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Theorem lspsneleq 16179
Description: Membership relation that implies equality of spans. (spansneleq 23064 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 16170 . . . . 5  |-  ( W  e.  LVec  ->  W  e. 
LMod )
42, 3syl 16 . . . 4  |-  ( ph  ->  W  e.  LMod )
5 lspsneleq.x . . . 4  |-  ( ph  ->  X  e.  V )
6 eqid 2435 . . . . 5  |-  (Scalar `  W )  =  (Scalar `  W )
7 eqid 2435 . . . . 5  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
8 lspsneleq.v . . . . 5  |-  V  =  ( Base `  W
)
9 eqid 2435 . . . . 5  |-  ( .s
`  W )  =  ( .s `  W
)
10 lspsneleq.n . . . . 5  |-  N  =  ( LSpan `  W )
116, 7, 8, 9, 10lspsnel 16071 . . . 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 643 . . 3  |-  ( ph  ->  ( Y  e.  ( N `  { X } )  <->  E. k  e.  ( Base `  (Scalar `  W ) ) Y  =  ( k ( .s `  W ) X ) ) )
13 simpr 448 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  ( Base `  (Scalar `  W ) ) )  /\  Y  =  ( k ( .s `  W ) X ) )  ->  Y  =  ( k ( .s
`  W ) X ) )
1413sneqd 3819 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  ( Base `  (Scalar `  W ) ) )  /\  Y  =  ( k ( .s `  W ) X ) )  ->  { Y }  =  { (
k ( .s `  W ) X ) } )
1514fveq2d 5724 . . . . . 6  |-  ( ( ( ph  /\  k  e.  ( Base `  (Scalar `  W ) ) )  /\  Y  =  ( k ( .s `  W ) X ) )  ->  ( N `  { Y } )  =  ( N `  { ( k ( .s `  W ) X ) } ) )
162ad2antrr 707 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  ( Base `  (Scalar `  W ) ) )  /\  Y  =  ( k ( .s `  W ) X ) )  ->  W  e.  LVec )
17 simplr 732 . . . . . . 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 707 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  ( Base `  (Scalar `  W ) ) )  /\  Y  =  ( k ( .s `  W ) X ) )  ->  Y  =/=  .0.  )
20 simplr 732 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  k  e.  ( Base `  (Scalar `  W )
) )  /\  Y  =  ( k ( .s `  W ) X ) )  /\  k  =  ( 0g `  (Scalar `  W )
) )  ->  Y  =  ( k ( .s `  W ) X ) )
21 simpr 448 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  k  e.  ( Base `  (Scalar `  W )
) )  /\  Y  =  ( k ( .s `  W ) X ) )  /\  k  =  ( 0g `  (Scalar `  W )
) )  ->  k  =  ( 0g `  (Scalar `  W ) ) )
2221oveq1d 6088 . . . . . . . . . . 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 2435 . . . . . . . . . . . . . 14  |-  ( 0g
`  (Scalar `  W )
)  =  ( 0g
`  (Scalar `  W )
)
24 lspsneleq.o . . . . . . . . . . . . . 14  |-  .0.  =  ( 0g `  W )
258, 6, 9, 23, 24lmod0vs 15975 . . . . . . . . . . . . 13  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (
( 0g `  (Scalar `  W ) ) ( .s `  W ) X )  =  .0.  )
264, 5, 25syl2anc 643 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( 0g `  (Scalar `  W ) ) ( .s `  W
) X )  =  .0.  )
2726ad3antrrr 711 . . . . . . . . . . 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 2471 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  k  e.  ( Base `  (Scalar `  W )
) )  /\  Y  =  ( k ( .s `  W ) X ) )  /\  k  =  ( 0g `  (Scalar `  W )
) )  ->  Y  =  .0.  )
2928ex 424 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  ( Base `  (Scalar `  W ) ) )  /\  Y  =  ( k ( .s `  W ) X ) )  ->  ( k  =  ( 0g `  (Scalar `  W ) )  ->  Y  =  .0.  ) )
3029necon3d 2636 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  ( Base `  (Scalar `  W ) ) )  /\  Y  =  ( k ( .s `  W ) X ) )  ->  ( Y  =/=  .0.  ->  k  =/=  ( 0g `  (Scalar `  W ) ) ) )
3119, 30mpd 15 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  ( Base `  (Scalar `  W ) ) )  /\  Y  =  ( k ( .s `  W ) X ) )  ->  k  =/=  ( 0g `  (Scalar `  W ) ) )
325ad2antrr 707 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  ( Base `  (Scalar `  W ) ) )  /\  Y  =  ( k ( .s `  W ) X ) )  ->  X  e.  V )
338, 6, 9, 7, 23, 10lspsnvs 16178 . . . . . . 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 1189 . . . . . 6  |-  ( ( ( ph  /\  k  e.  ( Base `  (Scalar `  W ) ) )  /\  Y  =  ( k ( .s `  W ) X ) )  ->  ( N `  { ( k ( .s `  W ) X ) } )  =  ( N `  { X } ) )
3515, 34eqtrd 2467 . . . . 5  |-  ( ( ( ph  /\  k  e.  ( Base `  (Scalar `  W ) ) )  /\  Y  =  ( k ( .s `  W ) X ) )  ->  ( N `  { Y } )  =  ( N `  { X } ) )
3635ex 424 . . . 4  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) ) )  -> 
( Y  =  ( k ( .s `  W ) X )  ->  ( N `  { Y } )  =  ( N `  { X } ) ) )
3736rexlimdva 2822 . . 3  |-  ( ph  ->  ( E. k  e.  ( Base `  (Scalar `  W ) ) Y  =  ( k ( .s `  W ) X )  ->  ( N `  { Y } )  =  ( N `  { X } ) ) )
3812, 37sylbid 207 . 2  |-  ( ph  ->  ( Y  e.  ( N `  { X } )  ->  ( N `  { Y } )  =  ( N `  { X } ) ) )
391, 38mpd 15 1  |-  ( ph  ->  ( N `  { Y } )  =  ( N `  { X } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598   E.wrex 2698   {csn 3806   ` cfv 5446  (class class class)co 6073   Basecbs 13461  Scalarcsca 13524   .scvsca 13525   0gc0g 13715   LModclmod 15942   LSpanclspn 16039   LVecclvec 16166
This theorem is referenced by:  lspsncmp  16180  lspsnel4  16188  lspdisj2  16191  lspexch  16193  lsmcv  16205  mapdpglem10  32416  mapdpglem15  32421
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-tpos 6471  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-3 10051  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-mulr 13535  df-0g 13719  df-mnd 14682  df-grp 14804  df-minusg 14805  df-sbg 14806  df-mgp 15641  df-rng 15655  df-ur 15657  df-oppr 15720  df-dvdsr 15738  df-unit 15739  df-invr 15769  df-drng 15829  df-lmod 15944  df-lss 16001  df-lsp 16040  df-lvec 16167
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