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Theorem lspsneleq 15868
Description: Membership relation that implies equality of spans. (spansneleq 22149 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 15859 . . . . 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 2283 . . . . 5  |-  (Scalar `  W )  =  (Scalar `  W )
7 eqid 2283 . . . . 5  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
8 lspsneleq.v . . . . 5  |-  V  =  ( Base `  W
)
9 eqid 2283 . . . . 5  |-  ( .s
`  W )  =  ( .s `  W
)
10 lspsneleq.n . . . . 5  |-  N  =  ( LSpan `  W )
116, 7, 8, 9, 10lspsnel 15760 . . . 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 3653 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  ( Base `  (Scalar `  W ) ) )  /\  Y  =  ( k ( .s `  W ) X ) )  ->  { Y }  =  { (
k ( .s `  W ) X ) } )
1514fveq2d 5529 . . . . . 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 5873 . . . . . . . . . . 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 2283 . . . . . . . . . . . . . 14  |-  ( 0g
`  (Scalar `  W )
)  =  ( 0g
`  (Scalar `  W )
)
24 lspsneleq.o . . . . . . . . . . . . . 14  |-  .0.  =  ( 0g `  W )
258, 6, 9, 23, 24lmod0vs 15663 . . . . . . . . . . . . 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 2319 . . . . . . . . . 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 2484 . . . . . . . 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 15867 . . . . . . 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 2315 . . . . 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 2667 . . 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 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544   {csn 3640   ` cfv 5255  (class class class)co 5858   Basecbs 13148  Scalarcsca 13211   .scvsca 13212   0gc0g 13400   LModclmod 15627   LSpanclspn 15728   LVecclvec 15855
This theorem is referenced by:  lspsncmp  15869  lspsnel4  15877  lspdisj2  15880  lspexch  15882  lsmcv  15894  mapdpglem10  31871  mapdpglem15  31876
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-tpos 6234  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-sbg 14491  df-mgp 15326  df-rng 15340  df-ur 15342  df-oppr 15405  df-dvdsr 15423  df-unit 15424  df-invr 15454  df-drng 15514  df-lmod 15629  df-lss 15690  df-lsp 15729  df-lvec 15856
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