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Theorem lspsneleq 17322
Description: Membership relation that implies equality of spans. (spansneleq 25145 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 17313 . . . . 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 2454 . . . . 5  |-  (Scalar `  W )  =  (Scalar `  W )
7 eqid 2454 . . . . 5  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
8 lspsneleq.v . . . . 5  |-  V  =  ( Base `  W
)
9 eqid 2454 . . . . 5  |-  ( .s
`  W )  =  ( .s `  W
)
10 lspsneleq.n . . . . 5  |-  N  =  ( LSpan `  W )
116, 7, 8, 9, 10lspsnel 17210 . . . 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 661 . . 3  |-  ( ph  ->  ( Y  e.  ( N `  { X } )  <->  E. k  e.  ( Base `  (Scalar `  W ) ) Y  =  ( k ( .s `  W ) X ) ) )
13 simpr 461 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  ( Base `  (Scalar `  W ) ) )  /\  Y  =  ( k ( .s `  W ) X ) )  ->  Y  =  ( k ( .s
`  W ) X ) )
1413sneqd 4000 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  ( Base `  (Scalar `  W ) ) )  /\  Y  =  ( k ( .s `  W ) X ) )  ->  { Y }  =  { (
k ( .s `  W ) X ) } )
1514fveq2d 5806 . . . . . 6  |-  ( ( ( ph  /\  k  e.  ( Base `  (Scalar `  W ) ) )  /\  Y  =  ( k ( .s `  W ) X ) )  ->  ( N `  { Y } )  =  ( N `  { ( k ( .s `  W ) X ) } ) )
162ad2antrr 725 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  ( Base `  (Scalar `  W ) ) )  /\  Y  =  ( k ( .s `  W ) X ) )  ->  W  e.  LVec )
17 simplr 754 . . . . . . 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 725 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  ( Base `  (Scalar `  W ) ) )  /\  Y  =  ( k ( .s `  W ) X ) )  ->  Y  =/=  .0.  )
20 simplr 754 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  k  e.  ( Base `  (Scalar `  W )
) )  /\  Y  =  ( k ( .s `  W ) X ) )  /\  k  =  ( 0g `  (Scalar `  W )
) )  ->  Y  =  ( k ( .s `  W ) X ) )
21 simpr 461 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  k  e.  ( Base `  (Scalar `  W )
) )  /\  Y  =  ( k ( .s `  W ) X ) )  /\  k  =  ( 0g `  (Scalar `  W )
) )  ->  k  =  ( 0g `  (Scalar `  W ) ) )
2221oveq1d 6218 . . . . . . . . . . 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 2454 . . . . . . . . . . . . . 14  |-  ( 0g
`  (Scalar `  W )
)  =  ( 0g
`  (Scalar `  W )
)
24 lspsneleq.o . . . . . . . . . . . . . 14  |-  .0.  =  ( 0g `  W )
258, 6, 9, 23, 24lmod0vs 17107 . . . . . . . . . . . . 13  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (
( 0g `  (Scalar `  W ) ) ( .s `  W ) X )  =  .0.  )
264, 5, 25syl2anc 661 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( 0g `  (Scalar `  W ) ) ( .s `  W
) X )  =  .0.  )
2726ad3antrrr 729 . . . . . . . . . . 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 2499 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  k  e.  ( Base `  (Scalar `  W )
) )  /\  Y  =  ( k ( .s `  W ) X ) )  /\  k  =  ( 0g `  (Scalar `  W )
) )  ->  Y  =  .0.  )
2928ex 434 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  ( Base `  (Scalar `  W ) ) )  /\  Y  =  ( k ( .s `  W ) X ) )  ->  ( k  =  ( 0g `  (Scalar `  W ) )  ->  Y  =  .0.  ) )
3029necon3d 2676 . . . . . . . 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 725 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  ( Base `  (Scalar `  W ) ) )  /\  Y  =  ( k ( .s `  W ) X ) )  ->  X  e.  V )
338, 6, 9, 7, 23, 10lspsnvs 17321 . . . . . . 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 1224 . . . . . 6  |-  ( ( ( ph  /\  k  e.  ( Base `  (Scalar `  W ) ) )  /\  Y  =  ( k ( .s `  W ) X ) )  ->  ( N `  { ( k ( .s `  W ) X ) } )  =  ( N `  { X } ) )
3515, 34eqtrd 2495 . . . . 5  |-  ( ( ( ph  /\  k  e.  ( Base `  (Scalar `  W ) ) )  /\  Y  =  ( k ( .s `  W ) X ) )  ->  ( N `  { Y } )  =  ( N `  { X } ) )
3635ex 434 . . . 4  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) ) )  -> 
( Y  =  ( k ( .s `  W ) X )  ->  ( N `  { Y } )  =  ( N `  { X } ) ) )
3736rexlimdva 2947 . . 3  |-  ( ph  ->  ( E. k  e.  ( Base `  (Scalar `  W ) ) Y  =  ( k ( .s `  W ) X )  ->  ( N `  { Y } )  =  ( N `  { X } ) ) )
3812, 37sylbid 215 . 2  |-  ( ph  ->  ( Y  e.  ( N `  { X } )  ->  ( N `  { Y } )  =  ( N `  { X } ) ) )
391, 38mpd 15 1  |-  ( ph  ->  ( N `  { Y } )  =  ( N `  { X } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2648   E.wrex 2800   {csn 3988   ` cfv 5529  (class class class)co 6203   Basecbs 14295  Scalarcsca 14363   .scvsca 14364   0gc0g 14500   LModclmod 17074   LSpanclspn 17178   LVecclvec 17309
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-tpos 6858  df-recs 6945  df-rdg 6979  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-nn 10437  df-2 10494  df-3 10495  df-ndx 14298  df-slot 14299  df-base 14300  df-sets 14301  df-ress 14302  df-plusg 14373  df-mulr 14374  df-0g 14502  df-mnd 15537  df-grp 15667  df-minusg 15668  df-sbg 15669  df-mgp 16717  df-ur 16729  df-rng 16773  df-oppr 16841  df-dvdsr 16859  df-unit 16860  df-invr 16890  df-drng 16960  df-lmod 17076  df-lss 17140  df-lsp 17179  df-lvec 17310
This theorem is referenced by:  lspsncmp  17323  lspsnel4  17331  lspdisj2  17334  lspexch  17336  lsmcv  17348  mapdpglem10  35684  mapdpglem15  35689
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