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Theorem lincext2 31107
Description: Property 2 of an extension of a linear combination. (Contributed by AV, 20-Apr-2019.) (Revised by AV, 30-Jul-2019.)
Hypotheses
Ref Expression
lincext.b  |-  B  =  ( Base `  M
)
lincext.r  |-  R  =  (Scalar `  M )
lincext.e  |-  E  =  ( Base `  R
)
lincext.0  |-  .0.  =  ( 0g `  R )
lincext.z  |-  Z  =  ( 0g `  M
)
lincext.n  |-  N  =  ( invg `  R )
lincext.f  |-  F  =  ( z  e.  S  |->  if ( z  =  X ,  ( N `
 Y ) ,  ( G `  z
) ) )
Assertion
Ref Expression
lincext2  |-  ( ( ( M  e.  LMod  /\  S  e.  ~P B
)  /\  ( Y  e.  E  /\  X  e.  S  /\  G  e.  ( E  ^m  ( S  \  { X }
) ) )  /\  G finSupp  .0.  )  ->  F finSupp  .0.  )
Distinct variable groups:    z, B    z, E    z, G    z, M    z, S    z, X    z, Y
Allowed substitution hints:    R( z)    F( z)    N( z)    .0. ( z)    Z( z)

Proof of Theorem lincext2
StepHypRef Expression
1 fvex 5808 . . . . . 6  |-  ( N `
 Y )  e. 
_V
2 fvex 5808 . . . . . 6  |-  ( G `
 z )  e. 
_V
31, 2ifex 3965 . . . . 5  |-  if ( z  =  X , 
( N `  Y
) ,  ( G `
 z ) )  e.  _V
4 lincext.f . . . . 5  |-  F  =  ( z  e.  S  |->  if ( z  =  X ,  ( N `
 Y ) ,  ( G `  z
) ) )
53, 4dmmpti 5647 . . . 4  |-  dom  F  =  S
65difeq1i 3577 . . 3  |-  ( dom 
F  \  ( S  \  { X } ) )  =  ( S 
\  ( S  \  { X } ) )
7 snssi 4124 . . . . . . 7  |-  ( X  e.  S  ->  { X }  C_  S )
873ad2ant2 1010 . . . . . 6  |-  ( ( Y  e.  E  /\  X  e.  S  /\  G  e.  ( E  ^m  ( S  \  { X } ) ) )  ->  { X }  C_  S )
983ad2ant2 1010 . . . . 5  |-  ( ( ( M  e.  LMod  /\  S  e.  ~P B
)  /\  ( Y  e.  E  /\  X  e.  S  /\  G  e.  ( E  ^m  ( S  \  { X }
) ) )  /\  G finSupp  .0.  )  ->  { X }  C_  S )
10 dfss4 3691 . . . . 5  |-  ( { X }  C_  S  <->  ( S  \  ( S 
\  { X }
) )  =  { X } )
119, 10sylib 196 . . . 4  |-  ( ( ( M  e.  LMod  /\  S  e.  ~P B
)  /\  ( Y  e.  E  /\  X  e.  S  /\  G  e.  ( E  ^m  ( S  \  { X }
) ) )  /\  G finSupp  .0.  )  ->  ( S  \  ( S  \  { X } ) )  =  { X }
)
12 snfi 7499 . . . 4  |-  { X }  e.  Fin
1311, 12syl6eqel 2550 . . 3  |-  ( ( ( M  e.  LMod  /\  S  e.  ~P B
)  /\  ( Y  e.  E  /\  X  e.  S  /\  G  e.  ( E  ^m  ( S  \  { X }
) ) )  /\  G finSupp  .0.  )  ->  ( S  \  ( S  \  { X } ) )  e.  Fin )
146, 13syl5eqel 2546 . 2  |-  ( ( ( M  e.  LMod  /\  S  e.  ~P B
)  /\  ( Y  e.  E  /\  X  e.  S  /\  G  e.  ( E  ^m  ( S  \  { X }
) ) )  /\  G finSupp  .0.  )  ->  ( dom  F  \  ( S 
\  { X }
) )  e.  Fin )
15 lincext.b . . . 4  |-  B  =  ( Base `  M
)
16 lincext.r . . . 4  |-  R  =  (Scalar `  M )
17 lincext.e . . . 4  |-  E  =  ( Base `  R
)
18 lincext.0 . . . 4  |-  .0.  =  ( 0g `  R )
19 lincext.z . . . 4  |-  Z  =  ( 0g `  M
)
20 lincext.n . . . 4  |-  N  =  ( invg `  R )
2115, 16, 17, 18, 19, 20, 4lincext1 31106 . . 3  |-  ( ( ( M  e.  LMod  /\  S  e.  ~P B
)  /\  ( Y  e.  E  /\  X  e.  S  /\  G  e.  ( E  ^m  ( S  \  { X }
) ) ) )  ->  F  e.  ( E  ^m  S ) )
22213adant3 1008 . 2  |-  ( ( ( M  e.  LMod  /\  S  e.  ~P B
)  /\  ( Y  e.  E  /\  X  e.  S  /\  G  e.  ( E  ^m  ( S  \  { X }
) ) )  /\  G finSupp  .0.  )  ->  F  e.  ( E  ^m  S
) )
23 elmapfun 7345 . . 3  |-  ( F  e.  ( E  ^m  S )  ->  Fun  F )
2422, 23syl 16 . 2  |-  ( ( ( M  e.  LMod  /\  S  e.  ~P B
)  /\  ( Y  e.  E  /\  X  e.  S  /\  G  e.  ( E  ^m  ( S  \  { X }
) ) )  /\  G finSupp  .0.  )  ->  Fun  F )
25 elmapi 7343 . . . . 5  |-  ( G  e.  ( E  ^m  ( S  \  { X } ) )  ->  G : ( S  \  { X } ) --> E )
264fdmdifeqresdif 30879 . . . . 5  |-  ( G : ( S  \  { X } ) --> E  ->  G  =  ( F  |`  ( S  \  { X } ) ) )
2725, 26syl 16 . . . 4  |-  ( G  e.  ( E  ^m  ( S  \  { X } ) )  ->  G  =  ( F  |`  ( S  \  { X } ) ) )
28273ad2ant3 1011 . . 3  |-  ( ( Y  e.  E  /\  X  e.  S  /\  G  e.  ( E  ^m  ( S  \  { X } ) ) )  ->  G  =  ( F  |`  ( S  \  { X } ) ) )
29283ad2ant2 1010 . 2  |-  ( ( ( M  e.  LMod  /\  S  e.  ~P B
)  /\  ( Y  e.  E  /\  X  e.  S  /\  G  e.  ( E  ^m  ( S  \  { X }
) ) )  /\  G finSupp  .0.  )  ->  G  =  ( F  |`  ( S  \  { X } ) ) )
30 simp3 990 . 2  |-  ( ( ( M  e.  LMod  /\  S  e.  ~P B
)  /\  ( Y  e.  E  /\  X  e.  S  /\  G  e.  ( E  ^m  ( S  \  { X }
) ) )  /\  G finSupp  .0.  )  ->  G finSupp  .0.  )
31 fvex 5808 . . . 4  |-  ( 0g
`  R )  e. 
_V
3218, 31eqeltri 2538 . . 3  |-  .0.  e.  _V
3332a1i 11 . 2  |-  ( ( ( M  e.  LMod  /\  S  e.  ~P B
)  /\  ( Y  e.  E  /\  X  e.  S  /\  G  e.  ( E  ^m  ( S  \  { X }
) ) )  /\  G finSupp  .0.  )  ->  .0.  e.  _V )
3414, 22, 24, 29, 30, 33resfsupp 7757 1  |-  ( ( ( M  e.  LMod  /\  S  e.  ~P B
)  /\  ( Y  e.  E  /\  X  e.  S  /\  G  e.  ( E  ^m  ( S  \  { X }
) ) )  /\  G finSupp  .0.  )  ->  F finSupp  .0.  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   _Vcvv 3076    \ cdif 3432    C_ wss 3435   ifcif 3898   ~Pcpw 3967   {csn 3984   class class class wbr 4399    |-> cmpt 4457   dom cdm 4947    |` cres 4949   Fun wfun 5519   -->wf 5521   ` cfv 5525  (class class class)co 6199    ^m cmap 7323   Fincfn 7419   finSupp cfsupp 7730   Basecbs 14291  Scalarcsca 14359   0gc0g 14496   invgcminusg 15529   LModclmod 17070
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 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481
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 2649  df-ral 2803  df-rex 2804  df-reu 2805  df-rmo 2806  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-int 4236  df-iun 4280  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-we 4788  df-ord 4829  df-on 4830  df-lim 4831  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-om 6586  df-1st 6686  df-2nd 6687  df-supp 6800  df-recs 6941  df-rdg 6975  df-1o 7029  df-oadd 7033  df-er 7210  df-map 7325  df-en 7420  df-fin 7423  df-fsupp 7731  df-0g 14498  df-mnd 15533  df-grp 15663  df-minusg 15664  df-rng 16769  df-lmod 17072
This theorem is referenced by:  lincext3  31108  lindslinindsimp1  31109  islindeps2  31135
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