Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lsslindf Unicode version

Theorem lsslindf 27403
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  =  ( Ws  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.
StepHypRef Expression
1 rellindf 27381 . . . 4  |-  Rel LIndF
21brrelexi 4745 . . 3  |-  ( F LIndF 
X  ->  F  e.  _V )
32a1i 10 . 2  |-  ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  ->  ( F LIndF  X  ->  F  e.  _V ) )
41brrelexi 4745 . . 3  |-  ( F LIndF 
W  ->  F  e.  _V )
54a1i 10 . 2  |-  ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  ->  ( F LIndF  W  ->  F  e.  _V ) )
6 simpr 447 . . . . . . . 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  =  ( Ws  S )
8 eqid 2296 . . . . . . . . 9  |-  ( Base `  W )  =  (
Base `  W )
97, 8ressbasss 13216 . . . . . . . 8  |-  ( Base `  X )  C_  ( Base `  W )
10 fss 5413 . . . . . . . 8  |-  ( ( F : dom  F --> ( Base `  X )  /\  ( Base `  X
)  C_  ( Base `  W ) )  ->  F : dom  F --> ( Base `  W ) )
116, 9, 10sylancl 643 . . . . . . 7  |-  ( ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  /\  F : dom  F --> ( Base `  X )
)  ->  F : dom  F --> ( Base `  W
) )
12 ffn 5405 . . . . . . . . 9  |-  ( F : dom  F --> ( Base `  W )  ->  F  Fn  dom  F )
1312adantl 452 . . . . . . . 8  |-  ( ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  /\  F : dom  F --> ( Base `  W )
)  ->  F  Fn  dom  F )
14 simp3 957 . . . . . . . . . 10  |-  ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  ->  ran  F 
C_  S )
15 lsslindf.u . . . . . . . . . . . . 13  |-  U  =  ( LSubSp `  W )
168, 15lssss 15710 . . . . . . . . . . . 12  |-  ( S  e.  U  ->  S  C_  ( Base `  W
) )
17163ad2ant2 977 . . . . . . . . . . 11  |-  ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  ->  S  C_  ( Base `  W
) )
187, 8ressbas2 13215 . . . . . . . . . . 11  |-  ( S 
C_  ( Base `  W
)  ->  S  =  ( Base `  X )
)
1917, 18syl 15 . . . . . . . . . 10  |-  ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  ->  S  =  ( Base `  X
) )
2014, 19sseqtrd 3227 . . . . . . . . 9  |-  ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  ->  ran  F 
C_  ( Base `  X
) )
2120adantr 451 . . . . . . . 8  |-  ( ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  /\  F : dom  F --> ( Base `  W )
)  ->  ran  F  C_  ( Base `  X )
)
22 df-f 5275 . . . . . . . 8  |-  ( F : dom  F --> ( Base `  X )  <->  ( F  Fn  dom  F  /\  ran  F 
C_  ( Base `  X
) ) )
2313, 21, 22sylanbrc 645 . . . . . . 7  |-  ( ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  /\  F : dom  F --> ( Base `  W )
)  ->  F : dom  F --> ( Base `  X
) )
2411, 23impbida 805 . . . . . 6  |-  ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  ->  ( F : dom  F --> ( Base `  X )  <->  F : dom  F --> ( Base `  W
) ) )
2524adantr 451 . . . . 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 959 . . . . . . . . . 10  |-  ( ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  /\  F  e.  _V )  ->  S  e.  U
)
27 eqid 2296 . . . . . . . . . . . 12  |-  (Scalar `  W )  =  (Scalar `  W )
287, 27resssca 13299 . . . . . . . . . . 11  |-  ( S  e.  U  ->  (Scalar `  W )  =  (Scalar `  X ) )
2928eqcomd 2301 . . . . . . . . . 10  |-  ( S  e.  U  ->  (Scalar `  X )  =  (Scalar `  W ) )
3026, 29syl 15 . . . . . . . . 9  |-  ( ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  /\  F  e.  _V )  ->  (Scalar `  X )  =  (Scalar `  W )
)
3130fveq2d 5545 . . . . . . . 8  |-  ( ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  /\  F  e.  _V )  ->  ( Base `  (Scalar `  X ) )  =  ( Base `  (Scalar `  W ) ) )
3230fveq2d 5545 . . . . . . . . 9  |-  ( ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  /\  F  e.  _V )  ->  ( 0g `  (Scalar `  X ) )  =  ( 0g `  (Scalar `  W ) ) )
3332sneqd 3666 . . . . . . . 8  |-  ( ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  /\  F  e.  _V )  ->  { ( 0g
`  (Scalar `  X )
) }  =  {
( 0g `  (Scalar `  W ) ) } )
3431, 33difeq12d 3308 . . . . . . 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 2296 . . . . . . . . . . . . 13  |-  ( .s
`  W )  =  ( .s `  W
)
367, 35ressvsca 13300 . . . . . . . . . . . 12  |-  ( S  e.  U  ->  ( .s `  W )  =  ( .s `  X
) )
3736eqcomd 2301 . . . . . . . . . . 11  |-  ( S  e.  U  ->  ( .s `  X )  =  ( .s `  W
) )
3826, 37syl 15 . . . . . . . . . 10  |-  ( ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  /\  F  e.  _V )  ->  ( .s `  X )  =  ( .s `  W ) )
3938oveqd 5891 . . . . . . . . 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 958 . . . . . . . . . . 11  |-  ( ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  /\  F  e.  _V )  ->  W  e.  LMod )
41 imassrn 5041 . . . . . . . . . . . 12  |-  ( F
" ( dom  F  \  { x } ) )  C_  ran  F
42 simpl3 960 . . . . . . . . . . . 12  |-  ( ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  /\  F  e.  _V )  ->  ran  F  C_  S
)
4341, 42syl5ss 3203 . . . . . . . . . . 11  |-  ( ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  /\  F  e.  _V )  ->  ( F "
( dom  F  \  {
x } ) ) 
C_  S )
44 eqid 2296 . . . . . . . . . . . 12  |-  ( LSpan `  W )  =  (
LSpan `  W )
45 eqid 2296 . . . . . . . . . . . 12  |-  ( LSpan `  X )  =  (
LSpan `  X )
467, 44, 45, 15lsslsp 15788 . . . . . . . . . . 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 } ) ) ) )
4740, 26, 43, 46syl3anc 1182 . . . . . . . . . 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 } ) ) ) )
4847eqcomd 2301 . . . . . . . . 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 } ) ) ) )
4939, 48eleq12d 2364 . . . . . . . 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 } ) ) ) ) )
5049notbid 285 . . . . . . 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 } ) ) ) ) )
5134, 50raleqbidv 2761 . . . . . 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 } ) ) ) ) )
5251ralbidv 2576 . . . . 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 } ) ) ) ) )
5325, 52anbi12d 691 . . . 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 5899 . . . . . . 7  |-  ( Ws  S )  e.  _V
557, 54eqeltri 2366 . . . . . 6  |-  X  e. 
_V
5655a1i 10 . . . . 5  |-  ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  ->  X  e.  _V )
57 eqid 2296 . . . . . 6  |-  ( Base `  X )  =  (
Base `  X )
58 eqid 2296 . . . . . 6  |-  ( .s
`  X )  =  ( .s `  X
)
59 eqid 2296 . . . . . 6  |-  (Scalar `  X )  =  (Scalar `  X )
60 eqid 2296 . . . . . 6  |-  ( Base `  (Scalar `  X )
)  =  ( Base `  (Scalar `  X )
)
61 eqid 2296 . . . . . 6  |-  ( 0g
`  (Scalar `  X )
)  =  ( 0g
`  (Scalar `  X )
)
6257, 58, 45, 59, 60, 61islindf 27385 . . . . 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 } ) ) ) ) ) )
6356, 62sylan 457 . . . 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 2296 . . . . . 6  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
65 eqid 2296 . . . . . 6  |-  ( 0g
`  (Scalar `  W )
)  =  ( 0g
`  (Scalar `  W )
)
668, 35, 44, 27, 64, 65islindf 27385 . . . . 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 } ) ) ) ) ) )
67663ad2antl1 1117 . . . 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 } ) ) ) ) ) )
6853, 63, 673bitr4d 276 . . 3  |-  ( ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  /\  F  e.  _V )  ->  ( F LIndF  X  <->  F LIndF 
W ) )
6968ex 423 . 2  |-  ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  ->  ( F  e.  _V  ->  ( F LIndF  X  <->  F LIndF  W ) ) )
703, 5, 69pm5.21ndd 343 1  |-  ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  ->  ( F LIndF  X  <->  F LIndF  W ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   _Vcvv 2801    \ cdif 3162    C_ wss 3165   {csn 3653   class class class wbr 4039   dom cdm 4705   ran crn 4706   "cima 4708    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874   Basecbs 13164   ↾s cress 13165  Scalarcsca 13227   .scvsca 13228   0gc0g 13416   LModclmod 15643   LSubSpclss 15705   LSpanclspn 15744   LIndF clindf 27377
This theorem is referenced by:  lsslinds  27404
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-sca 13240  df-vsca 13241  df-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506  df-sbg 14507  df-subg 14634  df-mgp 15342  df-rng 15356  df-ur 15358  df-lmod 15645  df-lss 15706  df-lsp 15745  df-lindf 27379
  Copyright terms: Public domain W3C validator