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Theorem lspsneleq 15884
Description: Membership relation that implies equality of spans. (spansneleq 22165 analog.) (Contributed by NM, 4-Jul-2014.)
Hypotheses
Ref Expression
lspsneleq.v  |-  V  =  ( Base `  W
)
lspsneleq.o  |-  .0.  =  ( 0g `  W )
lspsneleq.n  |-  N  =  ( LSpan `  W )
lspsneleq.w  |-  ( ph  ->  W  e.  LVec )
lspsneleq.x  |-  ( ph  ->  X  e.  V )
lspsneleq.y  |-  ( ph  ->  Y  e.  ( N `
 { X }
) )
lspsneleq.z  |-  ( ph  ->  Y  =/=  .0.  )
Assertion
Ref Expression
lspsneleq  |-  ( ph  ->  ( N `  { Y } )  =  ( N `  { X } ) )

Proof of Theorem lspsneleq
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 lspsneleq.y . 2  |-  ( ph  ->  Y  e.  ( N `
 { X }
) )
2 lspsneleq.w . . . . 5  |-  ( ph  ->  W  e.  LVec )
3 lveclmod 15875 . . . . 5  |-  ( W  e.  LVec  ->  W  e. 
LMod )
42, 3syl 15 . . . 4  |-  ( ph  ->  W  e.  LMod )
5 lspsneleq.x . . . 4  |-  ( ph  ->  X  e.  V )
6 eqid 2296 . . . . 5  |-  (Scalar `  W )  =  (Scalar `  W )
7 eqid 2296 . . . . 5  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
8 lspsneleq.v . . . . 5  |-  V  =  ( Base `  W
)
9 eqid 2296 . . . . 5  |-  ( .s
`  W )  =  ( .s `  W
)
10 lspsneleq.n . . . . 5  |-  N  =  ( LSpan `  W )
116, 7, 8, 9, 10lspsnel 15776 . . . 4  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  ( Y  e.  ( N `  { X } )  <->  E. k  e.  ( Base `  (Scalar `  W
) ) Y  =  ( k ( .s
`  W ) X ) ) )
124, 5, 11syl2anc 642 . . 3  |-  ( ph  ->  ( Y  e.  ( N `  { X } )  <->  E. k  e.  ( Base `  (Scalar `  W ) ) Y  =  ( k ( .s `  W ) X ) ) )
13 simpr 447 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  ( Base `  (Scalar `  W ) ) )  /\  Y  =  ( k ( .s `  W ) X ) )  ->  Y  =  ( k ( .s
`  W ) X ) )
1413sneqd 3666 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  ( Base `  (Scalar `  W ) ) )  /\  Y  =  ( k ( .s `  W ) X ) )  ->  { Y }  =  { (
k ( .s `  W ) X ) } )
1514fveq2d 5545 . . . . . 6  |-  ( ( ( ph  /\  k  e.  ( Base `  (Scalar `  W ) ) )  /\  Y  =  ( k ( .s `  W ) X ) )  ->  ( N `  { Y } )  =  ( N `  { ( k ( .s `  W ) X ) } ) )
162ad2antrr 706 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  ( Base `  (Scalar `  W ) ) )  /\  Y  =  ( k ( .s `  W ) X ) )  ->  W  e.  LVec )
17 simplr 731 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  ( Base `  (Scalar `  W ) ) )  /\  Y  =  ( k ( .s `  W ) X ) )  ->  k  e.  ( Base `  (Scalar `  W
) ) )
18 lspsneleq.z . . . . . . . . 9  |-  ( ph  ->  Y  =/=  .0.  )
1918ad2antrr 706 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  ( Base `  (Scalar `  W ) ) )  /\  Y  =  ( k ( .s `  W ) X ) )  ->  Y  =/=  .0.  )
20 simplr 731 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  k  e.  ( Base `  (Scalar `  W )
) )  /\  Y  =  ( k ( .s `  W ) X ) )  /\  k  =  ( 0g `  (Scalar `  W )
) )  ->  Y  =  ( k ( .s `  W ) X ) )
21 simpr 447 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  k  e.  ( Base `  (Scalar `  W )
) )  /\  Y  =  ( k ( .s `  W ) X ) )  /\  k  =  ( 0g `  (Scalar `  W )
) )  ->  k  =  ( 0g `  (Scalar `  W ) ) )
2221oveq1d 5889 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  k  e.  ( Base `  (Scalar `  W )
) )  /\  Y  =  ( k ( .s `  W ) X ) )  /\  k  =  ( 0g `  (Scalar `  W )
) )  ->  (
k ( .s `  W ) X )  =  ( ( 0g
`  (Scalar `  W )
) ( .s `  W ) X ) )
23 eqid 2296 . . . . . . . . . . . . . 14  |-  ( 0g
`  (Scalar `  W )
)  =  ( 0g
`  (Scalar `  W )
)
24 lspsneleq.o . . . . . . . . . . . . . 14  |-  .0.  =  ( 0g `  W )
258, 6, 9, 23, 24lmod0vs 15679 . . . . . . . . . . . . 13  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (
( 0g `  (Scalar `  W ) ) ( .s `  W ) X )  =  .0.  )
264, 5, 25syl2anc 642 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( 0g `  (Scalar `  W ) ) ( .s `  W
) X )  =  .0.  )
2726ad3antrrr 710 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  k  e.  ( Base `  (Scalar `  W )
) )  /\  Y  =  ( k ( .s `  W ) X ) )  /\  k  =  ( 0g `  (Scalar `  W )
) )  ->  (
( 0g `  (Scalar `  W ) ) ( .s `  W ) X )  =  .0.  )
2820, 22, 273eqtrd 2332 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  k  e.  ( Base `  (Scalar `  W )
) )  /\  Y  =  ( k ( .s `  W ) X ) )  /\  k  =  ( 0g `  (Scalar `  W )
) )  ->  Y  =  .0.  )
2928ex 423 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  ( Base `  (Scalar `  W ) ) )  /\  Y  =  ( k ( .s `  W ) X ) )  ->  ( k  =  ( 0g `  (Scalar `  W ) )  ->  Y  =  .0.  ) )
3029necon3d 2497 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  ( Base `  (Scalar `  W ) ) )  /\  Y  =  ( k ( .s `  W ) X ) )  ->  ( Y  =/=  .0.  ->  k  =/=  ( 0g `  (Scalar `  W ) ) ) )
3119, 30mpd 14 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  ( Base `  (Scalar `  W ) ) )  /\  Y  =  ( k ( .s `  W ) X ) )  ->  k  =/=  ( 0g `  (Scalar `  W ) ) )
325ad2antrr 706 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  ( Base `  (Scalar `  W ) ) )  /\  Y  =  ( k ( .s `  W ) X ) )  ->  X  e.  V )
338, 6, 9, 7, 23, 10lspsnvs 15883 . . . . . . 7  |-  ( ( W  e.  LVec  /\  (
k  e.  ( Base `  (Scalar `  W )
)  /\  k  =/=  ( 0g `  (Scalar `  W ) ) )  /\  X  e.  V
)  ->  ( N `  { ( k ( .s `  W ) X ) } )  =  ( N `  { X } ) )
3416, 17, 31, 32, 33syl121anc 1187 . . . . . 6  |-  ( ( ( ph  /\  k  e.  ( Base `  (Scalar `  W ) ) )  /\  Y  =  ( k ( .s `  W ) X ) )  ->  ( N `  { ( k ( .s `  W ) X ) } )  =  ( N `  { X } ) )
3515, 34eqtrd 2328 . . . . 5  |-  ( ( ( ph  /\  k  e.  ( Base `  (Scalar `  W ) ) )  /\  Y  =  ( k ( .s `  W ) X ) )  ->  ( N `  { Y } )  =  ( N `  { X } ) )
3635ex 423 . . . 4  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) ) )  -> 
( Y  =  ( k ( .s `  W ) X )  ->  ( N `  { Y } )  =  ( N `  { X } ) ) )
3736rexlimdva 2680 . . 3  |-  ( ph  ->  ( E. k  e.  ( Base `  (Scalar `  W ) ) Y  =  ( k ( .s `  W ) X )  ->  ( N `  { Y } )  =  ( N `  { X } ) ) )
3812, 37sylbid 206 . 2  |-  ( ph  ->  ( Y  e.  ( N `  { X } )  ->  ( N `  { Y } )  =  ( N `  { X } ) ) )
391, 38mpd 14 1  |-  ( ph  ->  ( N `  { Y } )  =  ( N `  { X } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557   {csn 3653   ` cfv 5271  (class class class)co 5874   Basecbs 13164  Scalarcsca 13227   .scvsca 13228   0gc0g 13416   LModclmod 15643   LSpanclspn 15744   LVecclvec 15871
This theorem is referenced by:  lspsncmp  15885  lspsnel4  15893  lspdisj2  15896  lspexch  15898  lsmcv  15910  mapdpglem10  32493  mapdpglem15  32498
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-tpos 6250  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-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506  df-sbg 14507  df-mgp 15342  df-rng 15356  df-ur 15358  df-oppr 15421  df-dvdsr 15439  df-unit 15440  df-invr 15470  df-drng 15530  df-lmod 15645  df-lss 15706  df-lsp 15745  df-lvec 15872
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