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Theorem lspsneleq 16115
Description: Membership relation that implies equality of spans. (spansneleq 22921 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 16106 . . . . 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 2388 . . . . 5  |-  (Scalar `  W )  =  (Scalar `  W )
7 eqid 2388 . . . . 5  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
8 lspsneleq.v . . . . 5  |-  V  =  ( Base `  W
)
9 eqid 2388 . . . . 5  |-  ( .s
`  W )  =  ( .s `  W
)
10 lspsneleq.n . . . . 5  |-  N  =  ( LSpan `  W )
116, 7, 8, 9, 10lspsnel 16007 . . . 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 3771 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  ( Base `  (Scalar `  W ) ) )  /\  Y  =  ( k ( .s `  W ) X ) )  ->  { Y }  =  { (
k ( .s `  W ) X ) } )
1514fveq2d 5673 . . . . . 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 6036 . . . . . . . . . . 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 2388 . . . . . . . . . . . . . 14  |-  ( 0g
`  (Scalar `  W )
)  =  ( 0g
`  (Scalar `  W )
)
24 lspsneleq.o . . . . . . . . . . . . . 14  |-  .0.  =  ( 0g `  W )
258, 6, 9, 23, 24lmod0vs 15911 . . . . . . . . . . . . 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 2424 . . . . . . . . . 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 2589 . . . . . . . 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 16114 . . . . . . 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 2420 . . . . 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 2774 . . 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 1649    e. wcel 1717    =/= wne 2551   E.wrex 2651   {csn 3758   ` cfv 5395  (class class class)co 6021   Basecbs 13397  Scalarcsca 13460   .scvsca 13461   0gc0g 13651   LModclmod 15878   LSpanclspn 15975   LVecclvec 16102
This theorem is referenced by:  lspsncmp  16116  lspsnel4  16124  lspdisj2  16127  lspexch  16129  lsmcv  16141  mapdpglem10  31797  mapdpglem15  31802
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-tpos 6416  df-riota 6486  df-recs 6570  df-rdg 6605  df-er 6842  df-en 7047  df-dom 7048  df-sdom 7049  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-nn 9934  df-2 9991  df-3 9992  df-ndx 13400  df-slot 13401  df-base 13402  df-sets 13403  df-ress 13404  df-plusg 13470  df-mulr 13471  df-0g 13655  df-mnd 14618  df-grp 14740  df-minusg 14741  df-sbg 14742  df-mgp 15577  df-rng 15591  df-ur 15593  df-oppr 15656  df-dvdsr 15674  df-unit 15675  df-invr 15705  df-drng 15765  df-lmod 15880  df-lss 15937  df-lsp 15976  df-lvec 16103
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