Theorem lsslindf 20169

Description: Linear independence is unchanged by working in a subspace. (Contributed by Stefan O'Rear, 24-Feb-2015.) (Revised by Stefan O'Rear, 6-May-2015.)

Hypotheses

Ref	Expression
lsslindf.u	|- U = ( LSubSp ` W )
lsslindf.x	|- X = ( W ↾s S )

Assertion

Ref	Expression
lsslindf	|- ( ( W e. LMod /\ S e. U /\ ran F C_ S ) -> ( F LIndF X <-> F LIndF W ) )

Proof of Theorem lsslindf

Dummy variables k x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rellindf 20147	. . . 4 |- Rel LIndF
2	1	brrelexi 5158	. . 3 |- ( F LIndF X -> F e. _V )
3	2	a1i 11	. 2 |- ( ( W e. LMod /\ S e. U /\ ran F C_ S ) -> ( F LIndF X -> F e. _V ) )
4	1	brrelexi 5158	. . 3 |- ( F LIndF W -> F e. _V )
5	4	a1i 11	. 2 |- ( ( W e. LMod /\ S e. U /\ ran F C_ S ) -> ( F LIndF W -> F e. _V ) )
6		simpr 477	. . . . . . . 8 |- ( ( ( W e. LMod /\ S e. U /\ ran F C_ S ) /\ F : dom F --> ( Base ` X ) ) -> F : dom F --> ( Base ` X ) )
7		lsslindf.x	. . . . . . . . 9 |- X = ( W ↾s S )
8		eqid 2622	. . . . . . . . 9 |- ( Base ` W ) = ( Base ` W )
9	7, 8	ressbasss 15932	. . . . . . . 8 |- ( Base ` X ) C_ ( Base ` W )
10		fss 6056	. . . . . . . 8 |- ( ( F : dom F --> ( Base ` X ) /\ ( Base ` X ) C_ ( Base ` W ) ) -> F : dom F --> ( Base ` W ) )
11	6, 9, 10	sylancl 694	. . . . . . 7 |- ( ( ( W e. LMod /\ S e. U /\ ran F C_ S ) /\ F : dom F --> ( Base ` X ) ) -> F : dom F --> ( Base ` W ) )
12		ffn 6045	. . . . . . . . 9 |- ( F : dom F --> ( Base ` W ) -> F Fn dom F )
13	12	adantl 482	. . . . . . . 8 |- ( ( ( W e. LMod /\ S e. U /\ ran F C_ S ) /\ F : dom F --> ( Base ` W ) ) -> F Fn dom F )
14		simp3 1063	. . . . . . . . . 10 |- ( ( W e. LMod /\ S e. U /\ ran F C_ S ) -> ran F C_ S )
15		lsslindf.u	. . . . . . . . . . . . 13 |- U = ( LSubSp ` W )
16	8, 15	lssss 18937	. . . . . . . . . . . 12 |- ( S e. U -> S C_ ( Base ` W ) )
17	16	3ad2ant2 1083	. . . . . . . . . . 11 |- ( ( W e. LMod /\ S e. U /\ ran F C_ S ) -> S C_ ( Base ` W ) )
18	7, 8	ressbas2 15931	. . . . . . . . . . 11 |- ( S C_ ( Base ` W ) -> S = ( Base ` X ) )
19	17, 18	syl 17	. . . . . . . . . 10 |- ( ( W e. LMod /\ S e. U /\ ran F C_ S ) -> S = ( Base ` X ) )
20	14, 19	sseqtrd 3641	. . . . . . . . 9 |- ( ( W e. LMod /\ S e. U /\ ran F C_ S ) -> ran F C_ ( Base ` X ) )
21	20	adantr 481	. . . . . . . 8 |- ( ( ( W e. LMod /\ S e. U /\ ran F C_ S ) /\ F : dom F --> ( Base ` W ) ) -> ran F C_ ( Base ` X ) )
22		df-f 5892	. . . . . . . 8 |- ( F : dom F --> ( Base ` X ) <-> ( F Fn dom F /\ ran F C_ ( Base ` X ) ) )
23	13, 21, 22	sylanbrc 698	. . . . . . 7 |- ( ( ( W e. LMod /\ S e. U /\ ran F C_ S ) /\ F : dom F --> ( Base ` W ) ) -> F : dom F --> ( Base ` X ) )
24	11, 23	impbida 877	. . . . . 6 |- ( ( W e. LMod /\ S e. U /\ ran F C_ S ) -> ( F : dom F --> ( Base ` X ) <-> F : dom F --> ( Base ` W ) ) )
25	24	adantr 481	. . . . 5 |- ( ( ( W e. LMod /\ S e. U /\ ran F C_ S ) /\ F e. _V ) -> ( F : dom F --> ( Base ` X ) <-> F : dom F --> ( Base ` W ) ) )
26		simpl2 1065	. . . . . . . . . 10 |- ( ( ( W e. LMod /\ S e. U /\ ran F C_ S ) /\ F e. _V ) -> S e. U )
27		eqid 2622	. . . . . . . . . . . 12 |- (Scalar ` W ) = (Scalar ` W )
28	7, 27	resssca 16031	. . . . . . . . . . 11 |- ( S e. U -> (Scalar ` W ) = (Scalar ` X ) )
29	28	eqcomd 2628	. . . . . . . . . 10 |- ( S e. U -> (Scalar ` X ) = (Scalar ` W ) )
30	26, 29	syl 17	. . . . . . . . 9 |- ( ( ( W e. LMod /\ S e. U /\ ran F C_ S ) /\ F e. _V ) -> (Scalar ` X ) = (Scalar ` W ) )
31	30	fveq2d 6195	. . . . . . . 8 |- ( ( ( W e. LMod /\ S e. U /\ ran F C_ S ) /\ F e. _V ) -> ( Base ` (Scalar ` X ) ) = ( Base ` (Scalar ` W ) ) )
32	30	fveq2d 6195	. . . . . . . . 9 |- ( ( ( W e. LMod /\ S e. U /\ ran F C_ S ) /\ F e. _V ) -> ( 0g ` (Scalar ` X ) ) = ( 0g ` (Scalar ` W ) ) )
33	32	sneqd 4189	. . . . . . . 8 |- ( ( ( W e. LMod /\ S e. U /\ ran F C_ S ) /\ F e. _V ) -> { ( 0g ` (Scalar ` X ) ) } = { ( 0g ` (Scalar ` W ) ) } )
34	31, 33	difeq12d 3729	. . . . . . 7 |- ( ( ( W e. LMod /\ S e. U /\ ran F C_ S ) /\ F e. _V ) -> ( ( Base ` (Scalar ` X ) ) \ { ( 0g ` (Scalar ` X ) ) } ) = ( ( Base ` (Scalar ` W ) ) \ { ( 0g ` (Scalar ` W ) ) } ) )
35		eqid 2622	. . . . . . . . . . . . 13 |- ( .s ` W ) = ( .s ` W )
36	7, 35	ressvsca 16032	. . . . . . . . . . . 12 |- ( S e. U -> ( .s ` W ) = ( .s ` X ) )
37	36	eqcomd 2628	. . . . . . . . . . 11 |- ( S e. U -> ( .s ` X ) = ( .s ` W ) )
38	26, 37	syl 17	. . . . . . . . . 10 |- ( ( ( W e. LMod /\ S e. U /\ ran F C_ S ) /\ F e. _V ) -> ( .s ` X ) = ( .s ` W ) )
39	38	oveqd 6667	. . . . . . . . 9 |- ( ( ( W e. LMod /\ S e. U /\ ran F C_ S ) /\ F e. _V ) -> ( k ( .s ` X ) ( F ` x ) ) = ( k ( .s ` W ) ( F ` x ) ) )
40		simpl1 1064	. . . . . . . . . . 11 |- ( ( ( W e. LMod /\ S e. U /\ ran F C_ S ) /\ F e. _V ) -> W e. LMod )
41		imassrn 5477	. . . . . . . . . . . 12 |- ( F " ( dom F \ { x } ) ) C_ ran F
42		simpl3 1066	. . . . . . . . . . . 12 |- ( ( ( W e. LMod /\ S e. U /\ ran F C_ S ) /\ F e. _V ) -> ran F C_ S )
43	41, 42	syl5ss 3614	. . . . . . . . . . 11 |- ( ( ( W e. LMod /\ S e. U /\ ran F C_ S ) /\ F e. _V ) -> ( F " ( dom F \ { x } ) ) C_ S )
44		eqid 2622	. . . . . . . . . . . 12 |- ( LSpan ` W ) = ( LSpan ` W )
45		eqid 2622	. . . . . . . . . . . 12 |- ( LSpan ` X ) = ( LSpan ` X )
46	7, 44, 45, 15	lsslsp 19015	. . . . . . . . . . 11 |- ( ( W e. LMod /\ S e. U /\ ( F " ( dom F \ { x } ) ) C_ S ) -> ( ( LSpan ` W ) ` ( F " ( dom F \ { x } ) ) ) = ( ( LSpan ` X ) ` ( F " ( dom F \ { x } ) ) ) )
47	40, 26, 43, 46	syl3anc 1326	. . . . . . . . . 10 |- ( ( ( W e. LMod /\ S e. U /\ ran F C_ S ) /\ F e. _V ) -> ( ( LSpan ` W ) ` ( F " ( dom F \ { x } ) ) ) = ( ( LSpan ` X ) ` ( F " ( dom F \ { x } ) ) ) )
48	47	eqcomd 2628	. . . . . . . . 9 |- ( ( ( W e. LMod /\ S e. U /\ ran F C_ S ) /\ F e. _V ) -> ( ( LSpan ` X ) ` ( F " ( dom F \ { x } ) ) ) = ( ( LSpan ` W ) ` ( F " ( dom F \ { x } ) ) ) )
49	39, 48	eleq12d 2695	. . . . . . . 8 |- ( ( ( W e. LMod /\ S e. U /\ ran F C_ S ) /\ F e. _V ) -> ( ( k ( .s ` X ) ( F ` x ) ) e. ( ( LSpan ` X ) ` ( F " ( dom F \ { x } ) ) ) <-> ( k ( .s ` W ) ( F ` x ) ) e. ( ( LSpan ` W ) ` ( F " ( dom F \ { x } ) ) ) ) )
50	49	notbid 308	. . . . . . 7 |- ( ( ( W e. LMod /\ S e. U /\ ran F C_ S ) /\ F e. _V ) -> ( -. ( k ( .s ` X ) ( F ` x ) ) e. ( ( LSpan ` X ) ` ( F " ( dom F \ { x } ) ) ) <-> -. ( k ( .s ` W ) ( F ` x ) ) e. ( ( LSpan ` W ) ` ( F " ( dom F \ { x } ) ) ) ) )
51	34, 50	raleqbidv 3152	. . . . . 6 |- ( ( ( W e. LMod /\ S e. U /\ ran F C_ S ) /\ F e. _V ) -> ( A. k e. ( ( Base ` (Scalar ` X ) ) \ { ( 0g ` (Scalar ` X ) ) } ) -. ( k ( .s ` X ) ( F ` x ) ) e. ( ( LSpan ` X ) ` ( F " ( dom F \ { x } ) ) ) <-> A. k e. ( ( Base ` (Scalar ` W ) ) \ { ( 0g ` (Scalar ` W ) ) } ) -. ( k ( .s ` W ) ( F ` x ) ) e. ( ( LSpan ` W ) ` ( F " ( dom F \ { x } ) ) ) ) )
52	51	ralbidv 2986	. . . . 5 |- ( ( ( W e. LMod /\ S e. U /\ ran F C_ S ) /\ F e. _V ) -> ( A. x e. dom F A. k e. ( ( Base ` (Scalar ` X ) ) \ { ( 0g ` (Scalar ` X ) ) } ) -. ( k ( .s ` X ) ( F ` x ) ) e. ( ( LSpan ` X ) ` ( F " ( dom F \ { x } ) ) ) <-> A. x e. dom F A. k e. ( ( Base ` (Scalar ` W ) ) \ { ( 0g ` (Scalar ` W ) ) } ) -. ( k ( .s ` W ) ( F ` x ) ) e. ( ( LSpan ` W ) ` ( F " ( dom F \ { x } ) ) ) ) )
53	25, 52	anbi12d 747	. . . 4 |- ( ( ( W e. LMod /\ S e. U /\ ran F C_ S ) /\ F e. _V ) -> ( ( F : dom F --> ( Base ` X ) /\ A. x e. dom F A. k e. ( ( Base ` (Scalar ` X ) ) \ { ( 0g ` (Scalar ` X ) ) } ) -. ( k ( .s ` X ) ( F ` x ) ) e. ( ( LSpan ` X ) ` ( F " ( dom F \ { x } ) ) ) ) <-> ( F : dom F --> ( Base ` W ) /\ A. x e. dom F A. k e. ( ( Base ` (Scalar ` W ) ) \ { ( 0g ` (Scalar ` W ) ) } ) -. ( k ( .s ` W ) ( F ` x ) ) e. ( ( LSpan ` W ) ` ( F " ( dom F \ { x } ) ) ) ) ) )
54		ovex 6678	. . . . . . 7 |- ( W ↾s S ) e. _V
55	7, 54	eqeltri 2697	. . . . . 6 |- X e. _V
56	55	a1i 11	. . . . 5 |- ( ( W e. LMod /\ S e. U /\ ran F C_ S ) -> X e. _V )
57		eqid 2622	. . . . . 6 |- ( Base ` X ) = ( Base ` X )
58		eqid 2622	. . . . . 6 |- ( .s ` X ) = ( .s ` X )
59		eqid 2622	. . . . . 6 |- (Scalar ` X ) = (Scalar ` X )
60		eqid 2622	. . . . . 6 |- ( Base ` (Scalar ` X ) ) = ( Base ` (Scalar ` X ) )
61		eqid 2622	. . . . . 6 |- ( 0g ` (Scalar ` X ) ) = ( 0g ` (Scalar ` X ) )
62	57, 58, 45, 59, 60, 61	islindf 20151	. . . . 5 |- ( ( X e. _V /\ F e. _V ) -> ( F LIndF X <-> ( F : dom F --> ( Base ` X ) /\ A. x e. dom F A. k e. ( ( Base ` (Scalar ` X ) ) \ { ( 0g ` (Scalar ` X ) ) } ) -. ( k ( .s ` X ) ( F ` x ) ) e. ( ( LSpan ` X ) ` ( F " ( dom F \ { x } ) ) ) ) ) )
63	56, 62	sylan 488	. . . 4 |- ( ( ( W e. LMod /\ S e. U /\ ran F C_ S ) /\ F e. _V ) -> ( F LIndF X <-> ( F : dom F --> ( Base ` X ) /\ A. x e. dom F A. k e. ( ( Base ` (Scalar ` X ) ) \ { ( 0g ` (Scalar ` X ) ) } ) -. ( k ( .s ` X ) ( F ` x ) ) e. ( ( LSpan ` X ) ` ( F " ( dom F \ { x } ) ) ) ) ) )
64		eqid 2622	. . . . . 6 |- ( Base ` (Scalar ` W ) ) = ( Base ` (Scalar ` W ) )
65		eqid 2622	. . . . . 6 |- ( 0g ` (Scalar ` W ) ) = ( 0g ` (Scalar ` W ) )
66	8, 35, 44, 27, 64, 65	islindf 20151	. . . . 5 |- ( ( W e. LMod /\ F e. _V ) -> ( F LIndF W <-> ( F : dom F --> ( Base ` W ) /\ A. x e. dom F A. k e. ( ( Base ` (Scalar ` W ) ) \ { ( 0g ` (Scalar ` W ) ) } ) -. ( k ( .s ` W ) ( F ` x ) ) e. ( ( LSpan ` W ) ` ( F " ( dom F \ { x } ) ) ) ) ) )
67	66	3ad2antl1 1223	. . . 4 |- ( ( ( W e. LMod /\ S e. U /\ ran F C_ S ) /\ F e. _V ) -> ( F LIndF W <-> ( F : dom F --> ( Base ` W ) /\ A. x e. dom F A. k e. ( ( Base ` (Scalar ` W ) ) \ { ( 0g ` (Scalar ` W ) ) } ) -. ( k ( .s ` W ) ( F ` x ) ) e. ( ( LSpan ` W ) ` ( F " ( dom F \ { x } ) ) ) ) ) )
68	53, 63, 67	3bitr4d 300	. . 3 |- ( ( ( W e. LMod /\ S e. U /\ ran F C_ S ) /\ F e. _V ) -> ( F LIndF X <-> F LIndF W ) )
69	68	ex 450	. 2 |- ( ( W e. LMod /\ S e. U /\ ran F C_ S ) -> ( F e. _V -> ( F LIndF X <-> F LIndF W ) ) )
70	3, 5, 69	pm5.21ndd 369	1 |- ( ( W e. LMod /\ S e. U /\ ran F C_ S ) -> ( F LIndF X <-> F LIndF W ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   \ cdif 3571   C_ wss 3574   {csn 4177   class class class wbr 4653   dom cdm 5114   ran crn 5115   "cima 5117   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   Basecbs 15857   ↾_s cress 15858  Scalarcsca 15944   .scvsca 15945   0gc0g 16100   LModclmod 18863   LSubSpclss 18932   LSpanclspn 18971   LIndF clindf 20143

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-int 4476  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-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-4 11081  df-5 11082  df-6 11083  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-sca 15957  df-vsca 15958  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-sbg 17427  df-subg 17591  df-mgp 18490  df-ur 18502  df-ring 18549  df-lmod 18865  df-lss 18933  df-lsp 18972  df-lindf 20145

This theorem is referenced by:  lsslinds  20170