36	34, 5, 35, 6	lmodvscl 18880	. . . . . . . 8 |- ( ( W e. LMod /\ r e. ( Base ` D ) /\ x e. V ) -> ( r ( .s ` W ) x ) e. V )
37	33, 13, 26, 36	syl3anc 1326	. . . . . . 7 |- ( ( ( W e. LMod /\ ( r e. ( Base ` D ) /\ x e. V ) ) /\ y e. V ) -> ( r ( .s ` W ) x ) e. V )
38		simpr 477	. . . . . . 7 |- ( ( ( W e. LMod /\ ( r e. ( Base ` D ) /\ x e. V ) ) /\ y e. V ) -> y e. V )
39		eqid 2622	. . . . . . . 8 |- ( +g ` W ) = ( +g ` W )
40	34, 39	lmodvacl 18877	. . . . . . 7 |- ( ( W e. LMod /\ ( r ( .s ` W ) x ) e. V /\ y e. V ) -> ( ( r ( .s ` W ) x ) ( +g ` W ) y ) e. V )
41	33, 37, 38, 40	syl3anc 1326	. . . . . 6 |- ( ( ( W e. LMod /\ ( r e. ( Base ` D ) /\ x e. V ) ) /\ y e. V ) -> ( ( r ( .s ` W ) x ) ( +g ` W ) y ) e. V )
42	3	fvconst2 6469	. . . . . 6 |- ( ( ( r ( .s ` W ) x ) ( +g ` W ) y ) e. V -> ( ( V X. { .0. } ) ` ( ( r ( .s ` W ) x ) ( +g ` W ) y ) ) = .0. )
43	41, 42	syl 17	. . . . 5 |- ( ( ( W e. LMod /\ ( r e. ( Base ` D ) /\ x e. V ) ) /\ y e. V ) -> ( ( V X. { .0. } ) ` ( ( r ( .s ` W ) x ) ( +g ` W ) y ) ) = .0. )
44	25, 32, 43	3eqtr4rd 2667	. . . 4 |- ( ( ( W e. LMod /\ ( r e. ( Base ` D ) /\ x e. V ) ) /\ y e. V ) -> ( ( V X. { .0. } ) ` ( ( r ( .s ` W ) x ) ( +g ` W ) y ) ) = ( ( r ( .r ` D ) ( ( V X. { .0. } ) ` x ) ) ( +g ` D ) ( ( V X. { .0. } ) ` y ) ) )
45	44	ralrimiva 2966	. . 3 |- ( ( W e. LMod /\ ( r e. ( Base ` D ) /\ x e. V ) ) -> A. y e. V ( ( V X. { .0. } ) ` ( ( r ( .s ` W ) x ) ( +g ` W ) y ) ) = ( ( r ( .r ` D ) ( ( V X. { .0. } ) ` x ) ) ( +g ` D ) ( ( V X. { .0. } ) ` y ) ) )
46	45	ralrimivva 2971	. 2 |- ( W e. LMod -> A. r e. ( Base ` D ) A. x e. V A. y e. V ( ( V X. { .0. } ) ` ( ( r ( .s ` W ) x ) ( +g ` W ) y ) ) = ( ( r ( .r ` D ) ( ( V X. { .0. } ) ` x ) ) ( +g ` D ) ( ( V X. { .0. } ) ` y ) ) )
47		lfl0f.f	. . 3 |- F = (LFnl ` W )
48	34, 39, 5, 35, 6, 22, 14, 47	islfl 34347	. 2 |- ( W e. LMod -> ( ( V X. { .0. } ) e. F <-> ( ( V X. { .0. } ) : V --> ( Base ` D ) /\ A. r e. ( Base ` D ) A. x e. V A. y e. V ( ( V X. { .0. } ) ` ( ( r ( .s ` W ) x ) ( +g ` W ) y ) ) = ( ( r ( .r ` D ) ( ( V X. { .0. } ) ` x ) ) ( +g ` D ) ( ( V X. { .0. } ) ` y ) ) ) ) )
49	10, 46, 48	mpbir2and 957	1 |- ( W e. LMod -> ( V X. { .0. } ) e. F )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   C_ wss 3574   {csn 4177   X. cxp 5112   -->wf 5884   ` 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  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-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

This theorem is referenced by:  lkr0f  34381  lkrscss  34385  ldualgrplem  34432  ldual0v  34437  ldual0vcl  34438  lclkrlem1  36795  lclkr  36822  lclkrs  36828