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Theorem lincext2 33310
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 5858 . . . . . 6  |-  ( N `
 Y )  e. 
_V
2 fvex 5858 . . . . . 6  |-  ( G `
 z )  e. 
_V
31, 2ifex 3997 . . . . 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 5692 . . . 4  |-  dom  F  =  S
65difeq1i 3604 . . 3  |-  ( dom 
F  \  ( S  \  { X } ) )  =  ( S 
\  ( S  \  { X } ) )
7 snssi 4160 . . . . . . 7  |-  ( X  e.  S  ->  { X }  C_  S )
873ad2ant2 1016 . . . . . 6  |-  ( ( Y  e.  E  /\  X  e.  S  /\  G  e.  ( E  ^m  ( S  \  { X } ) ) )  ->  { X }  C_  S )
983ad2ant2 1016 . . . . 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 3729 . . . . 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 7589 . . . 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 33309 . . 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 1014 . 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 7435 . . 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 7433 . . . . 5  |-  ( G  e.  ( E  ^m  ( S  \  { X } ) )  ->  G : ( S  \  { X } ) --> E )
264fdmdifeqresdif 33185 . . . . 5  |-  ( G : ( S  \  { X } ) --> E  ->  G  =  ( F  |`  ( S  \  { X } ) ) )
2725, 26syl 16 . . . 4  |-  ( G  e.  ( E  ^m  ( S  \  { X } ) )  ->  G  =  ( F  |`  ( S  \  { X } ) ) )
28273ad2ant3 1017 . . 3  |-  ( ( Y  e.  E  /\  X  e.  S  /\  G  e.  ( E  ^m  ( S  \  { X } ) ) )  ->  G  =  ( F  |`  ( S  \  { X } ) ) )
29283ad2ant2 1016 . 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 996 . 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 5858 . . . 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 7848 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823   _Vcvv 3106    \ cdif 3458    C_ wss 3461   ifcif 3929   ~Pcpw 3999   {csn 4016   class class class wbr 4439    |-> cmpt 4497   dom cdm 4988    |` cres 4990   Fun wfun 5564   -->wf 5566   ` cfv 5570  (class class class)co 6270    ^m cmap 7412   Fincfn 7509   finSupp cfsupp 7821   Basecbs 14716  Scalarcsca 14787   0gc0g 14929   invgcminusg 16253   LModclmod 17707
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-supp 6892  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-en 7510  df-fin 7513  df-fsupp 7822  df-0g 14931  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-grp 16256  df-minusg 16257  df-ring 17395  df-lmod 17709
This theorem is referenced by:  lincext3  33311  lindslinindsimp1  33312  islindeps2  33338
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