Theorem lkrscss 34385

Description: The kernel of a scalar product of a functional includes the kernel of the functional. (The inclusion is proper for the zero product and equality otherwise.) (Contributed by NM, 9-Oct-2014.)

Hypotheses

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

Assertion

Ref	Expression
lkrscss	|- ( ph -> ( L ` G ) C_ ( L ` ( G oF .x. ( V X. { R } ) ) ) )

Proof of Theorem lkrscss

Step	Hyp	Ref	Expression
1		lkrsc.v	. . . . . 6 |- V = ( Base ` W )
2		lkrsc.f	. . . . . 6 |- F = (LFnl ` W )
3		lkrsc.l	. . . . . 6 |- L = (LKer ` W )
4		lkrsc.w	. . . . . . 7 |- ( ph -> W e. LVec )
5		lveclmod 19106	. . . . . . 7 |- ( W e. LVec -> W e. LMod )
6	4, 5	syl 17	. . . . . 6 |- ( ph -> W e. LMod )
7		lkrsc.g	. . . . . 6 |- ( ph -> G e. F )
8	1, 2, 3, 6, 7	lkrssv 34383	. . . . 5 |- ( ph -> ( L ` G ) C_ V )
9		lkrsc.d	. . . . . . . 8 |- D = (Scalar ` W )
10		lkrsc.k	. . . . . . . 8 |- K = ( Base ` D )
11		lkrsc.t	. . . . . . . 8 |- .x. = ( .r ` D )
12		eqid 2622	. . . . . . . 8 |- ( 0g ` D ) = ( 0g ` D )
13	1, 9, 2, 10, 11, 12, 6, 7	lfl0sc 34369	. . . . . . 7 |- ( ph -> ( G oF .x. ( V X. { ( 0g ` D ) } ) ) = ( V X. { ( 0g ` D ) } ) )
14	13	fveq2d 6195	. . . . . 6 |- ( ph -> ( L ` ( G oF .x. ( V X. { ( 0g ` D ) } ) ) ) = ( L ` ( V X. { ( 0g ` D ) } ) ) )
15		eqid 2622	. . . . . . 7 |- ( V X. { ( 0g ` D ) } ) = ( V X. { ( 0g ` D ) } )
16	9, 12, 1, 2	lfl0f 34356	. . . . . . . . 9 |- ( W e. LMod -> ( V X. { ( 0g ` D ) } ) e. F )
17	6, 16	syl 17	. . . . . . . 8 |- ( ph -> ( V X. { ( 0g ` D ) } ) e. F )
18	9, 12, 1, 2, 3	lkr0f 34381	. . . . . . . 8 |- ( ( W e. LMod /\ ( V X. { ( 0g ` D ) } ) e. F ) -> ( ( L ` ( V X. { ( 0g ` D ) } ) ) = V <-> ( V X. { ( 0g ` D ) } ) = ( V X. { ( 0g ` D ) } ) ) )
19	6, 17, 18	syl2anc 693	. . . . . . 7 |- ( ph -> ( ( L ` ( V X. { ( 0g ` D ) } ) ) = V <-> ( V X. { ( 0g ` D ) } ) = ( V X. { ( 0g ` D ) } ) ) )
20	15, 19	mpbiri 248	. . . . . 6 |- ( ph -> ( L ` ( V X. { ( 0g ` D ) } ) ) = V )
21	14, 20	eqtr2d 2657	. . . . 5 |- ( ph -> V = ( L ` ( G oF .x. ( V X. { ( 0g ` D ) } ) ) ) )
22	8, 21	sseqtrd 3641	. . . 4 |- ( ph -> ( L ` G ) C_ ( L ` ( G oF .x. ( V X. { ( 0g ` D ) } ) ) ) )
23	22	adantr 481	. . 3 |- ( ( ph /\ R = ( 0g ` D ) ) -> ( L ` G ) C_ ( L ` ( G oF .x. ( V X. { ( 0g ` D ) } ) ) ) )
24		sneq 4187	. . . . . . 7 |- ( R = ( 0g ` D ) -> { R } = { ( 0g ` D ) } )
25	24	xpeq2d 5139	. . . . . 6 |- ( R = ( 0g ` D ) -> ( V X. { R } ) = ( V X. { ( 0g ` D ) } ) )
26	25	oveq2d 6666	. . . . 5 |- ( R = ( 0g ` D ) -> ( G oF .x. ( V X. { R } ) ) = ( G oF .x. ( V X. { ( 0g ` D ) } ) ) )
27	26	fveq2d 6195	. . . 4 |- ( R = ( 0g ` D ) -> ( L ` ( G oF .x. ( V X. { R } ) ) ) = ( L ` ( G oF .x. ( V X. { ( 0g ` D ) } ) ) ) )
28	27	adantl 482	. . 3 |- ( ( ph /\ R = ( 0g ` D ) ) -> ( L ` ( G oF .x. ( V X. { R } ) ) ) = ( L ` ( G oF .x. ( V X. { ( 0g ` D ) } ) ) ) )
29	23, 28	sseqtr4d 3642	. 2 |- ( ( ph /\ R = ( 0g ` D ) ) -> ( L ` G ) C_ ( L ` ( G oF .x. ( V X. { R } ) ) ) )
30	4	adantr 481	. . . 4 |- ( ( ph /\ R =/= ( 0g ` D ) ) -> W e. LVec )
31	7	adantr 481	. . . 4 |- ( ( ph /\ R =/= ( 0g ` D ) ) -> G e. F )
32		lkrsc.r	. . . . 5 |- ( ph -> R e. K )
33	32	adantr 481	. . . 4 |- ( ( ph /\ R =/= ( 0g ` D ) ) -> R e. K )
34		simpr 477	. . . 4 |- ( ( ph /\ R =/= ( 0g ` D ) ) -> R =/= ( 0g ` D ) )
35	1, 9, 10, 11, 2, 3, 30, 31, 33, 12, 34	lkrsc 34384	. . 3 |- ( ( ph /\ R =/= ( 0g ` D ) ) -> ( L ` ( G oF .x. ( V X. { R } ) ) ) = ( L ` G ) )
36		eqimss2 3658	. . 3 |- ( ( L ` ( G oF .x. ( V X. { R } ) ) ) = ( L ` G ) -> ( L ` G ) C_ ( L ` ( G oF .x. ( V X. { R } ) ) ) )
37	35, 36	syl 17	. 2 |- ( ( ph /\ R =/= ( 0g ` D ) ) -> ( L ` G ) C_ ( L ` ( G oF .x. ( V X. { R } ) ) ) )
38	29, 37	pm2.61dane 2881	1 |- ( ph -> ( L ` G ) C_ ( L ` ( G oF .x. ( V X. { R } ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   C_ wss 3574   {csn 4177   X. cxp 5112   ` cfv 5888  (class class class)co 6650   oFcof 6895   Basecbs 15857   .rcmulr 15942  Scalarcsca 15944   0gc0g 16100   LModclmod 18863   LVecclvec 19102  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-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-tpos 7352  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-3 11080  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  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-oppr 18623  df-dvdsr 18641  df-unit 18642  df-invr 18672  df-drng 18749  df-lmod 18865  df-lss 18933  df-lvec 19103  df-lfl 34345  df-lkr 34373

This theorem is referenced by:  lfl1dim  34408  lfl1dim2N  34409  lkrss  34455